Quantitative Analysis For Business

Data into Information

It is important to know the difference between data and information. In today's digital, connected, and online world, we are surrounded by tremendous amounts of data. There are numbers everywhere. It is so easy to find references that we are inundated with data; we are drowning in a sea of data. There are mountains of data in computer systems that run businesses. That data is easily accessible and should be very useful in helping guide our actions and decisions. "Should" is the key word. Data by itself can, as noted above, be overwhelming and thus confusing to us when we are determining a course of action or making other decisions. The goal of using spreadsheets is to mine that mountain of data and turn it into information that will help guide our actions and decisions. Let's look at a simple example of how we might turn data into information. Consider the School Absences - Test Score dataset from earlier in this module. This data in a spreadsheet does not give much information. We would really have to study and ponder it to glean any useful information from this small set of 13 observations. What might we do to improve this tabular data in a spreadsheet? For one, we could order the data set by either listing the School Absences or Test Scores in increasing order. Which should we choose? The underlying suspicion seems to be that as absences increase, test scores decrease. It makes sense to order this dataset by school absences. All spreadsheets have data filters that can be used to order one column, but keeping all the row data the same. Doing this on the dataset yields the following: With this simple re-ordering, this numerical display of the data set conveys information more easily than in the original presentation. Just looking at the above table indicates there might be a relationship between the two variables as we suspected. We can calculate the Pearson Correlation Coefficient, as in Module 4, and see that an r=−.96 shows a very strong linear correlation between these variables. We might want to calculate descriptive statistics for School Absences or Test Scores. Most spreadsheet packages have individual functions for each of the descriptive statistics listed below. Better yet, there are Data Analysis add-ins that provide the the base functionality of full blown statistical software. With electronic spreadsheet Data Analysis, all of the descriptive statistics can be calculated at once. It is easy to take even this simple set of raw data and create information that can guide both actions and decisions. Note: For example purposes, we are looking at a very simple dataset of 13 pairs of observations. On a dataset this small, many of the descriptive statistics are easily seen or, if not easily seen, we could almost guess the numbers and be relatively close. What if our dataset had 200 pairs of observations? We would seriously need to reorder the data and calculate the descriptive statistics to see what areas we would want to take the analysis. The display and manipulation of numerical data is very helpful indeed. However, do you remember the old adage "A picture is worth 1,000 words"? Graphs simply make things easier to see. After looking at the descriptive statistics, perhaps School Absences are an item of interest. It might be good to see a graph of this data. First, we have to group them into buckets of five absences (i.e. 0-5, 6-10, 11-15, and 16-20). The following table shows this grouping and count, which can be done manually or using the Data Analysis add-on. Notice that the percentages were also calculated for each grouping of five. We can see that there are four students each in the 0-5 and 6-10 categories. There are three students and two students in the next two categories. This is a visual display of what was shown in the Descriptive Statistics table. The median and mode at 8 and the mean at 9.23. The standard deviation is not usually apparent but we easily can see approximately the minimum, maximum, and range. The question that might arise looking at this histogram is why we have nine people that have five or more absences. In the histogram, this is easier to see than in the tabular data. We might want to use a pie chart instead of a histogram. The pie chart can be constructed from the same data. A simple pie chart could be constructed. There are options in Chart Tools to make the chart even easier to read and interpret containing all the data in the table. Notice how the categories and Percentage of Students are now labeled in each slice of the pie. Which should be used: pie charts or histograms? Both are used to display and reflect the descriptive statistics for one variable. The answer to the question is often a matter of preference which often is decided by the audience. Let's say you prefer histograms but your supervisor, director, of VP just loves pie charts. The best advice is both simple and obvious: Go with the boss's preference. Consider, looking at the scatter plot of the two variables. We can easily see that there appears to be a strong negative correlation of the two variables. A few clicks on the graph and we can select the option to include the best fit line. This shows the relationship between the variables with even more clarity than the scatter plot alone. Where r2 is Coefficient of Determination which is the % of variation in the Test Scores explained by this linear model. Our linear model explains 92% of the variation in test scores. This is an indication of a very good model. The Pearson Correlation Coefficient, which by the way is the square root of the Coefficient of Determination and therefore explains why one is r and the other is r2, show strong linearity. The overall conclusion here is that we have an excellent model to explain why it is better to have less absences. In the graph below, we are showing the r instead of r2.

In setting up a LP problem, constraints are considered. One particular type of constraint requires the constraint to be greater than or equal to 0. This is known as a ___ constraint.

Non-negative: These linear inequalities x>=0 and y>=0. These are included because x and yare usually the number of items produced and you cannot produce a negative number of items; the smallest number of items you could produce is zero.

What is a project?

That is a very good question that requires a specific answer in order to isolate the difference between project tasks and the other operational work tasks. Before defining what a project is, a further example should be considered regarding the routine tasks of individuals and of organizations. As individuals, various types of work consumes most of our lives. It begins in grade school and continues in college, never stopping. After completing our education, a career begins. In each of these endeavors, school, part time jobs, new career positions, and in our personal lives, things need to get done on a somewhat routine basis. At first, our skills may not be at an expert level but because these tasks are somewhat routine, improvement in both performance and confidence continues to grow. Organizational enterprises also have routine tasks, these are the operational tasks or activities referred to earlier. Accounts receivables, accounts payables, customer service, order processing, manufacturing, purchasing, and logistics are all routine business processes for which companies must be skilled and carry out day to day. These type of business processes and activities occur over and over again. 1. Accounting departments process transactions each and every day. 2. Cashiers at a clothing store check customers out: scan purchases, bag the goods, and take payments. 3. Manufacturing planners work on monthly, weekly, and daily routines to plan the production for the month, provide detailed schedules for the next week, and revise the detailed schedule each day depending on what was not completed the previous day or what cannot be immediately processed. To understand a project's characteristics and how it differs from operational characteristics, consider these points: 1. A project has a defined beginning and end. 2. A project creates a unique product, service, or event. 3. A project is much larger in scope than most routine tasks. 4. The timeline for completion of a project usually takes longer to complete. 5. A project could involve just one person but more often a team of people. 6. A project involves work activities that are in addition to the routine tasks. 7. Projects are considered strategic undertakings, whereas operational activities are routine. An activity is a unique unit of the project which can be described within limits of time. For the same operational activities as above, projects can also be incorporated, as an example: A cashier may be required to participate in an annual physical inventory count of the store merchandise. This would likely include activities not done on a routine basis. A manufacturing planner might be assigned to a one-year project to implement a new software system such as an enterprise resource planning system. This would be in addition to routinely developing the daily schedules. Considering personal or school projects, if a teacher assigned a "project" in middle school, we knew it would take several days or weeks to complete. It would be specialized work in addition to our normal routine homework. More often than not, the project was intended to expand our skills and knowledge horizons. In regards to school or personal projects, people generally fall into two categories. 1. Wait until the last possible day and then work furiously to complete the assigned project on time. 2. Break the large project into smaller tasks that could be worked on every day, thus having a higher likelihood of producing high quality work. People in the first category are mystified at how those in the second category stay focused. This could be that people in the second category are natural and well organized future project managers.

Statistics is a key methodology behind quantitative analysis and is a very robust and complex field. Statistics seeks to collect, analyze, and ___ data. A key advantage in using statistics derives from the fact that it can draw conclusions about a(n) ___ based on a smaller ___ , which is representative of the whole.

1. Interpret: The process of data interpretation follows the process of data analysis. Interpretation is a follow-on event and is a separate function in that interpretation leads to decision-making based on the analysis. 2. Population: A key advantage in using statistics is the ability to draw conclusions about an entire population by way of a smaller sample. This saves time and effort on the part of the researcher and yet maintains the validity of the conclusions. 3. Sample: A key advantage in using statistics is the ability to draw conclusions about an entire population by way of a smaller sample. This saves time and effort on the part of the researcher and yet maintains the validity of the conclusions.

Under what circumstances could quantitative analysis be used successfully?

When the problem being analyzed can be numerically represented and valid and reliable data are available There are many techniques available in quantitative analysis, but they all require valid and reliable numeric information to begin with.

When is quantitative analysis most useful?

When the problem being evaluated can be represented by discrete data that are valid, objective, and reliable. In order to use the tools of quantitative analysis, data must be valid, reliable, and objective. Quantitative data are number-based data.

Qualitative Analysis

Qualitative analysis is based on subjective factors such as opinions, thoughts, and judgments. Subjective reasoning is heavily influenced by personal experiences, such as upbringing and past employment. Opinions are usually based on someone's past experiences as well. The qualitative approach is best used when a business is faced with uncertainty. The uncertainty could stem from product quality problems, competition, new product considerations, or territory expansion. When managers or executives perceive that the uncertainty is impacting the business, subjective reasoning allows for exploration and in-depth research to eventually determine the specific problem. For example, when a company is looking to introduce a new product into the market, there is no historical sales data to rely upon and to help guide initial inventory requirements. A company must first depend on expert judgment to formulate its reasoning. Researchers must be careful about using expert judgment and subjective reasoning due to the negative association of introducing personal reflections and emotions.

Reorder Point (ROP)

Reorder point is determined by multiplying the daily demand for an inventory item by the lead time, the number of days between making an order and receiving the order. ROP = ((daily demand * lead time) + safety stock)

[True/False] Project scheduling can be done before the cost estimates are complete.

True: It is not necessary to have the cost estimates done first; in fact, the estimates can only be done after all tasks and subtasks are identified.

Which of the following equations will return the middle value in a range of cells?

=MEDIAN(A1:A25) The MEDIAN function finds the middle value of all the values in a selected range of cells.

Summary

Businesses use both qualitative and quantitative analysis to help solve problems. Qualitative analysis is used when in-depth subjective reasoning is required, such as understanding customer needs and expectations. Quantitative analysis is used when a business problem demands specifics. The first step in quantitative analysis is to identify the problem. Research questions help define the problem to be solved and leads to a testable hypothesis. A hypotheses is a declarative statement that predicts the relationship between a dependent and independent variable. Decision analysis involves evaluating each alternative, the corresponding risks and benefits, as well as the likelihood of occurrence. Expected monetary value analysis is a specific quantitative analysis technique that uses mathematics to determine the average of the most likely and least likely alternative scenarios. The alternative with the highest monetary benefit is usually the alternative the business will select. Quantitative analysis is used for inventory analysis to help business managers keep appropriate levels of inventory. Holding inventory can be one of the largest expenses of a firm so inventory analysis helps to minimize these expenses.Quantitative analysis can also be used to prepare forecasting models. Forecasting is a method or process used to make short range decisions where the actual outcomes have not yet occurred. Qualitative analysis is based on subjective factors such as opinions, thoughts, and judgments. Subjective reasoning is heavily influenced by personal experiences, such as upbringing and past employment. The qualitative approach is best used when a business is faced with uncertainty or is trying to understand the underlying causes of a problem. Subjective reasoning allows for exploration and in-depth research to eventually determine the specific problem. The outcome of qualitative research will reveal information that may have an influence on the quantitative analysis. Most business problems are related to profitability or optimization of resources and require quantitative analysis. Quantitative analysis provides methods to analyze numeric data to look for patterns, trends and relationships. Managers can use mathematics and statistics to find solutions to business problems. This course will focus on the tools needed for effective quantitative analysis. We have introduced the fundamental concepts of quantitative analysis. It is important to understand the business purpose of using quantitative analysis. Quantitative analysis applies mathematical models and statistics to help a business gain an understanding of underlying problems. Businesses make better decisions when they apply quantitative analysis to thoroughly identify and investigate alternatives and their consequences when solving problems. Quantitative analysis begins with problem identification. Using research questions will help state the problem that will be analyzed. This process defines what you want to learn or solve in regard to the problem. In quantitative analysis, a hypothesis is a declarative statement predicting the relationship between the dependent and independent variables. Researchers must determine the dependent and independent variables under study to determine the cause and effect in a problem. Recall, that a change in one or more factors of a business can produce an impact on another. An independent variable is the impacting variable, and the dependent variable responds to the independent variable. When a problem does not have a single solution, quantitative analysis will use mathematical models, measurements, and calculations to understand differences in numerical data. Analyzing numerical differences is used to help to resolve, reconcile, and understand the reasons behind data differences. Quantitative analysis uses models as mathematical representations of a situation or problem. The models use numbers, mathematical symbols, and expressions that form an equation. Mathematical models help managers find the optimal solution to a problem. Now that you understand what quantitative analysis is, and why it is critical to business decision making, we will review math concepts, statistics and probability, and graphing techniques you will need to use when using quantitative analysis. In the Big Picture, the business man is looking at a screen full of formulas. A solid review of math and statistics will make these formulas easier to understand and use when we analyze decision models, network diagrams, and inventory optimization models in later units. In the first module, we will review math concepts, such as working with positive and negative numbers, exponents, and solving equations. After reviewing the math concepts, we will put them to use by looking at electronic spreadsheet software to solve these math problems. The next module will cover descriptive statistics and probability. Techniques for quantifying uncertainty will be introduced to assess an event's outcome. Learning about time series patterns and trends will help you predict seasonal or cyclical business patterns. Finally, you will learn how using descriptive statistics to calculate mean, mode, median, and standard deviation can help find patterns and relationships in large amounts of data. Finally, in the last module of this unit, we will discuss how using graphs to summarize and display large quantities of data helps businesses understand relationships between variables and to see trends. Pictures are often easier to understand than large volumes of numeric data. A variety of graph types will be discussed, along with tips for when to select a certain type of graph to display your data.

Descriptive Statistics:

Descriptive statistics are used to support or summarize statements made about the data. Mean, mode, and median describe central tendencies of the data. Standard deviation and range describe the variation or dispersion of the data from the mean. . Descriptive statistics are meaningless or misleading if the sample population is too small. The size of the data sample will vary according to the problem being studied. The sample size must be large enough to truly represent the larger population of the group being studied.

Which of the following is an example of a linear programming example?

Determining the best employee work schedule given a set amount of minimum and maximum hours available. LP can be used to solve this type of problem as a scheduling problem.

Sampling

Sampling is the activity of obtaining representative samples. Sampling is both a science and an art. Of the many sampling methods and techniques, let us define four of the most basic and popular types:

Summary

A project is a task that is outside of the routine activities. Projects are a temporary endeavor and have a beginning and an end date. Every task in the project is specific to the objectives and goals of the project. Constraints, specific objectives that must be met, to the project will be related to time, cost, and/or performance. A work breakdown structure (WBS) lists a breakdown of all activities to complete the project. Using this list of project activities, determine which tasks must be completed before the next step can be started. A predecessor is a task that must be completed before the next step can be started. A successor is a task that follows an activity. After defining the project, listing the project tasks, describing task relationships and indicating which activities must precede or succeed others, draw the network diagram to connect the activities. The network diagram is used to help the decision maker plan, manage, and monitor the project throughout the project duration. Critical path management uses the network diagram to help decision makers to understand the sequence and duration of project tasks, as well as to understand which tasks must be completed before others are started. The network diagram shows where the decision maker does or doesn't have flexibility with decisions. Slack time allows for flexibility without jeopardizing the project completion by the deadline. The critical path has no slack time and as a result, the decision maker has no flexibility in adjusting those project tasks start and finish times. Using a network diagram to understand the task dependencies and the critical path will minimize risk of project failure. Project success is often determined by whether or not a project is delivered on budget and on time. A successful manager must know how to make accurate cost and time estimates for projects. The beta distribution method uses three time estimates to estimate task duration: pessimistic, optimistic, and most likely. The most likely time estimate is weighted (or multiplied by 4) in the formula since this time estimate is most likely to happen. This estimate, test, is calculated using a weighted average of A (optimistic), M (most likely), and P (pessimistic) as above using the following formula for each task. test=A+4M+P6 Managers often use the Critical Path Method (CPM) to determine a project's critical path. This method will require determining the task's earliest start and earliest finish time as well as the task's latest start and latest finish time. Earliest start time (ES) - The earliest an activity can start without starting before any predecessor activities. Earliest finish time (EF) - The earliest an activity can finish. Latest start time (LS) - The latest time an activity can start without delaying the entire project. Latest finish time (LF) - The latest time an activity can finish without delaying the entire project. A forward pass is made through the network diagram to determine the early start and finish times for each task. A backwards pass is used to determine late start and finish times. Slack time for an activity can be calculated by subtracting the late start from the early start of a project. Those tasks with zero slack time are on the critical path. PERT charts can be used to outline the critical path of a project. The critical path is often indicated by a red line connecting those activities. Strategies to ensure activities on the critical path do not get delayed is paramount to a project completing on time. When a project manager must rein in a deadline and is not able to decrease the scope of the project, the project manager can crash certain activities. Crashing the project is a metho d that will help the manager make decisions about the application of additional resources to work the necessary activities to meet the deadline. Only critical path activities can be crashed. By comparing the difference between the normal duration and the crash duration, and the normal and crash cost, the crash cost slope per week can be found. The formula to determine the crash cost per time period for a task is: CostSlope=CrashCost−NormalCostNormalDuration−CrashDuration Once each of the critical path's activity's crash costs has been determined, the manager would crash in the order of the least cost to the most cost. In this unit, you discovered how network diagrams help manage project tasks and costs to increase the likelihood of project success. Projects are best described in terms of a network of activities necessary to complete an objective. Each activity duration, as well as the activity predecessors and successors, must be identified. Once the logical relationships have been identified, the project activities can be organized into a network diagram. In this unit, we have seen how network diagrams help the decision maker allocate resources and schedules to ensure a project is completed on time and within the approved budget. Time estimates for each activity are assigned using the beta distribution method. The beta distribution method for estimating activity duration uses three time estimates: optimistic, most likely, and pessimistic. Using a weighted average with these three time estimates provides a method for estimating task duration. By using the beta distribution method, the estimation incorporates risk and uncertainty. The network diagram activity nodes contain the activity title, the early start, early finish, late start, and late finish, along with the task duration. The lines connecting the tasks indicate which task must be completed before others. The longest path from start to finish on a network diagram is the critical path. There is no slack time available for the tasks on the critical path. Several strategies to keep the critical path tasks on schedule are to use the more experienced workers on those tasks, or hire additional workers for critical path activities. If a project is running behind schedule and the scope outlined cannot be reduced, the only way to shorten the project is to crash tasks on the critical path. Using the formula to determine crash cost per time period for each activity can help a decision maker determine the order of activities on the critical path to crash.

Cyclical Trend

A cyclical trend is a pattern that tends to repeat itself every few years. A cyclical trend is observed over a long time period. Cyclical trends are harder to predict than seasonal and cover a longer period of time. Examples include economic depressions or recessions and the economic impacts of war.

Random Trends

Random trends are unpredictable trends in data. If there is a trend that cannot be classified as trend, seasonal, or cyclical, then it is considered random.

In estimating task durations in projects, many planners use the beta distribution technique, also known as the three-point estimate technique. This technique calls for using three separate time estimates for a task instead of using just one. The time estimate representing the normal amount of time required to perform a task is known as the ___ time. The quickest time in which the task may be completed is known as the ___ time. The final time to consider is the 'worst case' time, which is called the ___ time.

1. The normal time that a task should take is known as most likely time. 2. Optimistic time is considered the least amount of time that it would take to complete a task. 3. The longest time that a task should take is known as pessimistic time.

Decision Tree Continued

A decision tree moves from left to right and graphically illustrates the decisions to be made. Decision trees have decisions points and state-of-nature points. The squares represent the alternatives and the circles represent the outcomes or states-of-nature. First, the bakery owner must decide whether to build a new bakery, expand his current bakery, or lease new bakery space. These are the alternatives. The possible states-of nature are high demand or low demand. The next step is to estimate the payoffs and assign the probabilities for each alternative and state-of nature combination on the decision tree. Multiplying the probability by the potential revenue will provide the expected value of each of the three alternatives. The alternative with the best expected value is selected. The following video reviews the steps for creating a decision tree to identify a simple decision analysis problem and solution alternatives.

Beta Distribution

The beta distribution method is used to estimate task duration (the amount of time to complete an activity). This method uses three activity time estimates (pessimistic, optimistic, and most likely) and then takes a weighted average of these estimates.

In the example below, the nature of the problem is not disclosed. But assuming the issue is interpreted correctly, the initial expression would appear as:

The second step is to graph the linear representation of each regular constraint. In order to do that, some basic algebra is utilized. The conversion of the inequality to an equation ( "=" ) is accomplished by adding or subtracting a "slack" or "surplus" variable even if that variable is zero. This step will become clear at the end of the process described here. For now, we will simply convert the inequality to an equation and restate the expressions. For the first regular constraint, X≤6, we simply convert to x=6 and plot that on the graph as shown below, making sure to shade in the direction of the inequality ("≤") and bound our shading by the boundaries established by the non-negative constraints. This non-negative boundary will occur at every regular constraint forming a left and bottom boundary to the "to be established" feasible region (the area that satisfies all of the constraints). Continuing on to regular constraint 2, the inequality will be restated as: 2x+3y=19 Using basic algebraic substitution, the (x,y) intercepts (for purposes of graphing the line that represents this constraint) are obtained by setting x to zero and solving for y: 3y=19, therefore eliminating the coefficient 3 by dividing through both sides by 3 yields a y-intercept of 6.33. Following the same procedure for x (setting y to zero), yields an x intercept of 9.5. Therefore, installing a line that crosses (9.5, 0) and (0, 6.33) is the linear representation of constraint 2 as shown below. Following the direction of the less than inequality sign, the feasible area for this constraint is similar to regular constraint 1 except for the unique angle it represents. Continuing on to the third regular constraint and making sure that the non-negative constraints (x,y≥0) are also adhered to, the inequality is restated and the algebraic substitution yields coordinates of (0, 8) and (8, 0) with directional shading similar to regular constraints 1 and 2. Combining all three regular constraints, our graphical representation of the mathematical expression for this linear programming problem appears as follows: From this point, the graph can be a guide for the optimal solution. The results that are being sought are the best or optimal coordinates that satisfy the feasible area and provide the best (in this case the maximum) value of the objective function: 5x+7y=Z To obtain the Z value (largest in this case due to it being a maximization problem), we need to find the best and legitimate (on the feasible area) coordinates that produce the largest Z value. There are two logical choices. 1. Solve each regular constraint by simultaneous process (either substitution or elimination (an algebraic function). 2. Graph the objective function in linear form by using a sample Z value that lands the objective function inside the feasible region. Then, increase the Z value and track the linear representation of the objective function until it crosses the last possible corner (outermost in this case due it being a max problem). This will determine the best corner. Then solve for those two constraints ("binding") by simultaneous function until the optimal coordinates are located. Either way, the same corner or (x,y) coordinates that represent the optimal Z value are found. In this case that is (5, 3). By plugging x=5 and y=3 back into the objective function, the Z value is equal to 46. The two regular constraints that formed the optimal point yield zero, as the slack variables for regular constraints 2 and 3. The first regular constraint has a slack of 1 because the constraint limits x to 6, but the optimal solution has x at 5.

A project has the following characteristics:

1. The endeavor is outside of the routine or operational activities. 2. The assignment is a temporary endeavor. 3. The goal has a start and end date. 4. Every task in the project is specific to the objective or goal of the project.

In quantitative analysis, differences in numerical data are analyzed for what purpose?

To resolve, reconcile, and understand the reasons behind the data differences. Quantitative analysis looks at different types of data that may appear to be contradictory but are not after scrutiny.

Identify business problems that can be solved using qualitative analysis.

Qualitative analysis is based on subjective factors such as opinions, thoughts, and judgments. Subjective reasoning is heavily influenced by personal experiences, such as upbringing and past employment. Opinions are usually based on someone's past experiences as well. The qualitative approach is best used when a business is faced with uncertainty. The uncertainty could stem from product quality problems, competition, new product considerations, or territory expansion. When managers or executives perceive that the uncertainty is impacting the business, subjective reasoning allows for exploration and in-depth research to eventually determine the specific problem. For example, when a company is looking to introduce a new product into the market, there is no historical sales data to rely upon and to help guide initial inventory requirements. A company must first depend on expert judgment to formulate its reasoning. Researchers must be careful about using expert judgment and subjective reasoning due to the negative association of introducing personal reflections and emotions. To expand on the example, consider a toy company preparing to produce and market a new toy. Currently, no other products on the market are similar to the new toy. The company's manufacturing department must have plausible information about the initial inventory requirements. However, there is no past sales data or inventory analysis to rely upon. The managers of the company will need to use qualitative analysis. One approach the managers could take to explore the requirement is to prepare a subjective questionnaire and distribute it to a very targeted group. The respondents could be parents and grandparents who have children in the appropriate age bracket. Exploratory interviews can also be an excellent method the company could use to gather subjective data or opinions regarding this new product. The expert opinion of parents will be based on their experiences with their children and will be qualitative in nature. The toy company can use this input to assist in the manufacturing and marketing decisions. Quantitative analysis uses specific numeric or measurable data; however, managers cannot always ascertain as to why an event is occurring or how it will occur. Although product sales can be measured, it is not always simple to find out why sales are declining; this would involve qualitative analysis. By interviewing, asking questions, having focus groups, and surveying the appropriate audience, you can often determine what is motivating behavior and choices . Qualitative analysis can provide the subjective, descriptive information that would not show up in quantitative data. The goal of a qualitative analysis is to explore problems that have not been specifically determined or specifically identified. If a business cannot identify the problem, using quantitative analysis to prepare a mathematical model is not feasible. We can verify that qualitative analysis must first be used simply by asking, "What problem will the mathematical model solve, and what variables will the model contain?" If the business does not know what problem is being solved, they are not ready for the mathematical model; therefore, qualitative analysis is the necessary course of action. A significant goal of qualitative analysis is to increase the understanding of the problem. A qualitative analysis will reveal information that may have an impact on the quantitative analysis.

Linear programming (LP) is composed of a number of elements that together form the concept and the structure of linear programming. In setting up a LP problem, constraints are considered. One particular type of constraint that is used to apply the concept to a particular situation is called a(n) ___ constraint.

The problem constraint operationalizes the constraint concept in order to apply to a specific problem.

Identify the elements of the following expression, including constants, variables, and coefficients. 7x2+3xy+8 The number 8 in the above formula is a ___? The number 7 in the formula above is a __?

1. A constant is a term in a formula that contains only a number, with no variables. They never change in value. 2. Coefficients are the numbers in the terms of a formula preceding a variable letter. Any variable that is not preceded with a number is assumed to have a coefficient of 1. 3. Variables are terms in a formula that are letters and not numbers. They can change values.

First problem: 2^4∙5+62.

Enter the formula into your spreadsheet by typing an equal sign (=), then type the number 2, the caret (^), the number 4, an asterisk (*), the number 5, a plus sign (+), and the number 62. Here is what you should see when you click on the cell: 2^4 X 5+62 Following the order of operation, this formula was solve by completing the following passes: 1. Pass 1: Exponents: The exponent will be solved first, 2^4=16. Our expression now reads: 16∙5+62. 2. Pass 2: Multiply: Next the multiplication will be solved, 16∙5=80. Now our expression reads: 80+62. 3. Pass 3: Add: The answer is 142.

What are the components of linear programming?

1. Problem identification: Usually, an optimization problem surfaces when someone, often a manager, asks, "What is the best and most efficient way to accomplish this objective?" From such a question comes the framing of the problem. A problem statement must be formulated to clearly indicate the problem to be solved. Ask questions such as: - What exactly is to be optimized? - What is the scope of the issue? - What are the boundaries of the issue? Clearly stating the scope and boundaries clarify exactly what should be included in the problem solution. This is important to prevent scope creep and having the problem develop into something unsolvable or out of perspective. 2. Development of a mathematical model: The statement description of the problem must be translated in mathematics. Linear programs are solved mathematically; this is critical to getting an applicable and meaningful solution. The variables, objective function, and constraints must be defined. Determine if the problem is a maximization or a minimization problem. What are the coefficients of the variables in the linear objective function? Are the constraints greater than or equal to OR less than or equal to? Is the constraint simply equal to? What are the coefficients of the variables in each constraint? Consider a problem where the objective is to minimize the number of miles delivery trucks travel in a day. There is an easy solution to this minimization problem, yet it is unrealistic: The company could keep all the trucks in the garage, not assigning them any routes. The trucks would sit in the garage all day and accumulate zero miles. Zero is certainly a minimum number of miles for that day, but unrealistic for a company to pursue. The management or owners of the company expect to generate sales and service customers by fulfilling the orders. This requires the trucks not to sit idle all day but rather to be making deliveries to the customers so that the company can collect the revenue. There is a precondition to the objective of minimizing the number of miles the delivery trucks travel in a day. That precondition is that ALL the deliveries scheduled for the day must be made. This precondition is a problem constraint. This example problem can be restated as follows: Minimize the number of miles the company delivery trucks drive, subject to the constraint that all scheduled deliveries must be made. A problem constraint in a minimization problem sets the minimum activity that must occur and thus makes it so the objective does not just retreat to the state where all variables are zero. Consider the following maximization problem: A charitable organization is having a fundraising banquet and raffling a very expensive car. The objective is to maximize the revenue from the ticket sales of the banquet, which is $100, and the raffle, which is another $100 dollars. In addition, groups and individuals have two variants. Tables of ten for the banquet are $900, and if a banquet ticket and raffle ticket are both purchased, the cost for both is discounted. Therefore, the objective function is to optimize revenues given these four classes of ticket sales (four variables): banquet, banquet discount, raffle, banquet and raffle. Again, there is a maximum that could be sought; however, it is not realistic: The organization could sell a billion tickets in each of the four categories, thereby funding the charity for years and years to come. Reality quickly sets in with the physical limitations of unlimited revenue. The physical limitations and perhaps some minimum requirements are constraints to this problem. The venue for the banquet is not an unlimited space. The venue can only accommodate so many tables, and hence, there is an upper limit on the number of banquet tickets that can be sold. There may be a minimum number of banquet tickets that must be sold. For the raffle, state rules and regulations must be followed. Often there is a minimum number of tickets that must be sold and a maximum so that the probabilities of winning are bounded. The charity management may set an upper limit on the number of discount tables of ten and dinner/raffle tickets that can be sold. This maximization problem is now bounded. The solutions in this bounded region are the feasible solutions of the problem. The goal is to find the maximum revenue result for all points in the feasible region. The decision is made on the value of the objective on the set of feasible solutions established by the bounds of the constraints of the problem. The objective function sets the direction and goal of what is to be accomplished. The constraints bound the problem. As stated in the previous module, constraints define the microeconomic principle of scarcity in linear programs.

In defining a cause and effect relationship, the variable that drives some change in another variable is called what?

Independent variable The independent variable is the factor that causes change to the dependent factor. In other words, the movement of the dependent factor is driven by the movement of the independent factor.

John is trying to lose weight and is planning his dinner menu. His dinner menu must be limited to no more than 500 calories and contain at least 15 grams of protein. Sodium intake cannot exceed 750 milligrams. He is very hungry and wants to eat a lot for dinner, but must follow these strict guidelines. Which of the following statements might represent the objective function?

The objective function might be to maximize the quantity of food. John is very hungry and wants to eat as much food as possible (maximize) while staying within the provided constraints, such as calories, protein, and sodium.

LP can be used for transportation problems in order to minimize shipping costs. Sunnyland Co. can buy oranges from three different groves (A, B, and C) and send the oranges to three different processing plants (in Tampa, Ocala, and Plant City), each with varying distances between the groves and the processing plants. Each grove and processing plant has a set capacity. What is one constraint in this problem?

The total pounds of oranges that can be processed This is a fixed capacity at the plant's individual capacity, so the total is therefore a constraint.

Based on the activity node below, what does 2 represent? Using the diagram below, what is the late finish of Task D?

Early start is located in the upper left side of the activity node. Late finish is indicated in the bottom right of the activity node.

In setting up an LP problem, the concepts of minimization and maximization are critical. In context to LP, what do these concepts have to do with the objective function of the LP problem

Minimization and maximization are two possible goals of the objective function in an LP problem. The answer to an LP problem is characterized as either maximizing the creation of something of value, such as profit, or minimizing the expenditure of something of value, such as costs. This answer is known as the objective function of the LP problem, it is the optimal solution given the set of constraints and other values.

The Economic Order Quantity (EOQ)

The economic order quantity (EOQ) is the amount of inventory a business should order to minimize the total amount of inventory costs, including carrying, ordering, and shortage costs.

The expected monetary value for Alternative A in the table is $

Multiply the value of each state of nature by its probability of occurrence, then add the values together. Subtract the cost estimate from the revenue estimate to get the EMV. The calculation is as follows: Revenue calculation for alternative A is =(C4 * D4) + (E4 * F4) = $51,000. Cost calculation for alternative A is =(C3 * D3) + (E3 * F3) = $36,000. EMV calculation is Revenue - Cost = $51,000 - $36,000 = $15,000.

For business quantitative analysis, is it a fact that the only important data to be analyzed is financial data?

No, because all forms of performance metrics are used for quantitative analysis. Quantitative analysis uses financial and many other types of data relating to operations; only some of these are financial.

Maximin is also known by its other name, the ___ solution. Another method of determining expected value is based on selecting the minimum possible loss for a worst case. This is known as the ___ solution.

Pessimistic: This is another name for the maximin criteria. Minimax: A minimax decision is made by selecting the minimum possible loss of the worst alternative outcomes.

A Population:

A population refers to the entire set of data being evaluated.

Describe how to set up a linear programming problem for profit maximization.

Defining the problem is key to solving and initiating a tactic or strategy that benefits the firm. Because linear programming is an "optimization" model, the logical conclusion is that given the real constraints of either the business or the environment, the outcome is the best choice that is also feasible. When a business is not meeting its goals or is seeking to improve its performance, it often examines the conditions that define its operations. Obviously, firms always want to maximize money in (revenue) and minimize money out (expenses). Fundamentally, that is the standard unit of performance measurement. However, businesses operate in a multidimensional environment. Cost or expense reduction sounds like a natural tendency. However reducing one's labor force to zero would certainly reduce operating expense but greatly reduce its capacity to produce. Conversely, raising prices across the board on all products would have the potential to increase revenue, but probably would result in fewer units being sold and eventually a loss of customers. Therefore, understanding the problem and the corresponding solution strategy or directions is a key element in any model's success, including linear programming. Let's look at an example: A new, small company submits a design to an industry-based contest and wins the grand prize of $100,000. The new company now has choices to make as to what to do with sudden influx of cash. Prior to having the $100,000, the business was limited in its options. The company can deploy this new capital in several directions. It can put some of the money into a "money market" account or it can invest in a stocks and bonds. The money market account can achieve 6%, and the stocks and bond accounts can return a predictable 8%. However, the stock and bond markets has more fluctuations than the money market does. As a result, the company is inclined to invest at least as much into the money market account as the stocks and bonds accounts. When this scenario is reviewed, the issue or the "problem" should become clear. The company's challenge becomes its objective. When dealing with items like return on investment (ROI), the target is to maximize the ROI given the constraints whether those constraints are inherent, external, or self-imposed. Therefore, in this case, the problem definition is either a maximization of profit on a set investment level or a minimization on the amount of investment to make. Although the description of the problem is often indicative of whether a problem is a max or min issue, in real terms and in real business issues, one of the primary keys is to define not only the elements of the problem (decision variables, etc.) but also the "context" of the problem to be solved. This stage of setting up the linear programming components is often the purview of management versus analyst or decision scientist because experience becomes critical in gauging the nature and components of the challenge that faces the company. In fact, the mere recognition of such an issue is often more intuitive than analytical. In the above example the problem definition is a maximization problem. This particular maximization expression would be: Maximize: 0.08x+0.06y=Z The key is to not only know and understand the elements in the problem but also to be able to break down multiple elements that may be cost oriented or revenue oriented and determine the "objective" versus the constraint data. In another example: A manufacturing firm has a couple of sites, plant A and plant B. Each is capable of building four products: Product 1 Product 2 Product 3 Product 4 Each site has unique capacity as shown in the following table: Product Site A Site B Product 1 200 100 Product 2 60 200 Product 3 90 150 Product 4 130 80 The cost of operating each plant is different. Given a five-day workweek, Site A cost $40,000 per week, and Site B costs $55,000 per week. Obviously each product has potentially different price points in the market. Those price points are apt to change over time due to the following: - Market conditions - Macroeconomic activity - Competition However, the compelling feature of this type of situation is the cost of operations of the various plants or sites that are engaged. The mix optimization (objective) is clearly to minimize costs of operations as illustrated in the objective function: Minimize 40,000x+55,000y=Z Business problems, while often complex, require a skilled decision maker capable of understanding the issue at its core level, expressing the problem in a mathematical model (i.e., linear programming), and determining the optimal decision given a well-defined initiative.

What time series pattern does the following figure represent?

This represent seasonal variation, notice that there is a spike every Dec.-Jan. period. The overall trend can be determined; the bars within the graph continue to trend upward. Most graphs that show a long timeline are demonstrating where cyclical events occur. Downturns in economic situations like U.S. real estate sales are not seen in a single year.

Spreadsheets have many uses in a business, government, or educational setting:

1. Financial: Budgets, balance sheets, sales forecasting, payroll, taxes, investment proposals, mortgage calculations, and other accounting and financial applications 2. Scientific: Analyzing and comparing scientific data, tracking changes 3. Educational: Trends in enrollment, student grading, financial aid calculations, tracking courses 4. Sales and Marketing: Inventory tracking, sales forecasting, customer trends 5. Law Enforcement: tracking crime in areas, looking for trends, analysis of data ...as well as in many other areas.

Fred has three time estimates for his project activity. The optimistic estimate is 3 days, the most likely estimate is 4 days, and the pessimistic estimate is 11 days to complete the project activity. Use the beta distribution formula to determine the expected time to complete this project activity. Which of the following is NOT true about PERT?

(3+4*4+11)/6=(3+16+11)/6=30/6=5. PERT stands for Project Evaluation and Review Technique. PERT stands for program evaluation and review technique.

When multiplying fractions, multiply the numerators (top numbers) and multiply the denominators (bottom numbers). Simplify the result, if necessary. 1. (14)∙(23)=(212)=(16) 2. (23)∙−(17)=(−221) 3. 23)∙(−17)×(−18)=(2168)=(184)

1. Multiply the numerators, then multiply denominator. Simplify if necessary. 2. Because there is an odd number of negative signs in this product, the resulting product will be negative. 3. Because there are an even number of negative signs, the product will be positive.

When using expected monetary value calculations, careful thought must be given to better understand the risks involved with the decision alternatives. These are called states-of-nature and are scenarios, such as economic progress or economic downturn. Once the states-of-nature are determined, the ___ for each must be decided. Each alternative is ___ against the state of nature. Expected monetary value is ___ by determining the probability of all states-of-nature, and then multiplying that probability by the expected revenue for that alternative.

1. The probability or likelihood of each state-of-nature occurring must be determined. 2. Each alternative is evaluated against the state of nature. 3. Expected monetary values are found by calculating the probability of the state of nature, multiplied by the cost or revenue of the alternative.

An inspector wants to show the monthly defects on an assembly line for the year. A quality manager wants to show the monthly defects on an assembly line for the year for two sites. An analyst wants to see the percentage breakdown of types of defects on an assembly line, for all the defects over a month.

A bar chart is the easiest chart to display one variable. A stacked chart is suitable for showing two variables and month-to-month data. Pie charts can quickly represent a single variable as a percentage of all the defects as a pie chart.

Although there are many benefits to forecasting, the primary purpose of forecasting is to control the Hint, displayed below-Select-currentfuture-Select- situation in order to make better decisions.

Current Forecasting is used in order to help make better decisions in the current environment that may help the future, but this is not certain.

Sensitivity analysis is often used in conjunction with linear programming (LP). The key purpose for doing a sensitivity analysis is to:

Determine how sensitive the optimal solution is to changes. Sensitivity analysis identifies changes in the optimal solution when making changes to one or more of the variables used in the calculation.

Uncertain Outcome Scenario: We do not know which option will be best, there is no guarantee of an investment return. If we are optimistic about the investment, we will choose the option (perhaps bet on) the most potentially favorable positive outcome. If we are overly pessimistic, we will simply choose the option that has the least lost, which, in this case, is to do nothing.

Invest $10,000 a. Chance of gaining $20,000. - A 100% return on your initial investment. b. Chance of getting back $0. - You lost all of your initial investment. Invest $10,000. a. Chance of gaining $15,000. - A 50% return on your initial investment. b. Chance of getting back $5,000. - You lost 50% of your initial investment. Do nothing.

Which of the following is the term used for the outcome decision to be made that is governed by the constraints?

Objective Function: This is the linear function (equal sign) representing cost, profit, or some other quantity to be maximized or minimized subject to the constraints in the LP problem.

Describe the purpose of using activity cost crashing to manage project costs.

Project Management can be quite involved and complex. Throughout the life of the project, decisions are being made, primarily about: 1. Planning and breaking down tasks into smaller more manageable units. 2. Sequencing the tasks using logical preceding and succeeding task relationships and dependencies as well as the durations. 3. Managing, controlling, and completing the project in a timely manner using the critical path method and a Gantt chart. Closely managing a project is also about ensuring the cost effective completion of the scope. Project planning decisions should be made in order to best manage and minimize the resource time and budget. All projects should have a budget that is decided upon by the stakeholders. This might include the customer or the executive team. Budgets are established for the overall project and allocated to the necessary tasks. Minimal planning efforts can lead to underfunding activities and result in the jeopardizing of the overall scope, quality and effectiveness of the project. All projects should have a budget that is decided upon by the stakeholders. This might include the customer or the executive team. Budgets are established for the overall project and allocated to the necessary tasks. Minimal planning efforts can lead to underfunding activities and result in the jeopardizing of the overall scope, quality and effectiveness of the project. Budgets can be created and viewed in many different tables or chart configurations, demonstrating where the dollars are allocated. As an example, the budget can even be allocated by time period to create a Budget Gantt Chart. In this format, both time and budget are managed on the same chart. As the project progresses, the actual expenditures by time period are tracked to ensure appropriate spending has occurred. The table above helps demonstrate the following: In time period 5, $7K was spent versus the budgeted $5K.This can be seen by the reddish shading to both Design Activities. The difference in the budgeted versus planned may be due to overtime or additional consultants needed to work through an unanticipated issue. In period 6, there was a $1K excess spent in the Design Software Activity. Project budgets can be identified as on-budget, overspent, and under budget. When a project is on-budget, this demonstrates that the budgeted dollars for the activities is corresponding with the actual costs for the activities. When a project is overspent, this indicates that the actual cost for activities is exceeding the budget, and when a project is underspent, this indicates that the actual costs are less than the budgeted amount for each activity. Overspent budgets are a serious problem for a project manager. When budgets are overspent, the project manager must decide if there are ample savings in the project activities remaining to bring the project back to or even below the budget, or if the budget needs to be revised, increasing the funding to accommodate a continued overrun. While some organizations are very strict about increasing a budget, some do build contingency or funding reserves into projects. For this example project, performance is the constraint. Performance is the objective of the project, the scope of the work. All efforts would be made to adhere to the scope. The timeline of the project is the next priority, and lastly, management would be willing to accept some modest increases in the budget to make sure the performance of the new product is there. To understand this better, another way to review this table is that if cost were the constraint, management would most likely have to compromise on performance as extending the time usually grows the budget. A project manager has a few options when deadlines are in jeopardy of not being met. One option is to reduce the scope of the project. As demonstrated above, depending on management's priorities, this may not be possible. The other option is to "crash" the project with additional resources, which requires additional funding. The topic of crashing a project will be discussed in the next section. Project budget decisions are not an easy undertaking. A question such as, "Should the budget be increased to meet the scope?" is very typical, however, the complexity of such a question increases when the budget is increased. This is because the manager must now decide how the increased funding should be allocated. Additional questions such as, "Which tasks need additional resources in order to ensure timely delivery of the project objective?" need to be considered.

[True/False] For project scheduling, each subtask may include more than one type of resource.

True: Each subtask may involve multiple types of labor, such as plumbers and electricians. However, their costs are likely to be different.

What this means is that when a change is made in the ___ variable, then it causes some change in the ___ variable.

1. The independent variable causes the changes in the dependent variable. 2. The dependent variables change is caused by the change in the independent variable.

Continuing our example of H&S, we can review the appropriate data elements that would be used in a linear model. Follow through the previous information of the H&S marketing application: Marketing Application: Consider H&S again but from a cost minimization perspective. It was already decided to produce 225 hats and 100 scarves, which will produce a gross profit of $3,275. They are sure they can sell the hats and scarves but not without some advertising. H&S advertises on campus radio, the school newspaper, and social media as defined in the following table: MediaRadioNewspaperSocial MediaAudience (per ad)100200300Cost (per ad)20010050Maximum Per Week25710

Management could choose to approach this problem in a few ways. [1] One way might be to maximize the number of people reached in a week for a budget that cannot exceed $700. Let R = the number of radio ads N = the number of newspaper ads S = the number of social media ads Objective Function: Maximize Z=300R+200N+400S Subject to the following constraints: 100R+200N+300S≤$700 (Advertising Budget Constraint) R≤25 (Limit on number of Radio Ads) N≤7 (Limit on number of Newspaper Ads) S≤10 (Limit on number of Social Media Ads) R,N,S≥0 A typical setup of such a problem in Microsoft Excel would be as follows: The columns in the top table are for the decision variables. Note that only the coefficients of the objective function and constraint inequalities are presented. The right-hand side are the resources, in this case dollars, available to be used. The only numbers Solver will change are the cells in yellow. Solver sets up the problem as follows: (Note: Solver can be complicated. There are many resources, including videos, that can help learn how to use Solver.) The cells listed in the By Changing Cells in the following window are referring to the decision variables. Set Target Cell is referencing the objective function (making it a max or min value). When the "Solve" button is clicked, Solver provides the following answer in an instant: This solution shows the following Run 10 social media ads and two newspaper ads. 3,400 people will be reached. The full constraint of 10 social media ads is used. Two of seven newspaper ads were used. No radio ads were used. [2] Looking at the second approach to this problem, which is to minimize the budget need to reach a minimum of number of 5,000 people, the same data table is used. The formulation would be as follows. Objective Function: Minimize Z=200R+100N+50S Subject to the following constraints: 100R+200N+300S≥5,000 (Number of people reached constraint) R≤25 (Limit on number of radio ads) N≤7 (Limit on number of newspaper ads) S≤14 (Limit on number of social media ads) R,N,S≥0 Setting the problem up in Solver will look the same as example [1]: Solver provides the following solution: Exactly 5,000 people will be reached by running the following: Six radio ads (Only six of the available 25 radio ads were used.) Seven newspaper ads (All newspaper ads were used.) Ten social media ads (All social media ads were used.) This will cost $2,400. These two marketing applications use the same basic data. The data table applied to both examples. Note the difference between Examples 1 and 2: It is important to see that the objective of the first example became a constraint in the second example. The constraint in the first became objective in the second. This is not rare or isolated. The limits on the number of ads of each type were constraints in both problems. The key was in the problem statement. In Example 1, the goal was to maximize the number of people reached. In Example 2, the goal was to minimize the advertising dollars spent. In each case the objective was clear. In the real world, the story problem is not presented as it might be in a textbook. Examples 1 and 2 basically are solving the same problem. Which approach should be chosen? The answer is that it is up to the decision makers on what needs to be budgeted (constrained) and thus what needs to be optimized. Another approach might be to solve both and evaluate: [1] Reach 3,400 people at a cost of $700 or 20.6 cents per person reached. [2] Reach 5,000 people at a cost of $2,400 or 48 cents per person reached. Based on the results of the linear program, decisions can be made about the marketing solution that will reach the most people.

Read the following statement: In a business model, the most efficient manner to obtain the best or optimal output is through trial and error. Choose one of the following responses to support whether or not this statement is accurate.

No, statistical applications and models like linear programming point to the best alternative. Powerful decision-making tools like linear programming have not only been used in business, but they have also been successfully deployed in military operations where the offset is not merely the loss of money but the potential loss of life.

Calculate time estimates using the beta distribution method.

Project success is often determined by whether or not a project is delivered on budget and on time. A successful manager must know how to make accurate cost and time estimates for projects. There are several important steps to estimate project time accurately: 1. Identify all of the activities or work that must be done to complete the project. 2. Put the activities in the order that they need to be completed, noting any significant work that must be completed before another activity can begin. 3. Determine who needs to be involved in the activity time estimates, including groups that will be completing the work involved. 4. Estimate the time involved to complete each activity in the project. Build a buffer into the estimates for vacations, sick leave, meetings, and emergencies. Add the duration of each activity in a project to determine project duration.

Simplifying Radicals

Square roots can be simplified by factoring and removing perfect squares. √18=(3∙3∙2) = 3√(2)

What is the importance of summarizing spreadsheet data in a graph?

Summarizing data is usually one of the first steps in any kind of data analysis. From this summary, then the data set can be understood better. Frequently when summarizing data, categories of data are identified to enable a greater understanding. Relationships between the data categories can been understood quickly when looking at a picture rather than large tables of data.

Using the SQRT Function

The SQRT function returns the positive square root of a positive number. The following image shows the results of four equations. Notice what happens when a negative value is used. The SQRT function finds the square root of a positive number.

What is a key advantage in using closed-ended questions?

They are easier to quantify and categorize. This is because the answers themselves can act as the categories for data analysis purposes. For example, for a question on TV watching, there could be four specific time categories from which the respondent can choose. The researcher can make calculations from those responses.

Identify the type of objective for the following situation: Bell Technologies makes laptop and desktop computers. Each item has a different amount of labor to make it and a different amount of materials to construct it. They make twice the profit in selling laptops compared to desktops but must make some of both. What type of LP problem does this situation call for?

This objective seeks to maximize profits requiring a maximization objective.

The problem statements for the economic ordering quantity (EOQ) and economic production quantity (EPQ) problems are relatively easy to interpret and obtain the inputs necessary to run the models. This is clear in revisiting the problems from earlier in this module. Consider first the EOQ examples for purchased items:

[1] The ACME Motor Company assembles and sells electric motors of all kinds. For the Half-Horse (one half of a horsepower motor), they purchase the rotor assembly from another supplier. There is one rotor in each Half-Horse. The demand for the Half-Horse for the next year is 12,000. It costs $15 to place an order. It costs $1/year to hold a rotor in inventory. In this problem, the variable numbers are easily discernable: Demand = 12,000 per year Ordering cost = $15 per order Holding cost = $1 per item per year Note the cost of the item is not provided in this example. [2] Pens R US provides high-quality markers for whiteboards. They order their caps from a supplier. Each cap is the same color as the ink in the marker. The demand for black markers is projected to be 8 million for the next year. The cost of the caps in an order are $0.05 for orders up to 100,000 and $0.045 for orders over 100,000. The ordering cost is $9. It costs $0.015 to carry a cap in inventory per annum. Similar to the previous problem, the numbers are easy to get: Demand = 8,000,000 for the next year. Ordering Cost = $9 per order Holding Cost = $0.015 per item per year. In this case, we have unit pricing. It is provided because there is a price discount: $0.05 per cap up to 100,000 caps in an order $0.045 per cap for orders above 100,000 caps. For the EPQ or manufactured goods examples, the identification of the inputs is almost as easy. [1] ACME makes the housings for the Half-Horse themselves. Again, the demand for the Half-Horse for the next year is 12,000. Every time they run this housing, they incur $350 in setup costs, mostly in changing over the die. ACME can make 50 motors per hour, and it costs them $1.75 a year to keep a housing in stock. Demand = 12,000 housings per year Setup costs = $350 per run Rate of production = 50 housings per hour Holding cost = $1.75 per housing per year. The only issue here is that D and R are at different time periods. These need to be converted to daily rates for both as follows: Daily Demand = 12,000/365 = 32.87 ,round up to 33 housings per day Daily Production Rate = 50x8 = 400 housings per day Assume an eight-hour day (one production shift) unless otherwise stated. [2] Pens R US assembles their black markers at a rate of 3,000 per hour, and the factory runs three eight-hour shifts per day. The estimate for the coming year has been raised to 8.25 million. Because they have to change over the ink, each time they run the black markers the setup cost is $400. It costs them $0.16 to keep a marker in inventory for a year. Demand (D) = 8,250,000 markers per year Setup Costs (S) = $400 per run Rate of Production (R) = 3,000 markers per hour Holding Cost (H) = $0.16 per marker per year Again as in the previous case, the demand and rates of production have to be adjusted to daily demand and daily rates of production. Daily Demand =8,250,000/365 = 2,260.9 = 2,261 rounded-up. Daily Rate of Production = Hourly rate of production x 24 hours (the factory runs three eight-hour shifts a day) = 3,000x24 = 72,000 markers per day

[True/False] A purchasing agent wants to know the best quantity to order for a reorder of parts, and this should be modeled.

True: In this case there is an economic order quantity formula to model the decision-making.

[True/False] It does not matter which side (LHS or RHS) of the constraints you test for sensitivity.

False: Understanding and isolating the appropriate RHS constraints against the LHS objective is necessary to perform a correct sensitivity analysis.

[True/False] A stock broker is putting together the portfolio for a client and knows the client is very risk averse, and the portfolio should be modeled.

True: There are financial risk models that will help the stock broker to make the best decision for her client.

Variance

Variance is the measure of the distance between a specific data point and the mean of the data set.

When presented with a mathematical expression that involves multiple calculations, there is a particular order of operations that must occur.

1. Perform any calculations inside parentheses. (If there are parentheses inside parentheses, work with the inner parentheses first.) 2. Working left to right, perform all exponent calculations. 3. Working left to right, perform all multiplication and divisions, in order. 4. Working from left to right, perform all addition and subtraction, in order.

Project Tasks

Project tasks are scheduled in a rational sequence to meet the time objectives of the project. In preparing the schedule, the duration for each of the tasks and its subtasks is estimated, thus providing a total estimated duration for each task. The sum of the duration of all the tasks is the estimated duration for the whole project.

Describe how quantitative analysis is used in business decision-making.

Quantitative analysis applies mathematical computations and statistical modeling to a business scenario or problem in order to determine the results and to identify business alternatives and consequences. Therefore, quantitative analysis helps to determine results and solve problems for a company's strategic initiatives and regular business operations. Business managers spend much of their time on problem solving. Competitors can create problems when they introduce a new product. Lack of cash flow can create problems. Problems can be found in situations that involve maximizing profits, minimizing errors, increasing revenue, inventory analysis, and project scheduling. Good problem solvers will step back and look at at all of the issues surrounding a problem, not only the immediate problem, but all of the issues surrounding the problem. Quantitative analysis must be considered and planned for from the beginning of a project. This means that quantitative analysis must be considered at the time a problem is identified and, most importantly, before any decisions are made. Planning with quantitative analysis goals in mind from the beginning of the planning process allows the analysis to provide data that gain a deeper understanding of the problem and business decision alternatives. Instead of relying on instinct and past experience, quantitative analysis generates numerical data that are reported and interpreted through research that can then provide solid evidence to guide decisions supported by the results reported by the data. For example, if a manager wanted to increase production capacity with hopes of generating a desired profit, the manager would begin by evaluating the data to show how much production is needed to break even and then continue the analysis to determine the production increase required to produce the desired profit. In this example, quantitative analysis works with specific data to find solutions to production capacity problems. The resulting mathematical models used will interpret calculations to show if expanding capacity is attainable or if break even profit is attainable. To further the production example, mathematics and statistics can be used to evaluate quality assurance processes with the goal of reducing the amount of product being rejected due to errors during the production process. The data related to product quality would be collected at each stage in the production process for a predetermined period of time using identified measures. Mathematical computations and statistical models would then be used to conduct statistical analysis in order to analyze the data with the goal of revealing where in the production process the defects are occurring. Once the stage in the production process is identified, additional data may be collected to further analyze in an effort to provide evidence to assist with minimizing the defects and correcting the production process. This process will be repeated until the desired results are identified. Quantitative analysis helps identify the possible alternatives using metrics, a measurement used to gauge performance, instead of guesses. There are many business problems that can be solved using data metrics, such as operation expenditures, revenue, profit, and inventory ordering, all of which are considered specific numeric data. In medical research, a study might be performed to determine which treatment plan, surgery, or dietary changes has the highest survival rate or positive or negative influence for a particular disease over a set period of time. The quantitative analysis techniques are used by many professions to quantify ideas and practices. Nearly every industry or profession, for profit and not for profit, will use quantitative analysis to solve problems. The U.S. census keeps up with population in each age group and ethnic origin over time periods, helping government to better understand geographic movement and generational trends. Political parties analyze voter demographics, helping them to prepare and plan campaign strategies. Internet service providers track the number of visits to a company website over a period of time. Over and over again, businesses use quantitative analysis to analyze data. They use the appropriate quantitative methods to recognize trends and to understand potential causes of changes in data. Even a small change in data can represent a significant underlying problem. Analyzing numerical differences can stem from the use of mathematical models. Those models will incorporate common calculations such as addition, subtraction, multiplying, and dividing. Additional models that perform numerical analysis can include utilizing statistical averaging, finding the median or mode of a group of numbers, and the use of probabilities. Analyzing numerical differences in key business statistics or performance indicators may reflect and demonstrate a developing trend in one direction or another. For example, analyzing profit can reveal increasing or decreasing values or costs that appear to be increasing or decreasing. Quantitative analysis can make sense of various numbers using mathematical models and techniques. Analyzing numerical differences is used to help to resolve, reconcile, and understand the reasons behind the data differences. Numerical computation could be a simple math calculation, but usually more complex computation methods will provide a more accurate decision-making tool. Numerical data are the basis for quantitative analysis. Expert opinion, surveys, and open-ended questions do not provide the specific, numerical data needed for a quantitative analysis that ultimately leads to solving many business problems.

When a business is confronted with different types of numbers representing a business process, how can quantitative analysis be used?

Quantitative analysis can make sense of the various numbers using the best technique. Quantitative analysis has many techniques available for analyzing a wide range of data.

Using the MAX and MIN Function

The MAX function displays the highest cell value included in the argument. The MIN function displays the lowest cell value included in the argument. Similar to the MEDIAN function, the MIN and MAX functions find the lowest and highest values in a dataset. 1. Begin by clicking on an empty cell and then type in the equal sign (=). 2. Type "MAX" or "MIN," without the quotation marks to tell the spreadsheet to use this function. 3. Type an opening parenthesis. 4. Select the cell range. 5. Type the closing parenthesis. 6. Press the enter key on your keyboard to calculate.

Summary

In this unit, we discussed the importance of using probabilities and descriptive statistics to quantify an event outcome probability. Descriptive statistics help businesses describe important features of a data set. The summaries provided by statistics can sometimes leave out important details. But certainly, understanding the median, mode, mean, and variance of a data set is a powerful tool when trying to understand large volume of data. The mean of the data set is the average. A moving average is found by taking several months of data and dividing by the number of months under consideration. Then the next month, the most recent month is added in and the oldest month is dropped off. Simple moving averages can be helpful in identifying trend direction. A weighted moving average gives extra importance, or weight, to the more recent months of data because they are more relevant than data points in the distant past. Weights stated in percentages and should add up to 100 percent. The weighted average is the sum of the weighted numbers under consideration. Mode is the number that occurs most often in a set of numbers. If all numbers occur only one time, there is no mode. A data set can have more than one mode. To find the median, arrange the data in increasing sequence, then choose the middle term. If there are two middle terms, add the values and divide by 2. Variance is the measure of distance between data points and the mean. Probability provides a way to quantify uncertainty. Predicting uncertainty with the use of numerical data is beneficial to many organizations and helps managers plan for the future. In this unit, we discussed three types of probability: Conditional: an event is dependent on the outcome of the previous event. Joint: two events occur at the same time. Marginal: one event is not conditioned on another event. A time series is a resulting set of data points taken over specific time intervals, such as a day, week, month, or quarter. There are four possible types of time series: Trend: the general upward or downward movement of data over a relatively long period of time. Seasonal: an upward or downward movement in data that repeats at regular intervals. Cyclical: a pattern that tends to repeat itself every few years. Random: unpredictable trends in data. Time series analysis provides information regarding trends that helps businesses with forecasting. When it is impossible to study an entire population, it is necessary to take a sample of the population. Four methods of sampling are: Convenience: finding the easiest and most accessible way to get a sample. Random: every item in the population has an equal chance of being selected. Stratification: choosing a set number of subjects from each category or classification of data. Systematic: choosing every nth item in the population. From the sample study results, a business can infer the same characteristics about the entire population. Therefore, choosing a sample that is representative of the entire population of the items is critical. For quantitative analysis, correlation is used to compare two variables to see if there is a possible relationship between them. Correlation of data is demonstrated by graphing data points and then analyzing the pattern of the data. On a scatterplot, a downward trend in correlated data moves from upper left to lower right, and an upward trend would be indicated by points moving from lower left to upper right. Business managers must know how to summarize, interpret, explain, and present large volumes of data in order to make predictions, understand underlying problems, and make decisions. Having a fundamental understanding of descriptive statistics and probability is critical for business success.

Summary:

In this module, we learned the value and importance of visual displays of data or graphs. Graphs take raw data and transform them into visual information that we can literally see. We can then use that information to make more informed and better decisions on the actions we take. Making decisions and taking actions is, after all, the purpose for quantitative analysis in the business world. We looked at a variety of graphs: Histograms, which are used to display visually the central tendencies and spread of the data. We have a tendency to think of the number (central tendency) and not consider the variation in measures (spread) in data we collect. Scatter plots are used to compare two variables to determine if there is a possible relationship between them. If there is a possible relationship, we could use one variable to predict the performance of the other. Bar and pie charts, which can be similar to histograms, but are used to display categorical or qualitative data, such as colors, defect types, etc. The goal of graphing data is to present and summarize key information about the data set in a clear and effective manner. We learned about: which graph to use when. how to construct the graph. how to interpret the graph. We learned that these kinds of transformations of data into information are best done using spreadsheets. In using spreadsheets, there are two types of data activities: Organization of the data in the spreadsheet, including reordering of the data Transformations of the data into information. The manipulation of the numerical data in the spreadsheet could include, but is not limited to the following: - Calculating descriptive statistics - Calculating frequency percentages - Anything else needed to help transform the raw data into information Looking at the data and information as numbers in a spreadsheet is good. In fact, there are some people who prefer this presentation and are very good at interpreting this data. For the vast majority, a graphic display of the ordered and transformed data is much better. Graphs convey a totality in visual format that is simply easier to see and interpret. They can easily communicate areas of focus and trends and, thus, provide a basis for better decisions and action planning. Lastly, we looked at linear and nonlinear equations. For two variables, we defined what it meant for an equation to be linear or nonlinear. Linear equations have the graph of a line and can be written in the form y=mx+b, where m = the slope and b = the y-intercept.The only power of x is 1.There are now x's in any denominators (bottom part of the fraction). Nonlinear equations are, well, anything other than as described above. The graphs of these equations are not straight lines.There may be x's in the denominators.The powers of x can be bigger than 1 e.g. x2, x3, etc.The powers of x can be between 0 and 1 e.g.x1/2whichisx−−√orx1/3whichisx−−√3 Graphs of linear and nonlinear equations are important to determine trends and important points like maximums and minimums. In the business world, managers must have the ability to transform raw data into information that will be the basis for sound decisions and action. This requires the ability to both create and interpret data organization and transformation in spreadsheets. Furthermore, creating and interpreting graphs of the spreadsheet information is equally important whether you are presenting a scenario and recommending decisions and actions or the audience of such a presentation.

What is the critical path?

The longest path through the network diagram, indicating the shortest amount of time in which a project can be completed. The critical path is the path with the longest time duration and represents the shortest amount of time in which a project can be completed.

Mode

The mode of a set of numbers is the number that occurs the most in the list. The word mode almost sounds like the word "most"! Find the mode of the following set of numbers: 3, 2, 2, 1, 5, 4, 3, 2. Because 2 occurs the most, it is the mode.

Using the numeric values shown, write the appropriate formula to first multiply 6 times 5, then add that to the square of 3, then divide that by 16 divided by 4. You must include parentheses in your answer. (Use the actual numbers in your formula).

((6X5) + (2^2)) / (16/4) In the first part of this formula, a double parentheses must be used to ensure that all the functions were executed before the division of 16 by 4.

At first glance, each of these types of constraints may not appear to be linear. However, consider the two variables, x and y. The constraint is that the ratio of x to y must be less than or equal to 0.7. Expressed mathematically, this becomes:

(xy)≤.7, which does not look linear. Using an algebraic equation, the constraint becomes: x≤.7y or x−.7y≤0, which is linear.

A project is made up of tasks, or activities. A network diagram, introduced in the previous section, identifies each task and provides a time estimate, or duration, needed to complete that task. The steps for creating a network diagram are:

1. Define the project and all of the project tasks. 2. Determine the task relationships and which activities must precede or succeed others. 3. Draw the network diagram connecting the activities. 4. Assign time estimates or durations to each activity.

When making multiple criteria decision-making (MCDM) evaluations based on the given criteria, there are generally two types of techniques to make this evaluation: rating or ranking. Ranking assigns the alternatives in order of preference, as in the first, second, and third choices, moving from most preferred to least preferred. Rating, on the other hand, assigns a specific numeric score based on a scale. Study the graphic above. The Score column represents a ___ type of score. On the braking row, this is known as a

1. A ranking compares each of the alternatives against each other. Since there are three car manufacturers, each would receive a ranking of either 1, 2, or 3 for the identified criteria. 2. A rating requires a range of measurement. Each alternative is scored separately based on how it measures for the particular criteria.

Identify the elements of below formulas, including constants, variables, and coefficients. 3x2+2y+7xy+5 The number 2 in the above formula is a ___? The letter y in the above formula is a ___?

1. Coefficients are the numbers in the terms of a formula preceding a variable letter. Any variable that is not preceded with a number is assumed to have a coefficient of 1. 2. Variables are terms in a formula that are letters and not numbers. They can change values.

1. In the EOQ model, several numeric inputs are used in the formula. One input quantifies the forecast level of usage for a product over a period, usually a year. This input is known as a(n) ___. 2. One input for the EOQ model is the costs associated with maintaining the product on the shelves. This input is known as a(n) ___. 3. One input for the EOQ model identifies the costs associated with the procurement process. This input is known as a(n) ___. 4. In the EOQ model, several numeric inputs are used in the formula. What is the final mathematical step in using the EOQ formula? 5. In calculating the EOQ, one cost that is needed calculates the cost of all procurement actions for the year. This input is known as an annual 6. In calculating the EOQ, one cost that is needed calculates the cost of storing the entire product in the warehouse during the year. This input is known as an annual ___.

1. Demand is the total amount of product forecasted for consumption during a period, usually one year. 2. Holding (or carrying) costs are the cost associated for keeping the product in inventory for a year. 3. Ordering costs are costs associated with the ordering process, that is, the costs incurred during the reorder of a product. 4. Take the square root The final mathematical step in the EOQ formula is to take the square root. 5. The annual ordering cost is a function of the number of order per year and the ordering costs per order. 6. The annual carrying cost is the total of the cost to store the item in the warehouse for some period, usually one year.

There are a number of ways to determine expected value. One way is for the decision maker to choose the best maximum gain from among the alternatives, known as the ___ solution. This is also known by its other name, the ___ solution. Another way to determine expected value is for the decision makers to choose the alternative whose worst (maximum) loss is better than the least (minimum) loss, known as the ___.

1. Maximax: This decision is made by selecting the alternative with the maximum of the best outcomes of the alternative. 2. Optimistic: This is another name for the maximax criteria. 3. Maximin: This decision is made by selecting the alternative with the maximum of the worst outcomes of the alternative.

When dividing fractions, first find the reciprocal of the divisor. This is simply found by "flipping" the fraction: The numerator becomes the denominator, and the denominator becomes the numerator. Next, use the rule for multiplying fractions, which is to multiply across the numerators, then the denominators. Dividing Fractions 1. (23)(17)=(23)∙(71)=(143) 2. −27/34=−27*43=−821 3. (23)5=(23)(51)=(23)∙(15)=(215)

1. Notice the use of the reciprocal of the divisor, then follow the rules for multiplying fractions. 2. A negative sign in the dividend or the divisor will result in a negative. 3. When dividing a fraction by a whole number, remember the whole number is divided by one. First, rewrite the whole number as a fraction. Multiply the reciprocal of the divisor by the top fraction. (Multiply across the numerators and denominators.)

The primary objective in inventory management is to minimize inventory costs. Inventory costs have three categories:

1. Ordering costs (also called setup costs) represent the cost of replenishing inventory, including receiving logistics. 2. Shortage costs (also called stockout costs) pertain to costs resulting from not having an item on the shelf for sale, generally the unrealized profit per unit. 3. Carrying costs (also called holding costs) involve costs of storing inventory, insurance, and managing inventory risk due to damage or theft.

There are generally two types of measurement scales used in quantitative analysis: quantitative and qualitative. A quantitative scale measures by way of a numeric value, in which the numeric value is assigned to the item being measured (such as a ruler measuring an item in inches). A qualitative scale, on the other hand, measures by way of an arbitrary scale, such as small, medium, and large. Study the graphic above. The Style row represents an opinion scale for the evaluator, which is an example of a ___ measurement scale, whereas the Braking row is known as a ___ measurement scale.

1. The Style row is a qualitative measurement, as it does not define the measurable elements in metric or numeric values. 2. The Braking of each alternative is measured by a finite measurable element. This element is measured in the number of feet to stop the car from 60 mph. Therefore, this is a numeric, not subjective, value.

The standard deviation for number set "B" is 107, 119, 120, 101, 116, 70, 96, 110, 95, 80

15.77 The standard deviation is 15.77. The formula for calculating the standard deviation for a population is: SD=∑(x−x¯)2n Calculating standard deviation is the square root of the sum of the squared differences between the mean of the data set and the individual data points, divided by the number of data points in the data set.

Calculate the mean, median, and mode for the data set provided in the table. (Use whole numbers, no decimal points.) 298, 130, 130, 158, 118, 128, 136, 168, 146, 115 115, 118, 128, 130, 130, 136, 146, 158, 168, 298 The mean of the number set "B" is The median of the number set "B" is The mode of the number set "B" is

153 The mean is calculated by summing a range of numbers and then dividing by the count of the numbers in the data set. 133 The median is calculated by arranging the data set from lowest to highest. For an odd number of data points, the median in the data point in the middle of the data set. For an even number of data points, the median is the point halfway between the two numbers in the middle of the data set. The mode is 130. The mode is the data point that occurs the most often in the entire data set.

Safe Driver Car Insurance company is trying to determine the probability of accidents based on age categories. The data from the previous year have been captured below: Age Group Total Drivers Number of Accidents Under 20 500 20 20-39 495 30 40-59 655 40 60 or over 350 25 Totals 2,000 115 Based on this data, which age bracket has the highest probability of having an accident based on the number of drivers in the age group? Based on this data, what is the probability of an accident occurring when members are under 40? Based on this data, what is the probability that an accident will not involve the 60 and over age group?

60 or over Comparing the number of accidents for the age group with the number of drivers in the group will produce the probability for that group. 0.43 or 43% The answer is 0.43 or 43%. This probability is found by combining the number of accidents of the two age groups and comparing it to the total number of accidents. Based on this data, what is the probability that an accident will not involve the 60 and over age group? 0.78 or 78% The probability is 0.78 or 78%. Finding the probability of accidents that do not involve the 60 and over group will require eliminating those accidents from the total of all accidents.

Although spreadsheet software contains many functions, the most common include sum, average, median, maximum, square root, power, and minimum. When performing quantitative analysis, deciding which essential math expressions are going to be used depends on the problem being solved and the data available. Gaining knowledge and skill in regards to spreadsheet creation, the purpose of functions, and how to organize data will help the analyst save time and ensure that appropriate calculations are made to the data and in preparing the results. Which of the following spreadsheet equations correctly uses the "power" function for a base of 25, raised to the power of 3?

=POWER(25,3) The POWER function will raise a given base by a specific power.

Explore techniques for decision making under uncertainty in greater detail. A good decision is based on a rational decision making process as above. One technique for decision making is to choose the decision that maximizes the worst payoff, in other words, the worst possible loss for one choice is a smaller number than the least possible loss of the other choices. This is called the maximin criterion and is a very conservative and pessimistic approach. A pessimist thinks the worst will happen and wants to make the best decision to prepare for that worst case scenario. Maximin selects the alternative whose worst loss is better than the minimum loss of the other alternatives.

A minimin is the alternative that represents the minimum of all of the minimum alternative costs. Minimax criterion minimizes the possible loss by selecting the alternative with the minimum possible loss for a worst case outcome. Minimax is the minimum of all of the maximums. Maximax is a more optimistic approach, more appropriate for a risk taker. The approach maximizes the best payoff. This approach focuses on the large possible gains, and fails to focus on the possible losses. Maximax is not frequently used by a business because this method could bankrupt a company if losses were not evaluated. These criteria provide a continuum of optimism to pessimism as follows: Maximax - Most Optimistic Maximin - Less Optimistic Minimax - Less Pessimistic Minimin - Most Pessimistic To better understand these terms, let's review a few examples. There are risks of getting in an accident and being responsible for the deductible. Based on the following table, an person seeking insurance would need to choose which plan would meet their needs. Car Insurance Monthly Premium RatesMinimum Cost for Each PlanDeductible$100$500$750$1000Plan A775575300275275Plan B625400225165165Plan C600500270220220 Using the minimin technique, or minimum of the minimums, you would choose Plan B with $1,000 deductible. The decision maker in this case is betting on not having an accident and is a risk taker. Now look at another similar scenario. The following table shows maximum profits expected for an artist depending on the location and the weather on the day of the art show (each venue will require the artist to pay a participant fee which is unspecified): HotClearRainyColdMaximum ProfitConference hall500600300200600Outdoor art festival500650-100165650Sidewalk sale300500-50220500 The maximax optimistic approach will choose the alternative with the maximum profit. In the above scenario the maximax would be the outdoor art festival. Thus, the maximax choice is clearly a risk taking strategy in which this decision maker is only looking at where the most return can be made. HotClearRainyColdMinimum ProfitConference hall500600300200200Outdoor art festival500650-100165-100Sidewalk sale300500-50220-50 The pessimistic, maximin choice would look for the maximum of the minimum profits. In this case, the conference hall has the highest minimum profit. This is a risk-averse choice. The Conference Hall choice is the only option where the artist knows for certain that no loss will be incurred.

To add or subtract fractions, first find a common denominator between all of the fractions. Then, keep the denominator and add or subtract the numerators. Adding and Subtracting with Fractions 1. (14)+(24)=(312)+(612)=(912) 2. 23+17=1421+321=1721 3. 13+17+18=56168+24168+21168=101168

A quick way to find the common denominator is to multiply the denominator of one fraction by the denominator of the other. Sometimes this will create a fraction that needs to be simplified in the end.

A Subset:

A sample set is a subset of that population. Sometimes it is not realistic to evaluate an entire population, so a sample set must be acquired. A sample size is the number of data points used for the statistical analysis. It is important to have enough data points in the analysis to ensure the analysis performed is descriptive of the larger population.

The EOQ and EPQ formulas can be enhanced to cover the following kinds of twists on the basic scenario:

Adding probabilities to demand Adding lead times with and without probability distributions Considering if the item is perishable and in what time frame What percentage of the order or production run is defective historically Reliability of having inventory to ensure good customer service The basic EOQ and EPQ formulas of F. W. Harris just become more complicated with additional terms or factors.

Slack Time

After establishing the ES, EF, LS, and LF for each activity, one more calculation remains. This will be a calculation for how much slack time each task has. This is an important final determinant for establishing the project's critical path. Slack time is the amount of time a task can be delayed without impacting the overall project duration. Any task that is on the critical path will not have any slack time, it will be zero. Slack time (SL) is calculated using the following formula:

Decision-making is one of management's most important responsibilities. In order to make the best decisions, management often turns to quantitative methods. Inherent in the process of decision-making is the fact that a decision must be made on ___ solutions to the situation presented. When defining alternatives, there can be ___ alternatives at each decision point.

Alternative: If there were no alternatives to consider, then a decision is not necessary; there is only one course of action. Multiple: Although there are typically only two alternatives at any one decision point, there could be more.

Several formulas used in inventory management are:

Annual Holding Cost: Average inventory level * holding cost per unit per year Annual Ordering Cost: Annual demand / order quantity * cost per order Total Inventory Cost: Holding cost + ordering cost A careful balance must be made between ordering costs and holding costs. If orders are always small, the total ordering costs will be higher and the holding costs will be less. With large orders, the total ordering costs will be less but the holding costs will be higher. Inventory models are used to help make minimize the total inventory costs.

At what point during the decision-making process should quantitative analysis be used?

At the beginning before the decision is made The analysis is needed in order for the decision to be made in the first place.

Putting it to use: Use spreadsheet software to solve math problems needed to perform quantitative analysis.

Business analysts and accountants have used spreadsheets for many years in their paper format-that is when they were large sheets of paper used to organize business data and transactions so they could be used by businesses to make decisions. With the introduction of VisiCalc in 1978, electronic spreadsheets converted the business analyst's once large sheet of paper with columns and rows into a single electronic document that makes it possible to present large amounts of data. An electronic spreadsheet, or spreadsheet for short, is an interactive program that enables users to layout numeric data in a grid of columns and rows. The electronic data can be organized and manipulated with the use of math expressions to complete quantitative analysis that is then presented in reports and presentations as tables and graphs. All electronic spreadsheet software has similar basic functionalities. For the purpose of this module and course, you may use any spreadsheet software you prefer to develop your quantitative analysis skills and complete the exercises and problems.

Sensitivity analysis is often used in conjunction with linear programming (LP). How is sensitivity analysis performed?

By changing one of the constraints or change variables to determine the level of change in the optimal solution. Sensitivity analysis is done by changing one of the change variables or constraints and then seeing the change in the optimal solution.

Use spreadsheet software to calculate reorder points with a known demand.

Determining the appropriate time to reorder more inventory is important for managing inventory costs. The following video explains how demand, number of working days per year, and lead time are used to determine reorder points. Determining when to place an order requires understanding both the product demand and the lead time. Lead time is the amount of time it takes to receive products once they have been ordered. There must be enough inventory on hand to meet the demand during the time between ordering and receiving the inventory. The formula for the reorder point (ROP) is demand per day times lead time for a new order in days. Although the formulas for EOQ, EPQ, and others are taught in college courses, practitioners use software to make these calculations. Previously practitioners wrote their own software, now these calculations are usually done in SAP, Oracle, or other ERP systems. Planners perform these calculations for thousands of items as opposed to the one item at a time approach as shown in this module. Below is a simple spreadsheet EOQ calculator. It will be used to confirm the results achieved in our various examples. VariableDemandAnnual Demand of ItemDOrdering CostsCost per OrderOHolding CostsAnnual cost to hold one itemHEconomic Order Quantity=Orders per year=Average Inventory= Inputs are typed into the yellow highlighted cells, and the output are in the orange highlighted cells. Orders per year = the number of times the item is to be ordered = Demand/EOQ Average Inventory = the average inventory of the item = EOQ/2 Consider again the Sandy Beaches example: Sandy Beaches Inc. sells beach towels. The cost per order is $200, and carrying costs are $4 per unit. The store sells 10,000 beach towels each year. Delivery of an order takes 3 days. Assuming the store is open 200 days per year, the daily demand is 50 towels. The calculated reorder point is D * L = 50 * 3 = 150. The store should place another order for towels when inventory reaches 150. VariableDemandAnnual Demand of ItemD10,000Ordering CostsCost per OrderO$200Holding CostsAnnual cost to hold one itemH$4.00Economic Order Quantity=1,000Orders per year=10Average Inventory=500 Two other examples were presented but not solved. Here are those problems with solutions provided by the software. [1] The ACME Motor Company assembles and sells electric motors of all kinds. For the Half-Horse (one half of a horsepower motor), they purchase the rotor assembly from another supplier. There is one rotor in each Half-Horse. The demand for the Half-Horse for the next year is 12,000. It costs $15 to place an order. It costs $1/year to hold a rotor in inventory. In these kinds of problems, the variable numbers are easily discernable. D = Demand = 12,000 per year O = Ordering Cost = $15 per order H = Holding Cost = $1 per item per year. VariableDemandAnnual Demand of ItemD12,000Ordering CostsCost per OrderO$15Holding CostsAnnual cost to hold one itemH$1.00Economic Order Quantity=600Orders per year=20Average Inventory=300 Six hundred of the rotors should be ordered at time. This would result in 20 orders for the year and an average inventory of 300 rotors. To calculate the number of workdays in an order cycle, divide the number of workdays by the number of orders placed. [2] Pens R US provides high-quality markers for whiteboards. They order their caps from a supplier. Each cap is the same color as the ink in the marker. The demand for black markers is projected to be 8 million for the next year. The ordering cost is $9. It costs $0.015 to carry a cap in inventory per annum. (The price part of this problem has been left out in order to use the base EOQ formula). Similar to the previous problem, the numbers are easy to get. D = Demand = 8,000,000 for the next year O = Ordering Cost = $9 per order H = Holding Cost = $0.015 per item per year he EOQ is 97,980 caps. In a year, 82 orders will be placed. To calculate the number of workdays in an order cycle, divide the number of workdays by the number of orders placed. For example, if this company had 246 workdays each year, the number of days in an order cycle would be 246/82 or three days. Recall in the original problem that the cost of the caps in an order are $0.05 for orders up to 100,000 and $0.045 for orders over 100,000. Because the EOQ is so close to the price discount point of 100,000 caps, it is worth calculating the total cost at EOQ and at 100,001 caps. DemandOrder QuantityItem CostOrders per YearOrder Cost per YearAverage InventoryInventory Holding CostTotal CostEOQ8,000,00097,980$400,00082$73848990$735$401,473Quantity Discount8,000,000100,001$360,00080$72050001$750$361470 There is a $40,000 annual savings by increasing the order size 2,021 caps. It is well worth doing. Software can also calculate a variety of options and alternatives over a vast number of items. The planner only then has to review the results and look at the exceptions which can also be highlighted by the software.

Putting it to use: Use spreadsheet software to calculate the expected value for decision making.

Electronic spreadsheets can be used to draw decision trees and calculate the estimated value. Some electronic spreadsheet programs, such as Excel, have add-in programs that can be added to help create and evaluate decision models. For example, TreePlan is an available add-in program for Excel. Also, Excel QM add-in can solve decision analysis problems without probabilities, as well as solve problems using expected value criteria. The following diagram shows an example of a simple decision tree, using columns to represent alternatives, states-of-nature, potential revenue and probability.

Population Parameter

For each of the quantitative variables, we would have the opportunity to calculate descriptive statistics. For the qualitative variables, we can establish facts and opinions, which help with subjective analysis. These measures on population attributes are called population parameters or simply parameters. If and when we know these parameters, we can use them reliably in making decisions and taking actions.

Forecasting

Forecasting is a method or process used to make decisions about events where the actual outcomes have not yet occurred. Quantitative forecasting models should be used to help forecast future data as a function of past data. Forecasting models are most appropriate when past data is available to be quantified. A common example might be estimating customer demand for new products based on prior sales efforts of other new products. Forecasting methods are usually applied to short-range decisions, not exceeding one to two years.

Not all data points will have a linear relationship with a constant rate of change or slope. There are several methods used to determine if data points represent a linear equation or not:

Graph the data points. Look at the equation to see if all variables are expressed to the first power. (Remember that linear equations have variables expressed in the first power only). Evaluate the slope between points to determine if the slope is constant between all of the points given. Consider the following example: Suppose a television worth $1,200 decreases 20% of its value each year. YearValue in Dollars 0 1,200 1 960 2 768 3 491.52 If the value of the television over a 30-year period was graphed, it would look like this: As you can see, there is more decline in value over the first few years than in later years. The graph clearly demonstrates that this scenario does not represent a linear equation. Remember, in a linear equation the variables are in the first power. In a nonlinear equation, the power of x will be something other than 1. The following equations represent nonlinear equations: Equation x2+2x+4 Highest Power of x 2 Equation √x Highest Power of x 1/2 Another clue that the data table does not represent a linear equation, is that the slope between two points is not the same throughout the graph. Use this formula to determine the slope or rate of change between two points: y2−y1/x2−x1. The slope between the purchase and the end of the first year, (0, 1200) and (1, 960), is -240. The slope between the points from the end of the first year and the end of the second year, (1, 960) and (2, 768) is -192.

Identify the type of objective for the following situation: A company has seen a rise in its costs of scrap on the production line between making the Deluxe model, which gives a higher profit and the Standard model, which sells for a lesser profit. What type of LP problem does this situation call for?

In order to reduce production costs, a minimization objective is appropriate.

Solving linear programming (LP) problems involves making the best decision given a specific situation. This is known as the objective of the LP problem. Identify the type of objective for the following situation: A company has seen a rise in its shipping costs between its manufacturing plant and its distribution centers. What type of LP problem does this situation call for?

In order to reduce shipping costs, a minimization objective is appropriate.

Stationary Time Series

In some instances when graphing a series of data points over a period of time that the data points appear to be independent, or not related to, the variable of time. This is known as a stationary time series. A horizontal pattern fluctuates around a constant mean. There may be some variation but the data tends to fluctuate around a constant mean.

Identify the components of inventory costs.

Inventory is something that manufacturing, distribution, and retail businesses have to manage. Inventory is defined as any goods held in stock for immediate or future use. There is a relationship between inventory and cash: 1. It costs money to buy or create inventory. If the inventory is purchased from another company, the supplying company will expect to be paid. If the inventory was created from a manufacturing or assembly process, overhead expenses of utilities, worker salaries, and other building expenses including rent must be allocated to production of these products. The utilities, rent, and salaries must be paid. Thus, inventory consumes cash, or cash is transformed to inventory. 2. Generally, when inventory increases, cash decreases. When inventory decreases (i.e., sold off), cash increases. Inventory is a critical component of cash management in many companies.

Explain the outcome of a calculated probability for an event.

Life is full of uncertainties. Imagine if we could always be certain of what the stock market would do on a given day...we would never lose money on an investment! If we knew the weather forecast with certainty, we would never get caught in the rain again. Probability is used to quantify risk and help a business manager make the best possible decisions in spite of the risks involved. Probability is a mathematical statement about the likelihood of an event occurring. For example, if the weather forecaster tells us there is a 75 percent chance of rain, then we would probably decide that the risk is too high to go to the beach. If a manager knew there was an 80 percent chance that sales will increase over the next year, a business might adjust inventory accordingly. Probability is used as a tool to enable a business to quantify risk and increase the likelihood of favorable outcomes and diminish the impact of failures. There are several ways to determine probability. Objective probability is based on history. If a process is repeated many times, eventually the relative frequency of an event will approach a specific value. The relative frequency approach assigns a numerical value to the probability based on a formula: P(event)=(the number of occurrences of an event/number of trials) Suppose a car dealership looked at the past 160 days in order to evaluate how many cars were sold each day. The number of cars sold always varied from zero to five, and there was never a day when more than five cars were sold. The following table represents the number of days for each quantity of cars sold for the 160-day period. Based on the previous 160 days, the probability of selling two cars a day is 35 percent. To find the likelihood of selling two or three cars in a day, add the two probabilities together: Because P(2)=0.35 and P(3)=0.2,the P(2 or 3)=0.35+0.2=0.55 (or 55%). Subjective probability is used when historical data are not available. This type of probability is based on opinion, personal experience, or judgment. The probability determined or used is derived from an individual's judgment or experience about whether a specific outcome is likely to occur. The subjective probability determined does not use any formal calculations; instead, it only reflects the subject's opinions and past experience. For example, opinion polls are used to predict election results. Subjective probabilities are like personalities in which they differ from person to person and contain a high degree of bias. An example of subjective probability could be asking home team baseball fans, before the baseball season starts, the chances of their team winning the world series. A typical committed fan will most likely reply using an actual percentage, such as the team has a 55 percent chance of winning the world series; however, there is no mathematical equation or proof behind the answer! Although a business manager may utilize subjective probability, it would be far more precise to use objective, calculated probabilities. The business benefits by quantifying the risk, or uncertainty. This is best accomplished by preparing and using calculated probabilities.

A company is considering building a new plant. It is considering a small, medium, or a large facility. The health of the economy will directly affect the profitability of the new plant, such as whether the economy will expand or contract, thereby providing either a lot of potential profit or little potential profit for a contracted economy. For each of the identified locations A-C in the diagram below, choose the appropriate label for each part. Location A Location B Location C

Location A is an alternative, a point in which a decision maker decides to take a course of action over another course of action. An alternative precedes the state-of-nature under which it operates. Location B is an alternative, a point in which a decision maker decides to take a course of action over another course of action. The decision point precedes the state-of-nature under which it operates. Location C is the state-of-nature, which occurs after the alternative is made and must include all reasonable probabilities that add up to the total possible conditions under which the decision operates.

In most linear programming models it should be understood that the feasible region will not contain negative values. The non-negative constraints in a linear program would be: x,y≥0 In a manufacturing example where chairs and tables are produced, this non-negative constraint would mean that the company cannot make ___ amounts of chairs and tables.

Negative: The manufacturer cannot make negative chairs and tables.

The concept of minimization in linear programming is critical. Which of the following statements best describe minimization in relation to the objective function for a linear programming problem?

One of the possible goals of the objective function in an LP problem is to minimize something of value such as cost of operations. Minimization in LP refers to one of the possible goals of the objective function, or the solution, of an LP problem. Minimization seeks to minimize something of value such as the cost of operations. It is the opposite of the other possible goal of maximization.

In setting up an LP problem, the concept of maximization is critical. In context to LP and the objective function of the LP problem, maximization can be defined as what?

One of the possible goals of the objective function in an LP problem to maximize something of value such as profit. Maximization in LP refers to one of the possible goals of the objective function, or the solution, of an LP problem. Maximization seeks to maximize something of value such as profits. It is the opposite of the other possible goal of minimization.

This chart is a circle with six different sized sections labeled, from largest to smallest: Atlanta (blue), Miami (green), Orlando (purple), Tampa (orange), Savannah (aqua), and Columbus (red). This chart shows six columns. The x-axis shows the names of cities, from left to right: Atlanta, Columbus, Miami, Orlando, Savannah, and Tampa. The y-axis shows numbers in increments of 20 from 0 to 100, bottom to top. This chart shows six columns. The x-axis shows the names of cities, from left to right: Atlanta, Columbus, Miami, Orlando, Savannah, and Tampa. The y-axis shows numbers in increments of 50 from 0 to 200, bottom to top. Each column has two colors: blue for Series 1 and red for Series 2.

Pie charts can quickly represent a single variable as a percentage of a whole. A bar chart is the easiest chart to display one variable. A stacked chart is suitable for graphing two variables.

Moving Average

Random trend variations are hard to predict. Sometimes it is helpful to calculate a moving average. A moving average is found by adding several months of data together and dividing by the number of months under consideration. For example, to calculate the moving average for monthly sales over a three-month period, add the sales for each of the three months, and divide by 3. Using this approach, each month has the same level of importance. Suppose at the end of March a business wants to calculate average sales for the past three months. (January sales (2 months ago) + February sales (1 month ago) + March sales (last month))/3. Then at the end of April, the sales from January would drop off and the new moving average formula would be (February sales (2 months ago) + March sales (1 month ago) + April sales (last month))/3. At the end of each month when the moving average is calculated for the previous three months, the oldest month's sales drop off, and the newest month's data are added. The average is always moving to the most recent three-month period. Simple moving averages can be helpful in identifying trend direction.Calculate the three-month moving average for January through March from the following data set of production numbers. Round to nearest whole number (no decimal points):

The project manager in ABC Company determines the optimal mix of plastic resin to go into a new product line of cups. Given the constraints of labor and shipping time the manager needs to try to determine which is more sensitive to the optimal outcome, labor or shipping. The manager should:

Run a dual cost analysis in linear programming for each decision variable. Running a dual cost analysis in linear programming for each decision variable is otherwise known as changing the RHS of the binding constraints.

Simulation

Simulation, while not technically a part of linear programming, does have a number of similar characteristics. It is a quantitative tool or approach, it is model driven, it is key in the decision-making process, it uses mathematical expressions to compute output, and most importantly, it has capacity for sensitivity or risk analysis. However, there are some differences and benefits to using simulation vs. linear programming as a stand-alone function. The inputs in a classic linear programming model are considered, for the purpose of the model itself, to be certain. In the previous LP examples, we assumed the coefficients in the objective function and the values in the constraints were known. After the fact, we tested (sensitivity analysis) how much variation to these known coefficients or values the problem could tolerate before the original answer was no longer valid. That is different from a series of experiments run with controllable and probabilistic inputs. The probabilistic inputs are then randomly generated and the model generates a series of values. This is also known as a stochastic model. The other major difference is that simulation models are not considered optimization models like linear programming. It is more along the lines of assisting decision makers in designing systems for good performance rather than evaluating which choice(s) within a given system should be chosen. Simulation models typically have more flexibility and application because they inherently are not reliant on all data being known. In short, simulation is a way to model random events. Decision makers can see the outcomes of these experiments approximating real-world outcomes without the cost or expense of data mining or trial and error, and with increased applicability to other decision models like linear programming. Typically, the type of applications that use simulation models are: Inventory policy Transportation bookings (airline, train, etc.) New product development Waiting lines Traffic flow A key similarity between linear programming and simulation is risk analysis. Remember, risk analysis is the prediction of an outcome in the face of uncertainty. At the heart of simulation is a concept known as "what-if scenarios." These type of approaches generate values for the unknown or probabilistic inputs like labor requirements or demand and computing the results or output. The linear programming and simulation connection can most readily be found in risk analysis. The use of simulation as a part of the risk analysis "tool kit" in linear programming enhances the sensitivity analysis of linear programming. While the linear programming model provides an optimal answer given known or certain inputs, it does not account for random unknowns. If we examine the previously discussed linear programming sensitivity functions, the information is predicated on examining each component exclusively to all other components. Therefore, the test yields an output that defines which elements are most sensitive. This is valuable information but not the entire picture. We would need to simulate combinations of changes in a real-world setting. In addition, simulation in linear programming removes the bias in the manager or decision maker. One of the elements that is always uncertain is the effect of time on the known values. Most decisions that are significant enough to require linear programming are by nature strategic or consequential. For instance, one would not employ linear programming to optimize the amount of tea and coffee in the breakroom. The level of problem needs to be significant and perhaps have long-term impact. Once the decision is executed (buying a fleet of trucks, setting up a plant to manufacture hats vs. scarves, etc.), management will have a difficult time changing direction. Therefore, time is a component. To attempt to factor the effect of time in the decision model, the decision maker should attempt to factor in multiple changes to the input side without bias. Labor rates, inflation, raw material costs are just a few elements that can change unpredictably over time. Managers tend to have a preconceived idea of where these costs or inputs might be over time, and this leads to bias within the sensitivity analysis. Randomizing these values decreases bias and provides clarity to the decision-making process. Using linear programming and employing simulation (or what-if scenarios) provide for multi-dimensional sensitivity analysis while removing bias from the experiment. When linear programming tests the limits of any of the values, whether the RHS of the coefficients or LHS of the objective function, the sensitivity analysis is assuming uncertainty in that value. Hence the dependence of that value on the "independent" variable. As such, it is essentially a "what-if scenario" just like we have in a pure simulation. The uncertainty is tested within a range of acceptability and then, not only do we know the latitude or risk tolerance of that value, we understand the experiment or simulation itself. Previous Hats and Scarf example: Constraint 1: .4H+.3S≤120hours (knitting machine availability) Constraint 2: .4H+.2S≤110hours (finishing activities labor availability) In the previous example with hats and scarves, the LHS of constraint 1 was more sensitive to the profit objective than the LHS of objective 2. The "what if" analysis would yield to management that managing the hours of machine time would benefit the organization more than managing the labor availability in the manufacturing of hats and scarves. In other words, the risk is embedded in machine time more than labor time and the simulation allows us to measure the net effect. In addition, if we assume that one coefficient at a time is unknown or uncertain in the objective function LHS, we can simulate the net effect (what if) of each of these coefficients changing within the range of optimality. From that simulation, we can determine, over a period of time, areas of management concentration or locking up long-term raw material pricing or whatever variable will insulate the company from risk over a period of time. Taking this a step further, in the linear programming sensitivity analysis we did not test the RHS for randomness or uncertainty. Maximize Profits: 11H + 8S = Z Remember hats ("H") represented the "Y" axis or the dependent variable, and scarves ("S") were on the "X" axis or the independent variable. In a linear programming simulation, we would randomly change the values of the independent values to see the effect on the dependent variables. This allows us to run a series of what if scenarios and test the relationship through simulation of the dependent variable to the independent variable and then evaluate the resulting output to determine the effect on maximization. The key difference is that in typical linear programming sensitivity, the decision maker or manager has a built-in bias to testing, whereas a random number approach removes bias and sets the experiment up as unbiased. Once the decision maker has all the risk parameters assessed, long-term policy can be established with less risk of failing over time.

Using the AVERAGE Function

The AVERAGE function will find the average of the values included in the argument. It calculates the average of the cell range identified in the argument. Using the budget example, let's calculate the average monthly expenses using the AVERAGE function. Here is how to enter the AVERAGE function. 1. Begin by clicking on an empty cell and then type in the equal sign (=). 2. Type "AVERAGE," without the quotation marks to tell the spreadsheet to use this function. 3. Type an opening parenthesis symbol. 4. Use a cell range to identify the total monthly expenses B8 through F8. To do this you can type in B8, a colon (:) to identify this is a range, and then F8. Or you can use your mouse by clicking on cell B8 and holding as you drag over to cell F8—notice how the cells are highlighted as you select them. Either way, the cell range is the same. 5. Type the closing parenthesis symbol. 6. Press the enter key on your keyboard to calculate.

Weighted moving averages place a weighting on data points, giving more weight to more current data points because they are more relevant than data points in the distant past. The assigned weighting placed on the data points should add up to 1 (or 100 percent). For each data point, the weight is multiplied by the value of the data point. To find the weighted moving average of the data set, all calculated weighted values are summed. Using the following revenue data table, calculate the weighted moving average for all four months. Round to nearest whole number (no decimal points): Month Weight Revenue 1 0.1 200 2 0.25 400 3 0.3 175 4 0.35 300

The answer is 278. For this question, calculating a weighted moving average is computed by first multiplying each revenue value by its weight, then adding the calculated values together.

Linear Programming Model

The final topic in our Big Picture is linear programming. Linear programming seeks to optimize a business value, usually minimizing costs or maximizing profits. The problem to be optimized is the objective function. Constraints are restrictions that limit the degree in which a company can pursue its objectives. Linear programming is a mathematical optimization model. In this unit, you will use formulas to find optimal solutions for a business. Most often these problems involve minimizing expenses and maximizing company profits. Inventory is usually one of the biggest costs for businesses. Inventory expenses fall into one of three categories: ordering costs, carrying costs, and shortage costs. Because businesses strive to minimize expenses, inventory costs are of high importance.

When identifying and solving business problems, it is very important to understand the root cause of the problem. What is the risk of not finding the root cause of the problem prior to implementing a new process or supposed solution?

The issue may come back again. There is a very real risk that problems will re-occur unless the root cause of the problem is addressed.

The ___ time is the smallest latest start time of the activities that immediately follow a task. Based on the activity node below, what does 12 represent?

The latest finish is calculated during the backward pass. The late finish time is the lowest late start time of the successor activities. The bottom row of the activity node has the late start and late finish.

If a production capacity planner is preparing to make decisions about how to increase production, what type of a problem will the linear program help solve?

The production capacity planner wants to make decisions about how to increase capacity or maximize capacity. This is not a minimization problem.

The standard deviation for number set "C" is 165, 183, 161, 190, 189, 141, 125, 135, 127, 158

The standard deviation is 23.52. The formula for calculating the standard deviation for a population is: SD=√∑(x−x¯)^2 / n Calculating standard deviation is the square root of the sum of the squared differences between the mean of the data set and the individual data points, divided by the number of data points in the data set.

In setting up an LP problem, the concepts of minimization and maximization are critical. After the determination is made that the objective function has to be either minimized or maximized, how is the objective function actually calculated to determine that maximum or minimum value?

The values of the decision variables are multiplied by the values of the costs (for minimization problems) or profit (for maximization problems), subject to the constraint values. For example, if a company had to choose the right production mix between Pump A and Pump B, each with its own profit level, the number of each type of pumps based on their profit potential would be made by the LP program. This calculation will be constrained by the different parts used to construct each of the two types of the pumps.

What is the key purpose behind using the beta distribution technique to estimate task times? Which of the following is NOT one of the three time estimates used in the beta distribution method? Which of the following is the main distinguishing factor of beta estimation for time duration and traditional time duration estimation? Which of the following description correctly explains how to calculate the expected activity completion time for an activity using the beta distribution method?

To provide a more accurate method. The three estimates used in this technique provide a more accurate method. Average time: Average time is not used in the beta method. Beta estimation considers three time estimates for each task. Beta estimation considers three time estimates and includes the high and low estimates to consider time variation. Add the optimistic time, the pessimistic time, and 4 times the most likely time needed to complete an activity. Divide the result by 6. The correct formula is: test=A+4M+P6 A is the optimistic time, M is the most likely time, and P is the pessimistic time estimate to complete an activity.

[True/False] The crew chief of a race car is deciding on the type of tires to put on the race car for the day's race.This decision should be modeled.

True: There are specific factors relating to weather and track type that govern this decision, and this should be modeled.

[True/False] A commercial fisherman is deciding on the type of fish to focus on for the day's catch; he knows that this is contingent upon the weather to a great deal. This decision should be modeled.

True: There are weather forecasting models to help the fisherman with this decision.

Pie Chart

Using graphs makes understanding complex data much easier. Pie charts are used when evaluating one item of data and its components. Pie charts are circles divided into sections representing different categories. The size of the pie sector represents the proportion of each category to the whole. For example, Joe's Pet Store sells five breeds of dogs. Joe made a pie chart to show what percentage of each breed he sold over the past year. 38% of the dogs sold were labs. It is easy to see that labs are sold more than any of the other breeds in Joe's Pet Store because that piece of the pie is the largest. When the pieces of the pie are similar in size it is not always easy to make the data distinctions with a pie chart. If the actual percents were not being displayed on the pie chart above, it would be difficult to distinguish the differences between bulldog, dachshund, and beagle. When data differences are very small and it is important to see the relative data differences clearly, a bar chart would be the better option.

Using the COUNT Function

Using the COUNT function will display the number of cells that have been included in the argument. This function is useful for quickly counting items on the sheet. Let us count how many expenses were paid during the five months included in this budget spreadsheet. 1. Begin by clicking on an empty cell and then type in the equal sign (=). 2. Type "COUNT," without the quotation marks to tell the spreadsheet to use this function. 3. Type an opening parenthesis symbol, select the range B2 through F7, and then the closing parenthesis symbol. 4. Press the enter key on your keyboard to calculate.

Identify the elements of a problem situation in terms of the decision to be made, probability of each event, and an event's consequences.

We often have to make decisions where there are alternatives. Consider the following two scenarios. - Choose between these two alternatives: - Alternative 1: Kick in the shin. - Alternative 2: You get $10. - Choosing between these two alternatives is probably pretty easy. You would take the $10. Because you are controlling the outcome based on your choice, you would repeat this game as long as you know with certainty what the result will be. This is a fixed outcome scenario. - Consider this scenario: - Alternative 1: If the next customer who enters the store has blue eyes, you lose $50. - Alternative 2: Otherwise you get $50. - This is a more complicated outcome because of the introduction of uncertainty, i.e., the unknown eye color of the next customer. The outcome is not in your control. This is an uncertain outcome scenario.

Exponent in Spreadsheets

When you raise a base number by an exponent, the calculation is multiplying that number itself the number of times equal to the power. To calculate three to the fourth power, this means that 3 is multiplied by itself four times. That is the same as the math equation of 3∙3∙3∙3. To complete this calculation in a spreadsheet: 1. Click on the cell that will contain your formula to make the cell active. 2. Type an equal sign (=), the base number (in this example, 3), a caret sign (^), and then the exponent value (in this example, 4). Note, this equation can be typed into the empty cell or formula bar. 3. Then press enter on your keyboard to calculate the formula.

Which of the following questions is NOT one of the fundamental decisions a business must make when managing inventory?

Where will the inventory be stored? Having storage space to hold inventory is certainly important and would be a part of the carrying costs. But this is not one of the two fundamental decisions a business must make when managing inventory. Businesses must decide how much to order and when to order inventory.

When a project manager must rein in a deadline and is not able to decrease the scope of the project, there are few options available:

When a project manager must rein in a deadline and is not able to decrease the scope of the project, there are few options available. The primary option the manager has is to "crash" the project with additional resources, which requires additional funding. Crashing the project is a method that will help the manager make decisions about the application of additional resources to work the necessary activities to meet the deadline. After identifying the critical path, the project manager can calculate the cost of speeding up, or crashing, certain activities on the project. Only critical path activities can be crashed. For each activity, a crashing cost per unit, using per day or week, will be compared to its normal cost. Once this information is known, the manager would assign additional resources to the critical activities for which the crashing cost per unit is the lowest. Each activity's cost slope must be calculated to decide what order the critical tasks will be crashed in. Determining the cost slope helps the manager decide if allocating the funding to the task is a decision worth making. Cost Slope=(Crash Cost−Normal Cost)/(Normal Duration−Crash Duration) The calculated activity cost slope is the added cost of saving a unit of time. The project managers and stakeholders can then decide if the increased funding is worthwhile. When preparing to determine the crash cost of each activity, it is best to assemble a table of the activities that will include the normal and crash durations and the normal and crash costs. The table would look like the following: Duration (weeks) Total Activity Cost Crash Cost per Week (Cost slope) Activity Normal Crash Normal Crash X 4 2 $10,000 $14,000 $2,000 Y 6 5 $30,000 $42,500 $12,500 Z 2 1 $8,000 $9,500 $1,500 By comparing the difference between the normal duration and the crash duration, and the normal and crash cost, the crash cost slope per week can be found. Using activity X as the example, notice that the difference in the normal duration versus the crashed duration is two weeks. The difference in the cost is $4,000. By dividing the crash cost of $4,000 by the duration difference of two weeks, this produces the crash cost per week of $2,000 found in column D. Once each of the critical path's activity crash costs has been determined, the manager would crash in the order of the least cost to the most.

Frank purchased 100 acres of farmland. He will use linear programming to determine how many acres of corn to plant and how many acres should be peas in order to maximize his profits. He makes $10 per bushel for peas and $7 per bushel for corn. The yield per acre for corn is 75 bushels and the yield for peas is 60 bushels per acre. He must leave 10% of his land uncultivated. What are the decision variables in this problem?

number of acres planted of each crop (corn and peas) Frank wants to find the crop combination that will maximize his profits.

The Big Top Travel and Entertainment Company is considering either to upgrade its existing office space or to build a new office facility. In addition to deciding between upgrading or building new, the company must also consider whether or not it is going to have an increase in overall revenue for the upcoming year. This will be based on the level of expected tourist in the United States. The publications for tourism indicate that U.S. travel will be increasing based upon the expectation that there will be lower prices for both airlines and vehicles. There is a small expectation that U.S. travel will decrease. After the managers have met, they decided that they, too, feel very favorable about the increases in U.S. travel, and have placed a fairly high probability of 85% on the revenue increasing. What is the correct expected value for upgrading the current office? What is the correct expected value for building a new office? Which alternative should the Big Top Travel and Entertainment Company pursue?

$243.75 The table below shows the expected revenue for each state-of-nature. To calculate the expected value, multiply .85 by $300 and .15 by -$75. Add the two products together. $530.00 The table below shows the expected revenue for each state-of-nature. Row 3 represents the values for building a new office. To calculate the expected value, multiply .85 by $650 and .15 by -$150. Add the two products together. Build NewIt would be appropriate to select the alternative with the highest expected value. The alternative "Build New" has the highest value when considering the possible outcomes of tourism increases and decreases of the Big Top Travel and Entertainment Company revenues.

Recall the information about how to calculate an expected value using a payoff table: - Determine each alternative being considered. - Determine the various conditions that are possible that the alternative might encounter. - Determine the probability, or likelihood, for each condition. - Organize the information in a table or spreadsheet to evaluate the data. - To calculate, determine the expected value for each alternative. First, you would multiply the value of the first condition times the probability of that condition occurring. Next, repeat the calculation for each condition. Finally, add all of the results of the newly calculated values together to establish the expected value of the alternative. Based on the spreadsheet graphic below, answer the following questions: A B C Alternatives Favorable Market (75%) Unfavorable Market (25%) 1 Build large plant $650 -$100 2 Build small plant $275 $50 3 Do nothing $0 $0 What is the calculated expected value if the company builds a large plant? What is the calculated expected value if the company builds a small plant? If the company just got an updated forecast for the state of the future economy, and the likelihood of an unfavorable market increases to 35%, what will be the expected value for building a large plant? What is the formula for calculating an alternative's expected value?

$462.50 Both condition's probabilities must be multiplied by the values prior to adding the new values together. The correct formula in the spreadsheet would be =(B1*0.75) + (C1*0.25) $218.75 Both condition's probabilities must be multiplied by the values prior to adding the new values together. The correct formula in the spreadsheet would be =(B2*0.75) + (C2*0.25). $387.50 Both probabilities for the conditions must be changed. The total probability for all conditions will equal 1. The new formula considering both probability changes will be =(B1•0.65) + (C1•0.35). Notice that the sum of all probabilities for the states-of-nature is 1. To calculate the expected value of each alternative multiply the value of the first condition by the probability of that condition occurring. Next, repeat the calculation for each condition. Finally, add all of the results of the newly calculated values together to establish the expected value of the alternative. The formulas to calculate the expected value of each alternative in the above table are as follows: Conference hall: ($500∙.15)+($600∙.55)+($300∙.20)+($200∙.10)=$485 Outdoor Art Festival: ($500∙.15)+($650∙.55)+(−$100∙.20)+($165∙.10)=$429 Sidewalk Sale: ($300∙.15)+($500∙.55)+(−$50∙.20)+($220∙.10)=$332 (alternative value * probability of occurrence) + (alternative value * probability of occurrence) This is the correct formula for this example. Each alternative must have a value under the possible occurrence and each occurrence must have a probability that is multiplied against the value.

Using the the numeric values shown in the following spreadsheet graphic, build a spreadsheet equation that will first subtract 10 from 42, then divide the result by 4, and then raise that result to the third power. (Use the actual numbers in your formula). Using the following numeric values: 35, 5, 10, and 8, create a spreadsheet formula that will first subtract 5 from 35, then divide that result by 10, add 8 to that result, and finally raise that result to the third power. Using the numeric values shown in the following spreadsheet graphic, create the appropriate equation for the following: from 544 subtract 4 raised to the third power, and then divide the result by 10. Using the numeric values shown in the following spreadsheet graphic, create an equation that will divide 3072 by 3 first and then find the fifth root of that result. Using spreadsheet software, find the answer to the following equation: =5+6^2. The result is Using spreadsheet software, find the answer to the following equation: =((2*5)+6^2)+5. The result is

((42-10) / 4)^3 The first term is a subtraction expression that must be done first. Next, the result of the first term must be divided by 4; this requires a second set of parentheses. Finally, the result of the previous calculation was raised by the exponent of 3. (((35-5) / 10) + 8)^3 This problem required three sets of parentheses to fulfill the requirement of the equation. (544-(4^3))/10 The exponential calculation must be carried out first within parentheses, and then that result is subtracted from 544. Finally, that result is then divided by 10. (3072/3)^(1/5) The division calculation must be carried out first within parentheses, and then the fifth root of that result is found using the caret sign and the fraction of 1/5 to represent the fifth root. The result is 41. Including parentheses is not necessary in this equation. The order of operations will be applied using the spreadsheet software. The result is 51.The parentheses are necessary in this equation. The order of operations will be applied using the spreadsheet software.

Which of the following indicates a nonlinear equation?

(3, 0), (3, -4), and (3, 5) y = 3x2 + 2 An exponent of 2 in the equation indicates this is a nonlinear equation.

Instead, we look at a smaller, representative subset of all incoming college freshman in the country. This representative subset would be far more accessible and less costly to collect. A subset of a population is called a sample. We could look at the same attributes as in the population and calculate the descriptive statistics and percentages of the sample quantitative measures and qualitative attributes. The results would be approximations or estimates of the population parameters. We call them sample statistics or simply statistics. Using samples is the easier and less costly way to obtain and calculate statistics to draw conclusions or make inferences regarding the population parameters. To summarize:

(Set) - Population (Size) - All Items of Interest (Descriptive Statistics are Called) - Parameters (Set) - Sample (Size) - A Subset of the Population Descriptive Statistics are Called - Statistics

When using entire data sets of business results, such as monthly sales revenue, monthly profit, monthly expenses, and other financial metrics, it is appropriate to give more weight to more current data points because they are more relevant than data points in the distant past. For example, if a business started the year by introducing a new product, the sales revenue increase achieved from the new product would likely not be expected to make a significant difference until mid-year. That allows advertising and customer feedback to make an impact. When developing a time series analysis of the monthly increase in sales, the analysis would use a weighted moving average, giving more weight to the more recent months that should be more relevant to the new product gaining a bigger sales impact than the earlier months. Using the following sales data table, what would be the likely weight assigned to the fourth month? Month Weight Sales 1 0.2 200 2 0.25 400 3 0.25 175 4 ? 300 Complete this table for calculating the moving average. What is the correct weight to assign for month four? Month Weight Sales 1 0.15 350 2 0.2 140 3 0.25 400 4 ? 700 The correct procedure for calculating a weighted average is to assign a weight to each item in the data set, then multiply the weight times the value. All newly calculated weight values are added together. Calculate the moving average for the following four months of sales: 120 35 60 75 Using the following monthly operating expense data table, calculate the weighted moving average for month one through month four. Round to nearest whole number (no decimal points): 50 100 180 157.5 Using the following data table, calculate the weighted moving average for quarter 1 through quarter 4. Round to nearest whole number (no decimal points): Quarter Weight Price 1 0.05 200 2 0.10 400 3 0.25 175 4 0.60 300 10 40 43.75 180

0.3 The assigned weighting placed on the data points should add up to 1 (or 100 percent). Adding the first three months and then subtracting from 1 will give the value of 0.3 for the fourth month's weight. 0.4 A weighted average calculation will assign weights to the data in the set. The weights should add up to 1. 290 The weighted average is 290. For this question, calculating a weighted moving average is computed by first multiplying each sales value by its weight, then adding the calculated values together. The answer is 488. For this question, calculating a weighted moving average is computed by first multiplying each revenue value by its weight, then adding the calculated values together. 274 The answer is 274. For this question, calculating a weighted moving average is computed by first multiplying each revenue value by its weight, then adding the calculated values together.

Using cell references and not the numeric values shown, build a formula that spreadsheet software will accept to first add 24 and 16 and then multiply the result by 3. Using cell references and not the numeric values shown, build a formula that spreadsheet software will accept and that will first subtract 18 from 258 and then divide that by the square of 4. Using cell references and not the numeric values shown, build a formula (without using a function) that will add the first 4 values in column A.

1. (B2+C5) * D3 The addition cell references should be written in parentheses, then multiplied by the cell reference for the number 3. 2. (B8-B10)/(D10^2) The subtraction should be written in parentheses so that the term will be handled first and then that result will be divided by the the square of 4. 3. A1 + A2 + A3 + A4 All formulas and functions must begin with the = sign. If the value in one of these cells changes, the formula will automatically recalculate the new total.

1. Calculate the sum of the following figures: -6, -17, -11, 2, 15. The result is 2. Multiply the following figures: -10 and 31. The result is 3. Divide -8 by 2. The result is

1. .17 The answer is -17. The basic rule for adding positive and negative numbers is to use the sign of the larger of all of the positive and negative numbers, and then subtract the difference. 2. The answer is -310. If you are multiplying two numbers together and both numbers are positive, the product (resulting answer from a multiplication) is always positive. If you are multiplying two negative numbers, the product is always positive. When multiplying a positive number and a negative number, the product is always negative. 3. The answer is -4. If you are dividing two numbers and both numbers are positive, the quotient (resulting answer from a multiplication) is always positive. If you are dividing two negative numbers, the quotient is always positive. When dividing a positive number and a negative number, the quotient is always negative.

Male Female Totals Voted Yes 40 25 65 Voted No 12 13 25 Didn't Vote 8 2 10 Totals 60 40 100 Using this table as a guide as to how the males and females in the community voted, determine the following probabilities What is the probability that a male would vote YES? Out of 60 males, 40 voted YES in the last election What is the probability that a female will vote? Out of 40 females, only 2 did not vote. What is the probability of people either not voting or voting NO? Add the probabilities for each event. Out of 100 people, 10 did not vote, and 25 voted NO. So a total of 35 people either didn't vote, or voted NO and the probability is In this example, the possible outcomes are mutually exclusive, or disjointed, because there can only be one outcome (or one way to vote) for each person.

1. 4060=23 or 66.67% probability of a male voting YES. 2. 3840=1920 or 95% probability that the females will vote. 3. 35/100 = 35%

Calculate the sum of the following figures: 1. 34, 10, -8, 25, 14 2. -7, 8, -2, 15, 8 Multiply the following figures: 1. -5 and -20 2. Divide -25 by -5

1. 75 The answer is 75. The basic rule for adding positive and negative numbers is to use the sign of the larger of all of the positive and negative numbers, and then subtract the difference. 2. 22 The answer is 22. The basic rule for adding positive and negative numbers is to use the sign of the larger of all of the positive and negative numbers, and then subtract the difference. 1. 100 The answer is 100. If you are multiplying two numbers together and both numbers are positive, the product (resulting answer from a multiplication) is always positive. If you are multiplying two negative numbers, the product is always positive. When multiplying a positive number times a negative number, the product is always negative. 2. 5 The answer is 5. If you are dividing two numbers and both numbers are positive, the quotient (resulting answer from a multiplication) is always positive. If you are dividing two negative numbers, the quotient is always positive. When dividing a positive number and a negative number, the quotient is always negative.

Presenting the decision to be made, as well as the alternatives and expected value of each alternative, is best organized in either a payoff table or a decision tree. A ___ is best used when there is one decision to be made, and a ___ is more likely to be used when there is a sequence of decisions to be made.

1. A payoff table is best used when there is one decision to be made. 2. A decision tree is more likely to be used when there is a sequence of decisions to be made.

Recall that to find the critical path, the longest path in the network diagram must be determined. To do this the four data elements are necessary:

1. Earliest start time (ES) - The earliest an activity can start without starting before any predecessor activities. 2. Earliest finish time (EF) - The earliest an activity can finish. This can be determined by adding the task duration to the earliest start time. 3. Latest start time (LS) - The latest time an activity can start without delaying the entire project. 4. Latest finish time (LF) - The latest time an activity can finish without delaying the entire project. The latest finish time is the smallest latest start (LS) time of the activities that immediately follow a task. Each of these must be determined in order to find the critical path of the project. When configuring a network diagram, the calculated ES, EF, LS, and LF will be put into each task node as the example shows: Each task node will be developed and the entire network diagram can be configured. An example is found below: Using an example of a software project, the network diagram with all activities appropriately labeled might present as the following:

There are five basic guidelines to remember when creating formulas in spreadsheet software:

1. All formulas MUST start with an equals sign (=). This tells the spreadsheet that the cell contains a formula and that it (i.e., the cell) is equal to the value the formula in the cell calculates. 2. The answer to the formula displays in the cell into which the formula is entered. 3. Cells are referenced in a formula by their column-row identifier, also called a cell address (i.e., A1, B2, D5, E15...). 4. The standard operator symbols for calculations include the following: plus sign addition (+) hyphen for subtraction (-) asterisk for multiplication (*) forward slash for division (/) caret for exponents (^) 5. You do not have to enter capital letters in your formula because the software will automatically capitalize them. To enter data, you must begin by clicking on that cell to make the cell active. Then type the data (i.e., text, numbers, dates, formulas) in the cell. The data can also be entered or edited using the formula bar. Cell addresses are used in formulas to reference or use the data within a cell for a calculation. Formulas can reference a single cell or a range of cells. Format Tip: If a value, a number entered into a cell, is too large for the width of the cell, you may see a set of symbols such as #####. To fix this, change the column width so that the full number shows. To do so, click the right edge of the column heading and drag it to the right.

1. Production and inventory management are two related functions within a company. One of the concepts that they share has to do with determining the best size for production lots and for ordering materials and raw goods from suppliers. What type of metric is being described in the following situation? The purchasing agent is getting ready to place a reorder of raw materials. The metric behind this concept is known as a(n) 2. What type of metric is being described in the following situation? There has been a dramatic increase in the cost of accessing the new Web-based purchasing system, and the operations manager is concerned about how this will affect their inventory management. The metric behind this concept is known as a(n) 3. What type of metric is being described in the following situation? The production foreman is noticing a large stack of work-in-process (WIP) in front of a work station. The metric behind this concept is known as a(n) 4. Production and inventory management are two related functions within a company. One of the concepts that they share has to do with determining the best size for production lots and for ordering materials and raw goods from suppliers. What type of metric is being described in the following situation? The sales manager is noticing a delay in finished products coming off the assembly line at the time promised. 5. Production and inventory management are two related functions within a company. One of the concepts that they share has to do with determining the best size for production lots and for ordering materials and raw goods from suppliers. What type of metric is being described in the following situation? The operations manager has just been informed by the key raw goods supplier that due to recent process improvement in the supplier's organization, the processing time for future orders will be reduced from two weeks to one week. The metric behind this concept is known as a(n) ___. 6. Production and inventory management are two related functions within a company. One of the concepts that they share has to do with determining the best size for production lots and for ordering materials and raw goods from suppliers. What type of metric is being described in the following situation? The CEO got a call from the number 1 customer who was complaining about a rise in the frequencies of backorders from the CEO's plant. The CEO discussed this with her production manager who stated that they were getting more stockoutages recently.

1. Because the purchasing agent is ready to make the order, he will need to use the EOQ to complete. 2. This cost increase to order will be used in the EOQ calculation, so this change could affect the current EOQ. 3. The EPQ is used to calculate the size of the production batches or lots. It seeks to optimize the lot size considering the lost time for unfinished materials in process for each batch and the changeover time for the work station. This WIP could be reduced with a smaller batch size. 4. The EPQ is used to calculate the size of the production batches or lots. It seeks to optimize the lot size considering the lost time for unfinished materials in process for each batch and the changeover time for the work station. This WIP could be reduced with a smaller batch size. 5. The RP is the inventory level at which a reorder will be initiated. It seeks to optimize the ordering so that not too much material is on hand or on order at any one time. It considers the lead time to get the ordered materials and the usage levels in production. This change in the processing time changes the lead time in the RP calculation. 6. he RP is the inventory level at which a reorder will be initiated. It seeks to optimize the ordering so that not too much material is on hand or on order at any one time. It considers the lead-time to get the ordered materials and the usage levels in production. This change in the processing time changes the lead-time in the RP calculation.

Type/Definition: If the population is of size N, we want a sample of size n where n are used in the sample Example: All incoming freshman in the United States population.

1. Convenience: In this method, the easiest and most accessible ways to get n items are used in the sample. [1] A sample of 50 incoming freshman is desired. Go to orientation at the closest college or university and use the first 50 freshmen who agree to share their info.[2] You have access to the admissions database of XYZ State University. Your sample is all of their incoming freshmen. 2. Random: In this method, we choose our n items in a manner which every item in the population has an equal chance of being selected. We put every incoming freshman's name on a card and put the cards in a very large hat. We shake the hat and then choose n cards where n tends to be numbers like 30, 50, 100, 250, 500, or 1,000. 3. Stratification: This method is used if the population has natural clusters, groupings, or strata. Divide the number of strata by n and choose that number (rounded up) from each strata. A sample of 100 freshman are desired. There are 50 states, which can be considered as strata. Select two students from each state. 4. Systematic: Number all the items in the population from 1 to N. Choose a random number between 1 and N. This is the starting point; label it "k." Choose another number as the interval. This second number, label it "m," could be randomly chosen or simple selected. From the starting point, k, choose every mth item as part of the sample. Continue until n items are chosen. Assume there are 180,000, therefore N=180,000 incoming freshman in the United States. Choose one randomly: 166,732, labeled as "k.. This number is the first sample. Increment "k" by 180,000/500 = 360. Increment "k" by "m" = 360 until 500 freshmen are selected.

Using a function in a spreadsheet formula is helpful when performing calculations on specific values. Functions can make it easier to write formulas and return equation results when used with cell references and cell ranges. Although spreadsheet software contains many functions, one of the benefits of using common functions, including sum, average, median, maximum, square root, power, and minimum, when performing quantitative analysis is that they save time for the analyst building the data and summarizing the results. Which of the following spreadsheet equations correctly uses the "sum" function for a range of cells containing values in A1 through A25?

=SUM(A1:A25) The SUM function adds all of the values of the selected cells in the argument range.

1. When presented with an inventory management problem, the first step is to recognize each of the required data elements in order to calculate the EOQ and related calculations to managing the ordering process. Identify the correct data element in the following situation: A baker buys sugar in 25-pound bags. The bakery uses 1,215 bags a year, and ordering costs are $10 per order. It costs $75 to hold this in inventory for the year. What is the correct formula for finding the EOQ? 2. When presented with an inventory management problem, the first step is to recognize each of the required data elements in order to calculate the EOQ and related calculations to managing the ordering process. Identify the correct data element in the following situation: A baker buys sugar in 25-pound bags. The bakery uses 1,215 bag a year, and ordering costs are $10 per order. It costs $75 to hold this in inventory for the year. How many orders will have to be made over the year. 3. Which spreadsheet function is used to calculate the EOQ?

1. Correct. The formula is: 2DSH−−−−√=2DSH−−−−√ Where: D = yearly demand S = ordering costs H = carrying (holding) cost per unit 2. Correct. The formula is D/Q = 1,215 / 18 Where: D = Demand Q = economic order quantity 3. The last step in calculating the EOQ is taking the square root.SQRT

1. When presented with an inventory management problem, the first step is to recognize each of the required data elements in order to calculate the EOQ and related calculations to managing the ordering process. Identify the correct data element in the following situation: A wine distributor buys 4,500 bottles of red wine a year from its suppliers. It costs $10 per bottle to store this wine in a special cooler. It cost $36 to place orders. What is the correct formula for finding the EOQ? 2. When presented with an inventory management problem, the first step is to recognize each of the required data elements in order to calculate the economic quantities and related calculations to managing the ordering process. Identify the correct data element in the following situation: A baker buys flour in 25-pound bags. The bakery uses 1,000 bags a year, and ordering cost is $15 per order. It costs $75 to hold this in inventory for the year. How many orders will have to be made over the year? 3. When presented with an inventory management problem, the first step is to recognize each of the required data elements in order to calculate the economic quantities and related calculations to managing the ordering process. Identify the correct data element in the following situation: A baker buys sugar in 25-pound bags. The bakery uses 1,215 bags a year, and ordering cost is $10 per order. It costs $75 to hold this in inventory for the year. What is the total cost of carrying and ordering for this product?

1. Correct. The formula is: 2DSH−−−−√2(4500)*36/10−−−−−−−−−−−−√ This is the correct application of the formula to solve the EOQ Where: D = yearly demand S = ordering costs H = carrying (holding) cost per unit 2. Correct. The formula is D/Q Where: D = Demand Q = economic order quantity 3.

The different types of decisions businesses and individuals face fall into three categories:

1. Decisions under certainty - Decision makers know with certainty the exact outcome or consequence that will occur with a given alternative. 2. Decisions under uncertainty - There are several different outcomes that can occur with a given alternative and decision makers do not know the likelihood of each outcome occurring. 3. Decisions under risk - There are several different outcomes that can occur with a given alternative, and decision makers know the probability or likelihood of each potential alternative outcome. Following a structured decision-making approach will reduce the likelihood of making a decision based on emotions or judgments.

1. Which of the following is the most important and primary step in setting up a linear programming problem? 2. John Jones has $20,000 to invest in his stock portfolio. His portfolio is broken down into three categories: - High risk - Medium risk - Low risk He wants to maximize the returns on his investments subject to the risk. Therefore, he establishes that he will invest no more than $8,000 in high-risk, $14,000 in medium-risk, and $2,000 in low-risk stocks. The expected returns are as follows: - High risk = 11% - Medium risk = 9% - Low risk = 7% 3. John Jones has $20,000 to invest in his stock portfolio. His portfolio is broken down into three categories: - High risk - Medium risk - Low risk He needs to maximize profits subject to the risk. Therefore, he establishes that he will invest no more than $8,000 in high-risk, $14,000 in medium-risk, and $2,000 in low-risk stocks. The expected returns are as follows: - High risk = 11% - Medium risk = 9% - Low risk = 7% Given the above information, label the decision variables and then state the problem in mathematical terms or expression. 4. [True/False] Using Excel Solver (a linear programming computer solution) does not require a statement of the problem or identification of the variables. 5. The manager of a feed store wants to solve a mix optimization issue with the key ingredients of the feed. The manager chooses a linear programming approach. First he defines the problem. What does the manager do next to generate a computer solution? 6. A surfboard manufacturer makes small and large surfboards. There are three stages in the production process: assembly, finish, and inspection. Each small surfboard requires 3 hours to assemble, 2 hours to finish, and 1 hour for inspection. Each large surfboard requires 4 hours to assemble, 3 hours to finish and 1.5 hours for inspection. There are 140 hours available for assembly, 100 hours to finish, and 50 hours for inspection. The large surfboard makes $150 profit per board. The small surfboard makes $100 profit per board. Linear programming will be used to create the weekly production schedule. What would the variables in this problem be?

1. Define the problem. Stating the problem is typically the first step in linear programming and must be done correctly or the outcome is likely to be incorrect. 2. The problem is to maximize profit, and the variables are high-risk, medium-risk, and low-risk stocks. John wants to maximize the return on his investments, that is the objective. The variables are the types of stock he will be investing in. 3. X1 = high risk, X2 = medium risk and X3 = low risk; 0.11X1 + 0.09X2 + 0.07X3 = Z. The problem is to maximize profit given the risk associated with each investment category. 4. The first step in a linear programming problem is to define the problem and identify the variables. 5. Define the decision variables, and put them into an electronic spreadsheet table. After the problem is defined, the manager should determine the decision variables. 6. The production schedule will maximize profits by determining how many of each board to produce. S = Small surfboards produced L = Large surfboards produced

The decision-making process is an orderly and sequential model to help people come up with optimal choices. Although there are a number of models with some variation, they share common steps that are necessary in the process. It is important to understand the functions of the steps and their sequencing. In one of the steps, a review of the possible criteria for making a decision are considered. This step is called ___ the decision criteria. After criteria have been established, alternative solutions are considered. The first step in this part of the process is to ___ the alternatives. After implementation of the decision, the decision should be ___ to ensure its effectiveness.

1. Establishing the decision criteria must come before weighing the criteria, which is the next step. 2. Generating the alternatives must come before evaluating the alternatives, which is the next step. 3. The effectiveness of the decision must be determined after its implementation. No replication of the decision should be considered before determining its effectiveness.

Once the crash cost per unit for each activity has been calculated, the project manager will allocate the funding. The method of allocation will start with the least costly activity to crash. The manager will continue to allocate based on the next least costly activity until the new funding has been fully allocated and the project time schedule is back on track or reined in. For the example project above, the activity crashing order would be: C-A-B. This order was determined using the calculated crash cost amounts: Activity C: $1,500 Activity A: $2,000 Activity B: $6,000 Duration (weeks) Total Activity Cost Crash Cost per Week (Cost Slope) Activity Normal Crash Normal Crash A 8 4 $10,000 $18,000 $2,000 B 10 5 $30,000 $60,000 $6,000 C 2 1 $8,000 $9,500 $1,500 Based on the graphic below, which activity would you crash first? Based on the graphic below, which activity would you crash last? Based on the graphic below, what would it cost to crash this project by 8 days? Based on the graphic below, which activity would you crash last? Based on the graphic below, which activity would you crash first? Based on the graphic below, approximately what would it cost to crash this project by 9 days?

1. Generally, activities are crashed based of their cost per time period; that is, the least costly alternative per time period is crashed up to the maximum number of time reduction periods possible. If funds remain to spend on crashing more activities, then the next least costly activity per time period is crashed. In this case, Activity C costs the least amount to crash per time period at $750. 2. Generally, activities are crashed based on their cost per time period; that is, the least costly alternative per time period is crashed up to the maximum number of time reduction periods possible. If funds remain to spend on crashing more activities, then the next least costly activity per time period is crashed. In this case, Activity C cost the greatest amount to crash per time period at $1,333. 3. The activity with the cheapest cost per time period to crash is Activity A. However, you can only gain 7 days on the project and you need to gain 8 days. Therefore, we go to the activity with the next cheapest cost per time period, which is Activity C at $1000 per day. Calculation: 7 days * $857.14 plus 1 day * $1,000; then round answer to $7,000. 4. Generally, activities are crashed based on their cost per time period; that is, the least costly alternative per time period is crashed up to the maximum number of time reduction periods possible. If funds remain to spend to crash more activities, then the next least costly activity per time period is crashed. In this case, Activity B cost the greatest amount to crash per time period, at $2,000. 5. Generally, activities are crashed based on their cost per time period; that is, the least costly alternative per time period is crashed up to the maximum number of time reduction periods possible. If funds remain to spend to crash more activities, then the next least costly activity per time period is crashed. In this case, Activity A cost the least amount to crash per time period, at $857. 6. The cost to crash Activity A per day is $857.14. The cost to crash Activity B per day is $750. The cost to crash Activity C per day is $1,333. Activity B has the lowest crash cost per time period, but can only be crashed 4 days. Activity A has the next lowest crash cost per time period at $857.14 per day, so the remaining 5 days will be crashing Activity A. =4∙750+5∙857.14=$7,285.71 and is rounded to $7,286.

Your company has been asked to build a playground for an elementary school. The project must be completed before the first day of school. Which of the following steps would increase the likelihood of finishing the project on time? The critical path denotes the longest time to complete any path that is part of the project. Why might the critical path be the longest? You contract to have your house built. The first thing you do is hire an architect to design the building to specifications. After that, there are series of jobs with various time commitments associated. They include ordering materials, laying foundation, constructing the house, selecting the siding, selecting the interior, and finishing the work. Each step takes a different amount of time. Without determining the critical path, what are the elements that possibly make up the critical path? If a task is delayed, but it does not lie on the critical path, will the total time of project completion be extended? Does a critical path have to have a slack (time) equal to zero?

1. Have the most experienced workers complete tasks on the critical path. Any task on the critical path that has a delay will delay the entire project. Putting the most experienced workers on those tasks will increase the likelihood of finishing the project on time. 2. Due to predecessors, the path that is the longest may have to start later than other tasks. This is one possible explanation for why a critical path may be the longest in duration to complete. 3. The various jobs that are involved in constructing the house. It's not the personnel's time, but, rather, the task that is linked to predecessor or successor requirements and the amount of time that is allocated per task. 4. It depends on whether the delay in time exceeds the slack time that the task possesses in relation to the overall project timeline. 5. The critical path is the longest path. Not only is it the longest, but it is the most efficient sequence of steps. Therefore, there is no gap (time) in or in between each task.

1. Identify the type of costs based on the following scenario: Smith Fine Wines' business is doing well. In fact, they intend to open a large cooler in their retail store to include more premium beers to expand their market. The addition of the cooler will cost about $300,000 and add to the overhead of the store. This type of cost is known as a(n) ___ cost. 2. Identify the type of costs based on the following scenario: Green Frog Winery buys its grapes from a local cooperative. Before each buy, the procurement agent must negotiate the price and delivery schedules. This takes about three days every month for the procurement agent to complete. This type of cost is known as a(n) ___ cost. 3. Identify the type of cost from the following example: Powers Building Supply has a quarterly budget of $300,000 for buying 2" by 4" lumber. This type of cost is known as a(n) __ cost.

1. Holding costs are the costs associated with keeping an item in stock for a period of time, usually one year. 2. Ordering costs are the costs of ordering and receiving the inventory. 3. Purchase costs are the amount paid to buy the inventory. This is typically the largest of all inventory costs.

The two fundamental questions that must be asked when managing inventory costs are:

1. How much inventory should be ordered? 2. When should inventory be reordered?

There are two fundamental questions that must be answered to properly manage inventory:

1. How much needs to be ordered at a time? 2. How often should inventory be ordered?

Wes is building a new home and is using the beta distribution method to estimate the duration of adding the roof. He has the following estimates: Optimistic time = 8 days Pessimistic time = 17 days Most likely time = 11 days What is the estimated activity duration? Elle is planning a corporate holiday dinner party and is estimating the time needed to evaluate and select the venue. She has the following estimates: Optimistic time = 4 days Pessimistic time = 16 days Most likely time = 7 days What is the estimated activity duration? Jane is completing product research for a retail company. Using the beta distribution method and the following estimates, determine the estimated activity duration: Optimistic time = 3 days Pessimistic time = 12 days Most likely time = 6 days

1. The correct formula for this time estimate using beta distribution method is (8+(4*11)+17)/6=69/6=11.5. 2. The correct formula using the beta distribution method to estimate the task completion is (4+(4*7)+16)/6=48/6=8 days. 3. 6.5: (3+(4*6)+12)/6 is the correct formula to use to calculate the time estimate for Jane's research project.

Regardless of the type of decision you need to make, there is a series of steps that should be followed. There is no time limit to this process; each problem is different and should be given necessary attention.

1. Identify the problem. 2. Establish decision criteria. 3. Weigh decision criteria. 4. Generate Alternatives 5. Evaluate the Alternatives 6. Choose the best alternative 7. Implement the decision 8. Evaluate the decision. To provide a better understanding of these steps, consider Fred and his pursuit of finding a new location for his bakery First, clearly define the problem. This step includes identifying the root causes of the problem, any organizational limitations, and stakeholder issues. The focus must be on the problem and not the symptoms. The decision maker and the stakeholders should agree on the problem definition. The problem definition should include the problem, the existing conditions, and the desired conditions. Fred's problem definition is as follows: Fred's Bakery business is so successful that a bigger kitchen with more ovens is needed to keep up with product demand. Step two is to establish decision criteria. Clearly state what the solution must do to meet the requirements or to solve the problem. The decision maker should clearly list the criteria that would succeed in resolving the problem. Fred has listed decision criteria as more square footage, price of building, location of building, and the condition of the building. Step three is to weigh the decision criteria. By assigning weights to each criteria, the decision maker gains a clearer understanding of which criteria are the most important. For example, is it more important for Fred to have square footage or is the location of the building more important? The most important feature for Fred's business is the square footage, because he needs a much bigger kitchen for production. Price is also important because he is on a limited budget. (Hopefully having more square footage will increase company profits!) In the table below, Fred has given weights to each criteria, to indicate importance.The weights must add up to 1. Step four in the decision making process is to list the alternatives that are the choices to be evaluated for resolving the problem. In this step, consider every alternative without regard to the weighting of the criteria. Do not just focus on one alternative or option. Try to have a minimum of three alternatives to consider. Fred has decided to consider four different locations. Then in step five, the decision maker will use the criteria and the weights to analyze each alternative. The strengths and weaknesses will become evident as the alternative are compared to the weighted criteria. Fred rates each location on a scale of 1 to 5 for each of his criteria, with 5 being the best. This table provides a method for Fred to quantitatively see which option is best for him. In this example, each criteria has equal weight or importance. In the table below, the criteria is evaluated as before, but then the importance factor is applied. As you can see in the table below, applying the weights results in a different decision for this scenario. Step six is to choose the best alternative that yielded the highest score..

1. What is an advantage of a backorder to a business? 2. Which of the following products might be good options for planned inventory shortages? 3. Which of the following statements regarding backorders is true? 4. Which of the following inventory items might be a good candidate for a planned inventory shortage? 5. Which of the following costs are not included in the total inventory costs with backorders? 6. When would a business consider planned inventory shortages?

1. If the item is not ordered until the customer is buying the product, then the customer is removing the carrying cost from the seller. When the item is not ordered until the customer is purchasing, the business does not have to pay a carrying cost to store the item. 2. Low demand items: Inventory with low demand are good candidates for planned inventory shortages. 3. Backorder rate will usually increase as delivery time decreases because the wait time for delivery is reduced. If the delivery time is short, the business is more willing to risk the backorder. 4. A motorcycle: Customers are often willing to wait on expensive items or low demand items. 5. Lost Sales Costs: It is difficult to measure the loss of customer goodwill when an inventory item is not in stock. 6. If customers will not mind waiting a few days to receive a product. Backorders are considered if a business knows that their customers will not mind either receiving a substitute product or waiting a few days to receive a product.

One of the first tasks for a decision maker is to develop a set of decision criteria on which to base his or her evaluations. The individual criteria should be related specifically to management's objectives of the problem or issue under consideration. For example, if one of a firm's key objectives is to "Increase customer satisfaction in claim processing", then there should be at least one specific decision criteria that supports that intent such as, "Customer service personnel training." It is not appropriate to establish criteria to evaluate problem objectives that do not match. For example, if the problem objective was "increase profitability," the criteria for evaluation would not be "develop training." In the below table, match the appropriate type of decision criteria that should be considered with the key management problem objectives. Choice Customer Service and Product Improvement Criteria Financial and Accounting Criteria 1. Improve financial soundness. 2. Improve market position. 3. Enhance complaint tracking system.

1. In order to improve financial soundness or stability, financial or accounting criteria should be established. 2. Increasing market share of the products offered would need product improvement criteria. 3. Improving any element of customer satisfaction, such as reduced wait time, would improve customer relationship management and be evaluated with this type of criteria.

The graphic above represents an example of a MCDM problem. In this problem, the types of cars represent the ___ while the column showing Style, Comfort, etc., represent the ___. Although there are multiple criteria, there is no weighting. Therefore, all criteria are treated equally in importance.

1. In this example, Style, Comfort, etc., are the decision criteria. The alternatives are evaluated against each criteria. 2. In this problem, the possible alternative solutions (decisions) are the types of cars to choose between. The factors of Style, Comfort, etc., are the decision criteria.

Probabilities may be either marginal, joint, or conditional. Understanding their differences and how to recognize them is a key learning point of statistics. A probability that is marginal is not conditioned on another event. For example, each flip of a coin is ____ and event B is a(n) ____ probability. An example of this form of probability would be drawing a black card that is also the number four. The ____ probability of event B is the probability that the event will occur given the knowledge that event A has already occurred. An example of this form of probability is, given that the card drawn is red, what is the probability that it is also a four? It is conditional because the knowledge of red is already known; mathematically, the probability that the card is also a four can be determined.

1. Independent: Each time a coin is flipped, there is no dependency on the prior flip. The probability of a coin flip is marginal; it is not dependent on the prior flip result. 2. Joint probability is when two events occur at the same time, or sometimes called the intersection of two events. 3. This describes a conditional probability because there is knowledge that can be applied to the second event.

1. A manufacturer makes steel benches and tables. Each bench requires 2.5 hours for assembly, 3 hours for sanding, and 1 hour for crating. Each table requires 1 hour for assembly, 3 hours for sanding, and 2 hours for crating. The company can do only 20 hours of assembling, 30 hours of sanding, and 16 hours of crating per week. Based on the graphic below, identify the line for assembly. 2. A manufacturer makes steel benches and tables. Each bench requires 2.5 hours for assembly, 3 hours for sanding, and 1 hour for crating. Each table requires 1 hours for assembly, 3 hours for sanding, and 2 hours for crating. The company can do only 20 hours of assembling, 30 hours of sanding, and 16 hours of crating per week. Based on the graphic below, identify the line for sanding. 3. A manufacturer makes steel benches and tables. Each bench requires 2.5 hours to assemble, 3 hours for sanding, and 1 hour for crating. Each table requires 1 hours for assembly, 3 hours for sanding, and 2 hours for crating. The company can do only 20 hours of assembling, 30 hours of sanding, and 16 hours of crating per week. Based on the following graphic, identify the line for crating. 4. A small appliance manufacturer produces bread and bagel toasters. Long-term projections indicate an expected demand of at least 100 bread (x-axis) and 80 bagel (y-axis) toasters each day. Because of limitations on production capacity, no more than 200 bread and 170 bagel toasters can be made daily. To satisfy a shipping contract, a total of at least 200 toasters must be shipped each day. Based on the following graphic, identify the line that shows that no more than 170 bagel toasters can be made daily. 5. A small appliance manufacturer produces bread and bagel toasters. Long-term projections indicate an expected demand of at least 100 bread (x-axis) and 80 bagel (y-axis) toasters each day. Because of limitations on production capacity, no more than 200 bread and 170 bagel toasters can be made daily. To satisfy a shipping contract, a total of at least 200 toasters must be shipped each day. Based on the following graphic, identify the line that shows the expected daily demand of the bagel toasters. 6. A small appliance manufacture produces bread and bagel toasters. Long-term projections indicate an expected demand of at least 100 bread (x-axis) and 80 bagel (y-axis) toasters each day. Because of limitations on production capacity, no more than 200 bread and 170 bagel toasters can be made daily. To satisfy a shipping contract, a total of at least 200 toasters must be shipped each day. Based on the following graphic, identify the line that shows that the possible production to satisfy the shipping contract.

1. The formula for this constraint is 2.5x+y≤20. 2. The formula for this constraint is: 3x+3y≤30. 3. The formula for crating constraint is: x+2y≤16. 4. The formula for this constraint is y≤170. 5. The formula for this constraint is y≥80. 6. The formula for this constraint is x+y≥200.

1. What is a benefit of using EOQ? 2. EOQ is used to help manage inventory by calculating the best quantity to order for a given item based on several decision factors. What is one benefit of using EOQ? 3. What is one benefit of using EOQ? 4. Which of the following is a benefit of using EOQ? 5. What is one benefit of using EOQ? 6. Which of the following are two primary questions to answer when managing inventory?

1. It can minimize storage space used. By ordering the right quantity, the chances of overstock are lessened. 2. It can minimize procurement costs with less inventory ordered. By ordering the right quantity, the total procurement costs may be lessened overtime. 3. EOQ can save on ordering costs. Ordering the right quantity during any one buy will reduce the number of times that orders have to be made, saving ordering costs. 4. EOQ is used in conjunction with a continuous inventory review process, so a fixed quantity is ordered whenever needed. The EOQ is used as part of a continuous review inventory system in which the level of inventory is reviewed at all times so that a fixed quantity is ordered each time the inventory level reaches a specific reorder point. 5. EOQ can provide specific order numbers, unique to one's business. 6. How much inventory should be ordered? When should the orders be placed? The EOQ model helps businesses determine how often to reorder and the quantity of inventory to order.

One of the most commonly accepted ways to determine the impact of risk is by using the concept of the expected monetary value (EMV). To determine the EMV for each alternative below, subtract the estimated costs from the estimated revenues for each alternative. Using the chart above, determine the EMV for these situations. The expected monetary value for Alternative A in the table is $10,250

1. Multiply the value of each state of nature by its probability of occurrence, then add the values together. Subtract the cost estimate from the revenue estimate to get the EMV. -The calculation is as follows: -Estimated revenue calculation for alternative A is =(C4 * D4) + (E4 * F4) = $50,250. -Estimated cost calculation for alternative A is =(C3 * D3) + (E3 * F3) = $40,000. -EMV calculation is Revenue - Cost = $50,250 - $40,000 = $10,250. 2. The estimated revenue calculation for alternative A is =(C6 * D6) + (E6 * F6) = $68,500. The estimated cost calculation for alternative A is =(C5 * D5) + (E5 * F5) = $35,000. The EMV calculation is Revenue - Cost = $68,500 - $35,000 = $33,500. 3. The estimated revenue calculation for alternative C is =(C8 * D8) + (E8 * F8) = $22,800. The estimated cost calculation for alternative C is =(C7 * D7) + (E7 * F7) = $15,000. EMV calculation is Revenue - Cost = $22,800 - $15,000 = $7,800.

Quantitative analysis primarily uses objective, data analysis to solve problems, whereas qualitative analysis will emphasize the use of subjective expert opinion. When attempting to better understand and quantify uncertainty, both types of analysis will be sought. For example, if a weather expert was asked by an amusement park consortium to prepare a report about uncertain weather conditions and lightning safety for the upcoming Fourth of July, the analysis might include a(n) ___ analysis by researching the specific number of people struck by lightning in the United States over the past five years and divide that by the total U.S. population. The weather expert will also review and give a ___ ___.

1. Objective data are necessary for quantifying uncertainty. Business managers rely on quantitative analysis where mathematical models are used. 2. Although subjective information cannot be used in a mathematical model, it has value and significance when making decisions. 3. Predicting uncertainty with the use of numerical data is beneficial to many organizations. Although no manager can exactly predict the future, gaining a better understanding using quantitative analysis is vital.

1. Identify the elements of the following LP problem. Acme Corp. makes couches and chairs using three main components: wood, foam, and fabric. It takes 60 linear feet to make a couch and 35 to make a chair. It takes 30 hours of labor to make a couch and 15 to make a chair. Couches sell for $800 and chairs for $350. The selling price would be part of which element? 2. Identify the elements of the following LP problem. Acme Corp. makes couches and chairs using three main components: wood, foam, and fabric. It takes 60 linear feet to make a couch and 35 to make a chair. It takes 30 hours of labor to make a couch and 15 to make a chair. Couches sell for $800 and chairs $350. The company only has a total of 5,000 linear feet of wood in stock, what does this represent? 3. Identify the elements of the following LP problem. Acme Corp. makes couches and chairs using three main components: wood, foam, and fabric. It takes 60 linear feet to make a couch and 35 to make a chair. It takes 30 hours of labor to make a couch and 15 to make a chair. Couches sell for $800 and chairs for $350. The company can change the mix of couches and chairs it will make. What does this represent? 4. Acme Corp. makes couches and chairs using three main components: wood, foam, and fabric. It takes 60 linear feet to make a couch and 35 to make a chair. It takes 30 hours of labor to make a couch and 15 to make a chair. Couches sell for $800 and chairs for $350. State the mathematical representation to show that the total number of linear feet of wood for both couches (X) and chairs (Y) cannot exceed 5,000. 5. Solving LP problems involves changing the quantities of a set of decision variables in order to provide the best answer to meet the problem objective, subject to the constraints. Identify the elements of the following LP problem. Acme Corp. makes couches and chairs using three main components: wood, foam, and fabric. It takes 60 linear feet to make a couch and 35 to make a chair. It takes 30 hours of labor to make a couch and 15 to make a chair. Couches sell for $800 and chairs for $350. State the mathematical representation to show that the total number of labor hours used for both couches (X) and chairs (Y) cannot exceed 4,000. 6. Acme Corp. makes couches and chairs using three main components: wood, foam, and fabric. It takes 60 linear feet to make a couch and 35 to make a chair. It takes 30 hours of labor to make a couch and 15 to make a chair. Couches sell for $800 and chairs for $350. State the mathematical representation to show that the company must make at least 20 chairs to fill backorders.

1. Objective: The objective of this LP problem is to maximize the profits so, the selling price is needed as part of determining the objective. 2. Constraint: The total amount of lumber available is a constraint to the amount of items they can make. 3. The key in defining the decision variables is the process of understanding the elements that represent "levels of activity." The activity in this problem is making the couches and chairs. 4. 60X + 35Y <= 5,000 This adds 60 linear feet for each couch made to 35 linear feet for each chair made to arrive at total linear feet that cannot be exceeded. 5. 30X + 15Y <= 4,000: This adds 60 linear feet for each couch made to 35 linear feet for each chair made to arrive at total linear feet that cannot be exceeded. 6. Y >= 20: This term would have to be included in the setup of the problem.

1. Various types of costs are related to operations and inventory management. It is essential to understand the types of business costs in order to plan an operations and inventory management strategy. One type of cost identifies the monthly use of a new Web-based computer system for procurements. This type of cost is known as a ___ cost. 2. One type of inventory cost identifies the cost associated with the overhead costs of staffing and maintaining the warehouse. This type of cost is known as a(n) ___ cost. 3. One type of inventory cost identifies the price paid for the inventory itself. This type of cost is known as a(n) ___ cost.

1. Ordering costs are the costs of ordering and receiving the inventory. 2. Holding costs are the costs associated with keeping an item in stock for a period of time, usually one year. 3. Purchase costs are the amount paid to buy the inventory. This is typically the largest of all inventory costs.

Creating and building spreadsheets to calculate equations is a necessary tool when performing quantitative analysis. There are many built-in calculation features within a spreadsheet tool. Familiarizing yourself with the functions feature is an excellent skill that will make it easier to write formulas and return equation results when used with cell references and cell ranges. One of the most used functions is the "average" feature. It calculates the average, or mean of a range of cells in a quick fashion regardless of how many values are in the range. Which of the following spreadsheet equations correctly uses the "average" function for a range of cells containing values in A1 through A25?

=AVERAGE(A1:A25) The AVERAGE function finds the mean for all of the values of the selected cells in the argument range.

Which of the following equations will return the number of values in a range of cells?

=COUNT(A1:A25) The COUNT function will display the number of cells that have been included in the range of cells.

One type of cost related to operations and inventory management is the cost associated with the initial procurement of the part from a supplier for inventory. This type of cost is known as a(n) ___ cost. One type of cost related to operations and inventory management is the cost associated with losing a sale because the item was out of stock. This type of cost is known as a(n) ___ cost. Which of the following categories represents inventory being transformed from raw and packaging materials into finished goods, until released from manufacturing to the finished goods warehouse? Which of the following inventory costs would be a carrying cost? Identify the type of costs based on the following scenario: Acme Bicycle Co. recently ran out of their more popular bicycle model. Several customers were so disappointed that they indicated they will no longer buy their bicycles from Acme. This type of cost is known as a(n) ___ cost.

1. Ordering costs pertain to the costs of ordering and receiving inventory. 2. Shortage costs pertain to costs because of not having an item on the shelf for sale, generally the unrealized profit per unit. 3. Work-in-progress represents inventory being transformed from raw and packaging materials into finished goods, until released from manufacturing to the finished goods warehouse. 4. Utilities for the warehouse. Carrying costs include expenses necessary to hold the inventory, such as obsolescence, building costs, theft, spoilage, insurance, and utilities for the warehouse. 5. Shortage costs pertain to costs as a result of not having an item on the shelf for sale, generally the unrealized profit per unit. Lost goodwill can cost a company money.

Spreadsheets are primarily used for numerical calculations but also may be used as a data management tool. A few key points about spreadsheets follow:

1. Place numbers in easy to read rows and columns, with text or labels to identify the numbers. 2. Numeric formulas or equations are created within a spreadsheet to allow for rapid calculation and recalculation in order to conduct "what if" analysis. 3. Data can be sorted, managed, and re-organized to most effectively show information. 4. Results from calculations can easily be presented in tables, charts, or graphs that display all or portions of the spreadsheets data.

A network diagram is an identifiable set of project activities organized in a logical method. Determining the ___ for each activity is a necessary first step in creating the diagram. Olin is planning an open house for his business. The following table lists the activities and the immediate predecessors. Based on the table below, what is the earliest day activity D can start? Network diagrams are visual tools used to do what?

1. Predecessors and successors must be identified in order to organize the network diagram. 7 Network diagrams help plan, schedule, and control complex projects.

1. Simulation models are best used in applications that involve the following: Inventory policy Transportation bookings (airline, train, etc.) New product development Waiting lines Traffic flow Simulation models allow each of these applications to attempt to better understand its sensitivity to various changes. If a simulation model were to be developed to analyze sensitivity to airline bookings, the sensitivity tested for might include ___. 2. A key similarity between linear programming and simulation is risk analysis. Risk analysis is the prediction of an outcome in the face of uncertainty. What is another name for the running of a simulation model used to better predict or determine risk impacts? 3. The use of a simulation model should be a part of the risk analysis "tool kit." In conjunction with linear programming, sensitivity analysis is a necessary modeling technique. While the linear programming model provides an optimal answer given known or certain inputs, it does not account for random unknowns. Sensitivity analysis does account for various scenarios of the unknown. Using linear programming and sensitivity analysis is not necessary in each and every business decision. The level of problem needs to be significant and perhaps have long-term impact. Which of the following business examples would be an appropriate reason to use sensitivity analysis? 4. Simulation models are best used in applications that involve the following: Inventory policy Transportation bookings (airline, train, etc.) New product development Waiting lines Traffic flow Simulation models allow each of these applications to attempt to better understand its sensitivity to various changes. If a simulation model were to be developed to analyze sensitivity to traffic flow, the sensitivity tested for might include ___. 5. A key similarity between linear programming and simulation is risk analysis. Risk analysis is the prediction of an outcome in the face of uncertainty. Preparing a linear programming and using simulation analysis helps preview impacts that decisions may have. ___ is another name for the running of a simulation model that will be used to better predict or determine risk impacts. 6. The use of a simulation model should be a part of the risk analysis "tool kit." In conjunction with linear programming, sensitivity analysis is a necessary modeling technique. While the linear programming model provides an optimal answer given known or certain inputs, it does not account for random unknowns. Sensitivity analysis does account for various scenarios of the unknown. Using linear programming and sensitivity analysis is not necessary in each and every business decision. The level of problem needs to be significant and perhaps have long-term impact. Which of the following business examples would be an appropriate reason to use sensitivity analysis?

1. Price changes can definitely be tested for sensitivity in a simulation model. 2. What-if scenario: A simulation model is a series of "what-if" scenarios used to better predict the impacts of risk from decisions. 3. Determining the number of waiting lines necessary at an amusement park during various seasonal capacities. An amusement park of significant size would use sensitivity analysis in conjunction with a linear program to help determine the impact of seasonal capacities to the number of waiting lines necessary to accommodate. 4. The time of day can definitely be tested for sensitivity in a simulation model. 5. What-if Scenario: A simulation model is a series of "what-if" scenarios used to better predict the impacts of risk from decisions. 6. Determining the appropriate level of inventory in a national pool store chain during various peak months. Determining the appropriate level of inventory a national pool store chain would require. This is a complex decision that would likely use sensitivity analysis to help determine the impact of seasonal purchases.

1. What should a buyer of a goods do if the supplier increased the discount rate offered for quantity discounts? 2. The following table shows inventory costs for pressure cookers. The vendor is offering quantity discounts to Winner Dinner Supplies, Inc. During the year, the business will purchase 2,500 pressure cookers. Discount Code Unit Cost Order Quantity Annual Carrying Cost Annual Ordering Cost A $48 500 $500 $2,500 B $47 1,000 $1,500 $1,700 C $46 1,500 $4,000 $1,800 Which option provides the lowest overall inventory costs? 3. What is one of the disadvantages of using quantity discounting in the procurement of goods? 4. As part of the inventory management, ways to reduce costs are always under consideration. One consideration is quantity discounts. However, there are some considerations in using quantity discounting in the procurement of goods. What is one that the company financial director may point out? 5. As part of inventory management, ways to reduce costs are always under consideration. In deciding whether to use quantity discounting, the company is seeking to minimize the total cost as defined by the carrying cost, ordering cost, and what else? 6. The following table shows inventory costs for gas grills. The vendor is offering quantity discounts to Will's Grills, Inc. During the year, the business will purchase 1,000 gas grills. Discount Code Unit Cost Order Quantity Annual Carrying Costs Annual Ordering Cost A $478 50 $500 $2,500 B $472 100 $1,500 $2,000 C $470 500 $4,000 $1,400 Which option provides the lowest overall inventory costs?

1. The buyer should take advantage of this offer more often. If the supplier offers a higher discount rate, then the buyer is encouraged to buy in large quantities more often to take advantage of the situation. 2. Even though option C gives the greater discount, the increased carrying costs cause the inventory costs for option C to be higher than option B. The best option in the above table is option B because this option has the lowest total inventory costs of $120,700. 3. If a large amount of the product is on-hand, then it will take some time to deplete this stock. During the time it takes to deplete the stock, then it is possible that the price might go down or the product might become obsolete. 4. Buying in quantity ties up a lot of the company's cash.This could be a complaint from the financial director. The amounts saved and the time value of money for the tied up extra funds will have to be compared. 5. The goal of using quantity discounts is to minimize the total costs to include carrying cost, holding cost, and the total cost of the product. 6. Option C has total inventory costs of $475,400. Even though the carrying cost and ordering costs are higher for this option, the quantity discount makes this the best option.

In conclusion, we had some school absences and test score data. We suspected there might be a relationship between the two variables. We did the following analysis:

1. Rank ordered the data by school absences. 2. Calculated the descriptive statistics for both variables. 3. Plotted histograms and pie charts for school absences. 4. At this point, we wondered why the absence levels were so high that School Absences need to be tracked and reported on regular basis. 5. Scatter plot of school absences as the X-variable and test scores as the Y-Variable. a. The scatter plot visually confirmed a possible relationship. b. Adding the trend (best fit) line and reviewing the r instead of r2 confirmed this very clearly. 6. There is pretty solid evidence that school absences must be reduced. Actions to be taken: a. Develop a tracking method to keep reporting this data to determine if this relationship applies system wide. b. Develop communication program for all stakeholders (students, faculty, and perhaps parents) on attendance and performance. c. Track performance and repeat step b if the results are below expectations. This is a small example of how data can be transformed into information that leads to better decisions and actions. Again this dataset was only 13 paired observations of which the conclusions were clear without much analysis. It was an example to show how to proceed with the tools presented thus far for larger data sets. Is the analysis above a menu to follow? Not necessarily. There is an art to this kind of work. The art part of the quantitative analysis comes from experience. So, jump in, get started, and learn as you go! Rarely are the following wasted efforts: Calculating descriptive statistics for all variables Plotting histograms or pie charts for a single variable Plotting scatter diagrams of two variables to literally see if there is a potential relationship

Because all inventory is not the same, accounting has defined categories in which companies can measure and track where their inventory is. These accounts are also useful for companies to determine where their inventory is undersized, rightsized, and bloated, which can lead to continuous improvement projects. Here is a summary of the most popular of these accounts.

1. Raw and Packaging Inventory: This class of inventory is comprised of the raw ingredients, parts, subassemblies, and packaging materials that manufacturing and assembly companies use to make or assemble their products. For automotive companies, raw and packaging inventory consists of screws and fasteners, coils of sheet metal used to stamp out body panels, seats that come assembled from the seating supplier, and engines from the company's own engine plant. For restaurants, it is the meat and fish, eggs, potatoes, vegetables, buns and breads, coffees and teas, condiments like salt, ketchup, and pepper, canisters of soft drink syrup for the soda machines, and napkins. Typically, retail stores have minimal raw materials. But they do have packaging materials (i.e., bags, boxes, and wrapping materials) used to "package" customer purchases. 2. Finished Goods: Finished goods are the goods that a business sells. For automotive companies, the finished goods are the completely assembled, finished cars that are ready to be shipped to customers who are generally auto dealerships. For restaurants, finished goods might include the muffins and desserts prepared for the day's sales or the assembled hamburger that is about to delivered to a customer's table. For a retail establishment, the finished goods are all the items stocked, shelved, and ready to sell to customers. 3. Work-in-Process Inventory (WIP): When materials are released from the Raw and Packaging account, the physical inventory is in the hands of manufacturing or assembly who controls and is responsible for this inventory until the raw and packaging materials are transformed into finished goods and released from manufacturing to the finished goods warehouse. The goods in manufacturing are no longer raw and packaging and they are not yet considered finished goods, hence work-in-progress. Automotive companies have more WIP than restaurants do. Restaurants generally have more WIP than retail establishments do. Make-to-order companies, like aerospace manufacturers and shipbuilders, have the greatest majority of their inventory in WIP. 4. In-Transit Inventory: When goods are being supplied from the other side of the planet, significant portions of raw, packaging, and finished goods can be in transit for weeks. This inventory can be on the books for the supplier or customer company, depending how the purchase terms were negotiated. Whoever owns the inventory certainly must track these stocks in their accounting system to properly report inventory levels. No matter which company owns the inventory in-transit (or on-order), the customer company must track and account for these goods in the inventory planning and materials management processes. For any of the above categories, a company needs to know how much inventory is on order, in-transit, and on hand. Only then can they decide when next to order and how much of each item to order.

In reviewing sets of data covering a long period of time, it is often noted that there are changes in the behavior of the numbers, over and above the normal variation in data sets. This is usually attributed to changes over time, and time series analysis is called for. There are three primary types of influencing factors over time. ___ factors are those that can be seen repeated, usually on annual and quarterly fluctuations that tend to repeat themselves in a generally predictable manner. Another type of factor is ___ which is less predictable and sometimes dramatic or from unknown causes or upheavals in the economy. The third factor refers to the general direction of the overall movement of the data set over a long period of time, this is known as the ___.

1. Seasonal. The most common example of this is the spike in retail purchases from Thanksgiving through the Christmas holiday period. 2. Cyclical. These are less likely to be predictable, like the seasonal variations and are sometimes very large, expanding over several years. These can result from things like war or prolonged economic depression. 3. The overall movement over time is known as the trend. This could be up or down and is moderated between the cyclic ups and downs in a data set.

Dividing Positive and Negative Numbers 1. Here is an example of dividing a positive and negative number. (−132)÷12=−11 2. This is an example of dividing two negative numbers: (−132)÷(−12)=11 3. Here is an example of dividing a series of positive and negative numbers when there are more than two numbers. Divide the following numbers: 24÷(−3)÷(−4)=2

1. Steps: Divide 132 by 12. Because there is an odd number of negative signs in this problem, the answer is negative. 2. Steps: Divide 132 by 12. Because there is an even number of negative signs in this problem, the result is positive. 3. Steps: When there are more than two positive and negative numbers to divide, count the number of negative signs. In this example, there are two negative signs. This is an even number of negatives, so the result will be positive. Now that we know the sign of the answer, just start on the left and start dividing the numbers together without worrying about the signs. 24 divided by 3 is 8, 8 divided by 4 is 2.

Multiplying Positive and Negative Numbers 1. Here is an example of multiplying a positive and negative number: (−12)(11)=−132 2. This is an example of multiplying two negative numbers: (−12)(−11)=132 3. Here is an example of multiplying a series of positive and negative numbers when there are more than two numbers. Multiply the following four numbers together: (−2)(3)(−1)(4)=24

1. Steps: Multiply 12∙11. Because there is an odd number of negative signs in this problem, the answer is negative. 2. Steps: Multiply 12∙11. Because there is an even number of negative signs in this problem, the resulting product is positive. 3. When there are more than two positive and negative numbers to multiply, count the number of negative signs. In this example, there are two negative signs. This is an even number, so the result will be positive. Now that we know the sign of the product, just start on the left and start multiplying the numbers together without worrying about the signs. 2∙3=6,6×1=6,6∙4=24

Which of the following equations will return the highest value in a range of cells?

=MAX(A1:A25) The MAX function finds the highest value of all the values in a selected range of cells.

A decision maker must be able to identify the risks and benefits of each alternative course of action. Benefits are most often profit-related. However, risks can stem from many different sources. It is vital to understand the sources of risks in order to mitigate them. Each element or source of risk needs to be discovered and reviewed. This is a good way to ensure that quality thought has been applied. The following are the primary sources of risk that decision makers should consider: technical, organizational, environmental, schedule, safety, and suitability. Identify the source of risk with each example problem. The inability of the new servers to handle the unanticipated band width of data traffic ___. The inability of the organization and the employees to adapt to new process requirements under new ISO-9000-related certification ___. he unexpectedly high level of CO2 emissions coming from the newly redesigned automobile engine ___.

1. The inability of the new servers to handle the unanticipated band width of data traffic is a technical risk that could be encountered. 2. The inability of the organization and its employees to adapt to new process requirements under new ISO-9000-related certification puts an organization at potential risk. 3. The unexpectedly high level of CO2 emissions coming from the newly redesigned automobile engine represent an environmental issue.

The following are the primary sources of risk that decision makers should consider: technical, organizational, environmental, schedule, safety, and suitability. Identify the source of risk with each example problem. The inability to complete several predecessor tasks on time in the IT hardware upgrade project ___. The early failure of a hood latch on a newly designed automobile hood ___. The premature failure of a diode in a radio set, due to the high voltage requirements of the radio system ___.

1. The inability to complete several predecessor tasks on time in a project involving IT hardware upgrading is a scheduling problem. 2. The early failure of a hood latch on a newly designed automobile hood is a safety problem. 3. The premature failure of a diode in a radio set due to the high voltage requirements of the radio system represents a suitability problem.

The area "common" to both constraints represents the "feasible area" and we can deduce a couple of key items from this illustration:

1. The inequalities that represent each of these linear expressions or constraints is a ≤ sign due to the fact that the shading for each constraint starts at the linear border and moves toward the coordinates (0,0). 2. Because the problem is a maximization question we need to find out which of the extreme outer coordinates or "corners" of the feasible region represent the highest (maximum) value once those coordinates (variable) are plugged into the objective function corresponding variables. From the graph, the corner coordinates of the feasible region are (0,0), (0,20), (24,8), and (30,0). It is unclear (although the likely candidate is (24,8)) which coordinates or variables will produce the highest objective function output since we do not know the coefficients of said objective function. A much higher x variable coefficient might slant the outcome to the higher x variable value as in (30,0). In the case of minimization, the goal is to locate the set of coordinates that are: 1. Part of the feasible region. 2. Produce the least or "minimum" possible value to the objective function once those variable are plugged into the corresponding variables in the objective function. The area "common" to both constraints represents the "feasible area" and we can deduce a couple of key items from this illustration: The inequalities that represent each of these linear expressions or constraints is a ≥ sign due to the fact that the shading for each constraint starts at the linear border and moves away from the coordinates (0,0). Because the problem is a minimization question, we need to find out which of the extreme outer coordinates or "corners" of the feasible region represent the lowest (minimum) value once those coordinates (variable) are plugged into the objective function corresponding variables. In a minimization case, the outer bounds of the feasible area are unbounded due to the fact that the non-negative (x ≥ 0, y ≥ 0) no longer form "as much" of a border of the feasible area. From the graph, the corner coordinates of the feasible region are (0,8), (5,1.6), and (8,0). It is once again unclear which sets of coordinates or variables will produce the lowest (minimum) objective function output since we do not know the coefficients of said objective function. A much higher y variable coefficient might slant the outcome to the lower x variable value as in (0,8). In linear programming, the goal is either to maximize or minimize the value as stated in the mathematical expression of the problem objective in a manner that is within the boundaries (outermost or innermost) of the feasible area formed by the linear expression of the relevant constraints or limitations. Therefore, the terms maximization and minimization in linear programming defines: 1. The objective function. 2. The "direction" in the feasible region where the "optimal" point can be found. 3. The type of element or term (i.e., profit, cost, etc.) that is to be analyzed. In all cases in linear programming, the model seeks the best or optimal choice within a set of relevant constraints or parameters.

1. Add the following fractions: 1/6, 1/5, 4/5, 3/12. The reduced mixed number (not a decimal) result is 2. Subtract 2 1/6 from 4 7/12. The mixed fraction (not decimal) result is 3. Multiply the following fractions: 2 1/4 and 3 2/3. The mixed fraction (not decimal) result is 4. Divide 3 3/8 by 1 3/4.

1. The mixed number result is 1 5/12. By finding the lowest common denominator for 6, 5, and 12 will allow the correct calculations to be applied. 2. The result is 2 5/12. By finding the lowest common denominator for 12 and 6 will allow the correct calculations to be applied. 3. The result is 8 1/4. First, change each mixed fraction to an improper fraction. Next, multiply the numerators across and then the denominators. The initial result is 99/12. When simplifying, the final result is 8 1/4. 4. The result is 1 13/14. Applying the rule for dividing fractions, first change each mixed fraction to an improper fraction. Next, find the reciprocal of the divisor. Finally, multiply across the numerator and denominator. The initial result is 108/56. When simplifying, the final result is 1 13/14.

1. Add the following fractions: 1/2, 2/3, 5/7, 3/7. The mixed number result (not a decimal) is 2. Subtract 1/6 from 11/12. The fraction (not decimal) result is 3. Multiply the following fractions: 3/4 and 2/3. The fraction (not decimal) result is

1. The mixed number result is 2 13/42. Finding the lowest common denominator for 2, 3, and 7 will allow the correct calculations to be applied. 2. 3/4 The result is 3/4. By finding the lowest common denominator for 12 and 6 will allow the correct calculations to be applied. 3. The result is 1/2. By multiplying the numerators across and the denominators across, the initial result is 6/12. When simplifying, the final result is 1/2. 4. The result is 1 1/6. Applying the rule for dividing fractions, first find the reciprocal of the divisor, then multiply across the numerator and denominator. The initial result is 28/24. When simplifying, the final result is 1 1/6.

A commonly used function is the "square root" feature. It calculates the square root of a given value or the square root of a calculated value. The "average" function is another well-used feature in a spreadsheet. Although functions can be performed over a single cell or range of cells, functions can also be applied within the same equation. This can be done by adding, subtracting, multiplying, or even dividing the functions. If the problem is more complex, this can require the use of parentheses within the equation to ensure the order of operations are used appropriately to perform all expressions according to the requirement of the analysis. Which of the following spreadsheet equations correctly uses both the "square root" function and the "average" function to add the two calculated values?

=SQRT(25) + AVERAGE(A1:A25) The SQRT function expression will yield a value, and the AVERAGE function expression will be applied to a range of cells to formulate a value. The two values will be added together.

Raise the base number of 5 by the exponent 2. The result is Multiply 2^3 by 2^4. The result is For the dividend, you are given a base of 5 raised by an exponent of 4. The divisor is a base of 5 raised by an exponent of 3. The quotient result is Given a base of 3, raise it by an exponent of 2, then raise the new base by an exponent of 3. The result is Add the following radicals: 25-√+5-√. The result is Raise the base number of 7 by an exponent of 3. The result is Multiply 3^3 by 2^4. The result is Given a base of 4, raise it by an exponent of 3, then raise the new base by an exponent of 2. The result is Subtract the following radicals: 4√2 − √2

1. The result is 5^2. The problem is demonstrated as 52 or 5^2. This is the same as 5*5. 2. The result is 128. The problem is demonstrated as 23*24. We add the exponents because the base of 2 is the same. This results in 27. 3. The result is 5. The problem is demonstrated as 54/53. When dividing exponentials, if the base is the same, you can subtract the exponents before applying to the base. 4. The result is 729. The problem is demonstrated as (32)3. This was done by first calculating the base of 3 and exponent of 2. Next, using that figure as the new base, raise by the exponent of 3. 5. 3√5 Since both terms have the same radical, the square root of 5, they can be added. The second term is assumed to have a coefficient of one. 6. The result is 343. The problem is demonstrated as 73 or 7^3. This is the same as 7 * 7 *7. 7. The result is 432. The problem is demonstrated as 33*24. Each base and exponent must be calculated separately, then using each result, multiplied together. 8. The result is 4,096. The problem is demonstrated as (43)2. This was done by first calculating the base of 4 and exponent of 3, which results in 64. Next, using that figure as the new base and raised by the exponent of 2 results in 4,096. 7. The result is 3√2. Recognize that the number under both radicals is the same; they are called similar radicals. The coefficients can be subtracted to formulate the answer.

Find the answer to the following equation using the correct order of operations, 6+7∙8. The result is Find the answer to the following equation using the correct order of operations, (25−11)∙3. The result is The result is 42. The equation inside the parentheses must be applied first based on rule 1 of the order of operations. Next, the result of equation within the parenthesis can be multiplied by 3. Find the answer to the following equation using the correct order of operations: 16÷8−2. The result is Find the answer to the following equation using the correct order of operations, 9−5(8−3)∙2+6. The result is Find the answer to the following equation using the correct order of operations, 150(6+3∙8)−5. The result is

1. The result is 62. Using the order of operations, the problem is demonstrated as 6+(7∙8). The parentheses can be included to show that performing the multiplication section is accomplished first, then the addition. 2. The result is 42. The equation inside the parentheses must be applied first based on rule 1 of the order of operations. Next, the result of equation within the parenthesis can be multiplied by 3. 3. The result is 14. Several rules of the order of operation apply in this equation. It is helpful to rewrite equations by inserting sets of parentheses wherever necessary to help guide the order of operations. In this equation, it can be rewritten as: 3+((6∙(5+4))3)−7. Using the rule of parentheses, work from the inside out, then left to right. 4. The result is 0. Using the order of operations, the problem is demonstrated as (16÷8)−2. The parentheses can be included to show that performing the division section is accomplished first, followed by the subtraction. 5. The result is 13. Several rules of the order of operation apply in this equation. It is helpful to rewrite equations by inserting sets of parentheses wherever necessary to help guide the order of operations. In this equation, it can be rewritten as: 9−(5(8−3)∙2)+6. Using the rule of parentheses, work from the inside out and then left to right. 6. The result is 0. Several rules of the order of operation apply in this equation. It is helpful to rewrite equations by inserting sets of parentheses wherever necessary to help guide the order of operations. In this equation, it can be rewritten as: (150(6+(3∙8)))−5. Using the rule of parentheses, work from the inside out, then left to right.

During the decision-making process, decision makers are faced with many new and uncertain events. One of the first concepts a decision maker must understand is the difference between risk and uncertainty, which is considered in order to proceed with the evaluations. Risk can be calculated or measured based on prior results or other quantifying elements. Uncertainty is a complete unknown, with no ability to quantify it. From the table below, match up the description of an event with its correct identifier of either "risk" or "uncertainty." The weatherman on the nightly news is providing the forecast for tomorrow's weather. The bus is on time at a local bus stop. You catch a fish in a pond that you drove by last week. From the table below, match up the description of an event with its correct identifier of either "risk" or "uncertainty." The bet on a winning horse at a horse race. The repayment of a 15-year mortgage by a homebuyer.

1. This is a risk, as the past events and probabilities can be used to determine the likelihood of future events. 2. This is a risk, as the past events and probabilities can be used to determine the likelihood of future events. 3. This is an uncertainty, because there are no past events and probabilities to determine the likelihood of future events. 1. This is a risk, as the past events and probabilities can be used to determine the likelihood of future events. 2. This is an uncertainty, because there are no past events or probabilities that can be used to determine the likelihood of future events. 3. This is a risk, because the homebuyer has been qualified, signed the mortgage, and now is responsible for payments. The risk is whether or not the homeowner will pay the mortgage.

1. In solving LP problems, it is often helpful to visualize the problem using graphical means to get an understanding of the conceptual framework of the problem. Typically, this involves drawing lines to visualize the constraint lines and the corner points, from which the optimizing solution can be seen graphically. From the following graphic, identify the correct graphical representation for the following situation. Which chart below displays the requirement to make at least five couches? 2. In solving two decision variable LP problems, the direction of the inequalities of the constraints is important because of which of the following? 3. Because the shaded area or the feasible area is constructed mainly of the direction and crossover area of the "regular" constraints, is it true that graphing a non-negative constraint is not important? 4. In solving linear programming (LP) problems, it is often helpful to visualize the problem using graphical means to get an understanding of the conceptual framework of the problem. Typically, this involves drawing lines to visualize the constraint lines and the corner points, from which the optimizing solution can be seen graphically. From the following graphic, identify the correct graphical representation for the follow situation. Which chart below displays the upper limit of making not more than 30 tricycles 5. Given the following graphical representation, what can we conclude regarding which of the constraints are "binding?" 6. Looking at the following graphical representation, do the slack or surplus variables tell us anything about the optimal point?

1. This is the correct vertical line to indicate that at least five couches should be made. 2. The direction of the inequalities define the direction of the constraints and, therefore, the feasible area. The shading for each constraint and ultimately the construction of the feasible area are directly related to the direction of each inequality. 3. False: The non-negative constraints form a boundary (in full or in part) of most feasible areas. 4. This is a horizontal line indicating the number of tricycles should be at least 30. 5. Because the optimal solution is made up of the intercept of the two binding constraints, the solution is (5,3), which is the intercept of the second and third constraint. The optimal point is located at the intercept of the binding constraints. 6. Yes. The binding constraints will have zero as a slack (or surplus) variable. The "zero" value that makes the binding constraints work mathematically also confirms the optimality.

In the critical path method (CPM) of project management, the soonest time that a specific task can be begin is known as the ___ time. After this time is noted, then the ___ time can be calculated by adding on the task duration.

1. This is the earliest possible time a task can be started and is calculated by adding the total duration times of any preceding tasks to the task being calculated. This is done during the forward pass or the first part of the CPM calculation process. 2. The earliest finish time is calculated by adding the duration of a particular task to that task's earliest start time. This is done during the forward pass or the first part of the CPM calculation process.

Background: A company is considering building a new plant. It is considering a small, medium, or a large facility. The health of the economy will directly affect the profitability of the new plant, such as whether the economy will expand or contract, thereby providing either a lot of potential profit or little potential profit for a contracted economy. Recalling the exposition from previous sections and this section: A decision point is very specific. The decision maker has exactly the right information and has followed the decision-making techniques, resulting in the final step of making a decision. The expected payoff is a calculated value. Using a decision alternative's value and multiplying it by the probability will result in an expected payoff value. The state-of-nature is applied to an outcome. The probability represents the likelihood that an outcome will occur. For each of the following select the correct term for the scenario. Decision Point Expected Payoff State-of-Nature Constructing a large factory in an expanding economy will likely realize a $2,500,000 profit in 3 years. The company can install automated handling equipment or not, thereby adding to cost, but also adding to potential profitability. There is a 30% chance of a contracting economy (a recession) during the first 3 years after completion of the new plant.

1. This scenario describes an expected payoff calculation. 2. This scenario gives a specific decision. 3. This scenario gives the state-of-nature, "30% chance in a downturn economy."

Adding and Subtracting Radicals

1. To add or subtract two roots, the base must be the same. Notice that if there is no number in front of the radical, it is considered to be 1. 2√3 + 4√3 = 6√3 4√3 - √3 = 3√3 2. Sometimes a radical can be simplified first and then added (or subtracted). In this example, factor √12 as 2⋅2⋅3 or 2 √2√3-√. This can be rewritten as 2√3 3. These radicals cannot be combined because the base is different. √3+√2

Multiplying and Dividing Radicals

1. To multiply two square roots, multiply the roots and keep the product under the radical. Simplify if possible. Notice that 2∙√(2) has the same value as √(2∙2) 2. To divide two square roots, divide the roots and keep quotient under the radical. Simplify if possible. Notice that 2-√2 / √2 has the same value as √2÷2.

Several examples of linear programs are presented below. Notice that they are all maximizing or minimizing objectives and that the constraints are naturally related to the problem.

1. Type of problem (max or min): Maximize profits in manufacturing Possible constraints for such problems. There certainly could be more types. - Upper limit on machine/factory availability - Upper limit on materials critical to production - Upper limit on labor needed - Mix of products less than a certain ratio - Minimum requirements on one or more products 2. Minimize costs in manufacturing - Minimum requirement of products to be produced - Minimum requirement of materials to be used 3. Maximize reach of advertising - Maximum budget for advertising - Maximum number of any one kind of advertising that can be utilized - The ratio of one advertising medium to another must be less than or equal to a certain percentage 4. Maximize profit in a mixing problem (a company produces four raw food ingredients that go, in different proportions, to six different finished products) - Fixed amount in inventory of each raw ingredient, thus a maximum amount of each ingredient used - One or more products may have a minimum production requirement - Upper limit on machine/factory availability - Upper limit on labor needed * Upper Limit = Largest Quantity 5. Maximize the return on investments in N securities - Upper limit on how much can be invested in total - Upper limit on how much can be invested in a particular security either as a dollar amount or as a percentage of total investment - Upper limit on risk the investor is willing to assume

Consider the last example where the objective is to maximize the return on investment where one of the constraints is "risk." A related problem could be posed where the constraint on risk becomes the objective and the previous objective of maximizing return becomes a constraint as below.

1. Type of problem (max or min): Maximize the return on investments in N securities Possible constraints for such problems (There certainly could be more types.): - Upper limit on how much can be invested in total - Upper limit on how much can be invested in a particular security either as a dollar amount or as a percentage of total investment - Upper limit on risk the investor is willing to assume 2. Minimize the amount of risk over N possible investments - Upper limit on how much can be invested in total - Upper limit on how much can be invested in a particular security either as a dollar amount or as a percentage of total investment - Minimum amount of return required

Rational decision making follows a prescribed sequence of steps as part of the process. The steps are followed in order to allow the decision maker an opportunity to logically and rationally follow a process. There is not a time limit on how long each step is to take. However, there is validity in ensuring that quality thought has been applied to each step. What is the correct order for the following steps? The purpose of using a rational decision-making model is to ensure a logical process is followed. This leads to better decision making, rather than relying on subjective instinct. Rational decision making follows a prescribed sequence of steps as part of the process, shown above. The steps are followed in order to allow the decision maker an opportunity to logically and rationally follow a process. What is the correct order for the steps as indicated in the diagram above? What is the correct order for the steps shown above?

1. Weigh decision criteria, generate alternatives, and evaluate alternatives. Evaluating the alternatives must come after the generation of the alternatives, and the weighing of the decision criteria must come before the generation of the alternatives. 2. Choose the best alternatives, implement the decision, and evaluate the decision. Implementing the decision cannot come before the choosing of the best alternative, and the evaluation of alternatives must come before the choosing of the alternatives. 3. Identify the problem, establish decision criteria, and weigh decision criteria. Identification of the problem must be the first step in the decision-making process. Establishing the criteria must proceed the weighing of the decision criteria.

In the below table, match the appropriate type of decision criteria that should be considered with the key management problem objectives. Choice Customer Service and Product Improvement Criteria Financial and Accounting Criteria 1. Improve customer relationship management. 2. Improve corporate debt structure. 3. Train and develop new staff.

1. When a problem involves customer relationship matters, the evaluation criteria should be customer related. 2. Improving corporate debt structure should be evaluated with financial criteria. 3. Problems with training new staff should be evaluated using customer service criteria.

1. [True/False] In Excel Solver, the columns are used for decision variable values. 2. [True/False] If the cell references in Excel Solver are set up as rows on the data table in Excel, the results will be the same because Solver automatically interprets column and row data correctly. 3. Excel Solver is a useful tool in all linear programming problems, but is extremely useful under which of the following conditions? 4. [True/False] The data elements in a computer solution software package (i.e., Excel Solver) are set up in a row and column configuration. 5. The cells listed in the By Changing Cells in the Solver window are referring to which of the following? 6. A company produces three products and wants to maximize profits. Assume C4, E4 and G4 will contain the number of each product produced. Row 7 shows how much it cost to produce one unit of each product, with the expenditure limit in cell G7. Row 8 displays the labor required by hours to produce one unit of each product and G8 displays the limit for number of labor hours available. Row 10 displays the profit made for the sale of one unit of each category.

1. When setting up the Solver parameters, establish decision variable values in columns. Rows are for the type of elements in the linear programming problem and decision variable are in the columns. 2. The data table needs to set up correctly (columns and rows) and then referenced properly in the Solver drop down window for the correct results. 3. When there are three or more decision variables Once there are three or more decision variables, graphing becomes very difficult if not impossible. 4. The data must be accurately entered into the spreadsheet. Designate the cells for decision variables and write the formula to calculate the objective function. 5. Decision variables:" The By Changing Cells box in the Solver window is referencing the decision variables. The constraints are listed as inequalities in a constraint listbox in the Solver window. The objective function is the target cell. 6. (C4∙C7+E4∙D7+G4×E7)≤E14 If A, B, and C represent the quantity of each product produced then A*3000+B*2000+C*5000<=38000. When using a spreadsheet, use the cell references instead of the actual values.

15−x=4x,x= 3x+2x=12−x,x= 5y−3=3y+5,y= 5x=6+2x,x= 6x+7=13+7x,x= −14+6x+7−2x=1+5x,x=

1. x=3. By combining the variables, or isolating the x on one side of the equation, we find 15 = 5x. This now allows the solution for x to be found. 2. The answer is x=2. By combining the variables, or isolating the x on one side of the equation, we find 6x=12. This now allows the solution for x to be found. 3. The answer is y=4. By combining the variables, or isolating the y on one side of the equation, we find 2y=8. This now allows the solution for y to be found. 4. The answer is x=2. By combining the variables, or isolating the x on one side of the equation, we find 3x=6. This now allows the solution for x to be found. 5. The answer is x=−6. By combining the variables, or isolating the x on one side of the equation, we find x=−6. 6. The answer is x=−8. By combining the variables, or isolating the x on one side of the equation, we find x=−8.

Expected Monetary Value

: Many decisions we make are based on uncertainty. When you invest money in the stock market, you are not guaranteed a profit. When a business decides to develop a new product, there is no guarantee that the product will be successful. Both individuals and businesses must make important decisions often in the face of uncertainty. All problems have a root cause, however finding that cause is not easy. Attempting to implement solutions to problems without knowing the root cause of the problem can result in wasted money, time, and effort, as well as potentially making a situation worse. There are formal frameworks that can be used to analyze alternatives and have a higher probability of the best outcome. Courses of action represent the decision that are made, and state of nature represent things that impact the decision but are out of the decision maker hands. Examples of states of nature are weather and the stock market. Expected monetary value is a method used for decision making when the risks involved in a decision situation have several states of nature and the likelihood, or probability, of each state is known. Expected monetary value is obtained by determining the probability of all states of nature, then multiplying that probability by the expected payoff for that alternative. The expected mean value for an alternative is the sum of payoffs of the alternative, weighted by the probability of that event occurring. Alternative Build a store $7M Strong Economy Likelihood 65% $6.5M Weak Economy Likelihood 35% Alternative: Remodel current store $7.5M Strong Economy Likelihood 65% $6M Weak Economy Likelihood 35% There are several commonly used models to visually represent alternatives, states-of-nature, and the calculated expected values.

To calculate an expected value using a payoff table, you would first gain an understanding of each alternative being considered, the various conditions that are possible, and the probability, or likelihood, for each condition. Next, you would assemble the information in a table or spreadsheet to organize the data. Finally, you would create the necessary formulas and calculate the expected value for each alternative. For each alternative, first, you would multiply the value of the first condition by the probability of that condition occurring. Next, repeat the calculation for each condition. Finally, add all of the results of the newly calculated values together to establish the expected value of the alternative. Based on the spreadsheet graphic below, answer the following questions: ABC Alternatives Favorable Market (75%) Unfavorable Market (25%) 1Build large plant$650-$1002Build small plant$275$503Do nothing$0$0 1. If you were preparing a spreadsheet, what is the formula that will be used to calculate the expected value, if the company builds a large plant? 2. If you were preparing a spreadsheet, what is the formula that will be used to calculate the expected value, if the company builds a small plant? 3. If the company just got an updated forecast for the state of the future economy, and the likelihood of a favorable market drops to 60%, what will be the new formula for building a large plant?

=(B1*0.75) + (C1*0.25) Both condition's probabilities must be multiplied by the values prior to adding the new values together. =(B2*0.75) + (C2*0.25). Both conditions' probabilities must be multiplied by the values prior to adding the new values together. =(B1*0.6) + (C1*0.40) Both probabilities for the conditions must be changed. The total probability for all conditions will equal 1.

Gantt charts and PERT charts are visual tools used by business and project managers to plan required activities to complete a project.

A GANTT chart is a horizontal bar chart that shows the project tasks, start and end dates, and the duration of each task. Gantt charts can be constructed using the work breakdown structure (WBS) the manager developed. Gantt charts are very useful in showing overall progress of a project's status, however, they do not show enough detail for each task especially when managing a complex project. As presented in an earlier section, a PERT (Program Evaluation Review Technique) chart may be a better visual tool for managing complex projects. PERT charts, which are also called network diagrams, look more like a flowchart. PERT charts are helpful for scheduling and controlling the project activities. Each task has a separate box, called a node. Within each node is the task identifier, the task duration, the start and finish day. While project managers often will use both the Gantt and the PERT charts, each tool provides information to help manage through project complexities. The Gantt chart is typically considered less informative because of its inability to allocate resources and create real follow up pursuant to goal accomplishment. Gantt charts excel at showing relationships between tasks, but it is the PERT chart that clearly outlines the complex dependencies and logical relationships between the tasks. PERT charts are also used to outline the critical path of a project. Using the picture above, recall the project network diagram terminology introduced earlier: task, duration, predecessor, and successor. Notice in the picture, Task 1 has a duration of 20 days. The task starts on 03/16/09. Notice that Task 2 and 3 also start on 03/16/09. This shows that these three tasks can be worked on simultaneously and it is not necessary to complete one of those tasks before starting the next. Task 4 is dependent on the completion of Task 2, this task cannot start until the end of Task 2 duration. Task 2 is a predecessor for Task 4. Remember, a predecessor is a task that must be completed before the next step can be started. The task that comes after the predecessor is called the successor. In this example, Task 4 is the successor to Task 2. The start date for Task 4 is the end date of Task 2. The arrows represent dependencies.Task 6 cannot start until Task 4 is completed, and the final step is dependent on both Task 5 and Task 6.

Backorder

A backorder is a receipt of an order for a product when there is no inventory in stock. This is a type of inventory shortage where a customer makes an order, waits to receive the order, and when the product is available the order is filled. Often the inventory cost of backorders is reduced because the carrying cost is reduced by more than the additional backorder cost. But the risk is that the business will lose the sale completely; this occurs when customers go to a competitor that has the product available. Several types of products that are often set up using a backorder system are products with high perishable rates. Ordering on demand can avoid costs due to deterioration. Customers are also often willing to wait for delivery on very expensive items (such as cars or motorhomes) or low demand items such as specialty items. Backorder rates will usually increase as delivery time decreases since the wait time for delivery is reduced. Mathematical formulas can calculate inventory costs with allowed backorders. However, backorder costs are difficult to measure because it is hard to put a price on the cost of lost goodwill from the customer when a company does not have the product in stock. Total inventory costs will be the carrying costs + ordering costs + backorder costs. Backorders can be considered if the total cost with backorders is less than the total cost using the EOQ model. Remember, sometimes the backorder costs can create reduction in carrying costs. If there is a high risk of lost sales, the company may not allow backorders, even with the reduced costs.

Explain how a decision tree is used to identify a simple decision analysis problem and solution alternatives.

A decision tree is a graphical representation of the alternatives for a decision, along with possible states-of-nature or outcomes. All decision trees contain decision nodes and state-of-nature nodes. A decision point is a point where one of several alternatives can be selected. A state-of-nature node is a possible outcome that would be out of the decision maker's control, such as the weather or the state of the economy. To analyze a problem using a decision tree, first define the problem or decision to be made and list the alternative solutions. List states-of-nature that could impact the alternatives. Assign probabilities to each state-of-nature. Estimate the payoff for each alternative and state-of-nature combination. Multiply the probability by the potential revenue to determine the expected value. The alternative with the highest expected payoff will be selected.

Summary

A decision tree is a graphical representation of the alternatives for a decision, along with possible states-of-nature or outcomes. All decision trees contain decision nodes and state-of-nature nodes. A decision point is a point where one of several alternatives can be selected. A state-of-nature node is a possible outcome that would be out of the decision maker's control, such as the weather or the state of the economy. To analyze a problem using a decision tree, first define the problem or decision to be made and list the alternative solutions. List states-of-nature that could impact the alternatives. Assign probabilities to each state-of-nature. Estimate the payoff for each alternative and state-of-nature combination. Multiply the probability by the potential revenue to determine the expected value. The alternative with the highest expected payoff will be selected. Usually estimated values are related to profits, but not always. Electronic spreadsheets can be used to draw decision trees and calculate the estimated value. Some electronic spreadsheet programs, such as Excel, have add-in programs that can be added to help create and evaluate decision models. There are eight steps in the decision-making process: 1. Identify the problem; 2. Establish decision criteria; 3. Weigh decision criteria; 4. Generate alternatives; 5. Evaluate the alternatives; 6. Choose the best alternative; 7. Implement the decision; and 8. Evaluate the decision. Many decisions we make are based on uncertainty. There are formal frameworks that can be used to analyze alternatives, leading to a higher probability for determining the best outcome. Courses of action represent the decision that are made, and-state of-nature represent things that impact the decision, but are out of the decision maker's hands. Calculating expected monetary value is a widely used technique to help evaluate alternatives when the risks involved have several states-of-nature and the probability of each state is known. A decision tree is a graphical model used to visualize the important aspects of a problem, as well as show decisions, alternatives, actions, and outcomes. A payoff table is similar to a decision tree, but uses a table instead of the graphical representation. A payoff table is best used when there is one decision to be made, and a decision tree is more likely to be used when there is a sequence of decisions to be made. Uncertainty is not quantifiable because the future events are not predictable or known to exist in order to plan. Risks are known possibilities and can be measured by the probability of occurrence. Risk analysis involves quantifying the likelihood of an event occurring and the extent of that event's impact. Past events or models can be used to evaluate risks. Decisions made under risk and uncertainty should be identified and quantified when possible. Payoff tables provide a model to evaluate alternatives (along with states-of-nature) to determine estimated values of each alternative. Both payoffs and risks must be evaluated when making a decision.

A Decision Tree

A decision tree is a tool that can be used to visualize the important aspects of a problem. This graphical model shows decisions, alternatives, actions, and outcomes. The decision tree resembles a tree with roots and branches, but is horizontal. The decisions and conditions are given on the branches and the outcomes are shown on the far right. This visual model shows all possible combinations of decisions and alternatives.

Multiple Criteria Decision Making (MCDM)

A decision tree is more likely to be used when there is a sequence of decisions to be made. A homebuyer might have three alternatives, or home choices. The features, or criteria, of each alternative must be evaluated. The decision maker rates the importance of each feature to give weight to the more important criteria. In the below table, a homebuyer has narrowed the home choices to three homes. The criteria for each home is ranked from 1 to 5 with 5 being the best. Criteria Home 1 Home 2 Home 3 Square footage 5 3 2 Location 3 2 5 Price 3 4 3 Total Score 11 9 10 The next step is to apply the weights for each alternative. Location is the criteria given the most importance since it has a weight of .4. The sum of the criteria weights must equal one. CriteriaWeight Location A Location B Location C Square footage .25 5*.25 = 1.25 3*.25 = .75 2*.25 = .50 Location .40 3*.40 = 1.2 2*.40 = .80 5*.40 = 2.0 Price .35 3*.35 = 1.05 4*.35 = 1.4 3*.35 = 1.05 Total Score (weighted) 1.00 3.50 2.95 3.55 Using this structured multiple criteria decision-making model, the homebuyer can clearly see that home 3 has the highest score and is the best choice.

Putting it to use: Use spreadsheet software to solve math problems needed to perform quantitative analysis Continued:

A few of the commonly used functions that are available in most spreadsheet software tools include count, sum, maximum value, minimum value, and average, but there are also many more. The lists of available functions provide access to hundreds of functions for business and statistical analysis, financial calculations and predictions, and many other complex and advanced mathematical calculations. A key benefit is that because the entire formula does not have to be entered, functions can save time, especially when working in a large spreadsheet—time is saved both during data entry as well as during calculations. Functions are most commonly applied to a range of cells. Recalling the previous exposition on cell ranges, remember that a range of cells can be made up of adjacent or nonadjacent cells. Adjacent cells are grouped together with no gaps between cells. Nonadjacent cells consist of two or more separate blocks of one or more cells that can be separated by rows or columns. To use a function, insert the function into a sheet by typing an equal sign (=) in a cell and then the functions name. Once you begin typing the function name, a list will automatically appear identifying available functions. Depending on the spreadsheet software, there is usually a Function button, which allows access to a list of all available functions. You may have noticed that as you type in your formula, the spreadsheet software provides tips for the values that need to be entered into the formula. The format of these tips will vary depending on the spreadsheet software you use, but the requirements for the function will not change. In order to use functions correctly, you will need to understand the different parts of a function and how to create arguments in functions to calculate values and cell references.

Forward and Backward Pass

A forward and backward pass will be used to calculate the ES, EF, LS, and LF for each activity node. A forward pass refers to the first pass in calculating the total duration of the individual tasks in a project. As the typical flow chart moves from left to right when it is constructed, the forward pass moves from left to right. Start from the first task node on the left and proceed towards the last tasks in a project. A backward pass is done after the forward pass. Its intent is to find the latest start and finish times/dates for the project. This gives the planner visibility of those tasks that can be delayed while still not delaying the total project. The earliest times are found using the forward pass. This is done by starting at the beginning (or left side) of the network diagram and making a forward pass through the network. The start time of the entire project will be day 0 (day 1 can also be used). Remember that before an activity can be started, the predecessor activities must be completed. The earliest finish time can be calculated by adding the earliest start time to the activity's duration. The latest times are found using the backward pass. This is done by starting at the end (or right side) of the network diagram and making a backwards pass. The latest start time is the latest finish minus the duration of that task. The network diagram below contains 4 activity nodes. The first task identifier is Develop Specifications. Since this task has no predecessors, the early start is 1. The task duration is 2, by adding the duration of 2 to day 1, the earliest this task can finish on is day 3. There are three successors to the Develop Specifications task. These tasks can not start until the first task has been completed. The early start for the three successors will be the same as the early finish of the predecessor. To find the early finish, add the early start to the duration of each task. Add the task duration to the early start to calculate the early finish. The first node branches out to three nodes. The first node is a predecessor to Design hardware, Design Software, and Outline User's Manual. The early start date will be the earliest date the task can start, assuming all predecessor activities have been completed. So for all three of these next nodes, the early start will be 3. Note that the early finish for Design hardware is 13, calculated by adding the early start of 3 to the task duration of 10. Notice that Test software with hardware has two predecessors. The early start for this node is the largest early finish of the predecessors for a node. In this case, 14 is the largest predecessor early finish value. So 14 will be the early start for Test software with hardware. Add the task duration of 6 to 14 to get the early finish of 20 for this task. Continue this process from left to right, until reaching the final node in the network diagram. The finish time for this project will be 25. Completing the backward pass will establish the activity LS and LF. The late start and finish are entered in the bottom row of the activity node. Since the finish time for the entire project is 25, this is the latest finish of Implementation of System node. To determine the latest start, subtract the activity duration from the latest finish. LS = LF - duration = 25 - 5 = 20 There are two predecessors to the last node, Test Software with Hardware, and Create User's Manual. Both of these tasks will have a late finish of 20 (the smallest of the late start times). Subtract the duration of each task to calculate the late start. Continue this backwards pass from right to left through the network diagram. Notice that Develop Specifications has three successors. The latest finish time for Develop Specifications is the smallest of the latest starting times of the task predecessors. Since the smallest LS for the three successors is 3, this becomes the late finish time for Develop Specifications.

Summary

A good decision is based on the sound decision making techniques discussed in this module. This does not mean that the outcome will be guaranteed. It means that following sound decision techniques will allow decision makers to be better informed prior to making a decision. To calculate the expected value for each alternative being considered, use the following formula: ExpectedValue=pWrW+pLrL where p=probability r=return W=win L=lose Calculate this for each alternative under consideration to determine the option with the highest value. A payoff table lists the payoff for each decision-outcome pair. This is a tool for summarizing the advantages and disadvantages of a decision. Listing positive or negative returns (payoffs) with all possible combinations of alternative actions (these are under the decision maker's control) and external conditions (these are not under the decision maker's control). A payoff table shows the financial returns minus the costs for each alternative under consideration. One technique for decision making is to choose the decision that maximizes the worst payoff, in other words the worst possible loss for one choice is a smaller number than the least possible loss of the other choices. This is called the maximin criterion and is a very conservative and pessimistic approach. A pessimist thinks the worst will happen and wants to make the best decision to prepare for that worst case scenario. Maximin selects the alternative whose worst loss is better than the minimum loss of the other alternatives. A minimin is the alternative that represents the minimum of all of the minimum alternative costs. Minimax criterion minimizes the possible loss by selecting the alternative with the minimum possible loss for a worst case outcome. Minimax is the minimum of all of the maximums. Maximax is a more optimistic approach, more appropriate for a risk taker. The approach maximizes the best payoff. This approach focuses on the large possible gains, and fails to focus on the possible losses. Maximax is not frequently used by businesses because this method could bankrupt a company. These criteria provide a continuum of optimism to pessimism as follows: Criteria Optimism/Pessimism Maximax Most Optimistic Maximin Less Optimistic Minimax Less Pessimistic Minimin Most Pessimistic Payoff tables provide a model to evaluate alternatives, along with states-of-nature, to determine estimated values of each alternative. Both payoffs and risks must be evaluated when making a decision.

The network diagram will help to plan, manage, and monitor the project throughout the project duration. Which of the following is a key assumption underlying a network diagram? When drawing the network diagram, it is important to understand the item or duration of each task. It is also important to do what? Aside from determining the relationship of each task to its predecessor or successor, what else is important when building a network diagram? Network diagrams are often used to _______.

A network diagram can be created for the Bachelor of Science degree in Business and Economics project. When projects become complex, software is necessary to draw such diagrams. In this college degree example there are only 11 tasks. The convention is to show the flow of tasks horizontally from left to right as follows. An activity can begin only after all predecessors have been completed. A predecessor is an activity that precedes that current task and must be completed before the current task can begin. Determine whether the task precedes or supersedes another task in the project. You can only create network diagrams if you know the order of the elements. The duration of each activity. All projects have a begin and end date, so the time relationship of tasks in the network is critical. help plan and manage the project through the duration of the project. By having a design to accomplish necessary tasks in a time sensitive and proper order, the project can be managed to specific goals during the duration.

In statistics, one of the most fundamental procedures is that of sampling. This is where a subset of some entire population is taken and measured, and that measurement is used to infer some characteristic about the entire population. For example, an entire population would be "all grocery stores in the United States." However, for data collection and quantitative analysis, analysts will use a sample of the population. There are a number of types of sampling. The characteristic under study in the population is called the ___, whereas that same characteristic in the sample is called the ___. A "random" sample has one key characteristic whereby each member from the population has ___ chance of being selected from the population.

A parameter is some characteristic under study taken from the population, not the sample. A statistic is some characteristic under study taken from the sample, not the population. The chief characteristic of a random sample is that every member has an equal opportunity to be either selected for the sample or not selected for the sample.

Payoff Table

A payoff table is similar to a decision tree but uses a table instead of the graphical representation. All of the alternatives are listed on the left side of the table and all of the possible states of nature are listed across the top. The table contains the payoff values for all possible combinations of decision alternatives and states of nature. Payoff values can be expressed in terms of costs, profits, or any value appropriate for the decision being made. Decisions Strong Sales Likelihood 75% Weak Sales Likelihood 25% Expected Value Location 1 - $5,000 - $1,000 - $3,750 + $250 = $4,000 Location 2 - $4,000 - $2,000 - $3,000 + $500 = $3,500 A payoff table shows the consequences of each location for each state of nature that might occur. A payoff table is best used when there is one decision to be made and a decision tree is more likely to be used when there is a sequence of decisions to be made.

The cornerstone of the definition for project is the following:

A project is temporary, with a clearly established timeframe for the beginning and ending. A project creates a unique product, service, or event. Each task in the project is specific to the defined project goal or objective and can be described within limits of time. Consider a simplified, but applicable, personal project example: You are entering college and want to major in Business. The particular degree desired is a Bachelor of Science in Business Management and Economics. There are several core courses that must be taken before a degree major can be declared. Completing the degree is a top priority, time is money and the longer it takes the more it will cost. Project objective or goal: Complete the courses necessary to declare a major in Business and Economics. Finish the courses in the least amount of time possible. Receive the Bachelor of Science degree. This is a project because there will be an identified start and end date, and all courses that are taken will be specific to the degree. Completion of the project will result in a unique objective of receiving the degree. The types of courses that would need to be taken are within the areas of: - Eng Math Business While this is a breakdown of the types of courses, it is not detailed enough to plan and execute the project. Since a variety of courses in English, Math, and Business will need to be taken, specifics will need to be determined producing a more detailed breakdown of tasks. To accomplish this, a work breakdown structure (WBS) should be created to breakdown all of the activities necessary to complete the project. The right tasks at the appropriate management level must be determined in order to properly and effectively manage the project. The next steps would involve questions that answer how long the project will take and in what sequence should the tasks be done. Each activity must be assigned an activity duration, or length of time required to complete that task. For our example, each course is one term long. Finishing the project on our own timeline or in one year is not possible because there is a sequence in which the courses must be taken. Courses have prerequisites and logical order. From a project perspective, the prerequisites are called preceding activities, or predecessors. The preceding activities will help determine the sequence in which tasks must be completed. Courses that must follow another course are called successors. The sequence of activities will be placed into a visual tool called a network diagram. Network diagrams help plan, schedule, and control complex projects. After sequencing the courses that will be taken, each course will need a duration or length of time it will take to complete. Using the sequencing and duration for each course will eventually be used to determine the project duration, the time in which the project can be completed. The table below outlines which courses must be taken before others. A predecessor is a task that must be completed before the next step can be started. The task that comes after the predecessor is called the successor.

Another Alternate Description: Using the example of a Bachelor of Science in Business and Economics project model, a PERT Chart can be constructed.

A project's critical path is defined as the longest sequence of tasks, i.e. a path, indicating the minimum time in which the project can be completed. Finding the critical path of a project is an important step in determining not only how long the project will take, but also which tasks have float or slack time, and which tasks do not. Notice in the simplified table above for the degree project, that path five is the longest, therefore it is the critical path. In very small projects such as pursuing the Business and Economics degree, each task can be easily identified, as well as the path it is part of. In many business projects, determining a project's critical path is more complicated than this example, sometimes consisting of hundreds of tasks. Enumerating all the paths by hand, and placing them in a table is nearly impossible. There are other algorithms (a sequence of steps and activities) that will allow the manager to find the project's critical path, these are more efficient than enumerating all the paths by hand.

Seasonal Trend

A seasonal trend is an upward or downward movement in data that repeats at regular intervals. For example, a dentist in Vero Beach, Florida, will have increased dental cleanings during the winter months due to an influx of northern retirees. When the retirees return to their northern home in the spring, the cleanings will drop off. Another example may be that sales in toy stores are expected to be higher in November and December due to the holidays, and this trend would be expected to be repeated each year.

Explain time series analysis:

A time series is a resulting set of data points taken over specific time intervals, such as a day, week, month, or quarter. Examples include daily rainfall, weekly shipments of inventory, and quarterly sales. There are four possible types of time series: trend, seasonal, cyclical, and random. Gaining a more solid understanding of the underlying reasons for trends will help businesses with forecasting.

Trend

A trend is the general upward or downward movement of data over a relatively long period. For example, the stock market may have an upward trend over a period of a year. During that year, there may also be a few downward movements, but when observed over the year, there is clearly an upward trend in the data.

Describe a project as a network of activities.

All organizations, whether for profit or not for profit, move forward and grow through a combination of operational activity and project activity. Operational activity is considered the day-to-day functions of the organization. These activities are used to keep the business operational and can include accounting transactions, customer service, inventory processing, and any other tasks that are completed day in and day out. A project is a specific undertaking which involves many focused tasks that are intended only for the completion of the project. In the following exposition, the topic of projects will be explored in more detail.

Solve math equations needed to perform quantitative analysis.

An equation represents establishing that two mathematical expressions are equal. For example, in business we can use the element of profit and its relationship to income and expenses. As an equation this is written: Profit = Income - Expenses. This is a simplified business equation stating that by subtracting expenses from income it will be equal to profit. Equations often have variables. Variables are symbols that represent unknown numbers. For example, suppose a business has income of $10,000 and expenses of $7,500. Using the profit equation provided above, the variable P can represent profit in this equation: P=10000−7500 We can complete this calculation to solve for P. P=2500 Another example of an equation that uses variables is 4x−3=9. In this equation, x is the variable. The number in front of the variable is called the coefficient. A variable that does not have a number in front is assumed to have a coefficient of 1. The numbers that stand alone, 3 and 9, are constants. Coefficients are the numbers in front of a variable, while constants stand alone.

Histogram

An instructor wanted to graph the number of test scores that fell into certain grade intervals. Twenty-five test scores were recorded. The grade intervals were 0-9, 10-19, 20-29, 30-39, and so on, up to 100+. The following histogram shows how many scores fell into each category. A histogram is a graph that shows frequency of occurrence in each category. The x-axis displays the categories or ranges; the y-axis indicates the frequency. In the above graph, there is one student whose grade is in the range 0-9. There are two students with a grade in the range 10-19. To determine the number of students with grades in the range of 30 to 59, add the frequencies of the 30-39, 40-49, and 50-59 grades. There are two students in the 30-39 range, three students in the 40-49 range, and two students in the 50-59 range. There are a total of seven students with grades in the range of 30-59.

Exponents are a mathematical expressions that allow a base number to be multiplied by itself a given number of times.

As an example, consider an exponent expression of 34, the exponent is 4, and the base is 3. Manually writing this expression in an expanded form would result in multiplying the base of 3 four times (3∙3∙3∙3). The result is 81. The same expression can be calculated using a calculator or spreadsheet software. With these tools, use the caret symbol, ^, to represent an exponent. Using the same example, 34would be written 3^4. One exponent rule to always remember is that any base number raised to the 0 power is equal to 1 (except 00,which is called "undefined"). Exponents can be negative. A negative exponent indicates how many times to divide 1 by the base number. Using 5−2=(152)=15∙15=(125). A negative exponent does not indicate that the number is negative. When multiplying two numbers with the same base and each has an exponent, keep the base and add the exponents. Example: 23∙24=23+4=27=2∙2∙2∙2∙2∙2∙2=128 When dividing two numbers with the same base and each has an exponent, keep the base and subtract the exponents. Example: (3432)=34−2=32=3∙3=9 When raising an exponent to a higher exponent, multiply the exponents. Example: (32)3=32∙3=36=3∙3∙3∙3∙3∙3=729 The symbol, (x)−−−√, is read 'the square root of x', where x is the base, and √ is the radical. We use this symbol to take the square root of a number. For example, 9-√=3 because 32=9. When working with radicals, remember that it is not correct to take the square root of a negative number. This is because there is no number that can be squared and result in a negative number.

Use a payoff table to present information for a simple decision analysis problem:

As previously demonstrated, a payoff table lists the payoff, or expected value, for each decision-outcome pair. This is a tool for organizing, summarizing, and displaying the advantages and disadvantages of a decision. A payoff table lists positive or negative returns (payoffs) with all possible combinations of alternative actions, under the decision maker's control, and external conditions which are not under the decision maker's control. A payoff table shows the financial returns minus the costs for each alternative under consideration.

Complex Formulas Simplified with Parentheses

As you have seen, math expressions can become complex very quickly; this is especially true when conducting quantitative analysis to solve business problems. One strategy to keep a formula organized and the order of operation working correctly is to use parentheses. Correctly using one or more sets of parentheses in a formula will help control and organize the equation. Parentheses are used in mathematical expressions to identify an element within the formula that must be solved first-before any other calculation in the formula can be completed. The equation between the parentheses is treated like a nested formula or math expression-the answer to the formula between the parentheses replaces the math expression in the larger math expression. Take a look at a few examples of a complex formula that use one or more sets of parentheses: First problem: 40(4∙2)−7 Enter the formula into your spreadsheet by typing an equal sign (=), then type the number 40, a forward slash (/), open parenthesis, the number 4, an asterisk (*), the number 2, a closing parenthesis, a minus sign (-), and the number 7. Here is what you should see when you click on the cell: Following the order of operation, this formula was solved by completing the following passes: 1. Pass 1: Multiply: The multiplication located inside the parentheses must be completed first, 4∙2=8. Now our expression reads: 40/8−7. 2. Pass 2: Divide: Following the order of operations division is completed next, 40/8=5. Now our expression reads: 5-7 3. Pass 3: Subtract: The answer is -2 Second problem: 8+(7∙9)−12 Enter the formula into your spreadsheet by typing an equal sign (=), the type the number 8, a plus sign (+), open parentheses, the number 7, an asterisk (*), the number 9, closing parenthesis, a minus sign (-), and the number 12. Here is what you should see when you click on the cell: Following the order of operation, this formula was solved by completing the following passes: 1. Pass 1: Multiply: The multiplication located inside the parentheses must be completed first, 7∙9=63. Now our expression reads: 8+63−12. 2. Pass 2: Add and Subtract: Following the order of operations, both addition and subtraction will occur during this pass, reading the expression from left to right, 8+63=71−12=59. The answer is 59 Third problem: 17∙(63−54)+2613 Enter the formula into your spreadsheet by typing an equal sign (=), the type the number 17, an asterisk (*), open parenthesis, the number 63, a minus sign (-), the number 54, closing parenthesis, a plus sign (+), the number 26, a forward slash (/), and the number 13. Here is what you should see when you click on the cell: Following the order of operation, this formula was solve by completing the following passes: Pass 1: Parentheses: The subtraction expression will be solved first, 63−54. Our expression now reads: 17∙9+2613. Pass 2: Multiply and divide: There are no exponents, so the next pass completes the multiplication and division expressions moving from left to right. 17∙9=153 and 26/13=2. Now our expression reads: 153+2. Pass 3: Add. The answer is 155 The use of parentheses allows for quantitative analysis to apply the order of operations to complex mathematical expressions. This makes it possible for quantitative analysis to be used to make managerial decisions.

Population

Businesses take data and turn it into powerful information to make good decisions and choices. When a business has access to scores of data, the business can calculate meaningful and revealing descriptive statistics, turning the results into useful information that is helpful in making decisions. When working with data, a business either has access to the entire data set or all of the items of interest, or only a partial set of data. When accessing an entire set of data, it represents the population. As an example, perhaps we are interested in studying and marketing products to all of the college freshman in the United States. As these are all items of interest, this specific set of people represents that particular population. There are many things we want to know about a population. We want to know about quantitative and qualitative attributes of the population. Quantitative attributes or measures tend to be numerical, whereas qualitative attributes are more categorical. Using the population of all college freshman, we might be interested in the following attributes: Quantitative High School GPA ACT score SAT score Height Weight Household income Qualitive Needs Financial Aid? Yes/No Eye color Major (could included undecided) Ethnicity Sex Religion For each of the quantitative variables, we would have the opportunity to calculate descriptive statistics. For the qualitative variables, we can establish facts and opinions, which help with subjective analysis. These measures on population attributes are called population parameters or simply parameters. If and when we know these parameters, we can use them reliably in making decisions and taking actions. There is just one little problem. We often do not have access to all the data in the population. Even in our example, it is daunting to collect all the information we might want on the population of incoming college freshman across the entire country. It is daunting both in terms of time and cost to collect it all. Instead, we look at a smaller, representative subset of all incoming college freshman in the country. This representative subset would be far more accessible and less costly to collect. A subset of a population is called a sample. We could look at the same attributes as in the population and calculate the descriptive statistics and percentages of the sample quantitative measures and qualitative attributes. The results would be approximations or estimates of the population parameters. We call them sample statistics or simply statistics. Using samples is the easier and less costly way to obtain and calculate statistics to draw conclusions or make inferences regarding the population parameters. To summarize:

Summary

Businesses try to optimize their functional performance by maximizing profits or minimizing costs. Linear programming is a quantitative tool used to find the optimal solution for these problems. Optimization problems are often very complex. An objective function seeks to maximize or minimize some quantity. There is an economics concept of scarcity that states there is a limit to what can be produced and sold. Constraints are restrictions that limit the degree in which a company can pursue its objectives. A feasible region for a problem represents all of the points that satisfy all of the constraints. If the feasible region is formed by the intersection of the linear constraints, then the optimal solution to the problem is found at one or more of the corner points. Linear programming can be used in most any type of decision where there are viable choices that need to be examined under the aspect of optimization. Linear programming makes quick work of complex, integrated, and risk-oriented problems. Sensitivity analysis is a common post-optimization technique used in quantitative analysis. Virtually all systems, processes, or functions that utilize quantitative measure to predict or forecast results are subject to some form of sensitivity analysis. What a decision maker wants to know before the decision is implemented is, essentially, what is the chance that it will not work as predicted? This can be done through developing a series of "what-if" scenarios. By isolating each of the values of the regular constraints that originally formed the optimal intersection (feasible region), it is clear to see which requirement is more sensitive as it relates to maximizing profit. The new "optimal" value that results in changing either of the right-hand side values of the constraints (two constraints in this example) is referred to as the dual price. We can also change the left-hand side of each objective function coefficient until we reach a value where the slope of the objective function is no longer confined by the boundaries of the constraints and therefore cannot intersect at the optimal point. The range of optimality is the amount the objective function coefficient can change without changing the values of the optimal solution points. Graphing the regular constraints, locating the feasible area by interpreting the direction of the regular constraint inequality, and locating the optimal point (outermost "corner" of the feasible area) yields an optimal coordinate. In all cases in linear programming, the model seeks the best or optimal choice within a set of relevant constraints or parameters.

To calculate the standard deviation, use the following steps:

Calculate the mean of the data set. Subtract the mean from each value in the data set. Square the deviations. Add the squares of the deviations. Divide by the number of values. (Note that if the data do not represent the entire population, but instead represents only a sample of the population, then divide by n-1 instead of n). This value is the variance between the mean of the data sets and the data points. Take the non-negative square root of the result. This value is the standard deviation. Using these calculations the standard deviation of the Set 1 Test Scores is approximately 14.45, the standard deviation of Set 2 Test Scores is 25, and Set 3 Test Scores standard deviation is approximately 20.41. As you can see from looking at the individual values in each of the set of scores, Set 1 Test Scores tend to be clustered more around the mean value of 75. As a result, the standard deviation is lowest for that set of test scores. This is the mathematical formula for calculating the standard deviation described in the steps above: SD=√∑(x−x¯)^2/n SD stands for standard deviation. n represents the number of terms in the data set. x¯ represents the mean or average of the values in the data set. x represents each individual value in the data set. ∑ is the summation symbol and indicates that all of the terms will be added together. If the data does not represent the entire population, but instead represents only a sample of the population, then divide by n-1 instead of n). If most of the data are clustered around the middle of the range of values and the remaining values taper off symmetrically towards each end, this is considered normal distribution. The graph in the following picture represents a normal distribution of data points. Notice that the graph has a bell-shaped curve. The center line represents the mean, or most probable occurrence. The first line on either side of the mean represents data that are one standard deviation from the mean (see the darkest blue sections under the curve). Sixty-eight percent of the data points in normal distribution will fall within one standard deviation of the mean. Approximately 95 percent of the data will fall within two standard deviations of the mean, and 99.7 percent will fall within three standard deviations. The exact shape of a bell curve can vary, but the peak is always in the center, and the curves on both side of the midline are symmetrical.

Which of the following methods is NOT used to determine if an equation is linear or nonlinear?

Compare x and y intercepts. The x and y intercepts do not impact the location of the other points on the graph and do not determine whether an equation is linear or nonlinear.

Conditional Probability

Conditional probability is the probability of an event occurring after another event has occurred. Many business conditional probability situations are more complex than the card example given above and require a formula to solve. To demonstrate conditional probability, it is written as the following: P(B|A) means "The probability of Event B given Event A" In other words, event A has already happened; now what is the probability of event B happening? For example, consider if there was a handful of marbles in a sack. There are two blue marbles and three red marbles in the sack. What is the probability of getting a blue marble? This first probability is independent, or marginal and is 2 in 5, or ⅖. However, if you took the marble out, and did not replace it, the chances change, a condition has been created! Therefore, the next time selecting a marble, which is now event B, given event A has occurred, P(B|A) = If we got a red marble before, then the chance of a blue marble next is 2 in 4. If we got a blue marble before, then the chance of a blue marble next is 1 in 4.

When identifying decision criteria, organizational goals and corporate culture must be carefully considered. Often this means knowing the stakeholders (C-level executives and your immediate management) who will be approving the decision. It is a good idea to review both 1. Identify the Problem and 2. Establish Decision Criteria with the stakeholders. This could avoid having to rework the entire problem and thus save both time and money.

Consider, the following as possible decision criteria for the New Warehouse project: -Construction costs: The cost of land and to build the facility. -Operating costs: Labor, IT, and Overhead - Distance from the factory - Average and max number of orders that can be processed in the facility in a given 8-, 16-, or 24-hour workday.

Olds-Royce Motors wants to increase its sales of coupes and sedans; they believe they can accomplish more sales with more advertising. They can run TV ads for $600,000 each with 10 million viewers, or get magazine ads for $800,000 with 3 millions subscribers. They have a $5.6 million dollar budget. Olds-Royce Motors estimate they will increase coupe sales by 3% and reduce sedans sales by 1% for the TV ads. They estimate they'll increase sedan sales by 5% for sedans and decrease coupes sales by 2% for magazine ads. What is the constraint in this production problem, and what type of constraint is it?

Constraint: Advertising budget Constraint type: Cost The company has a fixed advertising budget. The budget is subject to the increases and decreases in the sales projections.

Wilson Company makes softballs and baseballs. Softballs cost $13 each, and baseballs cost $15 each to produce. It takes 5 oz. of leather for each softball produced and 4 oz. of leather for each baseball produced. What is the constraint in this production problem, and what type of constraint is it?

Constraint: Leather Constraint type: Materials Leather is the constraint in this problem, and it is a material constraint.

Constraints

Constraints of a project must be balanced and prioritized to achieve the specific objectives. These constraints will be related to time, cost, and/or performance. After clearly defining and understanding the objectives of a project, all of the tasks to complete the project are identified. Any task relationships must be known, as well as which tasks must be completed before other tasks are started. The network diagram is used to help the decision maker plan, manage, and monitor the project throughout the project duration. Each activity node on the diagram will indicate the name of the activity, its start and finish times, and the duration. The placement of the activity node in the diagram will indicate its predecessors and successors (tasks that must be completed before and after the activity). The critical path will be the longest path on the network diagram.

1. An appliance distributor expects to sell about 9,600 refrigerators next year. The annual carrying cost is $16 per refrigerator, and the ordering cost is $75. The distributor operates 288 days a year. What is the economic order quantity for refrigerators? 2. An appliance distributor expects to sell about 9,000 refrigerators next year. The calculated EOQ is 250. How many times a year will the distributor have to place orders? 3. An appliance distributor expects to sell about 9,600 refrigerators next year. The annual carrying cost is $16 per refrigerator, and the ordering cost is $75. The distributor operates 280 days a year. If the EOQ is 225, what is the length in workdays of an order cycle? 4. An automotive equipment manufacture uses about 3,200 armatures a year in production. Annual carrying cost is $4 per armature, and the ordering cost is $100. They operate 240 days a year. What is the economic order quantity? 5. An automotive equipment manufacture uses about 3,200 armatures a year in production. They operate 240 days a year and will be ordering 10 times a year. What is the number of workdays in an order cycle? 6. An automotive equipment manufacturer uses 300 armatures a day in production. Delivery of the armatures takes 7 days. What is the reorder point for the armatures?

Correct.The EOQ calculation is 2*(9600)*7516−−−−−−−−√=300 Where: D = yearly demand = 9,600 S = ordering costs = $75 H = carrying (holding) cost per unit = $16 2. Correct. Divide the annual demand by the quantity per order to find the number of times per year the distributor will place an order. = 9,000/250 3. Divide the annual demand by the EOQ to determine the number of orders. Because there are 280 workdays in a year, divide the number of workdays by the number of orders. = 280/40 4. The EOQ calculation is 2*(3200)*1004−−−−−−−−√=400 5. To calculate the number of workdays in an order cycle divide the number of workdays by the number of orders placed. 240/10 = 24 6. The formula would be = 300 * 7. To determine the reorder point, multiply the daily demand by the number of days between ordering and receiving the items.

Which of the following is the correct formula for calculating the crash cost per time period? Based on the graphic below, what would be the last choice for an activity the project manager would crash? Based on the graphic below, what would be the additional cost to crash Activity C by the maximum time reduction period? Activity A: Normal Time (Days) 12 Normal Cost $18,000 Crash Time (Days) 8 Crash Cost $22,000 Maximum Time Reduction: 4 Activity C: Normal Time (Days) 20 Normal Cost $32,000 Crash Time (Days) 12 Crash Cost $38,000 Maximum Time Reduction: 8 Based on the graphic below, what would be the first activity a project manager would crash in order to speed up the project?

Cost Slope = Crash Cost − Normal CostNormal Duration − Crash Duration Subtract the normal cost from the crash cost. Divide the result by the difference between the normal time and the crash time. The formula is (crash cost per period) * (# of time periods to crash). Substituting: ($1,000*2)=$2,000. This is the amount of costs added to this task in order to crash it by 2 days. Activity B has the most expensive cost per period to crash, at $3,000 per period. The formula is (Crash Cost Per Period * # of time periods to crash). Substituting: ($750×8)=$6,000. This is the amount of costs added to this task in order to crash it. Activity D has the cheapest cost per period to crash, at $667.

Consider a business that is expanding its territory. It is imperative that the business chooses a location with a high level of drive-by traffic, easy access, and a design layout that is easy to work with. Therefore, when using expected monetary analysis, these factors would be called ___. This same business must also evaluate how much revenue, and ultimately profit, will be possible for the various ___. By using the expected value method, a business can be more ___ about its alternatives, particularly when it comes to formulating decisions.

Criteria: Businesses choose new locations based on specific criteria that meets their organizational needs. Locations: Each potential location is likely to produce varying levels of revenue. This is an important determination for choosing a location. Specific: Expected monetary analysis requires the use of specific information to be helpful for making decisions.

Decision theory:

Decision theory uses a logical, analytical approach to making decisions. Following a rational decision making approach is critical to business success. Bad decisions can bankrupt a business and good decisions can bring profits to an organization. Good decisions must align with the overall goals of the business.

Describe the general steps used in decision-making models.

Decision theory uses a logical, analytical approach to making decisions. Following a structured decision-making approach will reduce the likelihood of making a decision based on emotions. It also ensures that decisions align with the overall goals of the business. This rational and structured approach is critical to business success. Many decisions we make are based on uncertainty. There are formal frameworks that can be used to analyze alternatives and have a higher probability for helping determine the best outcome. Courses of action represent the decision that are made, and state-of-nature represent things that impact the decision, but are out of the decision maker's hands. Decisions made under risk and uncertainty should be identified and quantified when possible.

In constructing a decision tree, there are several elements that must be first identified and then laid out in a logical order on a spreadsheet to enable the expected value calculation to be made. One element starts the whole process and all other elements follow in the decision tree set up. This element is the ___. A company wants to "increase production capacity" for its bicycle production line. This is an example of a ___. A construction project is considering the weather to plan for a rainy or a sunny day when it comes to laying the foundation of a new house. This is an example of what? There is a 85% chance of rain during the construction of the roof on a new house. This is an example of what? A manufacturing firm is considering building either a large or small plant in another state. This is an example of what? By building a large factory in an expanding economic situation, the company estimates that it will realize a profit of $4,500,000 in the first year. This is an example of what?

Decision: In constructing a decision tree, the first element is to specifically determine the decision or alternatives. For example, a business decision might be to "upgrade servers," which would lead to alternative choices to make the upgrade. Decision: In constructing a decision tree, the first element is to determine the specific decisions or alternatives. For example, a business decision might be to "upgrade servers," which would lead to alternative choices to make the upgrade. This is an example of a state-of-nature, a condition under which the decision operates. States-of-nature are out of the decision maker's control. This is an example of a probability, which is a specific number that represents the likelihood of an occurrence of some state-of-nature. This is an example of a decision point, which is a point at which a determination is made to move down one specific path among several alternative paths. This is an example of an expected payoff. First, the alternate possible decisions are considered, followed by the probabilities of the possible states-of-nature multiplied by the individual payoffs, and then the values are added together for the total expected payoff for each alternative decision.

John's saving account pays 1.5% interest annually. This is an example of what type of decision? Betsy owns a construction company. Her profit is greatly impacted by inclement weather. There is a 30% chance of rain tomorrow, and Betsy is trying to decide if she should hire a construction team to work on the construction site. This is an example of what type of decision? Which of the following is a benefit of using a rational decision-making process? Large companies sometimes have lawyers, accountants, and financial experts on call and ready to answer client concerns if the economy and or the stock market has a sudden crisis. This is an example of what type of decision? Fred reorders inventory when his store supply drops below 10 units. This is an example of what type of decision? Rational decision making has been used in many forms and has been effective. However, what is the key value behind using rational decision making compared to an unstructured or subjective form of decision making?

Decisions under certainty are when decision makers know with certainty the exact outcome or consequence that will occur with a given alternative. For decisions under risk, there are several different outcomes that can occur with a given alternative, and the decision makers know the probability of occurrence for each possible outcome. Removes emotional and judgmental influence.. The key value of using rational decision making compared to an ad-hoc, unstructured, subjective form of decision-making is the removal of the emotional and judgmental influence. Decisions under uncertainty. For decisions under uncertainty, there are several different outcomes that can occur with a given alternative, and decision makers do not know the likelihood of each outcome occurring. Decisions under certainty are when decision makers know with certainty the exact outcome or consequence that will occur with a given alternative. The answer should show a clear distinction between rational decision-making and other techniques such as emotion-based or judgmental decision-making. The narrative should refer to a more precise and structured method of arriving at a decision. The answer should refer in some way to preparation and use of decision criteria and refer to the need for objectivity of the part of the decision maker.

Items Produced In-House: Not all items are ordered from a supplier. They are also fabricated or produced in-house. Very similar questions apply in this case. How many should be produced in any given production run? How many times a year must the product be run (how many production runs a year)?

Demand (D): How many items will be needed? Again, this is usually done on an annual basis. This demand or forecast number is the basis for all inventory planning calculations. Setup costs (S): For any fabricated or assembled item, the factory has to set up to run the item. This could include changing fixtures in various machines, setting up the speed of the equipment for the particular item, and ensuring that the right software is loaded into any and all numerical control operations that are in the production sequence. This includes wages of the personnel doing the setup and the apportionment of the facility fixed costs for the time that the setup takes. Rate of production (R): How many of the items can be produced in a day? A week? A month? Or year? Hundreds if not thousands of paper clips can be produced in an hour. On the other hand, it takes up to four hours to assemble an automobile engine. Holding costs (H): This is exactly the same as above.

In the previous module, the concept of inventory and the cost of inventory was expanded to accommodate the strategic and fiscal importance most enterprises place on inventory in the modern era. Inventory was categorized depending on the kind of inventory (raw and packaging, work-in-progress, or finished goods) and what is called the velocity or how fast an item moves (A, B, C, or Slow and Obsolete). Tactically, it makes sense to manage how much of each item to order and keep in stock. Items Ordered from a Supplier: For items ordered from a supplier, the following are used to help determine how much to order and how often:

Demand (D): How many items will be needed? This is usually done on an annual basis. In other words, how many will be needed in the next year? This is more often a forecast than a rock solid number. Cost of the item (P): This seems very obvious because it is what purchasing departments focus on. Actually, in terms of how much to order, which determines how much to keep in stock, the purchase price of the item has no bearing unless volume discounts are involved. Ordering costs (O): How much does it cost to place an order? Most people do not consider this, but in business there are never zero costs for employee time and processing transactions even in a highly automated setting. Holding costs (H): The more inventory on hand, the higher the holding costs. The holding cost is the amount of cost to hold one unit of an item in the warehouse for year. Holding costs represent the amount of money a company spends to keep a certain level of inventory in stock and must include items such as cost of capital, storage, utilities, insurance, taxes, deterioration, obsolescence, and handling costs. Price is not a factor if the price is constant. If a company is going to order X amount of something, say 25,000 in a year, it doesn't matter how often they order and in what quantities. They will still spend 25,000 x Cost of the item for the goods. Ordering Cost and Holding Cost will determine how many orders they place in a year and in what quantity each order it. If there is a volume discount in ordering, then the Cost of the Item, along with Ordering Cost and Holding Cost, will determine the order size and frequency.

Which of the following is an example of a linear programming application?

Determining the best product mix for a retail store. LP can be used to solve this type of problem. Using the information to maximize profit based on the products offered.

Which of the following represents an application of LP?

Determining the most profitable inventory to keep in stock given different cost and profit levels. LP can be made to solve this type of problem as a profit-maximizing problem.

Second problem: 15∙2−5^3

Enter the formula into your spreadsheet by typing an equal sign (=), then type the number 15, the asterisk (*), the number 2, the minus sign (-), the number 5, the caret (^), and the number 3. Here is what you should see when you click on the cell: Following the order of operations, this formula was solved by completing the following passes: 1. Pass 1: Exponent: Following the order of operations, the exponent is solved, 5^3=125. Our expression now reads: 15∙2−125. 2. Pass 2: Multiply: Working left to right, the multiplication will be solved next, 15∙2=30. Our expression now reads: 30−125. 3. Pass 3: Subtract: The answer is -95.

A firm is attempting to make a decision between the alternatives of constructing a new plant, building either a large or small plant, or not building a plan at all. A payoff table was built to help with this decision. Column AColumn BColumn CFavorable Market (75%)Unfavorable Market (25%)Build large plant$650-$100Build small plant$275$50Do nothing$0$0 The total of the probabilities used for the market conditions must be The reason there is an alternative labeled "do nothing" is because a business must always consider that it does not need to It is important to ensure that the payoff table is kept well

Each state-of-nature, and the probability that the state will occur, needs to be carefully evaluated. All of the probabilities for the states-of-nature must add up to 1. A business must always consider that it does not have to make a choice. However, this too must be evaluated. Businesses cannot grow without taking additional risks and understanding the risks. Organized information in a payoff table allows both the analyst and the decision makers to clearly review the alternatives and the expected values. Review the following payoff table: Options Hot Clear Rainy Cold Max Min EV Conference hall 500 600 300 200 600 200 485 Outdoor art festival 500 650 -100 165 650 -100 429 Sidewalk sale 300 500 -50 220 500 -50 332 Probability 0.15 0.55 0.20 0.10 In this table, the artist's alternatives are: Conference hall Outdoor art festival Sidewalk sale The various weather outcomes are the states-of-nature: Hot Clear Rainy Cold A probability, or likelihood, is decided upon for each state-of-nature: Hot 15% Clear 55% Rainy 20% Cold 10%

In both cases, the goal is to produce an amount that minimizes the inventory costs. The two basic formulas are known as the Economic Order Quantity (EOQ) for purchased goods and as the Economic Production Quantity (EPQ) for manufactured goods. The formulas are as follows:

Economic Order Quantity EOQ = √ 2 ⋅ Annual Demand ⋅ cost to place an orderannual holding cost per unit or EOQ = √2 × D × O / H pens Where D = Annual Demand O = Cost to place an Order H = Annual Holding Cost per unit Economic Production Quantity EPQ = √ 2 ⋅ Demand ⋅ Fixed Setup Cost ⋅ Rate of Production / Holding Cost Per Unit ⋅ (Rate Of The Product − Demand For The Product) or EPQ = √2 x D x S x R / H x (R−D) Where D = Annual Demand S = Setup Costs R = Rate of Production H = Annual Holding Cost per unit Note: Demand and Rate of Production must both be over the same time period, either a year, or more commonly, daily. D = Annual Demand d = Daily production rate = D/365 S = Setup Costs r = Daily Rate of Production H = Annual Holding Cost per unit So the EPQ formula becomes EPQ = √ 2 x D x S x rH x (r−d) The production rate has to be greater than the demand. Otherwise, the factory will never be able to make enough product. The benefits of the EOQ and EPQ models are in answering the questions presented at the beginning of this section. How much to buy? How often? How much to produce? How frequently? These formulas take the guesswork out of the system and provide answers that minimize operational costs. Without such formulas, purchasing and production personnel would resort to experienced base estimation or educated guessing. These formulas or models guarantee minimal costs. In addition to lowering inventory costs, less warehouse space is used.

Third problem: 81/(−3)^3+9

Enter the formula into your spreadsheet by typing an equal sign (=), then type the number 81, a forward slash (/), open parenthesis, a minus sign (-), the number 3, closing parenthesis, the caret (^), the number 3, the plus sign (+), and the number 9. Notice that we are enclosing the negative three in parentheses—we do not want the software to think it needs to subtract; hence, using the parentheses will ensure it is treated as a negative number. This is another use for parentheses when developing complex expressions in a spreadsheet. Here is what you should see when you click on the cell: Following the order of operations, this formula was solved by completing the following passes: 1. Pass 1: Exponent: Following the order of operations, the exponent is solved, (−3)^3=−27. Our expression now reads: 81/(−27)+9. 2. Pass 2: Divide: Even though there is a set of parentheses in our expression, we move forward in the order of operations because in this case, the parentheses are used to tell the software the number is negative. The division is now solved, 81/(−27)=−3. Our expression now reads: (−3)+9. Pass 3: Add: The answer is 6. Before you move on to the next problem, go back and edit your formula to remove the parentheses. What happens when you delete one or both of them? How does the expression editor help or hinder your work? Re-enter the expression and watch for the guidance when you type in the parentheses or if you try to build the expression without the opening parentheses.

Decision criteria will help an organization determine which alternative will be the best option. Start by listing all of the criteria relevant to the decision. Then, select those criteria that are most important to the business (usually 5-7 criteria is appropriate). Finally, rank the criteria in order of importance.

Example criteria might be: -Will the solution be implemented by a certain deadline? -Will the solution be implemented within budget constraints? -Will the solution provide required space for all employees and equipment? -Will the solution meet all local, state, and federal regulations that may apply?

Operations and Finance Measures of Inventory:

Finance is primarily focused on the dollar value of the inventory being held. The higher the inventory levels, the less cash the company has to operate. While operations looks at inventory from a fiscal standpoint like finance, they are also interested in the counts and location of the physical goods. They need to know this to be able to source, produce, and deliver products. Having $10,000 in bottle cap inventory is not as meaningful to operations as knowing they have 203,782 bottle caps in inventory. Operations uses bottle caps, not dollars, to make product. If the count and location information is not accurate, production is disrupted. That can be quite costly. Neither the finance department nor operations management want inventory taking up too much space. Space is a cost. Warehouses have goods flowing in and out of them on a continual basis. A certain amount of empty storage locations is needed for a facility to be productive. A glutted warehouse where goods are stored in the aisles or sitting in trucks waiting to be unloaded is not very productive. The worst case is that the inventory overage is so high another facility must be built, bought, or leased to accommodate the excess. No one wants this, unless the inventory build up is truly due to an extreme amount of sales growth. Certainly sales is interested in inventory as they count on anything they sell being available for order fulfillment and delivery. Sales tends to only notice and complain when orders are being cut (i.e., there are shortages). Cut orders also gets the CEO's attention as well. This kind of "attention" is something operations strives to avoid. The sales department counts on any products sold to be available for order fulfillment and delivery. Cut orders due to shortages have a negative impact on sales.

When using Solver to solve a linear programming problem, it becomes quickly apparent that Solver does not provide any graphs. The reason for this is simple.

Graphics are very useful for problems with two variables. In fact, when there are only two variables, a graphic approach to the problem is a bona fide method to solving the problem. It is an excellent way to introduce students to linear programming. Real world problems, however, are much more complex. They will have three or more variables...usually much more. Feasible regions of linear programming problems with three or more variables are difficult to graph and to envision the feasible region, and simply impossible when there are more than three variables. Earlier in this module, the graphic solution method was introduced and used. The feasible region was easily depicted and shaded with the linear edges of the region being the various constraints. The corner points of the region could be read off of the graph, determined algebraically, or with the assistance of a graphic calculator or software such as desmos.com. As in this graphic, even the slack of variables could be determined. Consider a problem with three variables. Recall the second example in the previous module helps illustrate this point. Recall that the objective is to minimize the budget needed to reach a minimum of number of 5,000 people. The same table of data is used. The formulation would be as shown above. Objective Function: Minimize Z=200R+100N+50S Subject to the following constraints: 100R+200N+300S≥5,000 (Number of people reached constraint) R≤25 (Limit on number of radio ads) N≤7 (Limit on number of newspaper ads) S≤14 (Limit on number of social media ads) R,N,S≥0 There are four constraints and three variables. Each constraint is a plane. The planes intersect in lines. Graphing this is difficult, and probably requires computer-aided design software that requires some training to use and is not readily available to undergraduate business students. Here again is the problem set up in a spreadsheet to solve with Solver: Solver provides the following solution: Exactly 5,000 people will be reached by running the following: - Six radio ads (Only six of the available 25 radio ads were used.) - Seven newspaper ads (All newspaper ads were used.) - Ten social media ads (All social media ads were used.) This will cost $2,400. These two marketing applications use the same basic data. The data table applied to both examples. When Solver is run, the following screen pops up. Note that there are no options for graphics. There are, however, options for three reports. Answer Sensitivity Limits The reports can be highlighted. Upon selecting "OK," these reports are generated and saved in three separate worksheets. Experienced linear programming analysts use these reports in place of graphs to help understand the subtleties and sensitivities of the problem and solution much the way graphs were used in problems with two variables. Here is what the outputs of these reports look like for the example at hand. Of these, the answer report is the easiest to interpret. It basically summarizes the answers to the problem but in a tabular format. It provides which constraints are binding and thus would have to be expanded or modified in order to improve the results of the problem. In summary, graphics are very useful in problems with two variables. Unfortunately, these kinds of problems are used only in introducing the concepts of linear programming. Real-world problems have three or more variables, most often more. For these problems, graphs of the feasible region are not feasible, and when there are four or more variables, graphs are not even possible. In such cases, linear programming analysts will look at tabular reports like the sensitivity and limit reports presented above.

When multiplying two numbers together, three rules need to be considered.

First rule: If both numbers are positive, the product (resulting answer from a multiplication problem) is always positive. Second rule: If multiplying two negative numbers, the resulting product is always positive. Third rule: When multiplying a positive number and a negative number, the result is always negative. There is an easy way to remember these rules. If there is an even number of negative signs when multiplying, the resulting product will be positive. If there is an odd number of negative signs when multiplying, the resulting product will be negative.

Spreadsheet Formulas

Formulas are mathematical expressions that show the relationship between different variables located within a cell or cells of a spreadsheet. A simple formula might add the numbers in two cells, and a complex formula might add two cells then divide by the value in another cell. The active cell shows the result of the formula. The formula appears in the formula bar but not in the active cell.

Putting it to use: Use spreadsheet software to prepare graphical data.

Graphs are used to display data relationships in an easy to understand way, showing a picture instead of lots of numbers. Readers will often skip the detailed data and look at the pictures, or graphs, to understand the data. Some of the more common types of graphs are pie, bar chart, and stacking chart. Choosing the correct type of graph takes careful consideration. The pie and bar chart are great when graphing one variable. In particular, the pie chart shows the percentage each categories represents out of a whole. The bar chart can show values for each category but doesn't show the percentages of each category out of the whole. A pie chart should only be used if you need to show percentages of a whole, and the number of different categories being compared is small. When the pieces of the pie are similar in size it is not always easy to make the data distinctions with a pie chart. In this case, a bar chart would be the better option to clear see data relationships. Unlike a pie chart, a bar chart can show a trend over a period of time. Stacked charts can graph more than one variable. Each stack, or column, represents a total of several components. For example, in the stacked column chart below, each column represents quarterly sales. There are three product sales being shown for each quarter. With a quick glance at the chart, we can see that water was the top selling beverage during the second quarter. Also, it is easy to see that quarter 3 had the highest overall sales. With the stacked chart, we can see both a trend over time and the percentage each category represents within all beverages sold. Stacked charts can be columns or horizontal.

Fourth problem: 12.4−0.7∙4+5.5^2

Here you will notice that we added a few decimal points because seldom in business will our numbers all be whole numbers. Enter the formula into your spreadsheet by typing an equal sign (=), then type the number 12.4, the minus sign (-), the number 0.7, the asterisk (*), the number 4, the plus sign (+), the number 5.5, the caret (^), and the number 2. Here is what you should see when you click on the cell: Following the order of operations, this formula was solved by completing the following passes: Pass 1: Exponent: Following the order of operations, the exponent is solved, 5.5^2=30.25. Our expression now reads: 12.4−0.7∙4+30.25 Pass 2: Multiply: Next we multiply, 0.7∙4=2.8. Our expression now reads: 12.4−2.8+30.25. Pass 3: Subtract and Add: Working left to right, we subtract 12.4−2.8=9.6, and then we add, 9.6+30.25. The answer is 39.85.

If a manufacturing company produces two different products, a determination may need to be made as to how much of one product versus the other should be produced. Given various information about each product's profit, necessary materials, labor hours to produce, and labor hours available, a linear programming model can be developed that would help the company determine ___ of each product should be produced in order to maximize its profit.

How Many: In order to maximize profit, the manufacturing company would need to determine how many of each product should be produced. Determining the correct quantities will allow the company to maximize their profit.

Production has equivalent questions:

How many production runs should there be in a year? For each production run, how many items should be produced?

Critical Path

Identifying the critical path on the network diagram shows where the decision maker does or does not have flexibility with decisions. The activities on the critical path have no slack time and, as a result, the decision maker has no flexibility when it comes to adjusting the start and finish times. Tasks that are not on the critical path often have slack time. This allows the decision maker the flexibility to delay the start or completion of a task without impacting successful project completion by the deadline. (Of course, the delay cannot be longer than the slack time indicates on the network diagram!) Using a network diagram to understand the task dependencies and the critical path will minimize risk of project failure.

Complexity

If a company makes one item for sale and that product has two raw materials and three packaging materials, inventory and material planning and management is not so terribly hard. Consider a company, like many consumer packaged goods, that sells 10,000 to 20,000 different finished goods, called stock keeping units (SKUs), to their customers. Inventory and material management can be quite a challenge for such companies. A standard practice for dealing with complexity is an ABC classification. Companies use different planning and production strategies by classification: A items are fast movers. A SKUs make up the top 85% of sales. Generally, the A SKUs are small in number (e.g., 10-20% of all SKUs). These SKUs are produced almost daily and do not stay in inventory for long. B items are medium movers. The B SKUs comprise the next 10% of sales. They might comprise 20-30% of all SKUs C items are slow movers. The C SKUs are the vast majority, 50-70% of the total SKUs, that make up the bottom 5% of sales. These SKUs are only produced maybe once a month or every other month. There is an additional classification called Slow and Obsolete (SLOB) or Excess and Obsolete (E&O). The SKUs in this category have essentially zero sales and are most likely no longer in production. This inventory just sits in the warehouse and gathers dust. It is the cholesterol that clogs and diminishes the productivity of the warehouse. It is very hard to eliminate because it requires a write-off that directly and adversely impacts the bottom line.

A graph has diamond shaped points starting at approximately (150, 245) on the left and continuing in a downward direction to the right and ending at approximately (190,190). A graph has diamond shaped points starting at approximately (145,190) on the left and continuing in an upward direction to the right and ending at approximately (180, 250). A graph has diamond shaped points in a clustered pattern from approximately (145,145) on the left and to approximately (190, 300) on the right.

If the y-axis values decrease as the x-axis values increase, then the variables are negatively correlated. If the y-axis values increase as the x-axis values increase, then the variables are positively correlated. When there is no strong correlation, points appear randomly scattered on the graph.

Function Lists

If you have experience using any spreadsheet software to complete advanced mathematical calculations, you will quickly notice that the list of available financial, statistical, and data analysis functions are very similar. To access the function list in most spreadsheet software, click the Functions button; in some software, you may need to select a More functions... option to access the functions list. One commonly used and fun advanced function is the VLOOKUP function. Like most functions, VLOOKUP will work the same way in any spreadsheet software. Here are a few function lists for commonly used spreadsheet software: https://support.google.com/docs/table/25273?hl=en https://wiki.openoffice.org/wiki/Documentation/How_Tos/Calc:_Functions_listed_by_category https://www.excelfunctions.net/excel-functions-list.html Explore and enjoy the power and capabilities of these functions.

Expected Value Approach or Expected Monetary Value (EMV) criterion

If you recall, we can calculate the expected outcome of each option in the table below using the following formula: Expected Value=Pwrw+PLrL where p=probability r=return W=win L=lose Let's use the base payoff table introduced above. If we are calculating the expected value for each option, we will add another column to the right of the table for the expected value column. Option 1: (.75∗20000+.25∗0)=$15,000 Option 2: (.9∗15000+.1∗5000)=$14,000 Investment Option Gain Loss Expected Value Option 1 Probability 0.75 0.25 Return $20,000 $0 $15,000 Option 2 Probability 0.9 0.1 Return $15,000 $5,000 $14,000 Using this method, we would choose the Option with the highest expected value, i.e., Option 1. Is this a guarantee? By selecting Option 1 will it specifically result in you getting $15,000? No, it does not. If you select Option 1, you will either walk away with $20,000 or lose everything. The Expected Value Approach posits that if you make these investments repeatedly, the gains and losses will approach the expected value. Note that we can include any number of options and probabilities. In the above example, we only had two probability levels: Gaining and Losing. Let's look at a very similar example in which there are three levels of probability: High: Probability of the highest return Medium: Probability of a modest return or gain Low: Probability of losing part or all of what was originally invested Here is the table for this example: Investment Option High Medium Low Option 1 Probability 0.15 0.60 0.25 Return $20,000 $13,000 $0.0 Option 2 Probability 0.10 0.80 0.10 Return $15,000 $12,000 $5,000 Which would you choose? The following illustrates the Expected Value Approach: Investment Option High Medium Low Expected Value Option 1 Probability 0.15 0.60 0.25 Return $20,000 $13,000 $0 $10,800 Option 2 Probability 0.10 0.80 0.10 Return $15,000 $12,000 $5,000 $11,600 Option 3 $10,000 In this case, we see that Expected Value of Option 2 is the highest. Putting it all together, let's consider the artist example again, but assign probabilities to the different weather outcomes. (Note that the probabilities add up to 1.) Options Hot Clear Rainy Cold Conference hall 500 600 300 200 Outdoor art festival 500 650 -100 165 Sidewalk sale 300 500 -50 220 Probability 0.15 0.55 0.20 0.10 We can calculate all certainty and risk options on one decision table as follows:

Explain how linear programming can be used to solve business problems.

In linear programming we are modeling for optimization. We are trying to predict the best combination of elements that will provide the "best" output, given the constraints that are placed on the business or project. For example, consider the scenario where students are trying to achieve the highest grade in their classes. Assuming a student is enrolled in two or more classes and given the constraint of a limited number of available hours to study for these courses, the student needs to allocate his or her time to each class so as to achieve the highest GPA that is possible. Once again that objective is achievable within the confines of a limited number of hours to study. This is an example of a maximization problem in linear programming. The student is attempting to figure out how many hours to study for each of the classes so that optimal grades for all classes can be accomplished. Conversely, using the same example, the student may want to analyze and therefore optimize the available time to study. Let's assume that the student has an opportunity to get a part-time job while in school. However, the hours will reduce the hours available to study. Accepting the job will cut into the study time for each class, so instead of having an objective to obtain the best GPA with the available study time, the objective switches to studying the least or minimum amount of time with the constraints that the student must pass each class. This is an example of a minimization problem in linear programming. The student is attempting to figure out how to pass each class with the least amount of hours for study. Typically in business environments, decision makers are attempting to maximize items such as: Revenue Units sold They may also want to maximize items like: Profit Earnings The difference between these two groups can be found in the objective function. The objective function for items like revenue and # of units sold is fairly straightforward, and does not require an offsetting function. For instance, if I started with $0 in my cash register and, at the end of the sales day, I have $5,000 in cash, checks, and credit card receipts, then my revenue for that period equals $5,000. If, at the beginning of the day, I had 110 units on my shelf and, at the end of the sales day, I had 10 units on the shelf then I sold 100 units. However, an objective like profit and earnings requires an offsetting calculation(s) in order to be distilled sufficiently to become a part of the objective function. For instance, in the scarf and hat example of H&S Company, the company's objective was to maximize profit of the "best" or optimal combination (mix optimization) of each. The objective function reads: Maximize Profits: 11H+8S=Z Where the value 11 represented $11.00 in profit for each hat and the value 8 represented $8.00 in profit for each scarf. These profit values would have been deduced from the company's records reducing the revenue per hat and scarf by a corresponding cost value. This would not be the case if the values of 11 and 8 represented revenue or units sold that typically do not require an offsetting event. A graphical representation of a maximization problem might appear as:

As the following example illustrates we can set a maximization for profit given the parameters of the problem.

In linear programming, profit maximization and cost minimization are frequently assessed. These two objectives are also intertwined as profit maximization can occur if costs are minimized. That may appear as a dual function; however, in linear programming models, the decision maker needs to determine the objective as a singular function. In this case, the decision maker needs to set up the problem as either one of these business goals and then determine the data required to formulate the constraints. Profit Maximization Problem: The Kitchen Company is in the process of launching its own line of plates and saucers. The profit per setting for plates is $50 and per setting for saucers is $40. The company estimates that it will take 1 hour of labor per plate setting and 2 hours per saucer setting. It also estimates that the plates will take 4 units of material per plate setting and 3 units of materials per saucer setting. In addition it has no more than 40 hours in total labor and no more than 120 units in materials to devote to this new product line. The company is attempting to maximize profit by producing the best mix of plates and saucers to sell to its retail outlets. In setting up this problem it is beneficial to create a table that contains the key data: Plates Saucers Labor Units 1 2 Material Units 4 3 Profit 50 40 Reviewing the data table and the problem, the initial step is to define the decision variables: x = plates y = saucers Because we are attempting to maximize profit, the objective is to manufacture each item in the correct amount given their relative profit margin. The objective function is expressed as: 50x+40y=z (where z is the maximum profit) Additionally, the company is bound by constraints of both labor and materials. It has up to 40 hours total in labor given 1 hour per plate setting and 2 hours per saucer setting in labor to devote to this new product line and only 120 units of materials for which plate settings take up 4 units and saucers 3 units. These regular constraints are expressed as: x+2y≤40 4x+3y≤120 In addition, because the company cannot make negative amounts of plates and saucers, the non-negative constraints are: x,y≥0 Listing out the entire linear programming expression: - Objective function: 50x+40y=z - Regular Constraints: x+2y≤40 - 4x+3y≤120 - Non-negative constraints: x,y≥0 Graphing the regular constraints, locating the feasible area by interpreting the direction of the regular constraint inequality and locating the optimal point (outermost "corner" of the feasible area), yields an optimal coordinate of (24,8). Plugging our optimal points back into the objective function: 50(24)+40(8)=1,600 Therefore, if we manufacture 24 sets of plate settings to every 8 sets of saucer settings, we will maximize our profit given the constraints of labor and materials on the company.

Identify the type of objective for the following situation: A company has seen a rise in its production costs between use of several production plants, each with different costs of production What type of LP problem does this situation call for?

In order to reduce production costs, a minimization objective is appropriate.

Program and Evaluation and Review Technique (PERT) - Three Point Estimation / The Beta Distribution

In the 1950s, the US Navy embarked on the Polaris Missile Program. This was a massive project that involved coordinating the activities of several thousand suppliers. There was a great need for having a method to estimate the individual task times, the overall, and the uncertainty of both. Together with a consulting partner, the Navy developed many of the tools of modern project management. One such tool was the Critical Path Method (or CPM) which will be discussed in a later section of this unit, and the other was the Program Evaluation and Review Technique (PERT). PERT uses a statistical distribution called the beta distribution. This is sometimes referred to as the three-point estimation. The beta distribution estimating method is a powerful tool that helps predict project and activity duration while incorporating risk and uncertainty in the estimate. Three specific time estimates are used in the beta distribution calculation: 1. Optimistic -The best case scenario, or minimum length of time, to complete an activity. Let A stand for Optimistic. 2. Most likely - The duration of an activity given a normal level of effort and performance under normal working conditions. Since this time estimate is the most likely to occur, it is the mode of the beta distribution. Let M stand for Most Likely. 3. Pessimistic - The worst case scenario, or maximum length of time, to complete an activity. Let P stand for Pessimistic. Using the optimistic, most likely, and pessimistic time estimates will allow a project manager to more accurately estimate the expected completion time for a project or activity. This estimate, test, is calculated using a weighted average of A, M, and P as above using the following formula for each task. test=A+4M+P6 The most likely time is multiplied by four to weight it more heavily in the average because it is the most likely to occur. The weighted average will be one part optimistic, four parts most likely, and one part pessimistic, for a total of six parts. To find the average activity time, divide the sum by six.

Summary

In the business world, managers must have the ability to transform raw data into information that will be the basis for sound decisions and actions. In this unit, we reviewed important math concepts needed for quantitative analysis. Even though businesses use electronic spreadsheets and calculators, managers need to understand the concepts behind those calculations in order to understand the results. We discussed the importance of using probabilities and descriptive statistics to quantify an event outcome probability. Descriptive statistics help businesses describe important features of a data set. Understanding the median, mode, mean, and variance of a data set is a powerful tool when trying to understand a large volume of data. Business managers must know how to summarize, interpret, explain, and present large volumes of data in order to make predictions, understand underlying problems, and make decisions. Having a fundamental understanding of descriptive statistics and probability is critical for business success. Finally, we learned the value and importance of visual displays of data or graphs. Graphs take raw data and transform them into visual information that we can literally see. We can then use that information to make more informed and better business decisions. Making decisions and taking actions is, after all, the purpose for quantitative analysis in the business world.

Summary

In this module, you learned how decision theory uses a logical, analytical approach to making decisions. Following a structured decision-making approach will reduce the likelihood of making a decision based on emotions and ensuring decisions align with the overall goals of the business. This rational and structured approach is critical to business success. The three categories of decisions businesses and individuals face are: Decisions under certainty Decisions under uncertainty Decisions under risk There are eight steps in the decision making process: 1. Identify the problem; 2. Establish decision criteria; 3. Weigh decision criteria; 4. Generate alternatives; 5. Evaluate the alternatives; 6. Choose the best alternative; 7. Implement the decision; and 8. Evaluate the decision. Many decisions we make are based on uncertainty. There are formal frameworks that can be used to analyze alternatives and have a higher probability of the best outcome. Courses of action represent the decision that are made, and state of nature represent things that impact the decision, but are out of the decision maker's hands. Expected monetary value is a method used for decision making when the risks involved in a decision situation have several states of nature and the likelihood, or probability, of each state is known. A decision tree is a tool that can be used to visualize the important aspects of a problem. This graphical model shows decisions, alternatives, actions, and outcomes. A payoff table is similar to a decision tree but uses a table instead of the graphical representation. A payoff table is best used when there is one decision to be made and a decision tree is more likely to be used when there is a sequence of decisions to be made. Uncertainty is not quantifiable because the future events are not predictable. Because the event is unpredictable, past models are of little value. The future event would not even be known to exist in order to plan. However, risks are known possibilities and can be measured by its probability of occurrence. Risk analysis involves quantifying the likelihood of an event occurring and the extent of the impact of that event. Risks can be evaluated by looking at past events or models. With uncertainty, the possible outcomes are not known in advance. Decisions made under risk and uncertainty should be identified and quantified when possible.

Linear programming is made up of a number of elements that together form the concept and the structure of linear programming. One of these elements is a constraint. A constraint is a linear ___.

Inequality: Constraints are a system of linear inequalities. For example, in a problem where the number of hours worked by an employee every week has to be less than 40, the constraint would take the form <= 40 for the series of possible answers.

As top line measures, companies look at inventory turns (the number of times your inventory turns over per year) and inventory days coverage (the average number of days a company has its inventory before it is sold) as measures of inventory movement and management.

Inventory Turns=(Cost of G∞ds Sold (COGS)/Average Inventory) Inventory Days Coverage=(Average Inventory/Cost of Goods Sold(COGS))×365=(365/Inventory Turns)

Summary

Inventory is defined as any goods held in stock for immediate or future use. Inventory costs has three categories: ordering costs represent the cost of replenishing inventory; shortage costs pertain to costs resulting from not having an item on the shelf for sale, generally the unrealized profit per unit; and carrying costs involve costs of storing inventory and managing inventory risk due to damage or theft. The Economic Order Quantity (EOQ) is the amount of inventory a business should order to minimize the total amount of inventory costs, including carrying, ordering, and shortage costs. The EOQ model assumes the product arrives complete from another company that produces the product and arrives promptly after being ordered. The EOQ formula is 2*annualdemand*costtoplaceanorderannualholdingcostperunit−−−−−−−−−−−−−−−−−−−−−−√. An inventory shortage occurs when consumer demand exceeds the amount of inventory available. Once the amount to order has been determined, the business must decide when to order more inventory. This is defined as the reorder point (RP). Lead time is the number of days it takes from when the business orders the inventory to when the inventory is delivered. The formula for the reorder point is found by multiplying the demand per day by lead time. Because most businesses make profit as a result of selling inventory and carrying inventory is usually the largest business expenses, inventory costs must be carefully managed. Businesses try to optimize their functional performance by maximizing profits or minimizing costs. Linear programming is a quantitative tool used to find the optimal solution for these problems. An objective function seeks to maximize or minimize some quantity. Constraints are restrictions that limit the degree to which a company can pursue its objectives. A feasible region for a problem represents all of the points that satisfy all of the constraints. If the feasible region is formed by the intersection of the linear constraints, then the optimal solution to the problem is found at one or more of the corner points. Sensitivity analysis is a common post-optimization technique used in quantitative analysis. What a decision maker wants to know before the decision is implemented is, essentially, what is the chance that it will not work as predicted? This can be done through developing a series of "what-if" scenarios. In all cases in linear programming, the model seeks the best or optimal choice within a set of relevant constraints or parameters. When there are only two variables, solutions can be shown graphically. When there are more than two variables, these problems cannot be solved graphically. In this case, using software such as Microsoft Solver in Excel is the best option. Most real world linear programming problems have more than two variables. Inventory is defined as any goods that are held in stock for immediate or future use. Because most businesses make profit as a result of selling inventory, and carrying inventory is usually the largest business expenses, inventory costs must be carefully managed. The EOQ is the amount of inventory a business should order to minimize the total amount of inventory costs, including carrying, ordering, and shortage costs. The reorder point is calculated to determine when to order more inventory.

Until this point every calculation of EOQ or EPQ has been for one item. In the real world, planners, purchasing professional, and inventory analysts are responsible for doing such for hundreds if not thousands of items. They would use modern ERP software like Oracle, SAP, or others to do this. Smaller companies might just use a spreadsheet. Here is an example of such a spreadsheet with ten purchased items and the the EOQ, Average Inventory, and Ordering Costs calculate from the Demand, Ordering Costs, and Holding Costs. The Ordering Costs are different because each item is from a different supplier. The Holding Costs vary because the items take up different amounts of warehouse space. Here is the part of the spreadsheet with the data the analyst must collect and keep up to date in order to calculate the EOQ for the 10 items for which she is responsible:

Item #Annual DemandOrdering CostHolding Cost110,000$25$2.5025,000$50$4.003600100$3.50423,000$10$0.35542,500$40$5.00665$63$2.2571,100$25$0.7588,700$35$7.50911,500$10$3.00107,125$15$1.85 Here is the rest of the spreadsheet with the EOQ, Average Inventory, and Ordering Cost calculations: Item #Annual DemandOrdering CostHolding CostEOQAverage InventoryOrdering Cost110,000$25$2.50447224$11,18025,000$50$4.00354177$17,6783600100$3.5018593$18,516423,000$10$0.351,146573$11,464542,500$40$5.00825412$32,985665$63$2.256030$3,80171,100$25$0.75271135$6,77088,700$35$7.50285142$9,973911,500$10$3.00277138$2,769107,125$15$1.85340170$5,099 Here is the same spreadsheet showing the calculation formulas: BCDEFGHItem #Annual DemandOrdering CostHolding CostEOQAverage InventoryOrdering Cost110,000$25$2.50=SQRT(2*C2*D2/E2)=F2/2=F2*D225,000$50$4.00=SQRT(2*C3*D3/E3)=F3/2=F3*D33600100$3.50=SQRT(2*C4*D4/E4)=F4/2=F4*D4423,000$10$0.35=SQRT(2*C5*D5/E5)=F5/2=F5*D5542,500$40$5.00=SQRT(2*C6*D6/E6)=F6/2=F6*D6665$63$2.25=SQRT(2*C7*D7/E7)=F7/2=F7*D771,100$25$0.75=SQRT(2*C8*D8/E8)=F8/2=F8*D888,700$35$7.50=SQRT(2*C9*D9/E9)=F9/2=F9*D9911,500$10$3.00=SQRT(2*C10*D10/E10)=F10/2=F10*D10107,125$15$1.85=SQRT(2*C11*D11/E11)=F11/2=F11*D11

For the following questions, consider the following frequency table showing the breakdown of a group of employees in the engineering department and how many employees have a professional engineer (PE) designation: If you selected a female employee who was a PE, then placed that employee back into the total group, and were then asked what the probability is that the next employee selected is a PE, what form of probability is the question considering?

Joint: The probability of the intersection of the two events is called joint probability. Joint probability is when both events occur without any preconditions. A conditional probability is one that an event is dependent on the outcome of the previous event. This question is asking about selecting an employee from the entire group. Marginal probability is an unconditional probability; it is not conditional on another event occurring. By randomly selecting an employee, placing the employee back into the total group, and then making another selection, there is no precondition established.Using this table as a guide as to how the males and females in the community voted, determine the following probabilities

Acme Company builds two types of pumps, Model A and Model B, and the company wants to make the most profit possible. Acme wants to make the most out of the two primary parts used in their manufacturer: the vanes and the rotors. The Model A pump takes 30% more labor that Model B, but Model A is 25% more profitable. Identify a constraint in this problem.

Labor Hours: The amount of labor hours is a finite resource and therefore a constraint in production capacity.

Using the POWER Function for Exponents

Let us revisit the idea of calculating exponents in a spreadsheet because most spreadsheet software also has the POWER function available. Using the expression 34, here is how to use the POWER function. 1. Begin by clicking on an empty cell and then type in the equal sign (=). 2. Type "POWER," without the quotation marks to tell the spreadsheet to use this function. 3. Type an opening parenthesis symbol, the number to use as the base, a comma, the power or exponent to raise the base to, and then the closing parenthesis symbol. 4. Press the enter key on your keyboard to calculate. Your active cell and formula should now look like this: =Power(3,4) Notice that both the formula and the function returned the same value. However, as your formulas become more complex, there will be advantages to using functions over basic mathematical operators. Format Tips ~ Format spreadsheet cells containing the results of financial calculations as "Currency" or "Accounting." Large numbers produced by using the POWER function are displayed in scientific notation by default, even if you increase the size of the cell. If a cell displays a string of # symbols instead of actual numbers, increase the width of the cell. Now we will revisit a couple of the previous problems using the POWER Function: First problem: 2^4*5+62. As you type this expression into the formula bar, watch the formula bar for the guidance that is provided. Enter the formula into your spreadsheet by typing an equal sign (=), then type "POWER" without the quotation marks, opening parentheses, the number 2, a comma, the number 4, closing parenthesis, an asterisk (*), the number 5, a plus sign (+), and the number 62. Here is what you should see when you click on the cell: Following the order of operation this formula was solve by completing the following passes: Pass 1: Exponents: Once again, the exponent will be solved first, POWER(2,4)=16. Our expression now reads: 16*5+62. Pass 2: Multiply: Next the multiplication will be solved, 16*5=8016∗5=80. Now our expression reads: 80+62. Pass 3: Add: The answer is 142. Second problem: 15*2−5^3 Enter the formula into your spreadsheet by typing an equal sign (=), then type the number 15, the asterisk (*), the number 2, the minus sign (-), now type "POWER" without the quotation marks, opening parentheses, the number 5, a comma, the number 3, and closing parentheses. Notice in the following screen capture how there is some text that has been highlighted, and you will also see that there is a little arrow next to the explanation of the step where we are in the formula. Next, add the closing parentheses and press the enter key on your keyboard to complete the formula. Following the order of operation this formula was solve by completing the following passes: Pass 1: Exponent: Following the order of operations, the exponent is solved, POWER(5,3)=125. Our expression now reads: 15*2-125.15∗2−125 Pass 2: Multiply: Working left to right, the multiplication will be solved next, 15*2=3015∗2=30. Our expression now reads: 30-12530−125. Pass 3: Subtract: The answer is -95.

Summary

Linear programming is an optimization model to maximize or minimize a value. The quantity that is being maximized or minimized is the objective function. For example, an objective function might be to minimize the amount of freight costs for deliveries or to maximize company profits. The first step in linear programming is to define the problem, choose the decision variable, and express the objective in mathematical form. The key in defining the decision variables is the process of understanding the elements that represent "levels of activity". Once the objective has been expressed, linear programming captures and incorporates the "constraints". Constraints are restrictions that limit the degree to which a company can pursue its objectives. Constraints are linear equalities that mathematically define these limits. For example, let x represents the number of hours a week an employee works. If all employees are limited to work no more than 40 hours a week, this labor constraint would be expressed as x<=40. The feasible region will be the area formed on the graph that represents the total system of all possible linear inequalities, the points that satisfy all of the constraints. Only the points in the feasible region can be used as a solution. The optimal solution will be one of the corner points. When there are only two variables, solutions can be shown graphically. But when there are more than two variables, these problems cannot be solved graphically. Graphing in three dimensions is challenging, and graphing in four or more dimensions is not even possible. In this case, using Solver in Excel is the best option. Most real world linear programming problems have more than two variables.

Linear programming can be used to help decision makers assess considerations of profit maximization and cost minimization. These two objectives are commonly discussed in the same conversations. In order to maximize profit, the business must also minimize costs. However, in developing a linear program model, the decision maker needs to determine the objective as a ___ function.

Linear programming models must have a single objective function of either maximization or minimization.

A company is considering building a new plant. It is considering a small, medium, or a large facility. The health of the economy will directly affect the profitability of the new plant, such as whether the economy will expand or contract, thereby providing either a lot of potential profit or little potential profit for a contracted economy. For each of the identified locations D-F in the diagram below, choose the appropriate label for each part. Location D Location E Location F

Location D is the state-of-nature, which occurs after the alternative is made and must include all reasonable probabilities that add up to the total possible conditions under which the decision operates. Location E is expected value, which is calculated by multiplying the estimated benefit or cost of the preceding decision by the probability as defined in the states-of-nature. This is the last part in constructing the decision tree. Location F is expected value, which is calculated by multiplying the estimated benefit or cost of the preceding decision by the probability as defined in the states-of-nature. This is the last part in constructing a decision tree.

Here are two examples of how constraints are derived from word problems.

Manufacturing Applications: LPs are often used in manufacturing in production mix, capacity planning, and production scheduling. Remember our H&S example: Consider the H&S Company, a small business run by college students. H&S makes two products, hats and scarves, both in the colors and logo of the school. The hats and scarves are knitted from yarn on machines in a contract manufacturing knitter. This month H&S has been allocated 120 hours of machine time. It takes 24 minutes to make a hat and 20 minutes to make a scarf. After knitting, a lining and label is sewn into the hat and a college logo of the sewn on the front. For scarves, a label and logo are sewn on. Both products are then boxed and packed for shipping. These post knitting operations are done by the three students who run the business. These post knitting actives take 24 minutes per hat and 12 minutes per scarf. This month the three students can only allocate 110 hours to these finishing activities. Traditionally, they have sold all they produce with profit margins of $11 for hats and $8 for scarves. How many hats and scarves should H&S produce to maximize profit for the month? This is an example of a production mix problem. There was a limit on the capacity, and the goal of this problem was find the mix of production of hats and scarves to maximize profit. Maximize Profits: Z = 1H + 3SZ=1H+3S Subject to the following constraints 0.4+0.35≤120 (Knitting Machine Availability) 0.4H+0.25<100 (Finishing Activities Labor Hours) H, S > 0 (All variables are non-negative) Recall that the solution of this problem was to produce 225 hats and 100 scarves for a maximum profit of $3,275 given the factory and labor constraints. Consider the following twist on the problem using most of the same data: H&S makes two products, hats and scarves both in the colors and logo of the school. Both the hats and scarves are knitted from yarn on machines in a contract manufacturing knitter. In two months for the holidays, H&S is planning to sell 500 hats and 300 scarves. It takes 24 minutes to make a hat and 20 minutes to make a scarf. After knitting, a lining and label is sewn into the hat and a college logo of the sewn on the front. For scarves, a label and logo are sewn on. Both products are then boxed and packed for shipping. These post knitting operations are done by the three students who run the business. These post knitting actives take 24 minutes per hat and 12 minutes per scarf. An hour of Knitting at the contract manufacturer is $50 and an hour of finishing labor is $20. How many hours of knitting and finishing labor should H&S plan for to minimize costs? In this case, two new variables are introduced: - KH= hours of knitting time at the contract manufacturer for hats; each hour costs $50 - KS= hours of knitting time at the contract manufacturer for scarves; each hour costs $50 - FH= hours of finishing time at the contract manufacturer for hats; each hour costs $20 - FS= hours of finishing time at the contract manufacturer for scarves; each hour costs $20 Objective Function: Minimize Z=50KH+50KS+20FH+20FS Recall the number of hours required to knit and finish hats and scarves are: The hour numbers in the table are hours per hat. KH hours /0.4 hats/hour > 500 hats By multiplying both sides by 0.4: KH ≥ 200 In addition, KS ≥ 150 FH ≥ 200 FH ≥ 100 We do not need the sophistication of a computer linear programming solution to realize the optimal solution is when each constraint is at its minimal, i.e., binding.

A real estate developer is preparing to build a new commercial facility. In order to determine which type of facility will be built, research on the community must be completed. The research will focus on the floor and ceiling of the local rental rates, maximum occupancy of tenant, and likely growth in businesses that will not exceed a unit of 5. The developer is wanting to know what facility configuration will yield the highest rents. This is a type of ___ problem.

Maximization: The real estate developer is wanting to build a facility where the rents can be maximized. Problems that include revenue or profit, and in this case rental rates, are maximization problems.

Agri-Big makes custom feeds for livestock by mixing a certain amount of corn and grain with minerals for an optimal percentage of mix of these three commodities, which changes depending on the weather, pasture conditions, etc. Each commodity has a different cost, and Agi-Big tries to formulate the exact mix the farmer requires at the least cost. What is one constraint in this problem?

Meeting the minimum percentage of mix required by farmers using cheapest commodity This is done in setting up the problem such that the required minimum of each commodity is reached and then filled in with a cheaper commodity.

In setting up an LP problem, the concepts of minimization and maximization are critical. How does one determine whether the objective function in an LP problem will be solved as either a minimization or a maximization problem?

Minimization problems seek to minimize the expenditure of something of value, such as costs or driving distances. Maximization problems seek to maximize the creation of something of value, such as profits. For example, a company would set up a minimization LP problem in order to find the right production mix if the value calculation is based on costs to the firm, an expenditure. Conversely, a company would set up a maximization LP problem in order to find the most profitable mix of parts to make for a given set of differing profits from a potential mix in their line of products to make.

A plumbing supply manufacturer uses several different types of metals in manufacturing its products. Some of the combinations of metals result in more scrap than other combinations. The manufacturer is wanting to know what decisions need to be made to reduce the amount of scrap. This is a type of ___ problem.

Minimization: Scrap from the production process is a form of an expenditure of resources and therefore should be minimized. This is a minimization problem.

Inventory Myth

Most think that high customer service (customers get everything they order, in the right quantities, and on-time) requires high inventory. This is simply not true. Companies have very high inventories often have the wrong mix of inventory. They carry too much of the SKUs that are not selling and not enough of what is. High inventory in such cases results in very poor customer service. It is hard to reduce the inventory of SKUs that are not selling well or not selling at all. Companies can carry Slow and Obsolete inventory for years. Rarely do companies have such small amounts of inventory that they are unable to fill orders. This may occur in early days of startup. It might happen when a new product is a success beyond all expectations. It may happen as a company is about to go out of business. In the case of the majority of the companies, improving customer service actually results in a reduction of inventory. Improved service requires trimming the SKUs that are not selling and bolstering those that are.

Spreadsheet Order of Operations

Multiple operations can be combined in one formula, but ensure the use of parentheses where needed or results may not be correct. Recall the previous exposition on "order of operations." This ordering applies in spreadsheets as well. The order was Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. Ensure all operations are carried out from left to right. Here is how the order of operations are applied in a spreadsheet: 1. First pass, any math expression inside of parentheses is calculated. 2. On the second pass, all exponents are resolved. 3. During the third pass any multiplication OR division is performed. 4. The fourth pass, any addition OR subtraction is performed.

Putting it to use: Construct a decision-making model using spreadsheet software:

Now let's take a problem example and walk through the decision making steps. Fred's bakery sells made to order sandwiches but they are so popular that his customer lines are long, customer service is slow, and his customers don't like waiting in line. The first step in the decision-making model is to identify the problem and the decision-making criteria. This problem can be defined in terms of recognizing the current situation (slow customer service, long lines) and identifying the desired outcome (no customer waits in line more than 2 minutes). Fred's stated criteria are that any solution must maintain the outstanding quality he currently provides customers and profits must not decline. After identifying a problem, the next step in the decision-making model is to identify solution alternatives and identify the risks and benefits of each alternative. Review Fred's problem statement and decision-making criteria: - The problem can be defined in terms of recognizing the current situation (slow customer service, long lines) and identifying the desired outcome (no customer waits in line more than 2 minutes). Fred's stated criteria are that any solution must maintain the outstanding quality he currently provides customers and profits must not decline. Fred has identified 2 alternatives for his solution: - make sandwiches each morning before the store opens - hire more employees to reduce customer wait The risks of making sandwiches in advance are that he may make too few sandwiches, resulting in an unhappy customer, or he might make too many sandwiches, resulting in wasted products. The risks of hiring more employees is that his profit margin will be reduced due to the higher expense. Since Fred's criteria stated that the profits could not be reduced, he has decided on the alternative of making sandwiches each morning before the bakery opens for business. Next, Fred must decide how many sandwiches to make each morning. Each loaf of bread he bakes will make five sandwiches, so he will make sandwiches in increments of 5. Each sandwich cost $5 to make and he will sell the sandwiches for $7. Fred has tracked previous sandwich sales for 100 days and has provided the following information: Sandwiches Sold 40, 45, 50, 55 Number of days 10, 25, 50, 20 To provide better customer service, Fred has identified that he will make sandwiches in the morning before he opens. He has analyzed his previous daily sales to help him determine how many sandwiches to make each morning. Fred has made a decision table to help him quantify his analysis. In the left column of the table, he lists the alternative quantities of sandwiches to make each morning. The top of the table, he provides the previous daily sales history, along with the probability of occurrence. For each sandwich he makes and sells, he will earn $2 profit. However, if he makes a sandwich and doesn't sell that sandwich, he loses the amount he spent to make that sandwich ($5). For example, if Fred made 55 sandwiches and sold only 40 of them, he would make only $5 profit. The intersection of the rows and columns represent the profit made for the quantity made vs. the quantity sold. The far right column indicates the estimated value for each alternative. Multiply the profit by the probability for each state-of-nature for an alternative. Then add each product to determine the estimated value for that alternative. After analyzing the payoff table, Fred has decided to make 45 sandwiches each morning before the bakery opens. He will implement this solution and then evaluate the results after 30 days.

Spreadsheet Tips

One advantage of using a spreadsheet is that the software has the ability to add, subtract, multiply, and divide numerical data for you. This is a very convenient feature that saves time when conducting quantitative analysis. As we have previously discussed, using mathematical expressions called formulasmakes handling quantitative calculations easy. Most of the time, you will use a cell address (i.e., cell identifier) in the formula to reference another cell—this is called a cell reference. One advantage of using a cell reference it that the value can be changed in a referenced cell and any formula referencing that cell will automatically recalculate. Using cell references in your formulas ensures that the values in the formulas are accurate and current. Cell references are also useful because it means that you can update the numerical values in a cell without having to rewrite the formula. Cell references are commonly used for conducting what-if analysis, which is a process when the values within a cell are changed to see how that change will affect the formulas outcome within the spreadsheet. What-if analysis is extremely helpful when solving business problems. By combining a mathematical operator with cell references, you can create a variety of simple math formulas that include a combination of cell references and numbers. Let us walk through the simple math formula of =A1+B1, which adds the values of cells A1 and B1 in a spreadsheet. Fill your spreadsheet with some numbers and practice entering in the following simple math formulas: =B2-C2 This formula subtracts the value of cell B2 from the value of cell C2. =A3+B3+C3 This formula adds the value of the three cells identified. =E4*F4 This formula multiplies the numbers in cells E4 and F4. =G5/H5 This formula divides G5 by H5. = K15•1.08 This formula multiplies the value of K15 by 1.08. Here are some potential results based on the random numbers generated in columns A through C. When a complex problem requires a reference to a range of cells instead of a single cell, the range must be identified correctly. In the following picture, notice the data in column A starts in cell A1 and ends at A10. In the spreadsheet, this can be identified as a range, A1:A10. The colon that separates A1 from A10 indicates that all of the values in the range are to be included. A range can also expand over multiple columns. Consider the need to use all of the data in column A, starting with A1, and ending at B10. This range would be A1:B10. This range will include every value from A1 through B10. Using a large range will not skip any of the values.

Outcome Scenario with Risk:

Outcome Scenario with Risk: Sometimes, alternatives that are available or presented, the probabilities are known. While the specific outcome results are unknown, we can use probability to make more of an educated selection. Invest $10,000. a. 75% chance of gaining $20,000. - A 100% return on your initial investment. b. 25% of getting back $0. - You lost all of your initial investment. Invest $10,000. a. 90% chance of gaining $15,000. - A 50% return on your initial investment. b. 10% chance of getting back $5,000. - You lost 50% of your initial investment. Do nothing. What would you do? It is not as clear as in the first few scenarios. The addition of risk makes it harder in the following ways: The outcome is no longer certain and thus there is no clear-cut best decision. Any decision must be made with the risks in mind. In the first case, we knew what the return would be. $20,000 if you invested for ten years and $14,000 if you invested for four years. There was certainty about the outcomes. To make a decision in these scenarios, all you had to do was answer a few questions to determine the best course of action. In the second case, there are gains and losses in the first two options. Only the third option, Do Nothing, was certain. By not investing $10,000, there is no a chance of losing or gaining. There is no clear-cut best course of action because in each of the first two options you might win and you might lose. You just don't know. As the decision maker, it is best to have an understanding of your tolerance or intolerance for risk. Are you, as the decision maker, a risk taker or are you risk averse? This will have an impact on the decision of which option you choose to take. Option 1: There is a 75% chance of doubling your money but also a 25% of losing it all. Option 2: The gains and losses are more modest than in Option 1 with a greater probability of winning, 90%, and a lesser probability, 10%, of losing. Option 3: This is the only option where there is certainty. Risk Taker: If you are a risk taker, you would tend to only look at the upside of each alternative. In the new scenario, risk takers would only see the $20,000 payoff in Option 1 and go for that choice. Risk Averter: If you are risk averse, you would tend to focus on the possibility of losses. Option 1 is definitely out because all a risk averter would see is the chance of losing everything. A risk averter would probably choose Option 2. An extreme risk averter might even choose Option 3.

A key word used in the definition of a sample was the term representative. An important characteristic of using a representative subset is that although we want a smaller, subset of the population, we must have a true representation of the population. In the college freshman example, this would require a subset that is representative of the population, not a disproportionate representation. The subset would show things such as the following:

Percentages of males and females Breakdown of majors Type of colleges attended Ethnic and religious breakdown The subset breakdowns would match the population. We want to be cognizant of all of the above as we select our sample so that we do not end up with a sample containing only 60 women attending community colleges in Idaho or a sample of only 100 men attending Ivy League colleges. Clearly, these samples would not be representative of the entire population in which we are interested.

A business owner should recognize when to use quantitative decision techniques, such as expected monetary value. The owner will use analysis to determine outcomes and payoffs for each alternative associated with the potential decision of either moving forward with a new project, or program, or not. A business can use the method of expected monetary value when either the costs or revenue of a new project is anticipated, as well as the probability of occurrence under various conditions. Several business examples can be applied to better understanding of the expected monetary value method. Consider this: a year-long construction project needs to think about the impact the weather will have on the project's completion dates or expenses, as inclement weather could cause a delay. Therefore, weather would be assigned a ___ of occurring or impacting the project. Another business example of using expected monetary analysis is a project that requires specific labor or skill availability. If the skill is scarce, the availability may be tight at a critical time. Therefore, the availability of the specific skill for the project is an occurrence and would be assigned a(n) ___. By using the expected value method, a business can be more ___ about their alternatives and in formulating decisions.

Probability: In this example, the inclement weather would have a probability of occurrence. This would be measured against the cost of the project or the project's revenue. Cost: To use expected monetary analysis, the manager will need the cost or revenue of the alternative being evaluated and the probability of the occurrence. In this example, the question is asking for cost.

Other costs should be considered as well in the tactical management of inventory. These costs can be significant and are worth noting but will not be used in any calculations or examples in this module.

Quality costs: If goods and items are inspected at arrival into the warehouse, the inspection costs must be accounted for. If defects are found, the disposition of the defectives goods must be determined, and the costs associated must also be accounted for. Defective goods can be in the following categories: - Sorted and rework: The defects are repairable, the cost to repair are not outrageous, and it is worth doing the repairs to keep production and order fulfillment going. - Reject and return: If the defect rate reaches a certain threshold, many companies will simply reject the entire shipment and send it back to the supplier. - Scrap: The goods are unusable and cannot be reworked. The defective items simply have to be scrapped (i.e., discarded, destroyed, or sold for pennies on the dollar for the materials in the item). - Who pays for the defective goods? Often times this is set in the contractual terms of the purchase agreement. The larger the customer relative to the supplier company, the more likely it will be that the costs for all of the above are pushed to the supplier company. Obsolescence or spoilage costs: Some inventory is perishable. This applies to fruits, vegetables, and other food products. If the goods are no longer fit for use or consumption, they are sold for deep discount or discarded. The costs accumulate and inflate the cost of goods sold and thus directly, and negatively, influence profit. Perishable inventory is not restricted to food products. Other examples include the following: - Electronics: Old models of cell phones and computers are worth less or become completely worthless in very short order. They have the same behavior as fruits and vegetables albeit the "spoilage" time is longer for cell phones than for tomatoes. - Logo items: There are peripheral products that accompany blockbuster movies and sports teams. The movie studios and teams have very strict regulations on how long the goods can be sold to the public for. After this period ends, the product becomes unsalable. Either the product is written off and destroyed, or it is held indefinitely in finished goods inventory, which is bad because the productivity of the warehouse is compromised with goods that don't move. Holding goods indefinitely is not free. - Slow and obsolete: Customers and consumers may just not want the goods. In such cases, they can behave just like the logo goods in the previous point.

Explain how sensitivity analysis is used within linear programming.

Sensitivity analysis is a common post-optimization technique used in quantitative analysis. Virtually all systems, processes, or functions that utilize quantitative measure to predict or forecast results are subject to some form of sensitivity analysis. What a decision maker wants to know BEFORE the decision is implemented is, essentially, what is the chance that it will not work as predicted? Sensitivity analysis ask a series of "what-if" questions to determine shows how the optimal solution will change if there are changes in one or more of the assumptions or variables. In linear programming and, in particular, in two decision variable (x, y) linear programming, we need to examine the right-hand side (RHS) of the constraints and the left-hand side (LHS) of the objective function (or as it is also known, "optimizing objective"). The method employed in this "simple" version (only two decision variables) is to alter the right-hand side of each constraint by increasing it by one unit and each of the left-hand side of the objective function by one unit, until the objective function falls outside of the binding constraints. The key to this sensitivity analysis is the concept of exclusivity. Remember, the right-hand side of the constraints typically contains items like hours, shifts, or some other limitation. Each limitation needs to be examined for its own sensitivity to the output. Therefore, the process is exclusive: change one right-hand side value and examine the resulting effect to the output (objective function value), put the constraint back to the original level and change the next one, examining its effect and so on. It is worth noting that the non-negative constraints, while significant in the linear programming process and sometimes a part of the optimal point (the corner that provides the best outcome), is typically not examined for sensitivity. In the earlier hat and scarves example, the linear programming expression was: Maximize Profits: 11H + 8S Subject to the following constraints: .4H+.3S≤120hours (knitting machine availability) .4H+.2S≤110hours (finishing activities labor availability) H≧0 and S≧0(all variables are non-negative) An outcome of 225 hats and 100 scarves results in $3,275 in profits. Given the constraints of the business, this is the optimal mix of hats and scarf production. In this case, the right-hand side (RHS) of the "regular" and "binding" constraints is 120 and 110 hours. Consider that the H&S Company is a small business run by college students. H&S makes two products, hats and scarves. Both offered in the colors and logo of the school. The hats and scarves are knitted from yarn on machines in a contract manufacturing arrangement. This month H&S has been allocated 120 hours of machine time. It takes 24 minutes (.4) to make a hat and 20 minutes (.3) to make a scarf. After knitting, a lining and label is sewn into the hat and a college logo is sewn on the front. For scarves, a label and logo are also sewn on. Both products are then boxed and packed for shipping. These post-knitting operations are done by the three students who run the business. These post-knitting activities take 24 minutes (.4) per hat and 12 minutes (.2) per scarf. This month the three students can only allocate 110 hours to these finishing activities. Historically, they have sold all they produce with profit margins of $11 for hats and $8 for scarves. How many hats and scarves should H&S produce to maximize profit for the month? What if 120 hours of machine time was increased to 121 hours? What would the optimal solution convert to? Essentially, this problem question now asks, "What is the net effect leaving all other elements constant (absolute value change, % change, etc.)?" Let's use that example to examine the sensitivity of the number of hours it takes to "machine" hats and scarves. Restating the expression: Maximize Profits: 11H + 8S Subject to the following constraints: .4H+.3S≤121hours (knitting machine availability) .4H+.2S≤110hours (finishing activities labor availability) H≧0 and S≧0(all variables are non-negative) The resulting change to the number of hours equals optimal coordinates of (225, 110) vs. (225, 100). When we plug these values back into the objective function, the resulting equation and results is: 11(225) + 8(110) = $3,355 in profit This represents a change (increase) of $80 ($3,355 − $3,275) which is a 2.4% increase in the optimal value. Putting that into perspective: a 1-unit change in the hour's machine availability (from 120 hours to 121 hours) represents a .83% change (1/120), which corresponds to a 2.4% increase in the optimal solution. Now look at the right-hand side of the other constraint. First, put the machine hours back to 120 and then increase the finishing hours from 110 to 111. Restating the expression: Maximize Profits: 11H + 8S Subject to the following constraints: .4H+.3S≤120hours (knitting machine availability) .4H+.2S≤111hours (finishing activities labor availability) H≧0 and S≧0(all variables are non-negative) Using the previous technique to find the optimal point (optimal corner of the feasible region), the new optimal point coordinates are 232, 90. Plug those coordinates back into objective function: Maximize Profits: 11H + 8S And the resulting optimal value = $3,277.50 The difference between the original optimal value of $3,275 (original) and the new value of $3,277.50 equals an increase in profit of $2.50. This is equivalent to a .08% increase in profit for a 1-unit change (.91%) in the hours it takes for finishing labor. Analyzing these two sensitivity tests together for each right-hand side constraint change of 1 unit: Sensitivity ActionOriginal Optimal ValueNew Optimal Value$ Change% Change to Optimal Value% Change to HoursOne Unit change in Machine hours$ 3,275.00$ 3,355.00$ 80.002.4%0.8%One Unit change in Finishing hours$ 3,275.00$ 3,277.50$ 2.500.1%0.9% By isolating (exclusive treatment) each of the right-hand side values of the regular constraints that originally formed the optimal intersection (feasible region) or the optimal coordinates of (H, S), it is clear to see which "hours" requirement is more (or less) sensitive as it relates to maximizing profit. The new "optimal" value that results in changing either of the right-hand side values of the constraints (two constraints in this example) is referred to as the "dual price." Another name for dual price is shadow price. In this example the dual prices are $80.00 and $2.50 depending on which constraint is being tested. Dual price of a resource could tell a business how much more profit could be earned by increasing that resource. Certainly, given the data in this problem, a decision maker should be more cautious about the machine hour's requirement or historical data set for that parameter than finishing hours. How would this potentially translate into a decision maker's process? One very simple action might be that armed with this sensitivity analysis, the decision maker might ask that the data for machine hours be double-checked or compared to external data sources. But the decision maker might not be too concerned about the data relating to finishing. After all, of these two data points, the one that potentially has the greatest impact is machine (by % analysis). In addition to data recognition, management can use the outcome or "dual price" sensitivity to focus resources on improving that job function knowing all the while that improving the element that is most sensitive or creates the best dual price ($80.00 or $2.50) is going to return the highest gain. In this case it's clearly the dual price of $80.00 or machine time that is most sensitive to increasing profit. Remember since linear programming is equally adept at quantifying gain to revenue or profit, it is equally powerful at measuring reductions in costs or labor hours, etc. Therefore, when we are maximizing (for the sake of optimization) we would view large positive values of the dual price as a positive. Conversely, we would see large reductions (negative change) in dual price as a positive when we are attempting to minimize costs or overall labor hours, etc. There is one more significant sensitivity test used with linear programming when there are two decision variables: it is called the "Range of Optimality." The "Range of Optimality" considers the left-hand side of the objective function. That is the coefficients that make up the objective function. In the previous example with the H&S Company (makers of hats and scarves), we found out by analyzing the constraints that the optimal profit value for producing hats and scarves was 225 hats and 100 scarves. If H&S will produce that amount of each, given the constraints, the profit margins for hats and scarves will be $11.00 and $8.00, respectively. The objective function is: Maximize Profits: 11H + 8S This objective function has an optimal answer at 225H and 100S, which equals $3,225 in profit. We can convert this to a linear form and graph the objective function at optimality with the constraints (see below): Notice how the objective function intersects the binding constraints at the optimal point or "corner." This is always the case if in fact we have solved the problem correctly. In order to test the "Range of Optimality," examine the left-hand side (LHS) of the objective function, the values 11 and 8. We do not increase these values by a unit (like we did in the right-hand side (RHS) of the constraints or dual price). Instead, determine how much each value (11 or 8) can change up and down before the linear expression of the objective function no longer resides in the boundaries formed by the linear expression of the constraints. In other words, if the linear form of the objective function pivoted up and down at the optimal point, how far could it move before crossing over any constraint boundary? In the graphical representation, we see the gray line (objective function) sits like a "seesaw" in between the two linear expressions of the regular constraints. If we were to take that gray line and shift it along the point at which it pivots, or rests, (leaving the intercept points at 225,100) up and down as long as it remains in between the two constraint lines at all points, that would represent the "range" that the LHS of the objective function could alter before creating another or different optimal point. The concept is an interval test. You may recall that the formula of a line is: y = mx + b The slope of the line, which refers to the line's direction and steepness, is referenced as "m" with the y intercept as "b." In order to find slope "m," we need to solve for the values that are in the problem. With a few algebraic functions we can isolate the value of x and y (in this case, "H" represents hats and "S" represents scarves). Therefore, if we start at the end, knowing the coefficients of H and S (in this case 11 and 8), we realize that by changing either value we are altering the slope of the objective function: 11H + 8S = 11(225) + 8(100) = 2,475 + 800 = 3,275 The objective function is a linear form with a defined slope, and the slope is affected by any alteration to the coefficients of either H (11) or S (8). By changing the RHS of the objective function, we are changing the slope of the objective function line. Similar to the process of changing the RHS of each constraint exclusively, we can change the LHS of each objective function coefficient until we reach a value where the slope of the objective function NO longer is confined by the boundaries of the constraints and therefore cannot intersect at the optimal point. The range of optimality is the amount the objective function coefficient can change without changing the values of the optimal solution points. We can see by the previous illustration that the objective function line (gray line) does reside between the constraint lines that form the optimal point (aka: "binding" constraints): Further examination in this example shows that the gray line of the objective function does not reside in the exact center of the two binding constraint parameters. If it did, then the amount of change (up or down) of the objective function coefficients (11, 8) would be the same value (amount of change) before we break a constraint border. In this case however, that is clearly NOT the case. Performing some slope calculations of each constraint and the objective function range by slope, we can see how much slope tolerance there is in each of the RHS coefficients of the objective function. In this case the hat profits can go up from $11.00 to $16.00 and down $11.00 to $10.67 before the objective function line is pushed outside the limitations or the constraints of the problem. Just like in the LHS sensitivity analysis of the constraints or the dual price, we reset the hat coefficient back to $11.00 and test the range of the scarf coefficient ($8.00 profit margin). In this case, the upper range is $8.25 scarf profit and the lower range is $5.50 scarf profit. Therefore in testing the RHS of the binding constraints and the LHS of the objective function, both the variables and coefficients in the objective function have been tested for sensitivity: one for dual pricing and the other for range of optimality. In this example, as it relates to "Range of Optimality", a decision maker could look at the following results table and draw a variety of conclusions: As we can see by the % variation, the profit margin values for hats are slightly more sensitive (greater range) than scarves and as such, decision makers might want to re-verify the hat data set for accuracy. There is another possible conclusion. Since, in absolute terms, the situation is actually only 3.1% away from being the potential incorrect solution, the decision maker might want to reconsider executing the decisions if the risk is high. Without knowing more about the business, it appears that 3.1% range is relatively small and therefore, risky. If so, then the return on investment should also be high if one would choose to go ahead with the project. A 3.1% chance of the decision NOT being optimal might "deserve" a slightly higher return on investment since higher risk should typically result in higher returns.

Summary

Remember these basic rules: When adding two numbers that have the same sign, add the numbers and keep the sign. When adding two numbers with different signs, subtract the smaller number from the larger number. Keep the sign of the larger number. When multiplying positive or dividing with negative numbers, count the number of negative signs. If there is an even number of negative signs then the result will be positive. An odd number of negative signs in a multiplication or division problem will result in a negative result. To add or subtract fractions, find a common denominator for each fraction. Then keep the denominator, and add or subtract the numerators. Remember the acronym PEMDAS, which will help you remember the order of operations. First, simplify the inside of any parentheses. Then starting from left to right, complete all of the exponentiations. Next, start back on the left and complete all multiplication and division in the order you come to them. Finally, start back at the left side of the expression and complete any addition or subtraction in the order you come to them. When multiplying two numbers with exponents, if the bases are the same, keep the base and add the exponents. When dividing two numbers with exponents, if the bases are the same, keep the base and subtract the exponents. When raising a power (exponent) to a higher power, multiply the exponents. The distributive property states that multiplying a sum by a number is equal to multiplying each of the addends (numbers you are adding) by a number and then adding those values. An equation states that two mathematical expressions are equal. A basic approach to solving equations with variables on both sides is to move all the terms with the variables to one side of the equation and the terms with the numbers to the other side, and then solve for the unknowns. Electronic spreadsheets are widely used in businesses. Using cell references in formulas allow for automatic recalculation any time a value in the referenced cells change. Formulas and functions must begin with an =. Functions in spreadsheets are built-in shortcuts that help users perform complex computations by making frequently used mathematical operations available within the software. Popular electronic spreadsheet functions are SUM, AVERAGE, COUNT, MIN, and MAX. Following the function name, provide the range of cells to be used by the function. For example =SUM(A1:A25) will add the first 25 cells in column A of the spreadsheet.

Risk Continued

Risk is an issue for any business, large or small. Gaining an understanding of how risk impacts decisions is a skill many managers must achieve. There are techniques and methods we will review to frame the decision space in ways that make it easier to see options and then make a choice to match your organization's risk tolerance level with the best decision. The first step in evaluating risk and payoffs for decision options is to create a table in which the decision options are organized. There are many ways to organize this information. We will follow the format below where the options are listed vertically and the decision space is explored horizontally. These are called payoff tables. Here is an example of a base payoff table: Investment Option Gain Loss Option 1 Probability 0.75 0.25 Return $20,000 $0 Option 2 Probability 0.9 0.1 Return $15,000 $5,000 In the table above, each option available is listed, along with the possible outcomes: in this case, gain or loss. The probability of a gain and the probability of loss are both given, along with the payoff if that event occurred. Using this data, a manager can use formulas to calculate the expected value. This does not mean that the manager is guaranteed to make this exact amount for each option. But if the decision maker had to make the same decision over and over again with the same probabilities of outcomes, the average payoff would be the calculated expected value. Problems with multiple possible outcomes, with varying amounts of risk for each outcome, are good candidates for using expected value methods. Using payoff table, a business manager can select the option with the highest expected payoff.

Scatter Charts

Scatter charts use data points to show relationships between two sets of data. An x and y-axis are used to plot each point. There is usually not a line between the points, so it is easier to see where the clusters of data might be occurring. The relationship between the points represents the correlation between the two variables. If one variable increases as the other one increases, this is considered a positive correlation. If one variable decreases as another increases, this is a negative correlation. In the scatter plot below, the data points are the salary on the x-axis, and the home price on the y-axis. In this example, as salary increases, the home price also increases. This indicates that there is a positive correlation between an individual's salary and the purchase price of a home. Each of the points from the table have been plotted in the graph. The data points indicate an upward trend in home price as the salary increases. If a positive or negative correlation between the data sets exist, you can draw a line of best fit, or trend line, to estimate other values. The line should have close to the same number of points below the line as above. Trend lines can only be used if there is a positive or negative correlation between the two datasets. The table and graph below show a negative or inverse correlation between the number of work absences and sales. As the work absences increase, the sales go down. The line of best fit indicates the trend. Notice that there may be outliers that are not close to the line; but, in general, there is a negative correlation between work absences and sales. Sometimes there will be no correlation between the two variables or data points being considered. For example, the amount of water a person drinks does not impact their IQ. The scatterplot below does not indicate any correlation between drinking water and a person's IQ. The points in the graph appear randomly scattered without indicating any upward or downward trend.

What is sensitivity analysis actually measuring?

Sensitivity analysis measures the amount of change in the optimal solution as a result of a change in one of the constraints. Sensitivity analysis is measuring the amount or degree of change that results in the optimal solution when a decision variable or a constraint is changed.

Parts of a Function

Similar to how a spreadsheet calculates a formula based on how you enter it, the order in which you enter a function into a cell impacts the results. In order for a function to calculate properly, each function has a specific order (i.e., syntax) that must be followed. To create a formula with a function, use an equals sign (=), a function name (SUM, for example, is the function name for finding the total of identified values), and an argument. Arguments are enclosed by parentheses and contain the information you want the function to apply to, such as a range of cell references. = , Function Name, Argument Start all functions and formulas with an equal sign. All functions need an argument providing the cell range or values to be used by the function.

Once a problem has been defined, a business has a clear picture of the objective, as well as the criteria and limitations. The next phase will be to identify alternative solutions, along with risks and benefits of each alternative. In this phase, a business will learn a great deal of knowledge about the future system that will be developed.

Some of the questions to ask when identifying alternatives, risks and benefits are: -Can you test your solutions using low cost and low risk methods? -Has someone outside the organization implemented a similar project? -Can you learn from the experiences of others? -Have all stakeholders been considered in the problem analysis? Compare risks between the alternatives. If an alternative can be modeled, risks can be minimized and quantified. Compare costs and expected value between each alternative. State the benefits of each alternative, both quantitative and qualitative. Quantitative benefits usually relate to profit. An example of a qualitative benefit might be improvements in quality. Cost estimates can be based on previous history. If the project is a new idea, sometimes it is a good idea to hire an outside consulting company who has implemented a similar project. If possible, test the idea out on a small group. A prototype sometimes uncovers processes and risks before investing a lot of money.

The Critical Path Method (CPM)

The Critical Path Method (CPM) is a preferred method by many managers for determining a project's critical path. This method will require determining the task's earliest start and earliest finish time as well as the task's latest start and latest finish time. For both the earliest and latest time determinations, a network diagram similar to the one below will be used. The method of a forward and backward pass, which will be discussed in the next section, each will be used to calculate the ES, EF, LS, and LF for each. 1. Earliest start time (ES) - The earliest an activity can start without starting before any predecessor activities. 2. Earliest finish time (EF) - The earliest an activity can finish. 3. Latest start time (LS) - The latest time an activity can start without delaying the entire project. 4. Latest finish time (LF) - The latest time an activity can finish without delaying the entire project. PERT charts are used to outline the critical path of a project. The red line in the above picture outlines the critical path of this project, the series of steps in the project that will take the longest to complete and have zero slack time. The tasks on the critical path can not be delayed or the entire project will be delayed. Strategies to ensure activities on the critical path do not get delayed is paramount to a project completing on time. To reduce the time it takes to complete a project requires reducing the duration of one or more tasks on the critical path.

Based on the decision table below, which depicts three types of automobiles, answer the following questions: The alternative with the highest score for the fuel economy criteria is the ___. The alternative with the highest total score is the ___. The alternative with the highest total score had ___ of the highest criteria scores.

The Cruise-mobile has the highest fuel economy score compared to the other alternatives. The Super-comet has the highest overall score. Notice the scores for each criteria. It is interesting to see that the Super-comet was only the highest score in one of the criteria, braking. However, it was the highest overall. The alternative with the highest total score across all of the criteria only had one of the highest scores of the individual criteria. This is a good example of why a decision model is necessary and how weighting is purposeful. This is a decision table with three alternatives, which are types of automobiles: the Astro-tour, the Super-comet, and the Cruise-mobile. There are three criteria (style, fuel economy, and braking) based on 1-10 with 10 as the best. In the style category, which is weighted at 0.5, the Astro-tour was given an 8, which gives it a score of 4.00. The Super-comet was given a 6, which gives it a score of 3.00. The Cruise-mobile was given a 5, which gives it a score of 2.50. In the fuel economy category, which is weighted at .30, the Astro-tour was given a 4, which is a score of 1.20. The Super-comet was given a 7, which gives it a score of 2.10. The Cruise-mobile was given an 8, which gives it a score of 2.40. In the braking category, which is weighted at 0.20, the Astro-tour was given an 3, which gives it a score of 0.60. The Super-comet was given a 9, which gives it a score of 1.80. The Cruise-mobile was given a 7, which gives it a score of 1.40. The totals are listed in the final row, and they are: Astro-tour = 5.8, the Super-comet = 6.9, and the Cruise-mobile = 6.3

Explain how to apply the Economic Ordering Quantity (EOQ) model to additional inventory ordering scenarios.

The Economic Order Quantity (EOQ) is the amount of inventory a business should order to minimize the total amount of inventory costs, including carrying, ordering, and shortage costs. The EOQ model assumes the product arrives complete from another company that produces the product and arrives promptly after being ordered. EOQ = √ 2 ⋅ Annual Demand ⋅ cost to place an orderannual holding cost per unit Example: Sandy Beaches Inc. sells beach towels. The cost per order is $200, and carrying costs are $4 per unit. The store sells 10,000 beach towels each year. - Order size = √2*10000*200/4 = 1,000 towels per order - Number of orders = Annual demand / Order size = 10,000 towels /1,000 = 10 - Annual ordering costs = $200*10 = $2,000 The Economic Production Quantity (EPQ) is an extension of the EOQ model. The Economic Production Quantity model is used to calculate the size of production batches or lots. This model is typically used in a manufacturing environment. The model assumes the company is making its products or that products are being shipped incrementally from another company as they are manufactured. It seeks to optimize the production lot size in order to minimize costs, such as machine setup costs and carrying costs. Larger batches of products mean reduced setup charges but higher carrying costs. The EPQ model seeks to optimize, or minimize, these costs. EPQ = √ 2 ⋅ Demand ⋅ Fixed Setup Cost ⋅ Rate of Production / Holding Cost Per Unit ⋅ (Rate Of The Product − Demand For The Product) Example: A car manufacturer uses 24,000 parts for one of the products. The production line is available 200 days per year and can produce 1,000 parts per day. The carrying costs is $2 per unit per year, and the setup costs are $50. Rate of production is 1,000 parts per day. Annual Demand: 24,000 Days per year: 200 Daily demand: 24,000/200=120 EPQ = √2*24000*50*10002*(1000−120) = 1,167.75 or 1,168 Both EOQ and EPQ models assume the demand for inventory is known and constant over the year and that the holding, ordering, and purchases costs are known and constant. Once the amount to order has been determined, the business must decide when to order more inventory. This is defined as the reorder point (ROP). Lead time is the number of days it takes from when the business orders the inventory to when the inventory is delivered. The formula for reorder point is found by multiplying the demand per day by lead time. While waiting for new inventory to arrive, businesses will use remaining inventory on hand.

Using the MEDIAN Function

The MEDIAN function finds the center value of the cells in the argument. Using random numbers between 1 and 20, let us use the MEDIAN function to find the center value. 1. Begin by clicking on an empty cell and then type in the equal sign (=). 2. Type "MEDIAN," without the quotation marks to tell the spreadsheet to use this function. 3. Type an opening parenthesis symbol. 4. Now select the range A1 through C10. 5. Type the closing parenthesis symbol. 6. Press the enter key on your keyboard to calculate. Now change one of the values in your table, and see what happens to your median value. Consider the advantages of using functions to analyze your data as you experiment with the MEDIAN function. It is important to note that in order to create this table of values, the RANDBETWEEN function was used to randomly populate each cell with a value between 1 and 20.

Using the SUM Function

The SUM function adds all of the values of the selected cells in the argument. This function is useful for quickly adding values in a range of cells. Using the formula, let us add the simple equation of 4 plus 5 using the SUM function. 1. Begin by clicking on an empty cell and then type in the equal sign (=). 2. Type "SUM," without the quotation marks to tell the spreadsheet to use this function. 3. Type an opening parenthesis symbol, the number 4, a comma, the number 5, and then the closing parenthesis symbol. 4. Press the enter key on your keyboard to calculate. Take a close look at the guidance provided in regard to the formula format, as well as each of the values that can be included in the argument. See that square brackets [ ] can be used to group values together within the argument. Another format for including values in the argument is to use the cell reference for the value of a cell; for example, the value of B2 and C2 could be used to add the two cell values together. The use of cell references is more common because it allows the values of the cell to be without adjusting the formula. Cell ranges have the advantage that if a new row or column is added to the range, then the formula automatically updates to account for the additional data. See what happens when you add or remove rows to your spreadsheet. For the month of January, the SUM function has been entered to add up the currently entered expenses. We could type this formula into each of the total cells for each month, or we can use the shortcut of clicking on the square box in the lower right corner and dragging it across. Dragging the formula automatically updates the cell references to align with the next column. Give it a try and experiment a bit to see what happens when you copy the formulas. Dragging the square box in the lower left corner of B8 over to C8 will copy the function in B8 to cell C8. Cell references will be adjusted to indicate that values in column C are being added.

Calculate the three-month moving average for January through March from the following data set of production numbers. Round to nearest whole number (no decimal points): Jan 113, Feb 101, Mar 119 Using the following production number data table, calculate the three-month moving average for February through April. Round to nearest whole number (no decimal points): Feb 101, Mar 119, Apr 81 Using the following production number data table, calculate the three-month moving average for August through October. Round to nearest whole number (no decimal points): Aug 80, Sep 109, Nov 87 Using the following production number data table, calculate the four-month moving average for April through July. Round to nearest whole number (no decimal points): Apr 81, May 100, Jun 120, Jul 118 Using the following production number data table, calculate the five-month moving average for April through August. Round to nearest whole number (no decimal points): Apr 81, May 100, Jun 120, Jul 118, Aug 80 Using the following production number data table, calculate the seven-month moving average for February through August. Round to nearest whole number (no decimal points): Feb 101, Mar 119, Apr 81, May 100, Jun 120, Jul 118, Aug 80

The answer is 111. The basic rule for calculating a three-month moving average is to add the values for each of the three-months for which the moving average is being calculated and then divide by the total number of months desired. The answer is 100. The basic rule for calculating a three-month moving average is to add the values for each of the three-months for which the moving average is being calculated and then divide by the total number of months desired. The answer is 91. The basic rule for calculating a three-month moving average is to add the values for each of the three-months for which the moving average is being calculated and then divide by the total number of months desired. The answer is 105. The basic rule for calculating a four-month moving average is to add the values for each of the four-months for which the moving average is being calculated and then divide by the total number of months desired. The answer is 100. The basic rule for calculating a five-month moving average is to add the values for each of the five-months for which the moving average is being calculated and then divide by the total number of months desired. The answer is 103. The number of months in this example is seven, using February through August. The basic rule for calculating a moving average is to add the values for each of the months for which the moving average is to be calculated and then divide by the total number of months desired.

Working with Arguments

The argument tells the formula the data values (i.e., value, cell reference, or cell range) that will be used in the calculation. These values for the argument must be enclosed in opening and closing parentheses ( ) after the function name, and either a colon or comma is used to separate the values or cell references contained inside the parentheses. A colon ( : ) is used to create a reference to a range of cells. For example, =SUM(D2:D15) calculates the sum of the values in cell range D2 through D15. Notice how the range is highlighted as the formula is typed to show the values that will be included. A comma is used to separate values, cell references, and cell ranges. There may be more than one argument; therefore, a comma will separate each argument. For example, =COUNT(B2:B22,C2:C22,D5,D10,D13,D14:D22) will count all of the cells that have values and are included in the argument. The cell ranges include, B2:B22, C2:C22, and D14:D22, and individual cell references as part of the argument are D5, D10, and D13. Notice how each reference is a different color in the formula, this is to help you easily identify each individual reference. Spreadsheet software will not always tell you if your function contains an error, so it is up to you to double check all of your functions. A few of commonly used functions for quantitative analysis include SUM, AVERAGE, COUNT, MEDIAN, MAX, MIN, SQRT, and POWER. Let's take a closer look at each of these frequently used functions.

Bar Chart

The bar chart above is graphing the same data as the pie chart. The pie chart makes it easier to see the percentage of the of all sales that each breed represents. The bar chart allows for easier distinctions between some of the categories that are more closely matched in number. Bar charts are graphical displays that use bars of different heights to represent quantity. Pictorial representation of numeric data makes understanding relative data differences easier than analyzing a large table of numbers. Bar charts can display more than one item or range of data. In the chart below, Joe is able to compare his sales by breed for a 5-month period.

A certain type of sampling is characterized by the fact that the selected samples are readily available. This is known as ___ sampling and may not be representative of the entire population. At times, a researcher sees that there are natural groupings in the whole population, such as male and female, adults and children, that should be taken into effect and sampled separately so that the possible uniqueness of each grouping is not lost. The process to make this separation in sampling is known as ___. One of the simplest methods to ensure a truly random sample is to select the sample based on some natural ordering of the entire population. For example, an employee sampling survey could be done based on every 10th employee as sorted by the employee number. This is known as ___ Sampling.

The chief characteristic on selection of members from an entire population to a sample for a convenience sampling is that the selectees are readily convenient to the person conducting the sample. The process of separating the entire population into natural segments due to inherent differences is called stratification. A sampling technique designed to capture every 10th employee's response is called a systematic sample.

In the process of graphing constraints, there will be vertices where lines will intersect in the feasible region. This vertex is known as a ___.

The corner points occur only within the feasible region. The optimal solution will be at one of the corner points.

The Distributive Property

The distributive property states that multiplying a sum by a number is equal to multiplying each of the addends (numbers you are adding) by a number and then adding those values. For example, using the expression 3×(4+2), the answer is the same whether or not you add the numbers inside the parentheses and then multiply by 3 times the sum or if you multiply 3 times each of the numbers and then add the results. Add the numbers inside parentheses and then multiply 3*(4+2) 3∙6 18 Multiply each of the addends and then add 3∙4+3∙2 12+6 18 The distributive property also applies to expressions that contain variables. The following table demonstrates the distributive property, which states that multiplying x times the sum of 4 and 2 will produce the same result as multiplying x times 4 and multiplying x times 2 and then adding the two products together: Add the numbers inside parentheses and then multiply x∙(4+2) x∙6 6x Multiply each of the addends and then add x∙4+x∙2 4x+2x 6x The following example demonstrates how to use the distributive property in solving an equation: 4(3x−5)=16 Solve for x 12x−20=16 Use distributive property 12x=36. Add 20 to both sides of equation x=3. Divide both sides of equation by 12

The Economic Production Quantity

The economic production quantity (EPQ) is an extension of the EOQ model. The EPQ model is used to calculate the size of production batches or lots. It seeks to optimize the production lot size in order to minimize costs, such as machine setup costs and carrying costs. To determine when inventory should be reordered, we will use a reorder point formula.

There are several key factors in illustrating a graphical solution of a linear programming problem.

The first step is to breakdown the problem into its mathematical expression. The expression is set up as: 1. Objective function 2. Regular constraints 3. Non-negative constraints

The bar chart shows amounts by U.S. city. The x-axis lists six cities, from left to right: New Orleans, Louisiana; Chicago, Illinois; Los Angeles, California; Miami, Florida; Orlando, Florida; and New York, New York. The y-axis lists number in increments of 20, beginning at 0 and ending at 100. A bar goes to 90 for New Orleans. For Chicago, a bar goes to 85. For Los Angeles, a bar goes to 75. A bar goes to 35 for Miami. For Orlando, a bar goes to 30. A bar goes to 20 for New York. The bar chart shows amounts by U.S. city. The x-axis lists six cities, from left to right: Chicago, Illinois; Los Angeles, California; Miami, Florida; New Orleans, Louisiana; New York, New York; and Orlando, Florida. The y-axis lists number in increments of 20, beginning at 0 and ending at 100. For Chicago, a bar goes to 85. For Los Angeles, a bar goes to 75. A bar goes to 35 for Miami. A bar goes to 90 for New Orleans. A bar goes to 20 for New York. For Orlando, a bar goes to 30. The bar chart shows amounts by U.S. city. The x-axis lists six cities from left to right: Los Angeles, California; Miami, Florida; Orlando, Florida; Chicago, Illinois; New Orleans, Louisiana; and New York, New York. The y-axis lists numbers in increments of 20, beginning at 0 and ending at 100. A bar for Los Angeles goes to 75. For Miami, a bar goes to 36. A bar goes to 30 for Orlando. For Chicago, a bar goes to 83. A bar goes to 86 for New Orleans. For New York, a bar goes to 20.

The height of each bar indicates quantity and the bars are getting smaller from left to right on the graph. The x-axis represents the city and state. The names of the city are in alphabetical order. This indicates the sort order is ascending order of city. The state names are in alphabetic order as you move from left to right on the x-axis. Then, if there are cities with the same name, the city is be in descending order.

Calculate the mean, median, and mode for the data set provided in the table. (Use whole numbers, no decimal points.) 170, 136, 101, 171, 169, 102, 148, 104, 136, 115 101, 102, 104, 115, 136, 136, 148, 169, 170, 171 The mean of the number set "A" is The median of the number set "A" is The mode of the number set "A" is

The mean is 135. The mean is calculated by summing a range of numbers and then dividing by the count of the numbers in the data set. The median is 136. The median is calculated by arranging the data set from lowest to highest. For an odd number of data points, the median in the data point in the middle of the data set. For an even number of data points, the median is the point halfway between the two numbers in the middle of the data set. The mode is 136. The mode is the data point that occurs the most often in the entire data set.

Mean

The mean is the average of a set of numbers. To find the average of a set of numbers, add the numbers together and divide by the number of entries. Tom scored 80, 90, 100, and 78 on his four tests. What is the mean (average) of his scores? Add his scores together: 80+90+100+78=348. Divide the sum by the number of tests: 348/4=87 . The mean of Tom's test scores is 87. Sometimes the mean is implied in a statement. Stating, "The cost per member, if equally shared, is $50," implies that the average cost per member is $50.

"Pearson Correlation Coefficient."

The measure of this kind of linear correlation does not have to be subjective. As your experience with statistics and mathematics progresses, you can explore additional concepts of correlation. One concept is that determining whether there is correlation does not have to be a personal judgement, varying from one person to the next. In fact, Karl Pearson (1857-1936), an English mathematician who some refer to as the "Father of Mathematical Statistics," developed a measure for linear correlation. It is called the "Pearson Correlation Coefficient." A correlation coefficient is a measure of the amount a change in one variable will predict the change in another variable. The Correlation Coefficient, as it is sometimes called, can be identified simply by its variable name, r. The correlation coefficient relates two variables, X and Y, by the following formula: r = ∑i=1n (xi−x¯)(yi−y¯)∑ni=1 (xi−x¯)2√∑ni=1 (yi−y¯)2√ r is a number between -1 and 1. When r is negative, the best fit line is downhill from left to right or has a negative slope. When r is positive, the best fit line is going uphill from left to right or has a positive slope. The closer r is to -1 or 1, the more the scatter pattern resembles a line. When r is -1 or 1, the pattern is exactly a line. If there is a perfect positive correlation between two variables, r will equal 1. Similarly if there is a perfect negative correlation between two variables, r will equal -1. A coefficient of 0 indicates no correlation between the two variables. Note: The Xs and Ys are paired observations, (x,y), and thus the number of Xs and Ys are equal. It is not an easy formula to calculate on a simple calculator. More often, it is computed using the built-in function in a spreadsheet: =PEARSON(X data set, Y data set) We can demonstrate the calculated correlation coefficient for our original two data sets for the randomly selected college students: We see that for the first data set of 14 students, where we have their weights and heights, the correlation looks positive and quite good. This is confirmed by the correlation coefficient of 0.905. Likewise for the second data set of 12, where we captured the weights and ages, the correlation does not look nearly as good and appears to be negative. The correlation coefficient of -.142 confirms what was observed: that the trend line is downward from left to right and the data points do not fit tightly around a line of best fit. The correlation coefficient quantifies our observations. For this course, you should know what correlation is and recognize how correlated data would appear in a graph. In future mathematics or statistics courses, you may gain more exposure and a deeper understanding of the correlation coefficient formulas.

Recall that the basis for calculating descriptive statistics is finding the mean, median, and mode of a data set. When determining the median, look for the middle number in the data set. Recall that to find the median, first sort the numbers in increasing order. When there is an odd number of entries, the middle number in the list is the median. If there is an even number of entries, the median is equal to the average of the two middle numbers. For example, find the median in the following list of numbers: 16, 14, 18, 10, 2, 5. First put the numbers in increasing order: 2, 5, 10, 14, 16, 18. There are an even number of entries in the list, so add the two middle numbers and divide by 2: 10 +14 = 24, 24 / 2 = 12. The median is 12. When preparing data for a quantitative analysis, the analyst will seek to use more than four data points. Although an entire population of data may not be necessary, the analyst should produce at least 30 data points to ensure appropriate quantification.

The median is calculated by arranging the data set from lowest to highest. For an even number of data points, the median is the point halfway between the two numbers in the middle of the data set.

Median

The median is the middle number in a list. To find the median, first sort the numbers in increasing order. When there is an odd number of entries, the middle number is the median. If there is an even number of entries, the median is equal to the average of the two middle numbers. Find the median of the following list of numbers: 3, 2, 8, 4, 1, 2, 8. First put the numbers in increasing order. 1, 2, 2, 3, 4, 8, 8. Because there are an odd number of entries in this list, the middle number, 3, is the median. Find the median in the following list of numbers: 6, 4, 8, 10, 2, 5. First put the numbers in increasing order: 2, 4, 5, 6, 8, 10. There are an even number of entries in the list, so add the two middle numbers and divide by 2:5+6=11,11/2=5.5. The median is 5.5.

Recall that the basis for calculating descriptive statistics is finding the mean, median, and mode of a data set. When determining the mode, look for the number in the data set that occurs most often. Recall that to find the mode of a data set, arrange the values in numeric order, lowest to highest. Example: Find the mode of the following set of numbers: 3, 2, 2, 1, 5, 4, 3, 2. Because 2 occurs the most, it is the mode. If no number occurs more than the other, there is no mode. It is also possible to have more than one mode. When preparing data for a quantitative analysis, the analyst will seek to use more than four data points. Although an entire population of data may not be necessary, the analyst should produce at least 30 data points to ensure appropriate quantification.

The mode is the data point that occurs the most often in the entire data set.

Weighted Moving Average

The moving average gives equal weight, or level of importance, to each data item included in the average. In reality, what has happened recently is usually more relevant than older data. A weighted moving averagegives extra importance to certain data points. Typically the most recent data will be assigned more weighting than older data. In order to calculate a weighted moving average, each data point value is multiplied by an assigned weight, then each new value is summed for the average. Weights are usually stated in percentages, with all of the percentages adding up to 100 percent. For example, a company's most recent monthly sales might be given a weight of 50 percent, while the previous two months will split the remaining 50 percent equally, giving 25 percent weight to each. The weighted average formula would look like this: (25% * Sales two months ago) + (25% * Sales one month ago) + 50% * (Sales last month) Because the last month's sales were given a weight of 50 percent, that value is multiplied by 0.50. The remaining two months are splitting the remaining 50 percent, so they will each be multiplied by 0.25: Month Weight Sales Weighted Average 1 0.25*$1,000=$250 2 0.25*$1,200=$300 3 0.50*$1,500=$750 Total 1.00 $1,300 In the previous example, each month's sales is multiplied by the corresponding weight. The weighted average of the data set is $1,300. Think of the weighted average as being the sum of the weighted numbers.

Jane is considering opening a women's boutique. She is having a difficult time deciding how large her store should be. Her annual returns will depend on the size of her store and the state of the economy. If there is a poor economy, Jane knows that clients will not have as much income to spend on non-essentials. Jane has developed the following table: Size of Boutique Good Economy (60% probability) Weak Economy (40% probability) Small 45,000 20,000 Medium 70,000 10,000 Large 85,000 -10,000 From this table, Jane wants to create a decision tree. The alternatives for this decision tree are a small boutique, a medium boutique, and a large boutique. The possible states-of nature are a good economy and a weak economy. Remember, states-of-nature are outcomes that are beyond the decision maker's control.

The next step is to estimate the payoffs and assign the probabilities for each alternative and state-of nature combination on the decision tree. Multiplying the probability by the potential revenue will provide the expected value of each of the three alternatives. Jane has assigned a probability to each state-of-nature for this decision and has estimated her payoff for each combination of alternative/state-of-nature. She is now ready to build her decision tree. Multiplying the probability by the potential revenue will provide the expected value of each of the three alternatives and she will select the alternative with the best expected value.

Network Analysis

The next topic in our Big Picture is Network Analysis. In this unit, you will learn how to use and incorporate network diagrams for projects in order to make decisions about and manage logical relationships between predecessor and successor tasks. Network diagrams help decision makers allocate resources and schedules to ensure a project is completed on time and within the approved budget. You will also learn about the importance of managing a project budget, as well as how a project can be crashed to increase the project's budget, thereby increasing the likelihood of project success.

Graphical Data

The picture of a graph can clearly demonstrate relationships, ideas, and trends for a data set. When arranging data for a chart or graph, give careful consideration to the categories of data. The categories give us a way to present data in a meaningful way. Looking at data with a graph often provide more insight than studying a table of numbers. A few guidelines for creating a graphical display are: Add a meaningful title. Keep the graphs simple. Fancy is not always better. Use different colors for each category. When there are multiple categories, a legend should clearly identify category represented by each color. Choose the appropriate type of graph. Remember, pie charts show percent frequency, histograms show frequency distribution over data set category intervals, bar charts show relative frequency between categories, stacked bar charts for comparisons or percent frequency between two variables, and scatter plots show relationships between two variables. Typically, a bar chart will use the x-axis to identify the categories of data and the y-axis to identify the frequency. If the categories represent intervals, such as time or age, be sure to divide the intervals equally so the reader interprets the data correctly. For example, if one category represents 5 years, but if another category represent 10 years then the resulting graph will be difficult to understand. Another important consideration is the order of presented data. Sometimes data is presented in a certain order and the order should make the presentation easy to understand. For example, the categories might be sorted by city, age group, or years and displayed from left to right in that order. Perhaps within a major sort field, such as age, there will be a secondary sort field, such as gender. The order of presented data should add to the understanding, so keep it simple to avoid confusion.

Joint Probability

The probability of the intersection of two events is called joint probability. Joint probability is determining the probability that both events occur. For example, you would like to know the probability that the number three will occur on both dice when the a pair of dice are rolled at the same time. Each die has six possible outcomes, the probability of a three occurring on each die is ⅙. Probability of the first die is P(A), and the probability of the second die is P(B). This can be written as the following for a joint probability: P(A) = 1/6 P (B) = 1/6 P(A,B)=(16)×(16)=(136)or0.02778 The joint probability that a three will be rolled on both dice at the same time is 0.02777.

The Range

The range refers to the minimum and maximum values in a set of data. The mean, median, and range can tell us a lot about a set of numbers, but two other important pieces of the picture are variance and standard deviation, the amount the individual values deviate from the mean in a set of data. For example, in each of the following sets of test scores, the median is 75, the mean is 75, and the range is 50. Set 1 Test Scores 50, 74, 75, 75, 75, 76, 100 Set 2 Test Scores. 50, 50, 50, 75, 100, 100, 100 Set 3 Test Scores 50, 55, 60, 75, 90, 95, 100 However, there is a lot of variation between the tests scores in each set. If you only looked at the median, mode, and range, you might miss the distribution of values in the three sets of test scores. The greater the variability in the data set the greater the value of the standard deviation.

What are key benefits of quantifying uncertainty in a business setting?

The response should discuss quantifying uncertainty and producing better information to minimize uncertainty. As the level of uncertainty is identified and further quantified, decision makers are better able to make the best decision based on the available data. This fact acts to help minimize and mitigate risks for the firm and makes the probability of favorable outcomes greater.

Using spreadsheet software, find the answer to the following equation: =14+5(8−3). Using spreadsheet software, find the answer to the following equation: =(14+5)∙(8−3). The result is Using the numeric values shown, write the formula to first subtract 15 from 75, add that to 24, and then multiply the result by 3. You must use parentheses in your answer. Using spreadsheet software, find the answer to the following equation: =((5^2)+20)+5(7−3). The result is Using spreadsheet software, find the answer to the following equation: =(25/5)+(14+4^2). The result is

The result is 39. Recognizing the parentheses is important when building a formula in a spreadsheet. The parentheses will ensure that the calculation is correct. The result is 95. Recognizing the parentheses is important. First find the result inside the parentheses, then continue to use the order of operations. ((75-15) +24) In the first part of this formula, a double parentheses must be used to ensure that all the functions were executed before multiplying by 3. The result is 65. Recognizing the parentheses is important. Build your spreadsheet formula to include parentheses to focus on each expression. The result is 35. Recognize that parentheses are important. First find the result inside both sets of parentheses. Within the second set you must use the order of operations.

Find the answer to the following equation using the distributive method, 2(2x−4)=16. Solving for x, the result is Find the answer to the following equation using the distributive method, 6(x−3)=24. Solving for x, the result is Find the answer to the following equation using the distributive method: 14+5(2x−3)=39 Solving for x, the result is Find the answer to the following equation using the distributive method, 5(3−x)=10. Solving for x, the result is Find the answer to the following equation using the distributive method, 3x(5−4)=30. Solving for x, the result is Find the answer to the following equation using the distributive method, 26+4(2x−2)=42. Solving for x, the result is

The result is 6. The 2 must be distributed through (2x−4) to begin solving the equation. This would result in 4x−8=16. Solve for x by adding 8 to each side of the equation; this will isolate 4x. The result is 7. The 6 must be distributed through (x−3)to begin solving the equation. This would result in 6x−18=24. Add 18 to both sides of the equation. Isolating the coefficient and variable on one side will allow the solving of x. The result is 4. This equations requires using both the distributive method and the order of operations. The 5 must be distributed through (2x−3) to begin solving the equation. This would result in 14+10x−15=39. Isolating the coefficient and variable on one side will allow the solving of x. The result is 1. The 5 must be distributed through (3−x) to begin solving the equation. This would result in 15−5x=10. The answer could be arrived by adding 5x to each side of the equation, and subtracting 10 from each side, this will isolate the coefficient and variable, 5x. The result is 10. The 3x must be distributed through (5−4) to begin solving the equation. This would result in 15x−12x=30. Isolating the coefficient and variable on one side will allow the solving of x. The result is 3. Combining both the distributive method and the order of operations is required in this equation. The 4 must be distributed through (2x−2) to begin solving the equation. This would result in 26+8x−8=42. Isolating the coefficient and variable on one side will allow the solving of x.

Identify how decision-making techniques are used in a given business scenario:

The single hardest part (and most important step) of any decision making process it to clearly identify and frame the question. Note the first step in the Decision Making Model: Identify the Problem. If this is not done properly, the effectiveness of the remainder of the process will be impacted. Consider this example. Our current warehouse is too small. We need a new one. Is that enough of a problem identification? Do we need a new factory too? New offices? Or do we just need a new warehouse? It is these kinds of questions that have to be fully considered in framing the problem. How the problem is defined will influence how alternative solutions are developed. Evaluate a potential problem by looking at the difference between the current outcome and the desired outcome. Quantify the problem when possible. For example, if the problem is too many baked items are being thrown out at the end of the day, state how many are being discarded, along with how many are being sold. Determine if there are deadlines for solving the problem. Consider any constraints or limitations that will limit certain alternatives. Brainstorm with others to be sure all problem factors are uncovered. Why is this important? If the problem is not properly defined, organizational resources will be spent inefficiently. Minimally, it might simply be a waste of time and money in the decision making process. In the worst case, you might end up taking action that makes the problem worse rather than fixing the issue. Another important factor in properly identifying the problem is to get the scope of the problem properly defined. If the scope of the problem is too broad, the problem may not get resolved in a reasonable time. If the scope is too narrow, you might be able to solve it quickly, but it may have very little impact on the organization. Extra time spent on defining the problem is time well invested. It could result in getting a better more cost effective solution in less time.

The standard deviation for number set "A" is 76, 90, 93, 57, 71, 75, 60, 73, 73, 66

The standard deviation is 10.86. The formula for calculating the standard deviation for a population is: First calculate the mean (73.4). Then subtract the mean from each value in the data set. Square each of the differences and then add the squares. This total should be 1,178.4. Then divide the sum of the squares by the number of values (1,178.4/10=117.8)⬚. Take the square root of this quotient and round to two decimal places (10.86). Calculating standard deviation is the square root of the sum of the squared differences between the mean of the data set and the individual data points, divided by the number of data points in the data set.

Recall the steps necessary to calculate the standard deviation: Calculate the mean of the data set. Subtract the mean from each value in the data set. Square each of the resulting deviations. Add the squares of the deviations. Divide by the number of values in the data set. (Note that if the data does not represent the entire population, but instead represents only a sample of the population, then divide by n-1 instead of n). Calculate the non-negative square root of the result. Calculate the population standard deviation from the following data sets (use two decimal points for each answer): The standard deviation for number set "D" is 220, 271, 248, 262, 156, 275, 278, 209, 171, 248 Set E 470, 415, 352, 378, 579, 470, 482, 554, 464, 639 The standard deviation for number set "F" is 413, 394, 462, 382, 363, 450, 421, 463, 435, 459

The standard deviation is 41.27. The formula for calculating the standard deviation for a population is: Calculating standard deviation is the square root of the sum of the squared differences between the mean of the data set and the individual data points, divided by the number of data points in the data set. The standard deviation is 72.07. The formula for calculating the standard deviation for a population is: The standard deviation is 33.91. The formula for calculating the standard deviation for a population is:

Smith Manufacturing builds ornamental iron doors and gates. These products have two steps: cutting and finishing. The doors take 3.5 hours, and the gates take 6.5 hours each. The profit on the doors is $300, and the profit on the gates is $400. Smith can do a total of 500 hours of cutting a month and 650 hours of finishing a month. What is one constraint in this problem?

The total number of hours cutting This is a fixed capacity of the firm and is therefore a constraint.

Answer the following questions based on the decision table depicting three types of automobiles: Which criteria has the most weight? What is the formula used to obtain the score for "Style" criteria? Which car has the lowest "Style" score? The alternative with the highest total score is the ___.

The weight value of the decision criteria show that style is the most important factor, with a weighting factor of .50. The other two factors, fuel economy and breaking, are .30 and .20, respectively, meaning they are of lesser importance. The weight value of the decision criteria show that Style is a factor of .50, and the score for style is a value of 8. The calculation for the score would be to multiply the weight times the style score. The Cruise mobile has a lower score for style, a 2.5 versus a 3.0. Organized data is very important to help present results. This is a decision table with three alternatives, which are types of automobiles: the Astro-tour, the Super-comet, and the Cruise-mobile. There are three criteria (style, fuel economy, and braking) based on 1-10 with 10 as the best. In the style category, which is weighted at 0.5, the Astro-tour was given an 8, which gives it a score of 4.00. The Super-comet was given a 6, which gives it a score of 3.00. The Cruise-mobile was given a 5, which gives it a score of 2.50. In the fuel economy category, which is weighted at .30, the Astro-tour was given a 4, which is a score of 1.20. The Super-comet was given a 7, which gives it a score of 2.10. The Cruise-mobile was given an 8, which gives it a score of 2.40. In the braking category, which is weighted at 0.20, the Astro-tour was given an 3, which gives it a score of 0.60. The Super-comet was given a 9, which gives it a score of 1.80. The Cruise-mobile was given a 7, which gives it a score of 1.40. The totals are listed in the final row, and they are: Astro-tour = 5.8, the Super-comet = 6.9, and the Cruise-mobile = 6.3

The status of the welders' labor expenses for January is that they are ___? *January (45%) Welders: $125,000 The status of the non-labor for concrete expenses for January is that they are ___? * January (45%) Concrete: $280,000 The status of the divers' labor expenses for January is that they are ___? *January (45%) Divers: $35,000 The status of the non-labor for steel expenses for January is that they are __? *January (45%) Steel: 650,000 Quarterly: $1,500,000 The status of the labor expenses for welders for January and February combined is that they are __? *January (45%) Welders: $125,000 February (35%) Welders: $125,000 Quarterly: $300,000 The status of the workers' labor expenses for January is that they are ___? *January (45%) Workers: $90,000 Quarterly: $200,000

The welders' labor expenses should be $135,000 ($300,000 * .45 (for January %) = $135,000). Actual expenses are below that figure. The non-labor expenses for concrete should be $270,000 ($600,000 * .45 (for January %) = $270,000). Actual expenses were over that figure. The divers' labor expenses should be $33,750 ($75,000 * .45 (for January %) = $33,750). Actual expenses were above that figure. The non-labor expenses for steel should be $675,000 ($1,500,000 * .45 (for January %) = $675,000). Actual expenses were under that figure. The labor expenses for the welders should be $240,000 ($135,000 for January and $105,000 for February). Actual expenses are above that figure ($250,000). The workers labor expenses should be $90,000 ($200,000 * .45 (for January %) = $90,000). Actual expenses were on budget.

There are many ways to improve this graph. Add a legend to clearly describe the categories. Choose more distinct color variation between the data categories. Do not group several categories, such as Ford and Chevrolet, because it causes readers to misinterpret the results. The chart title should identify what this graph is displaying. Does the graph represent monthly sales, daily sales, or just customer preferences? Does the y-axis represent dollars or sales? Which answer below correctly describes the data arrangement or sort order of the bar chart? The company has three call centers in Houston, Orlando, and Chicago. Customers call to ask questions about credit card accounts. After the customer service is complete, the customer is asked to rank quality of service as excellent, good, or poor. Kevin wants to create one graph to evaluate the results, showing all three call centers and the percentage of each service rating received by each call center. This will help him determine which call center is providing the best overall service. Which of the following chart types would be the appropriate chart to use for this situation?

There are many ways to improve this graph. Add a legend to clearly describe the categories. Choose more distinct color variation between the data categories. Do not group several categories, such as Ford and Chevrolet, because it causes readers to misinterpret the results. The chart title should identify what this graph is displaying. Does the graph represent monthly sales, daily sales, or just customer preferences? Does the y-axis represent dollars or sales? The city names are in a descending sequence, moving from left to right on the graph. A stacked chart can graph two variables. This graph will need to show percentage of each rating (excellent, good, or poor) for the three cities.

A B C Alternatives Favorable Market (75%) Unfavorable Market (25%) 1 Build large plant $650 -$100 2 Build small plant $275 $50 3 Do nothing $0 $0 1. Column A represents the 2. Column B represents the 3. Column C represents the

This column displays the various alternatives available for a particular decision-making problem. This column shows states-of-nature, as well as the probability of the state. This column shows a state-of-nature, along with the probability of the state.

When developing more complex equations, there may be variables on both sides of the equation. To solve for the value of the variable, it is necessary to move all of the terms with the variable to one side of the equation and any terms that do not have a variable to the other side of the equation.

To find the value of x that makes this equation true, 4x−3=2x+3, subtract 2x from both sides of the equation. 4x−2x−3=2x−2x+3 2x−3=3 Then add 3 to both sides of the equation. 2x−3+3=3+3 2x=6 To solve for x, divide both sides of the equation by 2 to isolate the x. 2x / 2=6 / 2 x=3 To add two terms containing variables, both terms must contain the same variable. 2x+3y could not be combined because each term contains a different variable. If both terms contain the same variable, then they could be combined. For example, 2x+3x=5x.

In the process of graphing constraints, an area will be formed that represents the total system of all possible linear inequalities. This area is called ___.

This is the solution to the system of linear inequalities in the problem. That is, the set of all points that satisfy all the constraints. Only points in the feasible region can be used as a solution, which include the corner points.

Identify the type of objective for the following situation: Wilson Co. makes decorative windows and doors. Each item has a different amount of labor to make it and a different amount of lumber to construct. They make twice the profit in selling doors compared to windows, but must make some of both. What type of LP problem does this situation call for?

This objective seeks to maximize profits requiring a maximization objective.

For the following questions, consider the following frequency table showing the breakdown of a group of employees in the engineering department and how many employees who are men have a professional engineer (PE) designation: For the following questions, consider the following frequency table showing the breakdown of a group of employees in the engineering department and how many employees who are women do not have a professional engineer (PE) designation: For the following questions, consider the following frequency table showing the breakdown of a group of employees in the engineering department and how many employees who are women have a professional engineer (PE) designation: Men Women Total PE 288 36 324 Not PE 672 204 876 Total 960 240 1200

This question is looking for a randomly selected person from the entire group who is both a man and a PE. Using the table values, 288 men are PEs. The probability is found by dividing the 288 by the total of all subjects, 288/1,200 = 0.24. In this question, we are randomly selecting a person from the entire group who is both a woman and not a PE. Using the table values, 204 women are not PEs. The probability is found by dividing the 204 by the total of all subjects, 204/1,200 = 0.17. In this question, we are randomly selecting a person from the entire group who is both a woman and a PE. Using the table values, 36 women are PEs. The probability is found by dividing the 36 by the total of all subjects, 36/1,200 = 0.03.

Calculate descriptive statistics:

To better understand statistics, three terms are important: mean, median and mode.

Find the critical path of a project.

To find a project's critical path, the early start, early finish, late start, and late finish for each task must be determined. To prepare, each activity node on a network diagram will contain the task identifier or label, the task duration, early start, early finish, late start, and late finish. There are many conventions for creating an activity node. Here is an example: A forward and backward pass will be used to calculate the ES, EF, LS, and LF for each activity node. A forward pass refers to the first pass in calculating the total duration of the individual tasks in a project. As the typical flow chart moves from left to right when it is constructed, the forward pass moves from left to right. Start from the first task node on the left and proceed towards the last tasks in a project.

Quantity Discounts

Total inventory costs consist of product costs, carrying costs, and ordering costs. Many companies offer quantity discounts. Quantity discounts are offered to entice purchasers to increase the amount they order at a given time. A quantity discount will reduce the purchaser's overall inventory costs. However, placing too large of an order will increase carrying costs. Because the objective is to minimize inventory costs, the ideal quantity to order must evaluate the reduced product costs, along with the increased carrying costs. After using the EOQ model to determine the quantity to order, if this amount is less than the amount needed to receive the quantity discount, the buyer should consider adjusting the order quantity to the quantity required for the discount. Determine the total costs for both the original EOQ and the discounted quantity by adding the product costs, order costs, and carrying costs for each amount. Select the quantity with the lowest overall cost. The following table shows inventory costs for flashlights. The vendor is offering quantity discounts to Frank's Outdoor Adventures, Inc. During the year, the business will purchase 5,000 flashlights. Discount Code Unit Cost Order Quantity Annual Carrying Costs Annual Ordering Cost Annual Purchase Cost Total Inventory Costs A $5.75 500 $400 $400 $28,750 $29,550 B $5.65 1,000 $550 $300 $28,250 $29,100 C $5.55 1,500 $1,200 $200 $27,750 $29,150 To calculate the annual purchase cost, multiply the unit cost by 5,000. Add the carrying costs, ordering cost, and purchase cost to calculate total inventory costs. Even though option C gives the greater discount, the increased carrying costs cause the inventory costs for option C to be higher than option B. The best option in the above table is option B because this option has the lowest total inventory costs. Having too much inventory on hand has disadvantages that must be considered. In addition to increased carrying costs, there is the risk of product obsolescence and possibility of a shift in product demand. Businesses must determine the inventory level that balances the risk of running out of products with the risk of having too much inventory on hand.

Describe how risk and uncertainty are relevant to decision making.

Uncertainty is not quantifiable because the future events are not predictable. Because the event is unpredictable, past models are of little value. The future event would not even be known to exist in order to plan. However, risks are known possibilities and can be measured. Risk is measured by its probability of occurrence. Risk analysis involves quantifying the likelihood of an event occurring and the extent of the impact of that event. Risks can be evaluated by looking at past events or models. Several risks examples are a pair of dice and a multiple choice exam.Before we roll the dice, we know what the possible outcomes could be. We are able to state the probability of occurrence for each of those outcomes in advance. Since you know the probability of a positive outcome, you are able to manage your risks accordingly. With uncertainty, the possible outcomes are not known in advance. Our economy is an example of uncertainty. Quantitative methods can be used to recognize the impact of risk and uncertainty. Expected Monetary Value Analysis is a statistical tool to calculate an average outcome when the future outcomes have several scenarios that might happen. This tool is often used with decision trees. Sensitivity analysis evaluates how changes in the probabilities of the different payoffs or states of nature impact the outcome. What-if analysis considers worst case and best case scenarios by evaluating how changes in the variables, such as labor and cost of parts, impact the profit. Decisions made under risk and uncertainty should be identified and quantified when possible. Management should determine what costs would be incurred for each possible outcome. One of the most common ways to quantify the impact of risk is by using the concept of the expected monetary value (EMV). This is calculated by considering the likelihood of some occurrence multiplied by the monetary impact of that particular alternative under consideration. This is done for each alternative being considered. Next, the newly calculated value for each alternative are added. The one with the least total costs or most total profits is the best choice. In the following two tables, two alternatives are shown, Alternative A and B. Both costs and revenues are shown for a good and bad economy. In column G, the formula to calculate the estimated value of costs and revenues are shown. Since both revenues and costs are given, to calculate the EMV for each alternative, subtract the estimated costs from the estimated revenues. Since alternative B has the higher estimated value, then that would be the best alternative to select.

Putting it to use: Use linear programming software to construct a linear programming model and graph.

Up to now, most linear programming examples presented, formulated, and solved have had two variables, and a graphical approach was an easy and appropriate way to arrive at the solution. Variables were established. Objective function was written mathematically. Constraints were established and written mathematically. Feasible region was graphed and the corner points noted. Objective function was evaluated at the corner points to determine the optimal (maximum or minimum) solution. Most real world linear programming problems have more than two variables. In fact, they can have 10, 20, 50 variables or more. These problems cannot be solved graphically because graphing in three dimensions is challenging, and graphing in four or more dimensions is not even possible. Solving such linear programming problems with three or more variables graphically is not an option. There are methods for establishing a solution, credited to the work of George Danzig. In 1946, Danzig was a mathematician who formulated the general method for solving any linear program. His method is called the "Simplex Algorithm" and can be solved either by paper and pencil, or programmed into a computer. Because using paper and pencil gets very tedious, consumes many sheets of paper, and is prone to human errors in arithmetic, computer software is the way most serious linear programming problems are solved. Many analysts use a Microsoft Excel Add-In called Solver, which is used to solve linear programming problems. Please see this link for instructions to load the solver. Marketing Application: Consider H&S again but from a cost minimization perspective. It was already decided to produce 225 hats and 100 scarves, which will produce a gross profit of $3,275. They are sure they can sell the hats and scarves but not without some advertising. H&S advertises on campus radio, the school newspaper, and social media as defined in the table below: MediaRadioNewspaperSocial MediaAudience (per ad)100200300Cost (per ad)20010050Maximum Per Week25710 Management may approach this data by trying to minimize the budget needed to reach a minimum number of 5,000 people. MediaRadioNewspaperSocial MediaAudience (per ad)100200300Cost (per ad)20010050Maximum Per Week25710 The formulation would be as follows. Objective Function: Minimize Z=200R+100N+50S Subject to the following constraints: 100R+200N+300S≥5,000R≤25N≤7S≤14(NumberofPeopleReachedConstraint)(LimitonnumberofRadioAds)(LimitonnumberofNewspaperAds)(LimitonnumberofSocialMediaAds) R,N,S≥0 Note: 1. This problem has three variables and therefore having a graphical solution is not practical. Software, such as Solver in Excel, must be used to solve this problem. 2. The objective is a minimization problem, one of the constraints is a greater than or equal to constraint, and the other two constraints are less than or equal to. 3. Microsoft Excel Solver can solve maximization or minimization problems with a mix of "less than or equal to" and "greater than or equal to" constraints.

Recall that the basis for calculating descriptive statistics is finding the mean, median, and mode of a data set. For example: If a department spent (in thousands) 80, 90, 100, and 78 in each of four months, what is the mean (average) of the department's expenses? Add values together: 80+90+100+78=348. Divide the sum by the number of data points: 348/4=87. The mean or average of expenses (in thousands) is 87. When preparing data for a quantitative analysis, the analyst will seek to use more than four data points. Although an entire population of data may not be necessary, the analyst should produce at least 30 data points to ensure appropriate quantification. 96, 82, 80, 58, 59, 71, 85, 55, 86, 103, 57, 56, 73, 95,59, 101, 70, 66, 79, 91, 75, 62, 104, 86, 82, 91, 93, 101, 101, 93

Using the table above and rounding your answer to two decimal places, the mean of the data set is The answer is 80.33. The mean is calculated by summing a range of numbers and then dividing by the count of the numbers in the data set. For example, the mean of a three-number data set is calculated by summing the three numbers and dividing by 3.

Correlation

What is correlation? This word is bandied about quite a bit. However, when specifically discussing quantitative analysis results, quite simply, correlation is used to compare two variables to see if there is a possible relationship between them. By relationship, we will mean a linear relationship in this course. The more the data are linear, we would want to say the more correlated the data are. If the data are correlated, we might then be able to use one variable to predict the other. The key term here is "might." More on this in a bit. Using one variable to predict another is central to many science, engineering, social science, medical, and business fields and occupations. Let us look at some simple pairs of variables and data points. In the first example case, consider 14 college students. The variables are "Height in Inches" and "Weight in Pounds." The measures are summarized in the table below. It is a matter of opinion, but we conclude that these variables are moderately to highly correlated because of the fit of this line. Consider another example of students, viewing their ages and their weights.Notice that these variables do not seem to be correlated in a moderately or high fashion; instead, they lack correlation. Visualizing a line through the data points does not seem to fit as well for weight versus age as it did for weight versus height. In this case, we would be inclined to say the that the variables weight and height are more correlated or have a higher correlation than the variables weight and age.

Venn Diagram

Venn diagrams are helpful for visualizing events by diagramming all possible events in the interior of a rectangle. For example, suppose Joe's Grocery had 15 customers during a 30 minute time period, and each customer was observed making the following purchases: 5 customers purchased milk only, 3 bought bread only, 2 customers purchased both, and 5 customers bought neither. This series of events can be diagrammed in the following Venn diagram. A Venn diagram uses the two circles to include which items are being purchased. The circles are purposely "overlapped." The "overlap" or intersection of the two circles shows how many customers purchased both products. In this example, it is possible that a customer could purchase milk and bread; therefore, these two events are not mutually exclusive. Two events are mutually exclusive if and only if they cannot happen at the same time. The purpose of the Venn diagram is to visually show the relationship between the events. If this diagram is used to calculate the probability that a customer would purchase milk, then divide the total number of customers who purchased milk (7) by the total number of possible outcomes (15): 7/15 is the probability that a customer will purchase milk (with or without bread). The probability that a customer would purchase only milk is 5/15. The union of two events is the probability that one or the other event will happen. When calculating probability for two events that are not mutually exclusive, calculate the probability of each event, add those together, and then subtract the overlapping quantities. Using the grocery store example, 7 customers purchased milk, 5 purchased bread, and 2 purchased both. The formula to calculate the probability that a customer would purchase milk or bread is P(milk) + P(bread) - P(both milk and bread) =7/15+5/15−2/15=10/15 (or ⅔). By subtracting the customers that purchased both products you are eliminating the duplication of those customers. Some events are totally independent of one another. For example, no matter how many times a coin is flipped, the next flip is completely independent. The previous outcomes will not have an impact on the future flipping outcomes. This independent scenario uses marginal probability, which is an unconditional probability; it is not conditional on another event occurring. However, some events are dependent on another event occurring first. For example, assume we use a full deck of cards to determine the probability of drawing a red card. (Remember that a deck of cards has 26 red cards and 26 black cards for a total of 52 cards.) There is a 26 out of 52 (or 1/2) chance that a red card will be drawn. If the first card drawn is red, and not placed back into the set, then there are 25 red cards remaining and 51 total cards remaining. Now the chance that the second card drawn will be a red card is 25 out of 51 (25/51). The probability of the second card drawn being red is dependent on the outcome of the first event. (NOTE: If the first card drawn was returned to the set of cards before drawing again, then the two draws would be independent of one another.)

Consider this example:

We need a new warehouse. The current one attached to our factory is too small. This is the problem but it is too broad. There is not enough definition to the problem. The options we may look at with this general definition might solve the problem but they could easily add a lot of unnecessary operational expenses or create customer service issues and thus make the problem worse. We need a new warehouse. The current warehouse is 25,000 square feet. The new warehouse should be at least 35,000 square feet to solve our immediate need as we are leasing 15,000 square feet from a third party warehousing company. This problem definition is too narrow. If the business is growing and we spend 6 months to a year implementing a solution, the resulting "new" warehouse may just be too small, and we are faced with the exact same problem. The new warehouse should be 50,000 square feet to accommodate our growth projections for the next 5 years This problem definition seems to be just right. It solves our immediate problem and will cover us for 5 years. There may be other must-haves or constraints to be included here. They could include: - Upgrades to systems to make the operation more efficient - Must be a green facility to save on energy costs and to contribute to the company's image in the community - Must still be connected to the plant, assuming there is space to do this These all contribute to well-defined and well scoped problem statement.

Fixed-Outcome Scenario:

We often have to make decisions with multiple alternatives to choose from, and in some cases we know the outcomes of each of the alternatives. We are simply choosing the best outcome because there is no risk. Consider the following scenario, much like a guaranteed bond investment, with the following three options: 1. Invest $10,000 and get back $20,000 in 10 years. 2. Invest $10,000 and get back $14,000 in 5 years. 3. Do not invest anything. Since we know the outcomes of each alternative, making this decision is not very hard. However, consider how the decision might be made. First, let us assume that the above three options are the only alternatives, therefore, these define the Decision Space. Next, pursue the following questions: 1. A. Do you have or can you free up $10,000 to invest? If not, there is no decision to be made. 2. B. How long can you do without the $10,000? - a. If you have $10,000, can you do without it for 10 years? If so, you can double your investment. - b. If you can commit $10,000 for at least for four years, the return will be 40%. - c. If you need the cash within four years, again, you would have to do nothing. By answering the above questions, you can come to a best and most rational decision. Let's add another dash of realism to this scenario: Chance. Given the above scenario, instead of a guaranteed investment return, consider there is a chance your investment might grow larger than expected or there is a chance you might lose some or all of your investment. If the probabilities of gaining or losing are unknown, this is known as uncertainty. This decision would be made under uncertainty. If the probabilities of gaining or losing can be identified, they are known. This is know as risk. This decision would be made under the guidance of the known risks. Let us look at the same investment scenario of investing $10,000. The scenario is to make the investment for five years with two investment alternatives both of which have chances of gaining or losing.

In linear programming, like all other quantitative approaches to decision making, the most important and primary task is to "define the problem." There are a few questions to ask that reveal the problem definition:

What is the objective in terms of outcome? a. Maximize? b. Minimize? As we have seen earlier, it is logical to seek a solution that maximizes something that is beneficial when large, like revenue or profit, and minimize something that is beneficial to the business model when small, like costs or labor hours. As we analyze the problem, we categorize it as either a maximization issue or minimization issue, which starts the decision maker down the path of setting up the problem. Other key characteristics require our attention in selecting linear programming: - A choice between alternative action plans - Linear or proportional relationship - Restricted choice parameters In addition to determining the type of problem (maximum or minimum), as well as establishing other key characteristics of the problem, it is important to select the best quantitative analysis system. As we have seen with a brief review of simulation and other decision theory systems, using a particular method has relative advantages and disadvantages. Although there is crossover for a number of these methods, one method tends to be favored over others based on the nature of the issue itself. For its part, linear programming is typically favored in problems that are oriented as follows: - Product mix optimization - Allocation modelling - Investment modeling Although these are unique, they are all based on attempting to determine the best configuration for whatever objective function is formulated. This tool can be utilized over a wide variety of industries (healthcare, defense, finance, etc.) as long as the problem suits the linear programming elements. For this discussion, we will confine the problem to product mix optimization. After determining whether the problem is a maximum or minimum, the next step in setting up the problem is defining the decision variables. In simple linear programming problems, the decision variables are typically set up as the x-axis variable and the y-axis variable. In more complicated linear programming problems we may have more than two decision variables. In that case, we would use X1, X2, X3,... and use computer solutions to solve. In this discussion, we will confine our review to simple linear programming (two decision variables). The key in defining the decision variables is the process of understanding the elements that represent "levels of activity." If we go back to our original production mix problem of hats and scarves, the problem has activity that is being assessed. The elements that represent that activity is clearly hats and scarfs. H&S makes two products, hats and scarves, both in the colors and logo of the school. Both the hats and scarves are knitted from yarn on machines in a contract manufacturing knitter. This month H&S has been allocated 120 hours of machine time. It takes 24 minutes to make a hat and 20 minutes to make a scarf. After knitting, a lining and label is sewn into the hat and a college logo is sewn on the front. For scarves, a label and logo are sewn on. Both products are then boxed and packed for shipping. These post knitting operations are done by the three students who run the business. These post knitting actives take 24 minutes per hat and 12 minutes per scarf. This month the three students can only allocate 110 hours to these finishing activities. Traditionally, they have sold all they produce with profit margins of $11 for hats and $8 for scarves. How many hats and scarves should H&S produce to maximize profit for the month? Now that may seem obvious in this case, but let's assume that in the problem description, another element was introduced: H&S makes two products, hats and scarves, both in the colors and logo of the school. However, one of H&S's competitors makes gloves. Both the hats and scarves are knitted from yarn on machines in a contract manufacturing knitter. This month H&S has been allocated 120 hours of machine time. It takes 24 minutes to make a hat and 20 minutes to make a scarf. After knitting, a lining and label is sewn into the hat and a college logo of the sewn on the front. For scarves, a label and logo are sewn on. Both products are then boxed and packed for shipping. These post knitting operations are done by the three students who run the business. These post knitting actives take 24 minutes per hat and 12 minutes per scarf. This month the three students can only allocate 110 hours to these Finishing activities. Traditionally, they have sold all they produce with profit margins of $11 for hats and $8 for scarves. How many hats and scarves should H&S produce to maximize profit for the month? The question then becomes whether the "competitor's" product represents any of the activity that H&S is engaged in. In this step of linear programming, it can be quite easy to include elements that are NOT a part of the activity base that is under consideration. Therefore the decision variables are not only hats and scarves but the number of each. x−axisLet H= number of hats to produce this month (x) y−axisLet S= number of scarves to produce this month (y) The choice as to which is represented as the x value and the y value is discretionary as long as the logic is followed throughout the linear program process. Looking back at the problem the question, you need to ask: - What are we trying to solve for? - Is it a maximum or minimum question type? Because the objective is to maximize profit, it is a maximize type problem, and because we are looking for the relative profit of the mix of hats and scarves, then the coefficients of $11 in hat profits and $8 in scarf profit are relevant to our objective. Therefore, the objective function starts to take shape. Because we are combining, in an optimal mix, the profit of hats and scarves, we can assemble the decision variables (H, S or X, Y) with the associated profit coefficients to form the objective function: Maximize Profits: 11H+8S=Z (where Z is the "to be determined" optimal output value) Now that we have properly selected linear programming as our decision methodology, defined the problem, chosen the decision variable and expressed the objective in mathematical form, we can go on to capturing and incorporating the relative restrictions or "constraints". If we take another look at the problem, we can start to eliminate elements. Because we have incorporated the relative profit margin of hats and scarves ($11 & $8), we know based on the rules of linear programming that constraints limit the objectives so they can not act as both a constraint and an objective. Therefore, the constraints cannot be represented by profit of hats and scarfs or by their corresponding values. H&S makes two products, hats and scarves both in the colors and logo of the school. Both the hats and scarves are knitted from yarn on machines in a contract manufacturing knitter. This month H&S has been allocated 120 hours of machine time. It takes 24 minutes to make a hat and 20 minutes to make a scarf. After knitting, a lining and label is sewn into the hat and a college logo of the sewn on the front. For scarves, a label and logo are sewn on. Both products are then boxed and packed for shipping. These post knitting operations are done by the three students who run the business. These post knitting actives take 24 minutes per hat and 12 minutes per scarf. This month the three students can only allocate 110 hours to these Finishing activities. Traditionally, they have sold all they produce with profit margins of $11 for hats and $8 for scarves. How many hats and scarves should H&S produce to maximize profit for the month? The constraints therefore need to come from: 120 hours of machine time 24 minutes to make a hat 20 minutes to make a scarf Post knitting activities take 24 minutes per hat. Post knitting activities take 12 minutes per scarf. There is a limit of 110 hours for finishing activities. Looking at the problem and being careful to coordinate hats and scarves to the proper values, we can begin to assemble the constraints or work parameters that H&S has in the making hats and scarves in a logical manner. There is a limit of 120 hours of machine time. Convert 24 minutes to hours (24 minutes = 0.4 hour) and then convert of 20 minutes to hours (20 minutes = 0.3 hour). Associating the hats and scarves coefficients to the appropriate machine time a value (keeping consistent with the original H&S set up in the defined decision variables), we can write a constraint that is consistent with the problem and the data: .4H+.3S≤120 hours (Knitting machine availability) Notice that this is expressed as an inequality. All constraints by definition have to be expressed as inequalities because they represent limitations and not solutions. That leaves us with the second regular constraint. Following the previous procedure and logic, the other constraint is expressed as the following inequality: .4H+.2S≤110 hours (Finishing activities labor availability) Notice that in constructing these inequalities, the variable's in the problem are completely utilized. There are times during the problem definition phase that outside elements that are not a part of either the constraints or the objective function get included. These outside elements, if not filtered on the front end, will be graphed and found not to be a part of the graphical feasible region. Therefore, they are "redundant" and not to be included in the data gathering or decision making. Following the steps above, we can restate the problem mathematically: Define decision variables: x−axisLet H= number of hats to produce this month (x) y−axisLet S= number of scarves to produce this month (y) Express the objective (objective function): Maximize Profits: 11H+8X=Z (where Z is the "to be determined" optimal output value) Express the problem ("regular") constraints (as inequalities): .4H+.3S≤120 hours (Knitting machine availability) .4H+.2S≤110 hours (Finishing activities labor availability) Last, include the non-negative constraints. After all, it is not possible to make a negative amount of hats and scarves. H≥0andS≥0 (All variables are non-negative)

Crash

When a project manager must rein in a deadline and is not able to decrease the scope of the project, the project manager can crash certain activities. Crashing the project is a method that will help the manager make decisions about the allocation of additional resources to work the necessary activities to meet the deadline. Only critical path activities can be crashed. In this unit, you will learn the formula to calculate crashing costs for activities on the critical path. Once the crash costs have been determined, the activities are crashed starting with the order of least cost per time period to highest cost per time period. It is necessary for a decision maker to have resource and cost strategies to avoid delays on any activities in the critical path.

For purchased goods, models exist to answer two questions:

When or how often should the inventory be ordered? When an order is placed, how much inventory should be ordered?

Describe the difference between a linear equation and a nonlinear equation.

With graphs, we use pictures to describe the relationship between two variables. For a line graph, the x-axis represents the independent variable and the y-axis (or vertical axis) represents the dependent variable. The rate of change describes how one variable changes in relation to another variable. If the rate of change is the same between different intervals than the graph is a line. Slope is another name used to refer to the rate of change. To determine the slope, or rate of change, between two intervals, or points, use the following formula: y2−y1 / x2−x1 To calculate the slope between the first two amounts in the table, (4, 8) and (10, 20): 20−810−4=126=2 The slope of 2 means that for every increase in calls made (x variable), the sales will increase by two (y variable). If the slope is constant between any two intervals, then the graph represents a linear equation. If the slope is positive, then the line will slope in an upward direction as the independent or explanatory variable increases. This represents a positive correlation. If the slope is negative, then the line will slope in a downward direction, indicating negative correlation. Businesses can make predictions when the slope is constant. For the last day of the week, the supervisor wants to sell $50,000 and wants to know how many calls need to be made. Since the relationship between the calls and sales is constant, the slope formula can be used to determine the number of calls that will need to be made. Choose any point from the existing data representing (Calls, Sales). In the formula below, the two points are (20, 40), (Calls, 50). Solve for the number of calls that need to be made using the following steps: 1. Plug the data points into the slope formula 50−40 / Calls−20=2/1 2. Simplify the numerator 10 / Calls−20=2/1 3. Multiply both sides of the equation by (Calls - 20) 2(calls−20)=10 4. Use the distributive property 2calls−40=10 5. Add 40 to each side of equation 2calls=50 6. Divide both sides of the equation by 2 calls=25 Using the slope formula, the supervisor determines the company must make 25 calls on the last day of the week in order to reach the sales goal of $50,000. A linear equation that represents the relationship of the two variable in the table above (x = number of calls made and y = number of sales in thousands) is y = 2x. Note that in a linear equation, x will always be to the first power. Note: If no calls are made, then the sales will be 0. If we extended the graph, (0,0) would be another point on our graph. In algebra, you may have learned the slope intercept formula for a line, y = mx + b, where m is the slope of the line and b is the y intercept. In this case, our y intercept is 0 and slope is 2, giving us the equation for this line y = 2x.

Consider a few production examples:

[1] ACME makes the housings for the Half-Horse themselves. Again, the demand for the Half-Horse for the next year is 12,000. Every time they run this housing, they incur $350 in setup costs, mostly to changing over the die. ACME can make 50 motors per hour, and it costs them $1.75 a year to keep a housing in stock. [2] Pens R US assembles their black markers at a rate of 3,000 per hour, and the factory runs three eight-hour shifts per day. The estimate for the coming year has been raised to 8.25 million. Because they have to change over the ink, each time they run the black markers the setup cost is $400. It costs them $0.16 to keep a marker in inventory for a year.

Following are two typical purchased item scenarios:

[1] The ACME Motor Company assembles and sells electric motors of all kinds. For the Half-Horse (one half of a horsepower motor), they purchase the rotor assembly from another supplier. There is one rotor in each Half-Horse. The demand for the Half-Horse for the next year is 12,000. It costs $15 to place an order. It costs $1/year to hold a rotor in inventory. [2] Pens R US provides high-quality markers for whiteboards. They order their caps from a supplier. Each cap is the same color as the ink in the marker. The demand for black markers is projected to be 8 million for the next year. The cost of the caps in an order are $0.05 for orders up to 100,000 and $0.045 for orders over 100,000. The ordering cost is $9. It costs $0.015 to carry a cap in inventory per annum.

Identify the type of objective for the following situation: A farmer can use a mix of fertilizers and water for growing her crops of watermelons and cantaloupes. Cantaloupes are twice as profitable as watermelons. What type of LP problem does this situation call for?

his objective seeks to maximize profits requiring a maximization objective.

Shortage

he RP is the inventory level at which a reorder will be initiated. It seeks to optimize the ordering so that not too much material is on hand or on order at any one time. It considers the lead-time to get the ordered materials and the usage levels in production. This change in the processing time changes the lead-time in the RP calculation.

The formula to calculate an activity's duration using the beta distribution is

test=A+4M+P6 This formula adds the optimistic estimate, the pessimistic estimate, and 4 times the most likely time estimate. This sum is divided by 6 to determine the activity's duration. For example, Bob is redesigning a corporate website. His customer wants him to provide a time estimate to complete this project. Bob has determined the following three time estimates: Activity M = Most Likely A = Optimistic P = Pessimistic Website Redesign 5 weeks 4 weeks 12 weeks Beta Test of Redesign 2 weeks 1 week 3 weeks Train Support Desk 2 weeks 1 week 4 weeks Using the beta distribution formula to determine an accurate time estimate for Website Redesign: test=A+4M+P6 test=(4+4(5)+12)/6test=36/6=6 weeks The calculated estimate of duration for website redesign is 6 weeks. Here are the calculations for the entire project. Activity M = Most Likely A = Optimistic P = Pessimistic test Website redesign 5 4 12 6.0 Beta test of redesign 2 1 3 2.0 Train support desk 2 1 4 2.2 *All times in weeks Test 10.2

Apparent numeric differences can be ignored in business analysis because they tend to work themselves out over time.

No, because differences may be an indicator of a problem. The analyst must investigate apparent numeric differences.

The first step in seeing any difference in numeric values is simply to subtract one from the other to measure the level of significance.

No, because it is unlikely that simple subtraction will help understand the differences. Researchers look for trends as part of their analyses. Quantitative analysis often involves more than a simple subtraction to understand changes.

In analyzing numeric differences, averaging is the favored technique.

No, because there is no one favored technique over the other. The technique of analysis is driven by the situation; there are no favored techniques.

Inventory

Making decisions about inventory is often a large problem for retailers, manufacturers, and other firms with inventory-based operations. Decisions about the amount of inventory needed can range from seasonal situations, faced by many toy manufacturers, to keeping additional inventory on hand for disaster preparedness, faced by many building supply companies. Inventory analysis can also allow a business managers an opportunity to discover flaws in the system and improve operations. Quantitative analysis can help managers and executives decide upon appropriate inventory levels and physical flow of goods. To review an example, consider a hardware store that carries a very popular gas grill. The store owner, Fred, must decide when to reorder grills from the manufacturer so that he never runs out of grills. Being a small business, he does not have room to store a large quantity of extra grills. Fred must determine the minimum and maximum inventory quantity of grills he should have on stock at any given time to satisfy his customer's needs. In order to do this, he must manage his inventory and predict when the demand for gas grills will be higher or lower than usual. Because the demand for gas grills varies throughout the year, Fred must know what the minimum amount of inventory he should keep on hand during the lower demand months, and what the maximum number of grills he should keep on hand during the peak summer months. Fred could definitely use quantitative analysis to help him make his inventory decisions. In addition to determining how much inventory a business needs to order, businesses must also decide when to purchase inventory. This is usually expressed in terms of the reorder point (ROP). Reorder point is determined by multiplying the daily demand for an inventory item by the lead time, the number of days between making an order and receiving the order. When it is time to reorder, inventory managers must take care to balance the cost or reordering with the costs to hold the inventory, the best quantity to order considering these two factors is called the economic order quantity. If the demand or the number of days to receive replacement inventory is uncertain, a business will carry safety stock. Safety stock is additional inventory held to prevent a business from running out of stock. Safety stock ensures that a business will not run out of inventory when the demand is unusually high. When a business runs out of inventory stock this is called stockout. To allow for these unpredictable spikes in demand, a prudent manager will add the amount of safety stock needed to the reorder point: ROP = ((daily demand * lead time) + safety stock) Quantitative analysis can also be used to prepare forecasting models. Forecasting is a method or process used to make decisions about events where the actual outcomes have not yet occurred. Quantitative forecasting models should be used to help forecast future data as a function of past data. Forecasting models are most appropriate when past data is available to be quantified. A common example might be estimating customer demand for new products based on prior sales efforts of other new products. Forecasting methods are usually applied to short-range decisions, not exceeding one to two years. Consider the many tasks that must happen when building a new home. These tasks must be completed in the correct sequence and according to a schedule to ensure the construction is completed by a certain deadline. Businesses must have a system to manage and coordinate the many activities simultaneously. A work breakdown structure (WBS) is a common tool used to help break down and manage a project's deliverables and their supporting tasks. The benefit of using a WBS is to demonstrate in a chart the identification of the major deliverables of the project and each task that must be completed to achieve the deliverables. Depending on the complexities of the tasks, each can then be supported by subtasks. This lends a benefit to scheduling, which may involve multiple types of resources, for example a plumber and an electrician. By developing a WBS, work estimates can be more accurate as well as the eventual project schedule. Project scheduling is an important next step after the WBS has been finished. It is not necessary to have the cost estimates done prior to scheduling; in fact, the total cost estimates can only be done after all tasks and subtasks are identified. Scheduling is done by identifying each task, the resources necessary to complete the task, the duration or length of time to complete the task, and the relationship between the the current and previous tasks. Each of these is used to help a manager identify the total length of time it will take to complete a project. Project schedules are primarily done using scheduling software. They can also be created on paper; however, this takes a considerable amount of time to edit if changes occur. Project tasks are scheduled in a rational sequence to meet the time objectives of the project. In preparing the schedule, the duration for each of the tasks and its subtasks is estimated, thus providing a total estimated duration for each task. The sum of the duration of all the tasks is the estimated duration for the whole project. It is important to determine duration or time estimates in a project schedule to ensure that the schedule will be correct. When there is known to be variation or possible uncertainty in the time estimates of the tasks and subtasks, a project manager should consider using the beta distribution method to help determine the duration of each task. The beta distribution method is a calculation that uses a combination of three duration estimates based on experience or history: the most optimistic duration, a pessimistic duration, and the most likely duration in order to consider variation in the final duration estimate. This method is thought to give a better final estimate for a task's duration. The beta distribution method is equal to the sum of the optimistic time estimate for the task (o), four times the most likely time (m) estimate, and the pessimistic time estimate (p); this result is then divided by six. The formula is: Te=(o+4m+p)/6 Calculate duration estimates for the following situation: The store manager was in the process of remodeling the store interior and needed to know how long the store would be closed during the project. The construction manager believed that it could be done within 8 days if everything went according to plan. There is the possibility it could take up to 16 days if they ran into structural difficulties. The store manager asked if there was any way to decrease the number of days. The construction manager thought that if he focused all resources, the project could be done in as soon as 6 days. Using the beta distribution method, the estimated time to complete this task is (6 + 4(8) + 16)/6 = 9 days. A quantitative analysis process used with project scheduling is the identification of the project's critical path. The critical path method (CPM) is an algorithm that is used to calculate the longest sequence of tasks that must be completed on time for the project to be completed by the deadline. The critical path method determines which tasks are without any slack time and which tasks have extra time and can be delayed without delaying the entire project. The tasks that have zero slack time are considered critical; each of the critical tasks makes up a project's critical path. Delays on tasks that are on the critical path will negatively impact the completion of the project by the deadline. Detailed project scheduling analysis involves mathematical equations to determine quantities such as earliest finish time, latest start time, expected activity completion time, and value of work completed. Gantt charts are used to display tasks with duration and linkages to other activities in the project. Project scheduling analysis can help with cost analysis and provide insight to where additional costs may arise throughout the project.

Quantitative analysis is based on specific, not subjective, information. Businesses can obtain specific information through a method of questioning. However, in order for questions to be considered specific, each question must allow for closed answers. The answer is considered closed because the respondent can only select a specific answer. An example of a closed question follows:

"Customer service is responsive to your concerns." Strongly agree Agree Neither agree or disagree Disagree Strongly disagree Providing a set of closed answer choices allows for statistical analysis of a large number of responses. The choices must be carefully designed to avoid omitting possible response choices. Well-designed forced choice responses can be quantified and analyzed. Subjective questioning uses open-ended answers. The nature of an open-ended question is that it gives an almost endless variety of ways to respond to a question. Open-ended questions are questions that do not offer answer choices but instead encourage a narrative response. Open-ended questions allow the researchers to learn more about emotions or feelings in focus groups and to get new ideas that they might not get with closed-ended questions. Although open-ended questions can sometimes provide valuable information, this unstructured manner could also not force the respondent to pinpoint or discuss the researcher's key questions. An example of an open-ended question would be, "Describe how happy you are with our customer service." Executing an analysis of open-ended questions is very time consuming because the answers are subject to interpretation and not quantifiable. Examples of quantitative data gathering strategies can include surveys with close-ended questions, observing measurable events (how many people were in the coffee shop at a specific time of day), and performance data. These data gathering strategies are easy to summarize, analyze, and compare.

A business is impacted positively and negatively by many factors. These factors can be either internal or external. Problems stemming from internal factors occur within the organization due to issues such as quality control procedures, resource scheduling, or equipment failures. Problems stemming from external factors occur outside the organization and can be due to issues such as new competition, new territory developments, economic downturn, or customer requirements. Quantitative analysis will help businesses determine what is impacted and what are the contributing factors. The impacted factor is the dependent variable; the contributing factors are the independent variables. For example, consider a business that has seen a downturn in its profit, due to a new competitor. The profit is the ___ variable and the new competitor is the ___.

1. Because the profit was impacted by the new competitor, it is the dependent variable. Profit changes when the number of competitors changes. 2. The competitor is the impacting variable. The additional competitor impacted the profit.

One purpose behind business forecasting is to estimate future demand. Forecasting gives planners the ability to make prudent ___ among/for ___ in order to maximize their likelihood of success.

1. Choices: Forecasting proposes to present a method to create a number of choices and their likelihood of occurrence. This enables the manager to maximize their likelihood of success. 2. Several Alternatives: Forecasting generally views several possible future states and their likelihood, so there are multiple alternatives to consider. In forecasting, the planner will identify various criteria with which to provide some relative quantitative measures among the alternatives. Planners make their choice based on the weight of the evidence.

Steps in Quantitative Analysis

1. Define problem. 2. Develop mathematical model. 3. Prepare and input data. 4. Find best solution. 5. Test solution. 6. Analyze results. 7. Implement solution. Using the steps in quantitative analysis can be helpful in finding solutions thereby saving time and money by accurately representing a situation, solving complex problems, and providing insight to business decision makers.

Critical Path

A quantitative analysis process used with project scheduling is the identification of the project's critical path. The critical path method (CPM) is an algorithm that is used to calculate the longest sequence of tasks that must be completed on time for the project to be completed by the deadline. The critical path method determines which tasks are without any slack time and which tasks have extra time and can be delayed without delaying the entire project. The tasks that have zero slack time are considered critical; each of the critical tasks makes up a project's critical path. Delays on tasks that are on the critical path will negatively impact the completion of the project by the deadline.

Which of the following situations that need a foundational level of research before analysis can begin? Needs Foundational Research Does Not need Foundational Research 1. A large pizza chain is exploring changes to the recipe for its deep dish pie crusts. 2. A large pizza chain wants to consider adding breakfast items to its menu. 3. A TV station wants to expand to Spanish language broadcasts. 4. A local car dealership is considering keeping the parts department open all day Saturday instead of just four hours. 5. A state governor's office is considering opening up all state parks for free to all state residents. 6. A automobile maker is considering launching a new version of an old two-seater roadster.

1. Does Not need Foundational Research - This represents only a change to an existing product. 2. Needs Foundational Research: This is a radical departure for the menu, and extensive market research is needed. 3. Needs Foundational Research: This will take extensive research with a large costs involved to enter this market. 4. Does Not need Foundational Research: This can be piloted with little costs or risks to get the results. 5. Needs Foundational Research: This situation may result in financial loss for the state and is a radical departure from the current business model. 6. Needs Foundational Research: This decision will cost considerable funds to launch and may result in losses if the new roadster is not successful.

The advantage of using mathematical techniques in research and quantitative analysis are many. Among the key advantages are the following: Using numbers for research increases the level of ____ by more clearly defining the phenomena being evaluated. By using numbers, one also ___ the amount of ambiguity of the topic being evaluated. Finally, using numbers in research serves to better ___ the extent, breadth, and nature of the phenomena being evaluated.

1. Objectivity: By focusing the study around the numbers, there is a tendency to rely on objective criteria in lieu of subjective criteria for understanding the phenomena. This reduces subjectivity. 2. Reduces: Using numbers instead of textual responses in the analysis process enables a researcher to have a much more specific understanding of the issue. 3. Quantify: In studying many phenomena, a key part of the understanding is the quantitative nature of the phenomena. For example, questions can be the following: How much or how often does some event happen? What are the high and the low values of phenomena being evaluated, the range? What is the variation of a process being evaluated?

Adding Positive and Negative Numbers 1. Here is an example of adding a positive and negative number when the larger number is positive: 81-56=25 2. This is an example of adding a positive and negative number when the larger number is negative: -81+56=-25 3. Here is an example of adding positive and negative numbers when there are more than two numbers. Add the following five numbers together: −50+10+60−30+20 −50−30+10+60+20= −80+90= 10

1. Steps: Subtract smaller number from the larger. 81-56=25 Keep the sign of the larger number. Because 81 is greater than 56, the answer is positive. 2. Steps: Subtract smaller number from the larger. −81+56=−25 Keep the sign of the larger number. Because 81 is greater than 56, the answer is negative. 3. Steps: Group the negative numbers (adding them) and the positive numbers (adding them) and then add the two remaining numbers (one negative and one positive).

One of the first steps a researcher must do is to determine the dependent and independent variables under study. This is done in order to establish the cause and effect relationship. The ___ variable acts as the causal factor while the ___ acts as the affected variable.

1. The independent variable causes the change in the dependent variable. 2. The dependent variables change is caused by the change in the independent variable.

It is important to understand this relationship isn't always positive; it could be inverse. That is, when the ___ variable moves in an upward direction, the ___ variable will move in a downward direction.

1. The independent variable causes the changes in the dependent variable. 2. The dependent variables change is caused by the change in the independent variable.

Summary:

A key advantage in using statistics is the ability to draw conclusions about an entire population by way of a smaller sample. This saves time and effort on the part of the researcher and yet maintains the validity of the conclusions. Non-numeric data are a form of data that are in text, such as words, symbols, or letters, and are not used in calculations. Qualitative data are non-numeric. Non-numeric data often take the form of personal opinions, indicating likes or dislikes. Although they are not used in quantitative analysis, non-numeric data can still play a critical role in decision-making for a business.

What would be the best application of quantitative analysis for the situations below?

A marketing director wants to determine the average household income of the company's key customers. This question suggests the use of metrics to collect and evaluate data. This information could come from a variety of sources, such a customer survey or focus groups. The information is numeric in nature.

When arguing using subjective reasoning, on which of the following will the argument most likely be based?

A person's past personal experiences Subjective reasoning is heavily based on a person's personal experiences.

What is the best description of an open-ended question?

A question that can be answered multiple ways The nature of an open-ended question is that responses can be given in almost endless variety of ways. Sometimes such answers provide much valuable information, but this unstructured manner could also not force the respondent to discuss the researcher's key questions.

Variable

A variable is a characteristics of an object or person. Examples of variables might be expenditures, salary, revenue, profit, year, or quantity. Quantitative analysis will require the mathematical concepts using dependent and independent variables. These two variables are commonly found within economic and financial concepts. The dependent variable is often considered to be the Y variable. The term dependent implies it must be related or it is impacted by another variable. The impacting variable is called the independent variable. An independent variable is often denoted as X and is usually the variable that is changed or controlled in a mathematical model or calculation. For example, the amount of milk a convenience store stocks is dependent on the number of consumers who will buy the milk. In this example, the amount milk is the dependent Y variable; it is dependent on the consumers, which is the independent X variable. This can be understood by considering that the number of consumers impacts the amount of milk stocked; the milk does not impact the number of consumers. The independent variable is also referred to as the explanatory variable. The dependent variable can be referred to as the response variable. This is because the dependent variable, depends on, or responds to, the independent variable. The impact to the dependent variable is explained by the explanatory, independent variable. Consider another basic example. The number of hours we exercise each week might explain changes in our BMI. In this example, our BMI is the dependent variable, or response variable. It is explained by the independent variable, the number of hours we exercise per week. Using a business example to express independent and dependent variables, we could consider a salesperson and the amount of sales she generates. The number of hours the salesperson works impacts the total sales she produces for the month. The number of hours worked would explain the total sales. The hours worked is the independent variable. The response variable or dependent variable would be the total sales. Sales are dependent on the number of hours the salesperson worked. The independent variable causes a response in the dependent variable.

Work Breakdown Structure

A work breakdown structure (WBS) is a common tool used to help break down and manage a project's deliverables and their supporting tasks. The benefit of using a WBS is to demonstrate in a chart the identification of the major deliverables of the project and each task that must be completed to achieve the deliverables. Depending on the complexities of the tasks, each can then be supported by subtasks. This lends a benefit to scheduling, which may involve multiple types of resources, for example a plumber and an electrician. By developing a WBS, work estimates can be more accurate as well as the eventual project schedule. Project scheduling is an important next step after the WBS has been finished. It is not necessary to have the cost estimates done prior to scheduling; in fact, the total cost estimates can only be done after all tasks and subtasks are identified. Scheduling is done by identifying each task, the resources necessary to complete the task, the duration or length of time to complete the task, and the relationship between the the current and previous tasks. Each of these is used to help a manager identify the total length of time it will take to complete a project. Project schedules are primarily done using scheduling software. They can also be created on paper; however, this takes a considerable amount of time to edit if changes occur.

One purpose behind business forecasting it to estimate future demand. This is particularly helpful for an operations manager to manage the use of today's ___ in order to succeed in the future.

Assets The operations manager may need to adjust the current work plan to align the right amount of assets to meet the projected demand.

In order to quantify the variable used in a decision-making problem, another numeric value is needed when considering states of nature: the probability. All of the individual probabilities must ___ When setting up a decision-making problem, the alternative definitions must be ___ from each other for clarity.

Add up to one: This is required to consider 100 percent of the alternative states. Distinctly Different: The alternatives must be clearly different from one another; they must be mutually exclusive.

Multifaceted Problems

Businesses that are faced with complex, multi-faceted issues can utilize qualitative analysis to reveal significant information about the source or sources of a problem. A multifaceted issue will have many organizational aspects or phases to consider. Multifaceted problems typically span across the entire organization: marketing, manufacturing, sales, operations, and finance. Multifaceted issues have many unknowns; no single answer is going to solve a problem. Remember that the goal of solving problems or pursuing strategic initiatives is specific solutions, particularly developing specific solutions that can be implemented in the organization. Finding specifics is complicated the more multifaceted an issue is. Simple problems or issues do not need the same deep exploration as multifaceted issues. A complex issue will reveal itself the same as multifaceted, with many unknowns and significant uncertainty in specific solutions. When qualitative analysis is used on complex issues, the managers can move from initial research discovery through to the development of detailed, more specific information. Complex qualitative research emphasizes the need for depth and detail in the results. We can best emphasize a multifaceted issue by considering a pharmaceutical company that is embarking on a project to develop a medical method or drug to cure a disease. You can probably imagine the many unknowns that the company is faced with. The biggest unknown is most likely whether or not the eventual outcome will cure the disease. Starting the project with a qualitative analysis to discover additional details is a necessary approach. Having a seemingly great solution is not helpful if the business is solving the wrong problem. It takes significant research to get to the root cause and solution of complex business problems. Using qualitative research techniques such as group collaboration to brainstorm foundations of an issue may provide better information than an individual manager. Qualitative research allows a business to better discover the foundation that lies in not only what customers want, but why they want it, or how they want it. A car seat manufacturer might ask parents, "Besides safety, what are the most critical issues when purchasing a car seat?". Customers can give verbal or written descriptions of their wants, attitudes, beliefs, and needs through phone surveys, focus groups, questionnaires, and surveys. Observation can also be used as a qualitative research tool when evaluating the foundation of a problem. Using more than one approach or source to collect information will improve the credibility of qualitative research. This qualitative technique, known as triangulation, ensures that the research accurately reflects the situation.

Suppose a quantitative political survey asks the following question: "Do you plan on voting Democrat or Republican in the next election?" What is wrong with this question and why would a qualitative approach with an open-ended question be better?

By limiting the original quantitative answer to two options only, the respondent is limited to only two answers. This will force respondents to choose an answer that may not accurately reflect their opinions or voting plans, thus making the quantitative survey results inaccurate. There are other options for voting. Perhaps the voter plans to vote Libertarian or Green. An open-ended question (qualitative approach) might produce new trends that would not be discovered with the more rigid, quantitative question.

Identify business problems that can be solved using quantitative analysis methods:

Decision analysis involves finding the best solution when faced with an uncertainty. All possible alternatives must be evaluated when there are two or more events that might happen in the future. These alternatives or future events are called states of nature. In defining the alternatives, each must be distinctly different from the other for clarity. When using quantitative analysis, each state of nature must be evaluated as to its likelihood of occurrence and the monetary benefit that will be recognized by the alternative after costs. Quantitative analysis will use the alternatives' likelihood or probability of occurrence and multiply it times the monetary benefit, thus producing an expected value of that alternative. When evaluating the likelihood or probability of each alternative, the total probability must add up to one. For example, if alternative A has a 70% (or 0.7) probability of occurrence, alternative B will have a 30% (or 0.3) probability of occurrence. For example, when competition increases, an existing business can be faced with problems as to how the competition will impact their sales. This problem will require thoughtful analysis and presentment of decision alternatives. To combat the competition, should they lower their prices? Should they increase their product features? Should they advertise more? Often each of these decision alternative can be evaluated and determined through the use of quantitative analysis. In order to find solutions, the business must look for ways to optimize resources and improve processes in response to the problem. Decision alternatives might even lead to a possible lower profit margin. However, if it is the best alternative in comparison to the other potential solutions, the business must be prepared to respond. When the business is faced with these decisions, optimizing resources can involve scheduling more people, managing inventory, and better management of customer expectations. A business must recognize changing trends and quickly make decisions to adapt to these changes. Making decisions are not always simple. In other words, not all decisions are made and implemented or adopted by the business immediately. Sometimes decisions are sequential. In other words, a business may make a decision to develop a new product and then test that product on a small group. Depending on the results from the small group, the company will then make a decision on whether or not to sell this product on a larger scale.

In defining a cause and effect relationship, the variable that is changed as a result of the action of some other variable is called what?

Dependent variable The independent variable is the factor that causes change to the dependent factor. In other words, the movement of the dependent factor is driven by the movement of the independent factor.

Detailed Project Scheduling Analysis

Detailed project scheduling analysis involves mathematical equations to determine quantities such as earliest finish time, latest start time, expected activity completion time, and value of work completed. Gantt charts are used to display tasks with duration and linkages to other activities in the project. Project scheduling analysis can help with cost analysis and provide insight to where additional costs may arise throughout the project.

Identify situations where expected monetary analysis can be used. A banker wants to decide whether to approve a loan for a new customer. A lawyer is deciding on the monthly retainer fee for a new client A retired teacher is considering three alternative investments for her retirement fund. A real estate agent is considering investing in either a commercial strip mall or several single-family homes. A medical research group wants to determine the effectiveness of a new medicine. A marketing manager wants to determine the effectiveness of a new ad campaign.

Does Not Use Expected Monetary Analysis: This decision in not made using expected monetary but focuses on risk. Does Not Use Expected Monetary Analysis: This decision in not made using expected monetary analysis but considers profit potential and scope of work. Uses Expected Monetary Analysis: Expected monetary analysis can calculate what the alternative outcomes will be given likelihood of the alternatives. Uses Expected Monetary Analysis: The alternative investments can be compared using expected monetary analysis techniques. Does Not Use Expected Monetary Analysis: Expected monetary analysis is of no value in making this determination. Does Not Use Expected Monetary Analysis: Expected monetary analysis cannot be used to make this determination; this would compare the cost of the campaign with the cost of the sales increase.

Which of the following accurately describes quantitative analysis?

Ensuring the use of valid, reliable, and objectively measurable data in order to understand a phenomenon The hallmark of quantitative analysis is ensuring the use of only valid, objective, and reliable data to begin with. That is, we must ensure that the underlying data are a true and repeatable reflection of reality. Also, the data used must be measurable in nature: that is, objective and not subjective. The phenomenon being evaluated must be quantifiable.

Expected monetary value analysis:

Expected monetary value analysis is a specific quantitative analysis technique that utilizes mathematical calculations to determine the average of all potential alternatives being considered to solve a problem. The expected monetary value technique calculates an average of the most likely and least likely alternative scenarios. The calculation considers both the likelihood or probability of the alternative occurring and the monetary benefit assigned to the alternative after costs are considered. This analysis technique works best when the decision alternatives can be clearly identified and the likelihood of occurrence for each can be determined. For example, consider a business that is trying to make a decision about their information system network. The alternatives being evaluated are to either spend money to upgrade the current information systems network server or to purchase an entirely new network server. Choosing either alternative may have similar costs associated with the decision but the likelihood of success is different for each. Using expected monetary analysis, the business will consider each alternative as well as the likelihood, or probability of successful completion of implementing the alternative. Once the calculation is made, the business should select the alternative with the highest monetary benefit.

[True/False] Project scheduling is only possible by using project management software.

False: Project scheduling can be complex, but it can be easily handled using commonly available spreadsheet software.

[True/False] Project scheduling is the same as any other type of scheduling.

False: Project scheduling is similar but not the same; it involves identifying the tasks, subtasks, and linkages to the scheduling process.

[True/False] For project scheduling, project task times are estimated first and then the subtask times are derived after that.

False: Subtask times are calculated first and then added up to create the time estimate for the task over-arching the subtasks.

[True/False] A cake decorator is preparing an entry for a cake competition and is deciding on the color and style of the icing on the top layer, and this should be modeled.

False: This decision would be an artist decision with no specific mathematical model that would apply.

[True/False] A dance studio director is deciding on the routine for a children's recital. This is a good opportunity to test a quantitative model.

False: This decision would be an artist decision with no specific mathematical model that would apply.

[True/False] For project scheduling, it is essential to use only one time estimate for computing each task time.

False: When there is is known to be variation in the time estimates in the tasks and subtasks, the project manager should consider using a combination of three time estimates; the optimistic time, the pessimistic time and the most likely time.

In quantitative analysis, problems range from those that are very simple to those that are very complex. Essentially, simple problems involve few variables and limited alternatives to consider. For instance, if you only have an hour for lunch for an office of four, and there are only three restaurants to consider, that's very simple. However, if you had a special occasion for lunch for an office of 20 and could be out of the office for up to two hours, then you will have many more restaurants to go to and even different venues instead of restaurants, making that much more complex. Based on the descriptions below, identify whether simple or complex problem solving skills are most appropriate. A small retail store shift manager is planning staffing needs for the week. A buyer is choosing between several new suppliers for parts used in a new product line. A construction lead is planning the amount of lumber needed to frame in a standard 2200 square foot home. A buyer is making a calculation for the order quantity for a company's quarterly buy of raw goods from a preferred supplier. A researcher has to develop a research proposal for a new line of toothpaste for sensitive teeth using a new plant-based chemical. A marketing manager needs to make calculations involving sales from more than 100 locations to calculate commissions for more than 20 sales representatives.

Given that this is a small store and the time horizon is short, this is a simple problem. Because there are a number of variables and unknowns, then this is a non-routine problem needing more complex procedures. Because this situation suggest this is routine, then this is a simple problem. This calculation could be made with a standard economic order quantity model, reducing the complexity of the decision. This will definitely be a complex proposal because it is health related and involves new chemicals used for human application. Although there are hundreds of calculations to be made, the complexity of each is quite basic and standard desktop software makes this a non-complex task.

Safety Stock

If the demand or the number of days to receive replacement inventory is uncertain, a business will carry safety stock. Safety stock is additional inventory held to prevent a business from running out of stock. Safety stock ensures that a business will not run out of inventory when the demand is unusually high.

Like all approaches, subjective reasoning may have some inherent problems associated with it. What is one likely problem?

Personal reflections are often fallible. One indicator is when a researcher wants to obtain personal reflections and emotions.

Forecasting is used in order to help establish the ___ of a future event.

Probability Forecasting is not a guarantee of future results. Forecasting, though, seeks to identify the most and least probable outcomes of some future event.

Objective Analysis

Objective analysis focuses on facts and figures, eliminating any emphasis on opinions, emotions, and feelings. Objective data is not impacted by someone's personal opinion or prejudice. An objective analysis uses numerical, measurable facts that are observable. Objective or realistic analysis makes use of mathematical models to make decisions and set goals. The SMART acronym is often used as a guide to in objective analysis: Specific, Measurable, Achievable, Realistic, and Timely. It is not always practical to quantify goals and objectives; however, it is important to have a blend of subjective and objective analysis when making a business decision or solving problems.

When managers determine that there is a problem, it is important to recognize what specific information will be required to move toward a solution. Business problems that are specific in nature will use quantitative analysis to help identify the possible solution alternatives by using metrics, a measurement used to gauge performance. Examples of these business problems include increasing operational expenditures, diminishing revenue, and inventory ordering, all of which will need to consider specific numeric data in order to solve. Which of the following best explains why these particular examples require specific numeric data to help solve the problem?

Problems that are identified as specific in nature will require specific data to solve. Quantitative analysis is used in solving business problems. It requires the use of specific data to find solution alternatives and validate the alternatives.

Problem Identification

Quantitative analysis begins with problem identification. Using research questions help to identify the problem to be analyzed and helps define what you want to learn or solve in regards to the problem. Examples of research questions used to determine problems might be, "What impact would updating or creating a new company logo have on future company sales?" A research question should lead to a specific, testable hypothesis. The practice of preparing research always starts off with a research question. The research question is not the same as preparing a hypothesis. In quantitative analysis, a hypothesis is a declarative statement predicting the relationship between the dependent and independent variables. Typically the research question first makes a narrative statement or assertion about some phenomenon, followed by the hypothesis, which makes two differing statements: the null hypothesis statement and the alternative hypothesis statement. When writing a null hypothesis statement, it is written to state that there is not a relationship between the variables. The alternative hypothesis will state that there is a relationship between the variables. The job of the quantitative analysis researcher is then to collect evidence in order to reject the null hypothesis statement. Using the logo example, an analyst might have a theory, or hypothesis statement, "that a new company logo will generate more sales than the old logo did." The theory that the analyst wants to prove is the alternative hypothesis, and the status quo is the null hypothesis. The null hypothesis is "that the new logo will not generate more sales than the old logo did." The results of the alternative hypothesis must be significantly better for a change to be made. Therefore, using the new logo must generate significantly more sales than the old logo did before the company will be willing implement the logo change.

Describe how quantitative analysis is used in business decision-making continued:

Quantitative analysis models are used extensively by businesses to solve problems and assist in decision-making. Models are mathematical representations of a situation or problem, using numbers, mathematical symbols, and expressions. Mathematical models help managers find the optimal solution to a problem. Models usually contain controllable variables and parameters. Controllable variables, those inputs that are determined by the decision maker, are generally the decision variables. Production quantity is an example of a decision variable. Parameters are known quantities that are part of the problem and are not controlled by the decision maker. Examples of parameters are labor hours to produce a product, or the amount of corporate income taxes. A very simple example of a mathematical model is how we calculate the profit for a business making and selling pencils. To find profit in this example, the mathematical model is, Profit = Revenue - Expenses. The revenue would be the price per pencil multiplied by the number of pencils sold. The expense would be the cost to make each pencil multiplied by the number of pencils sold. The mathematical formula might look like this: Profit = (Price per pencil*number of pencils sold) - (cost to make pencil * number of pencils sold). If s represents selling price, c represent costs per pencil, X represents the number of pencils sold, the mathematical model might look like this: Profit=sx−(cx) The parameters are c and s since these inputs are not controlled by the decision maker in this model. The decision variable is X. Using this model, a manager can make inferences regarding expected profit for varying amounts of pencils sold. The decision maker might use this model to evaluate various scenarios of profits by changing the number of pencils sold to determine an optimal solution for the company. The decision maker is testing the model for the different scenarios. It is important to verify or test the model to ensure that the model is producing the appropriate results. Using a model is usually less expensive, less risky, and requires less time than experimenting with a real situation. Models can accurately represent reality and provide insight and information to decision makers. Using an electronic spreadsheet program, such as Microsoft Excel, a researcher can create a profit model or breakeven analysis model to determine how many pencils must be sold to cover all of the business costs to break-even, or how many pencils would need to be sold to generate $10,000 profit. Quantitative models are used extensively in business for a variety of applications, such as cost, revenue, profit, production volume, inventory, waiting line analysis, simulation, forecasting, and more.

In setting up a decision problem, in addition to alternative actions at the decision points, there is another factor to consider having to do with the context of the problem in the business setting. This concept is called ___ An example of ___ might be the assumed rate of growth in the economy, whether high, medium, or low growth because this will impact the outcomes of the alternative decisions.

States of Nature: This is a factor that brings the decision in context, such as an assumed rate of growth in the economy of high, medium, or low growth. States of Nature: This narrative describes a state of nature.

Subjective Analysis

Subjective analysis uses likes and dislikes, past experiences, personal opinions, and interpretations to determine an outcome. A manager using subjective analysis to make a decision is more likely to go with intuition or a feeling. If the manager has had a lot of experience with similar problems, this qualitative approach will be emphasized. It is not always practical to quantify goals and objectives. It is not always practical to quantify goals and objectives; however, it is important to have a blend of subjective and objective analysis when making a business decision or solving problems.

Define quantitative analysis, including its business purpose:

Technology gives businesses access to tremendous amounts of data—often more data than companies can manage. Sales figures, time management techniques, production costs, shipping routes—the possibilities for data are nearly limitless in today's business environment. Managers and workers may ask, can any of this data help move the company forward in helpful ways? Quantitative analysis provides methods to analyze large or small amounts of data to look for patterns, trends, and relationships. Mathematical analysis can help managers make strategic decisions and find statistically supported solutions to business questions. Data may be categorized as either subjective or objective. Subjective data, obtained through surveys and interviews, are considered non-measurable, though marketing research has developed ways to study the intensity of opinions that guide consumer behavior. Subjective data typically include personal perceptions, such as likes, dislikes, attitudes, and opinions. Objective data are measurable and typically arise from observation or testing in business areas like sales, operations, manufacturing, and logistics. Data must be valid: that is, the data must accurately represent the true business relationship at hand. Further, the data must be reliable: if we sought to characterize a particular business relationship by gathering data several different times (different samples), the data would reflect the relationship the same way with every sample. Quantitative analysis generally depends upon simple to complex mathematics and statistical modeling to discover important information about collected data. Companies analyze data to solve problems, and make decisions based upon objective data to gain an economic or competitive advantage. Examples of quantitative analysis include cost-benefit analysis, inventory analysis, logistical analysis, and forecasting revenue. The quantitative analysis approach defines a problem and then develops a mathematical model to represent the particular business situation. The model allows managers to make inferences regarding the data. For example, if Company XYZ makes $3 revenue from each book sold and p represents total revenue earned and q represents the number of books sold, then the mathematical model to determine the company profit would be p=3q. As long as the assumptions behind the model p=3q hold true, Company XYZ can use that equation to forecast its revenue from a particular book. But if the economic outlook changes and the company cannot say with confidence that each book contributes $3 to the bottom line, they will need to create a new model or equation. Perhaps the economy declines and students can no longer afford new books, or a competitor offers an alternative book with a more popular approach. The business world and the data available change constantly and thus data tell an ever-evolving story. For a particular period of time, then, mathematical models featuring valid and reliable data can guide managers to make rational decisions about pricing, distribution, manufacturing, and other functional areas. Understandably, gathering data for a complex problem can be time-consuming and difficult. Input data must be both relevant and accurate to achieve optimal results. The output from the model can then provide optimal solutions. Used correctly, model outputs can be predictive, helping managers understand future business outcomes if they change practices based on the results of sound analysis. Assumptions and solutions must be tested and analyzed prior to implementation. Cause and effect demonstrate a relationship between events where one event (effect) is the result of the other (cause). Cause and effect use independent and dependent variables. A dependent variable is the variable that is being measured, or affected. The independent variable is free to change in a given model. The dependent variable is affected by the changes in the causing independent variable. Although only one dependent variable is considered, many independent variables can have an effect. Quantitative research and analysis aim to determine if there is a relationship between one event (an independent variable) and another (a dependent variable). For example, when business managers are looking at quarterly performance that either exceeded expectations or fell short, they are attempting to find a relationship between events in order to explain or interpret the results. To find the relationship, it is necessary to determine the dependent and independent variables. If a pharmaceutical company wants to study the impact of a chemotherapy drug on a group of patients, the independent variable is the administration of the drug, how often, how much. The dependent variable is the result that the drug has on the cancer. In a math equation, such as y=3x+2, x is the independent variable, and y is the dependent variable. The y value is dependent on the x value. Cause and effect show how one event affects another. For example, when water is heated to a certain temperature (the independent variable or cause), the water boils (the dependent variable or effect). The following diagram provides an example graphical representation of the many different causes that could contribute to a problem. By using quantitative analysis, businesses can review a problem mathematically and statistically to determine the root cause(s). When a business has a serious problem (effect), it is important to evaluate and study potential causes before determining a solution. A fishbone diagram is sometimes used to determine the cause of a problem. It is important to look at all of the potential causes, not just the obvious. The problem statement is written on the diagram as the "effect." Brainstorming can take place for each category that might be causing a problem, allowing for a mathematical model to be constructed. Examples of problem areas might include equipment, process, management, materials, environment, and people. A completed fishbone diagram will help to draw out the probable root causes of a problem in business. Sometimes an additional variable can strengthen or weaken the impact of the relationship between the dependent and independent variable. A moderating variable is a third variable that changes the established effect of the independent variable on the dependent variable. In a moderating relationship, the relationship between the dependent and independent variables depends on the level of the moderating variable. For example, higher income levels are associated with higher levels of education; however, the effect of this relationship is even stronger for men than for women. Therefore, the strength of the relationship between income and education depends on gender as a third, moderating variable. A mediating variable explains the relationship between the dependent and independent variables. Mediating variables are intervening factors that can change the impact of the independent variable on the dependent variable. For example, having a personal trainer helps gym members stick with an exercise routine. The mediating variable might be that the trainer motivates the members, which in turn causes them to stick with their exercise routine. The motivation is the mediating variable and explains why people with personal trainers stick with their exercise routines more than those who do not. A very motivated person without a personal trainer would also stick with the routine. An additional example would be if a business was offering stock options as part of their employee compensation package. Each employee would receive between zero and 50 stock options based on their annual performance appraisal. The mediating variable is the awarding of stock options based on performance. The stock options are the motivation and explains why employees might receive more options than those underperforming employees. Scatter diagrams are used to graph pairs of numbers to determine the relationship. A coffee shop might keep track of how the temperature outdoors impacts the coffee sales. If the sales increased when the temperature decreases, then the business could assume that there is a negative linear relationship between these two variables. We could also say that the variables are correlated. The line that shows the general direction of the relationship of points over time is called the trend line. If the trend line moves downward as we progress from left to right, there is a negative correlation between the two variables. What happens today in business can help managers make predictions as to what might happen in the future. Researchers and analysts look for specific patterns in the current and past business data and then make forecasts. Forecasting helps businesses make adjustments to the current business environment to encourage better business outcomes in the future. Forecasting can help businesses make better or more accurate predictions about customer demand. For example, a fast-food restaurant can use forecasting to predict when more employees are needed to service customers. By studying previous and current weeks of customer demand and then making a prediction, businesses can then provide adequate staffing for faster, higher quality customer service. Another example of using mathematical forecasting is in manufacturing. Forecasting can assist manufacturing managers with decisions about increases or decreases in production capacity. Forecasting presents a number of choices and the likelihood of occurrence. When managers are presented with a forecast, they can select the option that will best maximize the likelihood of success. Although no one can be 100 percent sure of what will really happen in the future, forecasting helps to reduce the uncertainty and increase the likelihood of successful planning. Forecasting does not provide a guarantee of success in the future, but it certainly increases the likelihood of positive outcomes. Quantitative analysis uses math and statistics to make business decisions and evaluate business problems, strategies, and investments. Quantitative analysis is used to create models for pricing, risk management, inventory, stock trading strategies, credit analysis, and more. All of these activities involve numbers and formulas. Mathematical models can accurately represent business problems and help a decision-maker solve the problems. Mathematical models are helpful and are used to quantitatively analyze the impact of changes on business performance and the evaluation of risk. The models show mathematical relationships in the form of equations or inequalities, such as averages and totals. Statistics is the gathering, organizing, and interpreting of numerical data. It provides tools to analyze important numerical information through organizing, analyzing, and interpreting large amounts of data. There are two types of statistics: descriptive and inferential.

In determining relationships between cause and effect, what does it mean if one variable increases as the other variable also increases?

That there is a positive correlation between the variables The positive relationship means that the increase of the independent variable will cause the dependent variable to increase proportionately.

What is a key difficulty associated with using an open-ended questions?

The answers may not address the researcher's key question. It is possible for a respondent to speak at great length without addressing the concerns of the researcher. This is due to the unstructured nature of the question.

A researcher is performing quantitative analysis and notices a change in the data for a particular key performance indicator. What might this mean?

The key performance indicator may be reflecting a developing trend in one direction or another. This is common in quantitative analysis and researchers look for trends as part of their analyses.

A small retail pool supply business is planning for the upcoming busy summer months. They begin their inventory planning well in advance of the summer in order to communicate inventory needs to their suppliers. A pool supply company would not be successful if it did not have the right inventory offerings during peak seasons. Inventory planning will need to include quantitative analysis. Describe each of the key quantitative analysis terms and their importance that will be used by the pool company in preparation for inventory decisions.

The key terms and data that the small pool supply company will use in preparation for inventory decisions are: minimum inventory, maximum inventory, economic order quantity, and reorder point. Determining the minimum amount of inventory the pool company should have on hand will help keep the necessary stock available to customers during peak seasons. The maximum amount of inventory must be determined in order to not have over stock, which then must be warehoused due to lack of shelf space. Finding the appropriate economic order point will help the pool company balance their cost of additional storage. The reorder point will help the pool company determine the inventory level when new inventory should be ordered, taking into consideration inventory lead times.

Which of the following would be the most likely beneficiary of a researcher's quantitative analysis directed at adopting new operational efficiencies, such as introducing a "paperless" entity?

The organization and its customers When an organization is able to improve their operational efficiencies, there is an opportunity to include the customer in the benefits. This could be done by possibly passing along a price decrease for the products or services.

A quantitative analysis can benefit by a researcher incorporating closed-ended questions to gain specific information. Research questions would be considered closed when the respondent can only select a specific answer. By using closed-ended questions, the researcher can ensure data validity and consistency. It is common for researchers to qualify their survey questions beforehand by validating that the responses will be closed ended. Using the qualified closed-ended questions will then help ensure that the actual responses from the universe of respondents will be credible and consistent. How is this possible?

The questions and responses can be qualified or validated by the researcher beforehand. It is common for researchers to qualify their survey questions beforehand to validate the responses. This helps ensure that the actual responses from the respondents will be credible and consistent.

There are several advantages to using closed-ended questions, saving time is one. How is this possible?

The questions limit the range of possible answers available. The respondent has only limited options to be concerned with, making the task quicker.

Consider the following scenario: You are a clinician working in a hospital setting. You are tasked with determining whether Medicine A is more effective in preventing pain when administered intravenously or in pill form. Develop a research question and a hypothesis. What are the independent and dependent variables in this research?

The research question is a statement identifying the population being studied and the problem to be solved. The research statement for this scenario might be "Is there a difference in pain between hospital patients receiving Medicine A in pill form and those who receive Medicine A intravenously?" The independent variable, or consequence, is Medicine A, and the dependent variable is the pain. The hypothesis might say, "There is a significant reduction in pain for hospital patients receiving Medicine A intravenously compared to those patients receiving Medicine A in pill form."

f there is a linear relationship between variables, we can say they are correlated. Determining that variables are correlated is very important; however, to draw specific conclusions about the cause and effect relationship will take additional advanced analytical techniques to further test the relationship in order to exclude other contributory factors that could have caused the change. When we have established a positive relationship between two variables, what can we conclude?

There is an association between the variables but not proof that a change in one causes a change in the other. Establishing the basic relationship does not, in itself, establish proof of the effects of the relationship. It takes other advanced analytical techniques to further test the relationship in order to exclude other contributory factors that could have caused the change.

A TV executive wants to determine how many hours of television a person watches each week. A restaurant owner wants to know what customers think of a new item on the menu. An automotive engineer wants to know how stylish a new sports car is to potential buyers. A physician wants to know if a new blood pressure medicine results in lowering his patients' blood pressure. A city manager wants to know if fluoride in the drinking water is helping to prevent cavities. A hotel manager wants to know if a recent remodel improved a hotel's ambiance.

This is an easily quantifiable measure by simply asking the respondents a closed-ended question based on various ranges of television watching as numbered in hours. The way this question is worded suggests that the manager is asking an opinion about a feeling customers have; this suggests using a qualitative-type answer to solicit customers' "feelings" about the new menu item. "Stylishness" is not something that is objectively measurable; hence, a qualitative, non-numeric response is expected like "very stylish" or "not very stylish." The is a very straight-forward correlation-type question. A researcher would measure the amount of medicine and the resultant change in blood pressure to establish a correlation. This can be done in several quantitative ways. One is to compare the numeric data from cities that do and do not have fluoride in their drinking water. Another way is to measure the city's average number of cavities both before and after adding fluoride to the drinking water. "Ambiance" is not a characteristic that lends itself to metrics. Hence, a qualitative, non-numeric response is appropriate.

A TV reporter was covering a three-car pile-up at an intersection of a city. The reporter asked some passersby what they thought of the accident, and the two passersby said that this was the worst accident ever at that intersection. Mary ate at her favorite pizzeria last night. At work the next day, a coworker asked how her meal was. Mary said that the pizza is the best in the city. An automotive engineer wrote an article in an automotive magazine about the results of a comparison test he just completed. He indicated that the new model Cheetah sports car had the best braking of all the cars he tested. A geologist working for a mining company came back to the company headquarters with the results of his field tests. He said that site 47 had the best chance of yielding iron ore in the whole county. A business owner had a great idea for a new product, an electronic Gizmo. He said that because he lived in Chicago, a very large city, he was sure he could sell at least 100,000 new Gizmos. Several college students were talking in the commons and one mentioned that her cell phone battery has a short life. Another student retorted that his cell phone had the worst performance of all.

This is subjective for many reasons. The passersby may not have even lived in that neighborhood, or they may have just moved there. Also, their interpretation of "the worst" may be very different from someone else's definition. This is subjective. The word "best" in this context is nebulous. For instance, Mary may like pizzas that are very heavy in garlic, whereas the average pizza customer only likes a modest amount of garlic. Also, there is no indication that Mary has eaten at every pizzeria in the city. This is objective. Notice how the engineer was only writing about those cars that he actually tested, so he limited the scope. Also, braking is a performance characteristic that is clearly measurable. Because the geologist's work is based on results of field tests, and not a subjective opinion, this is objective in nature. This is subjective. There is no indication that the business owner did any type of market research in order to ascertain potential market demand. Without this type of data, the owner is only guessing and is as likely to be wrong as he is right, not a wise business decision to make. This is subjective. First of all, there is no mention about the use of the cell phones and this could make a huge difference in the battery performance. For example, a student who streams videos all day long will wear out the battery charge much sooner than a student who does not stream any videos.

Correlations between dependent and independent variables can be graphically displayed in a scatter diagram using spreadsheet software. Trend lines show as either an upward movement or a downward movement. If the line shows upward movement, from the bottom left to the upper right of the chart,there is a positive linear relationship. If there is a downward movement of the trend line, from upper left to lower right, then there is a negative linear relationship. Which of the following graphs best demonstrates a trend line with a positive correlation?

This upward movement to the right side of a chart represents the basic positive relationship between an independent variable and a dependent variable. If there is a downward movement of the trend line from left to right, then this represents a negative relationship.

Why do decision-makers use quantitative analysis?

To identify and quantify alternatives and consequences Quantitative analysis helps define the universe of alternatives using metrics instead of guesses.

Calculate math expressions needed to perform quantitative analysis.

Today, when we make mathematical calculations, we primarily use computer software or a calculator. However, there is occasionally a need to perform a calculation manually, without the assistance of these tools. Basic calculations, like adding or subtracting two numbers, are a good example. If both numbers are positive, the numbers can simply be added together. If both numbers are negative, the two numbers are added together and the negative sign remains in front of the sum. Another calculation involves adding a positive and negative number together. The correct method is to subtract the smaller number from the larger number, keeping the sign of the larger number in front of the sum.

Which of the following are some primary influencing factors involving subjective reasoning on a topic?

Upbringing and past employment Subjective reasoning is heavily influenced by personal experiences, such as upbringing and past employment.

Appropriate and useful quantitative analysis ensures the use of Hint, displayed below-Select-budget and financialsamples and sampling techniquesvalid and measurableobjective and subjective-Select- data to understand a business situation.

Valid and Measurable The hallmark of quantitative analysis is ensuring the use of only valid data to begin with. That is, there needs to be some assurance that the underlying data are a true reflection of reality. Also, the data used are measurable, that is, objective and countable. Subjective data that rely on opinions and tastes require different kinds of analysis. For quantitative analysis to be meaningful, it must rest on the ability to quantify the phenomenon being studied, whether Internet hits, dollars, miles, minutes, or some other discrete information.

Holding inventory can be one of the largest expenses of a firm so quantitative methods are used to minimize these expenses. One of the underlying premises of inventory management is that demand generally ___ over a period of time for most products. This means that the prudent inventory manager will identify both a(n) ___ level of inventory for each item stocked. Inherent in making these calculations is identifying the quantity on-hand at which point another procurement action must be made. This is typically called the ___ When a procurement action is initiated, there is a delay for the administrative time needed to make the purchase and the shipping time; this is known as the ___ . Prudent inventory managers also keep a certain small amount of inventory on-hand of critical items, and this is known as ___ to lessen the chance of stockouts. When it is time to reorder, inventory managers must take care to balance the cost of reordering with the costs to hold the inventory. The best quantity to order considering these two factors is called the ___ quantity.

Varies: The variation of demand typically occurs throughout the supply chain and business cycle. Minimum and Maximum: These are the correct terms for these quantities and represent the low- and high-level stockage objectives. Reorder Point: This is the point at which the next procurement action is taken. Lead Time: This is the amount of time from the date reorder to receipt of materials. Safety Stock: This is the extra stock to handle fluctuations in demand. Economic Order: This reorder quantity balances those two factors out to make the most economical decision.

Stockout

When a business runs out of inventory stock this is called stockout.

Economic Reorder Quantity

When it is time to reorder, inventory managers must take care to balance the cost or reordering with the costs to hold the inventory, the best quantity to order considering these two factors is called the economic order quantity.

In studying cause and effect relationships, what can we say when there is a positive relationship between the variables?

When the independent variable increases, the dependent variable will increase at a proportionate rate. A positive relationship is defined as one in which the dependent variable will increase as the independent variable increases. Conversely, as the independent variable decreases, the dependent variable will decrease proportionately.

What would be the best reason for not wanting to use quantitative analysis?

When the problem being evaluated lacks the presence of objective, valid, reliable, and measurable data to use for the analysis Without quantitative data, quantitative analysis cannot be done. In addition, it is assumed that the quantitative data being used are credible, that is, believable, and that the data represent a true reflection of reality.

Under what circumstances might a researcher want to use closed-ended questions and not open-ended questions?

When the research knows the reasonable field of alternative answers In order to construct meaningful closed-ended questions, the researcher must reasonably know the field of possible answers that may apply. Otherwise, the researcher would in fact be better off using open-ended questions to understand the phenomenon better.

Beta Distribution Method

When there is known to be variation or possible uncertainty in the time estimates of the tasks and subtasks, a project manager should consider using the beta distribution method to help determine the duration of each task. The beta distribution method is a calculation that uses a combination of three duration estimates based on experience or history: the most optimistic duration, a pessimistic duration, and the most likely duration in order to consider variation in the final duration estimate. This method is thought to give a better final estimate for a task's duration. The beta distribution method is equal to the sum of the optimistic time estimate for the task (o), four times the most likely time (m) estimate, and the pessimistic time estimate (p); this result is then divided by six. The formula is: Te=(o+4m+p)/6 Calculate duration estimates for the following situation: The store manager was in the process of remodeling the store interior and needed to know how long the store would be closed during the project. The construction manager believed that it could be done within 8 days if everything went according to plan. There is the possibility it could take up to 16 days if they ran into structural difficulties. The store manager asked if there was any way to decrease the number of days. The construction manager thought that if he focused all resources, the project could be done in as soon as 6 days. Using the beta distribution method, the estimated time to complete this task is (6 + 4(8) + 16)/6 = 9 days.

There are times when subjective reasoning are particularly appropriate for use. What are some indicators of when to use subjective reasoning?

When topic involves personal reflections of events One indicators is when a researcher wants to obtain personal reflections and emotions.

Do quantitative analysis techniques for business and for medical research share many commonalities?

Yes, because they share many of the same techniques and research practices. This is a true statement, many techniques are the same. Almost all types of industries use quantitative analysis to solve problems.

There are two types of statistics:

descriptive and inferential: Descriptive statistics is the analysis of data to describe, interpret, and summarize data in a meaningful way in order to find patterns. Descriptive statistics describes the basic features of the data and presents quantitative data in a manageable form. An example of descriptive statistics is a student's grade point average. The grade point average is a single number that describes an overall performance indicator. It does not provide details on how the student did in every course but provides a summary that can be used to compare this student with other students. Inferential statistics tries to reach conclusions, generalizations, and estimations based on a smaller sample of the population. Sometimes it is not possible to examine every member of a population. With inferential statistics, you are trying to draw conclusions, or make inferences, about populations based on samples of data. An example of inferential statistics would be a statement such as, "43% of people between 24 and 30 years old prefer oranges to apples." This statement relies on inferential statistics because it would be impractical to question everyone between the ages of 24 and 30 about their fruit preferences, so instead, a representative sample of the population would have been surveyed with the goal of making an inference about the entire population of 24 to 30 year olds. Descriptive statistics is only concerned with the observed data, but inferential statistics makes predictions or inferences about a population larger than the observed set of data.

The practice of research always starts off with a research question. This research question ___ identical to the hypothesis. Typically the ___ makes a narrative statement or assertion about some phenomenon while the ___ makes two differing statements. When writing a null hypothesis statement, it is written to state that there ___ a relationship between the variables, while the alternative hypothesis states that there ___ a relationship between the two variables. The job of the researcher is to collect evidence in order to reject the ___ hypothesis statement.

is not: They are related statements but not identical statements. Research Question: There are typically two hypothesis statements written at the same time, one asserting a relationship between the variable and the other asserting no relationship between the variables. Hypothesis: This is correct, the null and the alternative statements. is not: This is the function of the null hypothesis. is: This defines the alternative hypothesis. Null: The hypothesis statement must be stated in this manner.

Typically the research question first makes a narrative statement or assertion about some phenomenon, followed by the hypothesis, which makes two differing statements:

the null hypothesis statement and the alternative hypothesis statement. When writing a null hypothesis statement, it is written to state that there is not a relationship between the variables. The alternative hypothesis will state that there is a relationship between the variables. The job of the quantitative analysis researcher is then to collect evidence in order to reject the null hypothesis statement.