MGMT 101
Dual values are ___ for non-binding constraints because they have slacks
0
Infeasibility means that the number of solutions to the linear programming model that satisfies all constraints is
0
Note that slack variables have coefficients of
0 in the objective function
Let A, B, and C be the amounts invested in companies A, B, and C. If no more than 50% of the total investment can be in company B, then
0.5B -0.5A - 0.5C <= 0
What constant for alpha would mimic the naive method?
1
What do you set the first and last node of a shortest route model to?
1
Activities G, P, and R are the immediate predecessors for activity W. If the earliest finish times for the three are 12, 15, and 10, then the earliest start time for W is
15
The number of units expected to be sold is uniformly distributed between 300 and 500. If r is a random number between 0 and 1, then the proper expression for sales is
300 + 200(r)
If a basic transportation problem (with no capacity constraints for arcs) has four origins and five destinations, the LP formulation of the problem will have
9 constraints
The assignment problem constraint x31 + x32 + x33 + x34 ≤ 2 means A)agent 3 can be assigned to 2 tasks. B)agent 2 can be assigned to 3 tasks. C)a mixture of agents 1, 2, 3, and 4 will be assigned to tasks. D)there is no feasible solution.
A
How do you solve a decision tree?
Any decision tree can be solved by rolling it back Start at the end of the tree and proceed towards the beginning Every time you encounter a chance node (circle), replace it with its expected value Every time you encounter a decision node (rectangle), choose the best alternative
Exponential smoothing
Applies differential weights to different time periods Pretty much does what weighted moving averages does except you only need to apply 1 weight.
Assignment problems
Assign one job to one machine, one person to one job, one salesperson to one territory. The key feature is that is involves one on each side of the assignment
What does a maximin approach assume?
Assume the worst and make the best of it (worst of d1 and d2, then the best between them)
Rounding the solution of an LP Relaxation to the nearest integer values provides A)a feasible but not necessarily optimal integer solution. B)an integer solution that is optimal. C)an integer solution that might be neither feasible nor optimal. D)an infeasible solution.
C
Let x1 and x2 be binary (0-1) variables whose values indicate whether projects 1 and 2 are not done or are done. Which answer below indicates that project 2 can be done only if project 1 is done? A)x1 + x2 = 1 B)x1 + x2 = 2 C)x1 − x2 ≤ 0 D)x1 − x2 ≥ 0
D
The options from which a decision maker chooses a course of action are a) called the decision alternatives. b) under the control of the decision maker c) not the same as the states of nature. d) All of the alternatives are true.
D
Multiple choice (choose 1 and only 1)
D1 + D2 + D3 = 1
K out of N constraint (want to choose 2; at most want to choose 2)
D1 + D2 + D3 = OR ≤ 2
Mutually exclusivity (choose one or the other but not more than 1, but you don't have to choose 1 at all)
D1 + D2 + D3 ≤ 1
Corequisite constraint (if you do D1 you have to do D2, and if you don't do D1 you can't do D2)
D1 = D2
Conditional constraint (can only do D1 if you do D2)
D1 ≤ D2
What happens when slack variables are added to each constraint and to the objective function?
Doing so transforms a linear program into its standard form
(EVPI/EVSI) uses Bayes Theorem
EVSI
Minimax regret
First translate consequences into opportunity losses (regrets) by comparing each consequence to the best for each state of nature Identify the highest regret for each alternative and ignore the rest/choose the alternative with the lowest of the highest regrets
The relationship d = 5000 − 25p describes what happens to demand (d) as price (p) varies. Here, price can vary between $10 and $50. How many units can be sold at the $10 price? How many can be sold at the $50 price?
For p = 10, d = 4750 For p = 50, d = 3750
Consider prices of $20, $30, and $40. Which of these three price alternatives will maximize total revenue? What are the values for demand and revenue at this price?
For p = 20, d = 4500, TR = $90,000 For p = 30, d = 4250, TR = $127,500 For p = 40, d = 4000, TR = $160,000 (maximum total revenue)
Moving averages
Forecast the next time period by averaging the preceding k time periods, and then move that average along the time series
What are the steps to problem solving?
Identify and define problem Determine alternative solutions Determine evaluation criteria Evaluate alternatives Choose best alternative implement the decision Evaluate the results IDDECE
Weighted moving averages
Instead of simply averaging, apply different weights to different time periods. Weights are positive numbers that will add to 1.
The critical path
Is the longest path
There are basically two ways to balance expected return and risk
Maximize expected returns subject to some maximum risk Minimize risk subject to some minimum expected return
What are the key objectives of an linear program of an optimal portfolio?
Maximizing expected rate of return and minimizing investment risk
A snack food manufacturer buys corn for tortilla chips from two cooperatives, one in Iowa and one in Illinois. The price per unit of the Iowa corn is $6.00 and the price per unit of the Illinois corn is $5.50. Develop an objective function that would minimize the total cost.
Min 6x1 + 5.5x2
Model the expression for total revenue.
TR = p(5000 − 25p)
Maximizing expected values
Takes all consequences into consideration at their probability
Transportation problems
The most effective way to move products from a place of origin (supply) to a destination (demand)
How to develop a linear program?
Understand the problem Describe the objective Define the decision variables Describe each constraint Write the objective function in terms of decision variables
How to model seasonal patterns?
Using dummy variables in multiple regression models, treating the seasons as categorical varaibles
What is the formula for exponential smoothing?
What happened * alpha + (1 - alpha)*forecast for that time period
High values give a higher weight to the most recent time periods so are a) less smooth b) more smooth
a
What is management science?
a collection of quantitative methods to support managerial decision making based on the scientific method
The approach to determining the optimal decision strategy involves
a forward (left to right) pass through the decision tree
Uncertainty of a simulation must be described with
a particular probability distribution
What is a way to model the objective of a max flow problem?
add an additional arc from the destination node back to the origin node, which would have no capacity constraint, and maximize flow through this arc
Revenue management methodology was originally developed for
an airline
Applications are usually based on
an iterative process that involves managers and management science
Fixed costs
are independent of the decision variables
Decision criteria
are the ways to evaluate the choices faced by the decision maker
Inputs to a quantitative model
are uncertain for a stochastic model.
A decision tree
arranges decision alternatives and states of nature in their natural chronological order
How are chance events often modeled?
as mutually exclusive and collectively exhaustive outcomes (states of nature)
A negative dual price for a constraint in a minimization problem means
as the right-hand side increases, the objective function value will decrease
Static models
assume each trial is independent
How can we address the problem with forecasting errors?
average absolute errors or average squared errors
How do we adjust for scale effects?
average absolute percentage errors
All of the following are true about time series methods except a) They discover a pattern in historical data and project it into the future. b) They involve the use of expert judgment to develop forecasts. c) They assume that the pattern of the past will continue into the future. d) Their forecasts are based solely on past values of the variable or past forecast errors
b
Forecast errors a) are the difference in successive values of a time series. b) are the differences between actual and forecast values. c) should all be non-negative. d) should be summed to judge the goodness of a forecasting model.
b
Management science and operations research both involve a) operational management skills. b) quantitative approaches to decision making. c) scientific research as opposed to applications. d) qualitative managerial skills.
b
Seasonal patterns a) cannot be predicted. b) are regular repeated patterns. c) are multiyear runs of observations above or below the trend line. d) reflect a shift in the time series over time.
b
In a transshipment problem, shipments
can occur between any two nodes
What are the two scenarios of traditional sensitivity analysis?
changes in objective function coefficients and changes to the right hand side values of constraints
What-if analysis
changes some of the input parameters to study the impact of these changes on the outputs
What is sensitivity analysis?
changes to the objective function or changes to the constraints and how these changes affect the optimal solution
Chance nodes are depicted with
circles
Fixed setup costs
costs you incur only if you decide to produce something and are independent of production quantity
Cyclical impacts are often based on
cyclical changes in the economy like recessions
Times series data
data that have been obtained in regular intervals
Relevant cost
depend on the decision variables
Index funds
designed to mimic the performance of the stock market as a whole
Consequence/payoff nodes are depicted with
diamonds
Waiting time (queueing models) are dynamic/static
dynamic
The word "uniform" in the term "uniform random numbers" means
each number has an equal probability of being drawn
The difference between the transportation and assignment problems is that
each supply and demand value is 1 in the assignment problem
Expected value of perfect information (EVPI)
expresses how much better we would be if we knew what was going to happen (uncertainty is known)
A four-period moving average forecast for period 10 would be found by averaging the values from periods 10, 9, 8, and 7 (T/F)
false
A path through a project network must reach every node (T/F)
false
A transportation problem with 3 sources and 4 destinations will have 7 decision variables (T/F)
false
All quarterly time series contain seasonality (T/F)
false
Alternative optimal solutions occur when there is no feasible solution to the problem. (T/F)
false
An alpha (α) value of 0.2 will cause an exponential smoothing forecast to react more quickly to a sudden drop in demand than will an α equal to 0.4 (T/F)
false
Any change to the objective function coefficient of a variable that is positive in the optimal solution will change the optimal solution (T/F)
false
Computer-generated random numbers are normally distributed (T/F)
false
Cyclical impacts are possible to predict (T/F)
false
Decision alternatives are structured so that several could occur simultaneously (T/F)
false
Decision variables limit the degree to which the objective in a linear programming problem is satisfied. (T/F)
false
Frederick Taylor is credited with forming the first MS/OR interdisciplinary teams in the 1940's. (T/F)
false
In an all-integer linear program all objective function coefficients and right-hand side values must be integer (T/F)
false
Increasing the right-hand side of a nonbinding constraint will not cause a change in the optimal solution (T/F)
false
Inputs to a quantitative model must all be deterministic if the problem is to have a solution. (T/F)
false
Long term changes can only be linear (never nonlinear) (T/F)
false
Maximax, maximin, and minimax regret require probabilities (T/F)
false
Maximizing expected values accounts for risk (T/F)
false
Modern project management depends on either PERT or CPM (T/F)
false
Only binding constraints form the shape (boundaries) of the feasible region. (T/F)
false
Payoff tables can be used for more complicated, sequential decision situations (T/F)
false
Revenue management methodology was originally developed for the banking industry (T/F)
false
Rounding the solution of a regular linear program is the same as integer linear programming (T/F)
false
Sensitivity analysis cannot be used to examine how changes in any value impact the results of the analysis (in particular the preferred alternative) (T/F)
false
Simulation is an optimization technique (T/F)
false
The dual value associated with a constraint is the change in the value of the solution per unit decrease in the right-hand side of the constraint (T/F)
false
The earliest start time for an activity is equal to the smallest of the earliest finish times for all its immediate predecessors (T/F)
false
The field of management science concentrates on the use of qualitative methods to assist in decision making. (T/F)
false
The most successful quantitative analysis will separate the analyst from the managerial team until after the problem is fully structured. (T/F)
false
The parameters of a simulation model are the controllable inputs (T/F)
false
The terms 'stochastic' and 'deterministic' have the same meaning in quantitative analysis. (T/F)
false
The variance in the project completion time is the sum of the variances of all activities in the project (T/F)
false
There is a dual value for every decision variable in a model (T/F)
false
Transshipment problems are less complicated (T/F)
false
Uncontrollable inputs are the decision variables for a model (T/F)
false
Using minutes as the unit of measurement on the left-hand side of a constraint and using hours on the right-hand side is acceptable since both are a measure of time (T/F)
false
You must specify variables as binary in a shortest route problem (T/F)
false
You must specify variables as binary in an assignment problem (T/F)
false
If you increase expected return requirements (%), variance will probably
go up (correlated)
Media selection problems usually determine
how many times to use each media source
What is the risk premium
how much the company is willing to forgo in order to get rid of the risk
LP relaxation
ignore the integer constraints for a moment and treat the program as a regular linear programming situation
Range of feasibility is ___ for non-binding constraints
infinity
Cyclical impacts
irregular deviations from a trend
Slack
is the amount by which the left side of a <= constraint is smaller than the right side
Deterministic
known for sure
A simulation cosists of
many what-if analysis performed with different values for the uncertain input parameters
A model that uses a system of symbols to represent a problem is called
mathematical
For a maximization problem, the conservative approach is often referred to as the
maximin approach
Blending problems arise whenever a manager must decide how to
mix two or more resources to produce one or more products
Maximizing expected values is based on
monetary payoffs and probabilities
Low values give a higher weight to
more distant time period and are more smooth
What are the three smoothing techniques?
moving averages, weighted moving averages, and exponential smoothing
Infeasibility
not every constraint can be simultaneously satisfied. Thus, there is no feasible region and no possible (let alone optimal) solution
Unbounded situations
occurs if the constraints are not restrictive enough to prevent the objective function from becoming infinitely high or infinitely small
Sensitivity analysis information in computer output is based on the assumption of
one coefficient changes
Maximax
optimistic; Identify the best consequence for each alternative and ignore the others/choose the alternative with the best of the best consequence
Maximin
pessimistic/conservative; Identify the worst consequence for each alternative and ignore others/choose the alternative with the best of the worst consequence
American airlines
pioneered the use of linear programming to determine how many tickets should be sold at discount vs. full price
The measure of risk most often associated with the Markowitz portfolio model is the
portfolio variance
What is the problem with simply averaging forecast errors?
positive and negative errors cancel out and it can only give information about systematic bias (over/under predicting)
What is another term for sensitivity analysis?
post-optimality analysis
Arcs in a project network indicate
precedence relationships
A quantity that is difficult to measure with certainty is called a
probalistic input
Dual values will remain true as long as the hours remain within
range of feasibility
Decision nodes are depicted with
rectangles
The amount the objective function coefficient of a decision variable would have to improve before that variable would have a positive value in the solution is the
reduced cost
Dynamic models
refer to simulations that change over time where trials are not independent
Long term changes are modeled using
regression analysis
Sesonal impacts
regular recurring patterns
Should fixed and/or relevant costs be reflected in the objective function?
relevant costs only
How to make the gamble of the decision maker indifferent
repeatedly change p
The overall goal of portfolio models is to create a portfolio that provides the best balance between
risk and return
Utility function: concave
risk aversion
Utility function: linear
risk neutraility
Calculate risk premium
risk premium = EV -certainty equivalent
Utility function: convex
risk taking
What is dual price also known as?
shadow price
Risk profiles
show the actual possible payoffs/consequences along with their corresponding probabilities
When analytical solutions are not available, ___ are used
simulations
Quantitative is for
solving the problem
Whenever all the constraints in a linear program are expressed as equalities, the linear program is said to be written in
standard form
In what state do we analyze dynamic models?
steady states, after some time has elapsed
Qualitative is for
structuring the problem
A cost that is incurred no matter what values the decision variables assume is
sunk cost
Decision variables
tell how much or how many of something to produce, invest, purchase, hire, etc.
What is the market measured by in index funds?
the S&P 500
Dual value
the change of the objective function for a 1-unit increase in the right hand side of a constraints
Supply chain models
the collection of all parties and activities that are involved in producing and distributing a product. The objective is typically to minimize cost
What will happen if we change both simultaneously?
the condition is no longer met and the optimal solution will change
Forecast error
the difference between the actual value (Yt) and the predicted value/forecast (Ft) for a certain time period t et = Yt - Ft
The higher the expected return of an investment
the higher the risk
Critical path is (slack)
the path consisting of all activities with zero slack
What is an expected value
the probability weighted average of all possible consequences; is not the actual payoff, just what we expect the payoff to be
Break-even point
the production volume where profit would be zero (revenues = cost)
Range of feasibility
the range of right hand side values for which the dual value is valid
Reduced cost
the reduced cost of a decision variable is the dual value of the non-negativity constraint (how the objective function would change if we needed one, not zero, units)
The optimal solution will not change (even though the value of the objective function might) as long as
the slope of the objective function does not make a different extreme point optimal
Utility measures
the total worth of a consequence, including attitudes towards risk that our decision maker may have
To solve our decision dilemma we will replace the original payoffs with
their corresponding utilities
What is a major problem with decision rules?
they ignore all but one of the consequences for each alternative. We could average them, but this would assume that each consequence is equally likely to occur, which is not true
What is Bayes Theorem used for?
to update prior probabilities into posterior probabilities; prior probabilities refer to the situation before information is obtained and posterior probabilities refer to the situation after information is obtained
Utility function
translates payoffs into their corresponding utilities
The problem which deals with the distribution of goods from several sources to several destinations is the
transportation probelm
A critical activity can be part of a noncritical path (T/F)
true
A larger value of k means smoother (T/F)
true
A nonlinear optimization problem is any optimization problem in which at least one term in the objective function or a constraint is nonlinear (T/F)
true
A risk neutral decision maker will have a linear utility function (T/F)
true
After all probabilities and payoffs are placed on a decision tree, the decision maker calculates expected values at state of nature nodes and makes selections at decision nodes (T/F)
true
All activities on a critical path have zero slack time (T/F)
true
Any of the numbers that go into a decision analysis are subject to some amount of imprecision (T/F)
true
Chance events are beyond your control (T/F)
true
Convex function types are easier to solve than others (T/F)
true
Decision alternatives are choosen from (T/F)
true
Dual prices cannot be used for integer programming sensitivity analysis because they are designed for linear programs (T/F)
true
EVPI is always greater than or equal to EVSI (T/F)
true
Each simulation run provides only a sample of how the real system will operate (T/F)
true
For any constraint, either its slack/surplus value must be zero or its dual price must be zero (T/F)
true
Forecasts can be based on judgements and opinions (qualitative) or data (quantitative) (T/F)
true
Harry Markowitz won the Nobel prize in economics for his path-breaking work in quantifying this tradeoff to come up with optimal portfolios (T/F)
true
Higher values of R indicate less risk aversion and will flatten the function (T/F)
true
If the range of feasibility indicates that the original amount of a resource, which was 20, can increase by 5, then the amount of the resource can increase to 25 (T/F)
true
If x1 + x2 ≤ 500y1 and y1 is a binary (0-1) variable, then if y1 is 0, x1 and x2 will be 0 (T/F)
true
In an all-integer linear program all decision variables must be integer (T/F)
true
In general, rounding large values of decision variables to the nearest integer value causes fewer problems than rounding small values (T/F)
true
In portfolio models, risk is minimized by diversification (T/F)
true
In the case of functions with multiple local optima, most nonlinear optimization software methods can get stuck and terminate at a local optimum (T/F)
true
Integer LP allows for tremendous flexibility in modeling (especially in situations involving binary variables) (T/F)
true
Integer and binary programs are much more difficult to solve; algorithms much more complicated (T/F)
true
Integer linear programs are harder to solve than linear programs (T/F)
true
Maximizing expected values and maximizing expected utilities require probabilities (T/F)
true
Possible dependencies among uncertainties are difficult to model, we assume independency (T/F)
true
Program Evaluation and Review Technique (PERT) was developed in the 1950's by the Navy (T/F)
true
Ranges of optimality require that changes in objective function coefficients are made one at a time (T/F)
true
Revenue management methodology can be applied in the case of nonperishable assets (T/F)
true
Shortest route problems are a special case of transshipment models with one supply node(origin), one demand node (destination), and several transshipment nodes in between (T/F)
true
Simulation is an excellent technique to use when a situation is too complicated to use standard analytical procedures (T/F)
true
Simultaneous changes are much harder to handle (T/F)
true
Smoothing methods are more appropriate for a stable time series than when significant trend or seasonal patterns are present (T/F)
true
The Critical Path Method (CPM) was developed independently at DuPont and Remington Rand (T/F)
true
The assignment problem is a special case of the transportation problem in which all supply and demand values equal one (T/F)
true
The constraint 5x1 - 2x2 <= 0 passes through the point (x1 = 20, x2 = 50) (T/F)
true
The degree of risk is associated with the probability or magnitude of loss (T/F)
true
The direction of flow in the shortest-route problem is always out of the origin node and into the destination node (T/F)
true
The earliest finish time for the final activity is the project duration (T/F)
true
The exponential smoothing forecast for any period is a weighted average of all the previous actual values for the time series (T/F)
true
The field of management science approaches decision making rationally, with techniques based on the scientific method. (T/F)
true
The field of management science is another name for decision science and for operations research. (T/F)
true
The first step in the decision making process is to identify the problem. (T/F)
true
The length of time an activity can be delayed without affecting the project completion time is the slack (T/F)
true
The shortest-route problem is a special case of the transshipment problem where each supply = 1 and each demand = 1 (T/F)
true
The smoothing techniques can only forecast the very next time period (T/F)
true
The standard form of a linear programming problem will have the same solution as the original problem.( T/F)
true
The utility function for a risk avoider typically shows a diminishing marginal return for money (T/F)
true
Time series data can exhibit seasonal patterns of less than one month in duration (T/F)
true
Time series methods base forecasts only on past values of the variables (T/F)
true
To assign utilities, consider the best and worst payoffs in the entire decision situation (T/F)
true
Trials of a simulation show what would happen when values of the probabilistic input change (T/F)
true
Uncertainties and risks can result in good decisions leading to bad outcomes (T/F)
true
Utility measures account for risk (T/F)
true
When a route in a transportation problem is unacceptable, the corresponding variable can be removed from the LP formulation (T/F)
true
the value of the optimal solution of the LP relaxation is an upper/lower bound (depending on if you want to maximize or minimize) for the value of the the optimal solution of the mixed linear program (T/F)
true
Max flow problem
try to maximize the amount of flow that can move through a network
Stochastic/probabilstic
unknown
What bound does EVPI provide
upprt
Naive forecasting
uses the most recent values to predict the next time period
Alternative optimal solutions
when more than one combination of decision variables leads to the same, optimal value of the objective function
What is dynamic pricing?
when prices change depending on supply and demand
When are smoothing techniques popular?
when there are no significant trends or seasonal patterns
Where is the optimal solution of a linear program?
where the objective function is just about to leave the feasible region at an extreme point
A constraint with a positive slack value
will have a dual value of zero
The manufacturer needs at least 12000 units of corn. The Iowa cooperative can supply up to 8000 units, and the Illinois cooperative can supply at least 6000 units. Develop constraints for these conditions.
x1 + x 2 ≥ 12000 x1 ≤ 8000 x2 ≥ 6000