BUS 104 - Linear Programming.
single-criterion decision problems.
Problems in which the objective is to find the best solution with respect to one criterion are referred to as
The decision-making process
Problems in which the objective is to find the best solution with respect to one criterion are referred to as single-criterion decision problems. Problems that involve more than one criterion are referred to as multicriteria decision problems.
multicriteria decision problems.
Problems that involve more than one criterion are referred to as
Operations Management Applications
Production Scheduling Problem Product Mix Problem Blending Problem
PERT
Program Evaluation and Review Technique
A resource cost is a relevant cost if the amount paid for it is dependent upon the amount of the resource used by the decision variables.
Relevant costs are reflected in the objective function coefficients.
Mathematical models
Represent real-world problems through a system of mathematical formulas and expressions based on key assumptions, estimates, or statistical analyses
Standard form is attained by adding__________ to "less than or equal to" constraints, and by subtracting__________ from "greater than or equal to" constraints.
Slack Variables, Surplus Variables
Decision analysis
can be used to determine optimal strategies in situations involving several decision alternatives and an uncertain pattern of future events.
All LP problems have __________
constraints that limit the degree to which the objective can be pursued.
Decision Variables
controllable inputs; decision alternatives specified by the decision maker, such as the number of units of a product to produce.
Managers should focus on those objective coefficients that have a narrow range of optimality and coefficients near the
endpoints of the range.
Uncontrollable Inputs
environmental factors that are not under the control of the decision maker
The corners or vertices of the feasible region are referred to as the
extreme points
An optimal solution is a
feasible solution that results in the largest possible objective function value when maximizing (or smallest when minimizing).
General guidelines
for LP model formulation are illustrated on the slides that follow.
Linear functions are
functions in which each variable appears in a separate term raised to the first power and is multiplied by a constant (which could be 0).
Project scheduling: PERT and CPM
help managers responsible for planning, scheduling, and controlling projects that consist of numerous separate jobs or tasks performed by a variety of departments, individuals, and so forth
Waiting line (or queuing) models
help managers understand and make better decisions concerning the operation of systems involving waiting lines.
Deterministic Model
if all uncontrollable inputs to the model are known and cannot vary
Stochastic (or Probabilistic) Model
if any uncontrollable inputs are uncertain and subject to variation
Analytic hierarchy process
is a multi-criteria decision-making technique that permits the inclusion of subjective factors in arriving at a recommended decision.
Linear programming
is a problem-solving approach developed for situations involving maximizing or minimizing a linear function subject to linear constraints that limit the degree to which the objective can be pursued.
Goal programming
is a technique for solving multi-criteria decision problems, usually within the framework of linear programming.
Simulation
is a technique used to model the operation of a system. This technique employs a computer program to model the operation and perform simulation computations.
Integer linear programming
is an approach used for problems that can be set up as linear programs with the additional requirement that some or all of the decision recommendations be integer values.
Graphically, a shadow price (also called dual Price)
is determined by adding +1 to the right hand side value in question and then resolving for the optimal solution in terms of the same two binding constraints
A model with 50 decision variables and 25 constraints could have
over 1300 data elements!
Analog models
physical in form, but do not physically resemble the object being modeled
Iconic models
physical replicas (scalar representations) of real objects
We will enter the__________ in the top portion of the spreadsheet.
problem data
Generally, experimenting with models (compared to experimenting with the real situation):
requires less time is less expensive involves less risk
A__________ such as Microsoft Excel can be used to perform a quantitative analysis of Ponderosa Development Corporation.
spreadsheet software package
A linear program in which all the variables are non-negative and all the constraints are equalities is said to be in
standard form.
the shadow price for a nonbinding constraint is 0, and a negative shadow price indicates
that the objective function will not improve if the RHS is increased.
The more closely the model represents the real situation,
the accurate the conclusions and predictions will be.
Information about the constraints:
the amount of slack or surplus the dual prices right-hand side ranges (ranges of feasibility)
The reduced cost for a decision variable whose value is 0 in the optimal solution is:
the amount the variable's objective function coefficient would have to improve (increase for maximization problems, decrease for minimization problems) before this variable could assume a positive value.
Slack and surplus variables represent
the difference between the left and right sides of the constraints.
Sensitivity analysis (or post-optimality analysis) is used to determine how the optimal solution is affected by changes, within specified ranges, in:
the objective function coefficients the right-hand side (RHS) values
The range of feasibility for a change in the right hand side value is the range of values for this coefficient in which
the original shadow price remains constant.
Problem formulation or modeling is
the process of translating a verbal statement of a problem into a mathematical statement.
The range of optimality for each coefficient provides
the range of values over which the current solution will remain optimal.
Information about the decision variables:
their optimal values their reduced costs
A graphical solution method can be used to solve a linear program with
two variables.
Sensitivity analysis allows a manager to ask certain
what-if questions about the problem.
due to cost and ease of solution considerations.
Frequently a less complicated (and perhaps less precise) model is more appropriate than a more complex and accurate one
Three forms of models are:
Iconic models, Analog models, Mathematical models
infeasible
If the alternative does not satisfy all of the model constraints, it is rejected as being__________, regardless of the objective function value.
feasible
If the alternative satisfies all of the model constraints, it is__________ and a candidate for the "best" solution.
Stochastic models are often more difficult to analyze.
In our simple production example, if the number of hours of production time per unit could vary from 3 to 6 hours depending on the quality of the raw material, the model would be stochastic.
Alternative Optimal Solutions
In the graphical method, if the objective function line is parallel to a boundary constraint in the direction of optimization, there are alternate optimal solutions, with all points on this line segment being optimal.
Software packages such as Microsoft Excel and Lingo provide the following LP formulation:
Information about the objective function: Information about the decision variables: Information about the constraints:
Marketing Applications
Marketing Research Problem
Quantitative Analysis Process
Model Development Data Preparation Model Solution Report Generation
Infeasibility
No solution to the LP problem satisfies all the constraints, including the non-negativity conditions. Graphically, this means a feasible region does not exist.
until solutions are generated.
Often, goodness/accuracy of a model cannot be assessed
When looking for the optimal solution, you do not have to evaluate all feasible solution points. You have to consider
only the extreme points of the feasible region.
7 Steps of Problem Solving
(First 5 steps are the process of decision making)1. Define the problem.2. Determine the set of alternative solutions.3. Determine the criteria for evaluating alternatives.4. Evaluate the alternatives.5. Choose an alternative (make a decision).6. Implement the selected alternative.7. Evaluate the results.
Slack and surplus variables have objective function coefficients equal to
0
Infeasibility Causes include:
A formulation error has been made. Management's expectations are too high. Too many restrictions have been placed on the problem (i.e. the problem is over-constrained).
A variety of software packages are available for solving mathematical models.
CPLEX AMPL LINDO MOSEK Microsoft Excel SOLVER LINGO
Changes in Constraint Coefficients
Classical sensitivity analysis provides no information about changes resulting from a change in the coefficient of a variable in a constraint. We must change the coefficient and rerun the model to learn the impact the change will have on the solution.
If inaccuracies or potential shortcomings inherent in the model are identified, take corrective action such as:
Collection of more-accurate input data Modification of the model
CPM
Critical Path Method
due to the time required and the possibility of data collection errors.
Data preparation is not a trivial step
QuestionWhat is the breakeven point for monthly sales of the houses? [Spreadsheet]
Spreadsheet Solution: One way to determine the break-even point using a spreadsheet is to use the Goal Seek tool. Microsoft Excel 's Goal Seek tool allows the user to determine the value for an input cell that will cause the output cell to equal some specified value. In our case, the goal is to set Total Profit to zero by seeking an appropriate value for Sales Volume.
A resource cost is a sunk cost if it must be paid regardless of the amount of the resource actually used by the decision variables.
Sunk resource costs are not reflected in the objective function coefficients.
optimal solution.
The "best" output is the
Range of Optimality and 100% Rule
The 100% rule states that simultaneous changes in objective function coefficients will not change the optimal solution as long as the sum of the percentages of the change divided by the corresponding maximum allowable change in the range of optimality for each coefficient does not exceed 100%.
maximization or minimization
The __________ or__________ of some quantity is the objective in all linear programming problems.
The body of knowledge involving quantitative approaches to decision making is referred to as
The body of knowledge involving quantitative approaches to decision making is referred to as
Potential Reasons for a Quantitative Analysis Approach to Decision Making
The problem is complex The problem is very important The problem is new The problem is repetitive
A managerial report, based on the results of the model, should be prepared. The report should include:
The report should be easily understood by the decision maker. the recommended decision other pertinent information about the results (for example, how sensitive the model solution is to the assumptions and data used in the model)
Resource Cost is Sunk
The shadow price is the maximum amount you should be willing to pay for one additional unit of the resource.
Resource Cost is Relevant
The shadow price is the maximum premium over the normal cost that you should be willing to pay for one unit of the resource.
Unbounded
The solution to a maximization LP problem is unbounded if the value of the solution may be made indefinitely large without violating any of the constraints. For real problems, this is the result of improper formulation. (Quite likely, a constraint has been inadvertently omitted.)
Linear programming has nothing to do with computer programming.
The use of the word "programming" here means "choosing a course of action."
Guidelines for Model Formulation
Understand the problem thoroughly. Describe the objective. Describe each constraint. Define the decision variables. Write the objective in terms of the decision variables. Write the constraints in terms of the decision variables.
Consider a simple production problem. Suppose x denotes the number of units produced and sold each week, and the firm's objective is to maximize total weekly profit.
With a profit of $10 per unit, the objective function is 10x.
Objective Function
a mathematical expression that describes the problem's objective, such as maximizing profit or minimizing cost
Graphically, the range of feasibility is determined by finding the values of
a right hand side coefficient such that the same two lines that determined the original optimal solution continue to determine the optimal solution for the problem.
Constraints
a set of restrictions or limitations, such as production capacities
Qualitative Analysis
based largely on the manager's judgment and experience includes the manager's intuitive "feel" for the problem is more of an art than a science
A feasible solution satisfies
all the problem's constraints.
An optimal solution to an LP problem can be found at
an extreme point of the feasible region.
Quantitative Analysis
analyst will concentrate on the quantitative facts or data associated with the problem analyst will develop mathematical expressions that describe the objectives, constraints, and other relationships that exist in the problem analyst will use one or more quantitative methods to make a recommendation
Models
are representations of real objects or situations
Network models
are specialized solution procedures for problems in transportation system design, information system design, project scheduling,
Forecasting methods
are techniques that can be used to predict future aspects of a business operation.
Inventory models
are used by managers faced with the dual problems of maintaining sufficient inventories to meet demand for goods and, at the same time, incurring the lowest possible inventory holding costs.
Markov-process models
are useful in studying the evolution of certain systems over repeated trials (such as describing the probability that a machine, functioning in one period, will function or break down in another period).
The feasible region for a two-variable LP problem can be nonexistent, a single point, a line, a polygon, or an unbounded area. Any linear program falls in one of four categories:
is infeasible has a unique optimal solution has alternative (multiple) optimal solutions has an objective function that can be increased without bound A feasible region may be unbounded and yet there may be optimal solutions. This is common in minimization problems and is possible in maximization problems.
Information about the objective function:
its optimal value coefficient ranges (ranges of optimality)
Linear constraints are
linear functions that are restricted to be "less than or equal to" (≤), "equal to" (=), or "greater than or equal to" (≥) a constant.
Linear programming involves choosing a course of action when the mathematical model of the problem contains only__________
linear functions.
If both the objective function and the constraints are linear, the problem is referred to as a
linear programming problem.
The bottom of the spreadsheet will be used for
model development.
Cost/benefit considerations
must be made in selecting an appropriate mathematical model.
Decision Analysis and Management Science had its early roots in World War II and is flourishing in business and industry due, in part, to:
numerous methodological developments (e.g. simplex method for solving linear programming problems) a virtual explosion in computing power