LINEAR PROGRAMMING SENSITIVITY and INTERPRETATION

Chapter 8 of Quantitative Methods for Business by Anderson, Sweeney and WilliamsRead sections 8.1, 8.2 8.3, 8.4 (omitting “Simultaneous Changes”), 8.5, and appendix 8.1

Chapter 9Read 9.1, 9.2, 9.3, and appendix 9.1

Chapter 10Read 10.1 and 10.2

First section: Objective Value of optimal solution (Z*)

Second section: Optimal values of the decision variables (X*)

Third section: Status of constraints at optimal solution

Interpreting the Excel Answer Report

Before the next class, you should complete the following homework problems in chapter 8

#5a and #5b, #7a and #7b, #20a and #20b Be sure to solve the Chapter 8 problems using

Excel and answer the questions using your Excel reports

Homework status

What happens when problem data change? Recall that:

X* denote the optimal values of the decision variables and

Z* denote the optimal objective function value

Note: We will consider only “one at a time” changes

Interpreting the Excel Sensitivity Report

Consider the Glass Manufacturer example. Note that:

X* = (2,6) Z* = $36

Suppose unit profit on windows becomes $7 (It was originally $3)

OBJECTIVE FUNCTION CHANGES—”range of optimality” (top half of report)

Observe (on graph) that the feasible region is unchanged when only the objective function changes

So the only question is whether X* is still the best feasible solution

OBJECTIVE FUNCTION CHANGES—”range of optimality” (top half of report)

How far would unit profit on windows have to change before optimal solution would change?

OBJECTIVE FUNCTION CHANGES—”range of optimality” (top half of report)

The Excel Sensitivity Report tells us all these things!• Is the new coefficient within “allowable” range?

If yes, X* is unchanged (obviously Z* changes)

If no, we must re-solve problem

OBJECTIVE FUNCTION CHANGES—”range of optimality” (top half of report)

Allowable range for unit profit on windows is

__________________________

Any unit profit in that range leaves (2,6) as the optimal solution

OBJECTIVE FUNCTION CHANGES—”range of optimality” (top half of report)

Before the next class, you should complete the following homework problems in chapter 8

#8a and #8b, #20c Be sure to solve the Chapter 8 problems using

Excel and answer the questions using your Excel reports

Homework status

What about the “Reduced Cost” column? For decision variables with optimal value =

0, the reduced cost tells:• How much better (in this problem that’s “more

profitable”) the objective coefficient would need to be in order for the variable to take a positive value in the optimal solution. (In this class we’ll ignore the sign of the reduced cost)

OBJECTIVE FUNCTION CHANGES--reduced cost (top half of report)

For decision variables with upper bounds, and optimal value = upper bound, the reduced cost tells • How much worse (less profitable in this

problem) the objective coefficient would need to be in order for the variable to take a lower value at optimality.

OBJECTIVE FUNCTION CHANGES--reduced cost (top half of report)

Observe (on graph) that changing the RHS of a constraint may change the feasible region

RHS CHANGES—”range of feasibility” (bottom half of report)

Consider the Excel Sensitivity Report• Is the change in the “allowable” range?

If yes, the “shadow price” times the amount of the change tell the change in Z* (the optimal objective value)

If no, we must resolve the problem

RHS CHANGES—”range of feasibility” (bottom half of report)

Shadow price for a constraint is the change in Z* for each unit increase in its RHS (i.e. the marginal value of one additional unit of the resource).◦ (the sign does matter here)

RHS CHANGES—”range of feasibility” (bottom half of report)

Before the next class, you should complete the following homework problems in chapter 8

#5d, #6d, #7d, #8c, #20d Be sure to solve the Chapter 8 problems using

Excel and answer the questions using your Excel reports

Homework status

Chapters 9 and 10 – read several examples to get comfortable with the process of formulating.

Here are some useful examples in Chapter 9:

Media Selection (p. 359) Market Research (p. 362) Portfolio Selection (p. 365) Make-or-Buy Decision (p. 372) Blending Problems (p. 388)

Formulation

Formulate, solve using Excel, and answer the questions in these problems◦ Chapter 9:

#1, #3, #8, #15◦ Chapter 10:

#3, #5

Homework status

Another Optimization Example (IP): Capital Budgeting A company has 20 potential projects under

consideration. They wish to decide which of the projects to fund in order to maximize NPV.

They have already compiled a list of projected annual costs for each project

They have also computed the expected NPV of each project.

The data are provided in the worksheet called Capital Budget

Which projects should be selected to maximize NPV?

The Formulation What are the decision variables?

What is the objective?

What are the constraints?

Capital Budgeting Use Solver to solve this problem. Can’t use the Simplex Method since the problem is not linear

◦ You can try the Evolutionary solver May have to “solve” twice*

◦ You can also try the GRG Nonlinear solver. It will find the answer, but it will take a while!

◦ If you’re going to solve many of these integer or other non-linear problems, I suggest you get the Risk Solver Platform Solver by Frontline Systems (or another add-in)

◦ Note: The Sensitivity Report for Integer (or other non-linear problems) won’t be like the ones we looked at for LPs.

*The Evolutionary solver keeps trying until 30 seconds have passed without finding any improvement. Sometimes if you hit “solve” again, starting with the solution it found on the first try, it will find the answer.

Study guide, #9-9 (transportation) Text, #8-20 (funds allocation)* Text, #9-15 (blending) Study guide #9-5 (scheduling) Text, #8-24 (portfolio selection) American Airlines Revenue management (video)

* Assigned as homework

Formulation Examples