Chapter 8 of Quantitative Methods for Business by Anderson, Sweeney and Williams Read sections 8.1, , 8.4 (omitting “Simultaneous Changes”), 8.5, and appendix 8.1 Chapter 9 Read 9.1, 9.2, 9.3, and appendix 9.1 Chapter 10 Read 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

 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

 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

 Consider the Glass Manufacturer example. Note that:  X* = (2,6)  Z* = $36  Suppose unit profit on windows becomes $7 (It was originally $3)

 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

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

 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

 Allowable range for unit profit on windows is __________________________  Any unit profit in that range leaves (2,6) as the optimal solution

 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

 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)

 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.

 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

 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)

 Formulate, solve using Excel, and answer the questions in these problems ◦ Chapter 9: #1, #3, #8, #15 ◦ Chapter 10: #3, #5

 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?

 What are the decision variables?  What is the objective?  What are the constraints?

 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