Chapter 8 of Quantitative Methods for Business by Anderson, Sweeney and Williams
Read sections 8.1, 8.2 8.3, 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
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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
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Before the next class, you should complete
the following homework problems in chapter
8
 #5a and #5b, #7a and #7b, #20a and #20b
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Be sure to solve the Chapter 8 problems using Excel
and answer the questions using your Excel reports
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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
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Note: We will consider only “one at a time”
changes
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Consider the Glass Manufacturer example. Note
that:
 X* = (2,6)
 Z* = $36
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Suppose unit profit on windows becomes $7 (It
was originally $3)
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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
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How far would unit profit on windows have to
change before optimal solution would
change?
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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
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Allowable range for unit profit on windows is
__________________________
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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
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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)
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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.
RHS CHANGES—”range of
feasibility” (bottom half of report)
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Observe (on graph) that changing the RHS of a
constraint may change the feasible region
RHS CHANGES—”range of
feasibility” (bottom half of report)
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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)
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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)
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Before the next class, you should complete
the following homework problems in chapter
8
 #5d, #6d, #7d, #8c, #20d
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Be sure to solve the Chapter 8 problems using Excel
and answer the questions using your Excel reports
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Chapters 9 and 10 – read several examples to
get comfortable with the process of
formulating.
Here are some useful examples in Chapter 9:
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Media Selection (p. 359)
Market Research (p. 362)
Portfolio Selection (p. 365)
Make-or-Buy Decision (p. 372)
Blending Problems (p. 388)
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Formulate, solve using Excel, and answer the
questions in these problems
◦ Chapter 9:
#1, #3, #8, #15
◦ Chapter 10:
#3, #5
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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?
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What are the decision variables?
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What is the objective?
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What are the constraints?
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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.
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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