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5S: Chapter Decision Theory
Problems
1. A small building contractor has recently experienced two successive years in which work
opportunities exceeded the firm's capacity. The contractor must now make a decision on
capacity for next year. Estimated profits under each of the two possible states of nature are
as shown in the table below. Which alternative should be selected if the decision criterion is
a. Maximax?
b. Maximin?
c. Laplace?
d. Minimax regret?
2. Refer to Problem 1. Suppose after a certain amount of discussion, the contractor is able to
subjectively assess the probabilities of low and high demand: P(low) 5 .3 and P(high) 5 .7.
a. Determine the expected profit of each alternative. Which alternative is best? Why?
b. Analyze the problem using a decision tree. Show the expected profit of each alternative
on the tree.
c. Compute the expected value of perfect information. How could the contractor use this
knowledge?
3. Refer to Problems 1 and 2. Construct a graph that will enable you to perform sensitivity
analysis on the problem. Over what range of P(high) would the alternative of doing nothing
be best? Expand? Subcontract?
4. A firm that plans to expand its product line must decide whether to build a small or a large
facility to produce the new products. If it builds a small facility and demand is low, the net
present value after deducting for building costs will be $400,000. If demand is high, the firm
can either maintain the small facility or expand it. Expansion would have a net present value
of $450,000, and maintaining the small facility would have a net present value of $50,000.
p. 228
If a large facility is built and demand is high, the estimated net present value is $800,000. If
demand turns out to be low, the net present value will be $10,000.
The probability that demand will be high is estimated to be .60, and the probability of low
demand is estimated to be .40.
a. Analyze using a tree diagram.
b. Compute the EVPI. How could this information be used?
c. Determine the range over which each alternative would be best in terms of the value of
P (demand low).
5. Determine the course of action that has the highest expected payoff for this decision tree.
6. The lease of Theme Park, Inc., is about to expire. Management must decide whether to
renew the lease for another 10 years or to relocate near the site of a proposed motel. The
town planning board is currently debating the merits of granting approval to the motel. A
consultant has estimated the net present value of Theme Park's two alternatives under each
state of nature as shown on the following page.
p. 229
What course of action would you recommend using?
a. Maximax.
b. Maximin.
c. Laplace.
d. Minimax regret.
7. Refer to Problem 6. Suppose that the management of Theme Park, Inc., has decided that
there is a .35 probability that the motel's application will be approved.
a. If management uses maximum expected monetary value as the decision criterion, which
alternative should it choose?
b. Represent this problem in the form of a decision tree.
c. If management has been offered the option of a temporary lease while the town
planning board considers the motel's application, would you advise management to sign
the lease? The lease will cost $24,000.
8. Construct a graph that can be used for sensitivity analysis for the preceding problem.
a. How sensitive is the solution to the problem in terms of the probability estimate of .35?
b. Suppose that, after consulting with a member of the town planning board, management
decides that an estimate of approval is approximately .45. How sensitive is the solution
to this revised estimate? Explain.
c. Suppose the management is confident of all the estimated payoffs except for $4 million.
If the probability of approval is .35, for what range of payoff for renew/reject will the
alternative selected using the maximum expected value remain the same?
9. A firm must decide whether to construct a small, medium, or large stamping plant. A
consultant's report indicates a .20 probability that demand will be low and an .80 probability
that demand will be high.
If the firm builds a small facility and demand turns out to be low, the net present value will
be $42 million. If demand turns out to be high, the firm can either subcontract and realize the
net present value of $42 million or expand greatly for a net present value of $48 million.
The firm could build a medium-size facility as a hedge: If demand turns out to be low, its
net present value is estimated at $22 million; if demand turns out to be high, the firm could
do nothing and realize a net present value of $46 million, or it could expand and realize a net
present value of $50 million.
If the firm builds a large facility and demand is low, the net present value will be $20
million, whereas high demand will result in a net present value of $72 million.
a. Analyze this problem using a decision tree.
b. What is the maximin alternative?
c. Compute the EVPI and interpret it.
d. Perform sensitivity analysis on P(high).
10. A manager must decide how many machines of a certain type to buy. The machines will be
used to manufacture a new gear which for there is increased demand. The manager has
narrowed the decision to two alternatives: buy one machine or buy two. If only one machine
is purchased and demand is more than it can handle, a second machine can be purchased
at a later time. However, the cost per machine would be lower if the two machines were
purchased at the same time.
The estimated probability of low demand is .30, and the estimated probability of high
demand is .70.
The net present value associated with the purchase of two machines initially is $75,000 if
demand is low and $130,000 if demand is high.
The net present value for one machine and low demand is $90,000. If demand is high,
there are three options. One option is to do nothing, which would have a net present value of
$90,000. A second option is to subcontract; that would have a net present value of
$110,000. The third option is to purchase a second machine. This option would have a net
present value of $100,000.
How many machines should the manager purchase initially? Use a decision tree to
analyze this problem.
p. Determine the course of action that has the highest EMV for the accompanying tree
230 diagram.
11.
12. A logistics provider plans to have a new warehouse built to handle increasing demands for
its services. Although the company is unsure of how much demand there will be, it must
decide now on the size (large or small) of the warehouse. Preliminary estimates are that if a
small warehouse is built and demand is low, the monthly income will be $700,000. If
demand is high, it will have to either expand the facility or lease additional space. Leasing
will result in a monthly income of $100,000 while expanding will result in a monthly income of
$500,000.
If a large warehouse is built and demand is low, monthly income will only be $40,000,
while if demand is high, monthly income will be $2 million.
a. Construct a tree diagram for this decision.
b. Using your tree diagram, identify the choice that would be made using each of the four
approaches for decision making under uncertainty.
13. The director of social services of a county has learned that the state has mandated
additional information requirements. This will place an additional burden on the agency. The
director has identified three acceptable alternatives to handle the increased workload. One
alternative is to reassign present staff members, the second is to hire and train two new
workers, and the third is to redesign current practice so that workers can readily collect the
information with little additional effort. An unknown factor is the caseload for the coming year
when the new data will be collected on a trial basis. The estimated costs for various options
and caseloads are shown in the following table:
Assuming that past experience has shown the probabilities of various caseloads to be
unreliable, what decision would be appropriate using each of the following criteria?
a. Maximin.
b. Maximax.
c. Minimax regret.
d. Laplace.
14. After contemplating the caseload question (see previous problem), the director of social
services has decided that reasonable caseload probabilities are .10 for moderate, .30 for
high, and .60 for very high.
a. p. 231
Which alternative will yield the minimum expected cost?
b. Construct a decision tree for this problem. Indicate the expected costs for the three
decision branches.
c. Determine the expected value of perfect information using an opportunity loss table.
15. Suppose the director of social services has the option of hiring an additional staff member if
one staff member is hired initially and the caseload turns out to be high or very high. Under
that plan, the first entry in row 2 of the cost table (see Problem 13) will be 40 instead of 60,
the second entry will be 75, and the last entry will be 80. Assume the caseload probabilities
are as noted in Problem 14. Construct a decision tree that shows the sequential nature of
this decision, and determine which alternative will minimize expected cost.
16. A manager has compiled estimated profits for various capacity alternatives but is reluctant to
assign probabilities to the states of nature. The payoff table is as follows:
a. Plot the expected-value lines on a graph.
b. Is there any alternative that would never be appropriate in terms of maximizing expected
profit? Explain on the basis of your graph.
c. For what range of P(2) would alternative A be the best choice if the goal is to maximize
expected profit?
d. For what range of P(1) would alternative A be the best choice if the goal is to maximize
expected profit?
17. Repeat all parts of Problem 16, assuming the values in the payoff table are estimated costs
and the goal is to minimize expected costs.
18. The research staff of a marketing agency has assembled the following payoff table of
estimated profits:
Relative to the probability of not receiving the contract, determine the range of probability for
which each of the proposals would maximize expected profit.
19. Given this payoff table:
a. Determine the range of P(1) for which each alternative would be best, treating the
payoffs as profits.
b. Answer part a treating the payoffs as costs.

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