44) An operations manager’s staff has compiled the information below for four manufacturing
alternatives (E, F, G, and H) that vary by production technology and the capacity of the machinery. All
choices enable the same level of total production and have the same lifetime. The four states of nature
represent four levels of consumer acceptance of the firm’s products. Values in the table are net present
value of future profits in millions of dollars. Forecasts indicate that there is a 0.1 probability of acceptance
level 1, 0.2 chance of acceptance level 2, 0.4 chance of acceptance level 3, and 0.3 change of acceptance
level 4.
States of Nature
1
2
3
4
Alternative E
50
50
70
60
Alternative F
30
50
80
130
Alternative G
70
80
70
60
Alternative H
–140
–10
150
220
Using the criterion of expected monetary value, which production alternative should be chosen?
45) A toy manufacturer has three different mechanisms that can be installed in a doll that it sells. The
different mechanisms have three different setup costs (overheads) and variable costs and, therefore, the
profit from the dolls is dependent on the volume of sales. The anticipated payoffs are as follows.
Light Demand
Moderate Demand
Heavy Demand
Probability
0.25
0.45
0.3
Wind-up action
$325,000
$190,000
$170,000
Pneumatic action
$300,000
$420,000
$400,000
Electrical action
-$400,000
$240,000
$800,000
a. What is the EMV of each decision alternative?
b. Which action should be selected?
c. What is the expected value with perfect information?
d. What is the expected value of perfect information?
46) Steve Gentry, the operations manager of Baja Fabricators, wants to purchase a new profiling machine
(it cuts compound angles on the ends of large structural pipes used in the fabrication yard). However,
because the price of crude oil is depressed, the market for such equipment is down. Steve believes that
the market will improve in the near future and that the company should expand its capacity. The table
below displays the three equipment options he is currently considering, and the profit he expects each
one to yield over a two-year period. The consensus forecast at Baja is that there is about a 30% probability
that the market will pick up “soon” (within 3 to 6 months) and a 70% probability that the improvement
will come “later” (in 9 to 12 months, perhaps longer).
Profit from Capacity Investment (in Dollars)
Equipment Option
Manual Machine
NC Machine
CNC Machine
a. Calculate the expected monetary value of each decision alternative.
b. Which equipment option should Steve take?
47) Miles is considering buying a new pickup truck for his lawn service firm. The economy in town seems
to be growing, and he is wondering whether he should opt for a subcompact, compact, or full-size pickup
truck. The smaller truck would have better fuel economy, but would sacrifice capacity and some
durability. A friend at the Bureau of Economic Research told him that there is a 30% chance of lower gas
prices in his area this year, a 20% chance of higher gas prices, and a 50% chance that gas prices will stay
roughly unchanged. Based on this information, Miles has developed a decision table that indicates the
profit amount he would end up with after a year for each combination of truck and gas prices.
States of Nature
Alternatives
Lower gas prices
Higher gas prices
Probability
.3
.2
Subcompact
$16,000
$23,000
Compact
$15,000
$22,000
Full size
$18,000
$6,000
Calculate the expected monetary value for each decision alternative. Which decision yields the highest
EMV?
48) Earl Shell owns his own Sno-Cone business and lives 30 miles from a beach resort. The sale of Sno–
Cones is highly dependent upon his location and upon the weather. At the resort, he will profit $110 per
day in fair weather, $20 per day in foul weather. At home, he will profit $70 in fair weather, $50 in foul
weather. Assume that on any particular day, the weather service suggests a 60% chance of fair weather.
a. Construct Earl’s payoff table.
b. What decision is recommended by the expected monetary value criterion?
c. What is the EVPI?
Profit
Fair weather
Foul weather
Probability = 0.6
Probability = .4
Sell at the resort
Sell at home
49) The campus bookstore sells stadium blankets embroidered with the university crest. The blankets
must be purchased in bundles of one dozen each. Each blanket in the bundle costs $65, and will sell for
$90. Blankets unsold by homecoming will be clearance priced at $20. The bookstore estimates that
demand patterns will follow the table below.
a. Build the decision table.
b. What is the maximum expected monetary value?
c. How many bundles should be purchased?
Demand level
Probability
1 bundle
10 percent
2 bundles
30 percent
3 bundles
50 percent
4 bundles
10 percent
Profit
Probability
Order 1
Order 2
Order 3
Order 4
50) Daily sales of bread by Salvador Monella’s Baking Company follow the historical pattern shown in
the table below. It costs the bakery 50 cents to produce a loaf of bread, which sells for 95 cents. Any bread
unsold at the end of the day is sold to the parish jail for 25 cents per loaf. Construct the decision table of
conditional payoffs. How many loaves should Sal bake each day in order to maximize contribution?
Demand
400
500
600
700
800
Probability
.20
.20
.40
.15
.05
211
217
51) The campus bookstore sells highlighters that it purchases by the case. Cost per case, including
shipping and handling, is $200. Revenue per case is $350. Any cases unsold will be discounted and sold at
$175. The bookstore has estimated that demand will follow the pattern below
Demand level
Probability
10 cases
20 percent
11 cases
20 percent
12 cases
40 percent
13 cases
15 percent
14 cases
5 percent
a. Construct the bookstore’s payoff table.
b. How many cases should the bookstore stock in order to maximize expected profit?
c. How would your answer differ if the clearance price were not $175 per case but $225 per case? (It is not
necessary to re-solve the problem to answer this.)
52) The EMV of a decision with three states of nature is $33,000. If the profit/value under the states of
nature A, B, and C is $10,000, $20,000, and $50,000, respectively, and states B and C have equal
probabilities, determine the likelihood of state of nature A.
53) The EMV of a decision with three states of nature is $50. If the profit/value of A is 1/3 of B and B is 1/3
of C, determine the profit from A if all three states of nature are equally likely to occur.
54) Suppose a manufacturing plant is considering three options for expansion. The first one is to expand
into a new plant (large), the second to add on third-shift to the daily schedule (medium), and the third to
do nothing (small) . There are three possibilities for demand. These are high, medium, and low with each
having an equal likelihood of occurring. Suppose that the profits for the expansion plans are as follows
(respective to high, medium, low demand). The large expansion profits are $100000, $10000, -$10000, the
medium expansion choice $40000, $40000, $5000 and the small expansion choice $15000, $15000, $15000.
Calculate the EMV of each choice. Which of the expansion plans should the manager choose?
Section 5 Decision Trees
1) Decision trees and decision tables can both solve problems requiring a single decision, but decision
tables are the preferred method when a sequence of decisions is involved.
2) In a decision tree, the expected monetary values are computed by working from right to left.
3) When solving decision trees, what phrase represents the act of dropping an alternative from
consideration because it is less favorable than another available option?
A) cut the leaf
B) open the hatch
C) shake the tree
D) punt the ball
E) prune the branch
4) Decision trees:
A) give more accurate solutions than decision tables.
B) give less accurate solutions than decision tables.
C) are especially powerful when a sequence of decisions must be made.
D) are rarely used because one needs specialized software to graph them.
E) are too complex to be used by decision makers.
5) A decision tree is a(n):
A) algebraic representation of alternatives and states of nature.
B) behavioral representation of alternatives and states of nature.
C) matrix representation of alternatives and states of nature.
D) graphical representation of alternatives and states of nature.
E) tabular representation of alternatives and states of nature.
6) All EXCEPT which of the following steps are taken to analyze problems with decision trees?
A) Define the problem.
B) Structure or draw the decision tree.
C) Assign probabilities to the alternatives.
D) Estimate payoffs for each possible alternative/state of nature combination.
E) Solve the problem by computing expected monetary values for each state–of-nature node.
7) A primary advantage of decision trees compared to decision tables is that decision trees:
A) are more accurate.
B) are faster.
C) are smaller.
D) are cheaper.
E) can be used for sequential problems.
8) A problem that involves a sequence of decisions:
A) cannot be analyzed with expected monetary value.
B) can be better analyzed with a decision tree than by a decision table.
C) must be analyzed in the same order that the decisions are made.
D) cannot be analyzed with decision tree software.
E) can only be analyzed using decision making under certainty.
9) A(n) ________ is a graphical means of analyzing decision alternatives and states of nature.
10) A branch of a decision tree that is less favorable than other available options may be ________.
11) What limitation(s) do decision trees overcome compared to decision tables?
12) A toy manufacturer makes stuffed kittens and puppies that have relatively lifelike motions. There are
three different mechanisms which can be installed in these “pets.” These toys will sell for the same price
regardless of the mechanism installed, but each mechanism has its own variable cost and setup cost.
Profit, therefore, is dependent upon the choice of mechanism and upon the level of demand. The
manufacturer has in hand a forecast of demand that suggests a 0.2 probability of light demand, a 0.45
probability of moderate demand, and a probability of 0.35 of heavy demand. Payoffs for each
mechanism-demand combination appear in the table below.
Demand
Wind-up action
Pneumatic action
Electronic action
Light
$250,000
$90,000
-$100,000
Moderate
400,000
440,000
400,000
Heavy
650,000
740,000
780,000
Construct the appropriate decision tree to analyze this problem. Use standard symbols for the tree.
Analyze the tree to select the optimal decision for the manufacturer.
A-31
13) A local business owner is a bit uncertain of the demand forecast, and he is timidly approaching the
capacity decision for a business he is about to open. Here’s how he describes the decisions that confront
him over the next two years.
“First, I have to choose between building a large plant initially and building a small one that has room to
expand. Or I could rent now and decide whether to build next year. That one, too, could be the large
version or the small. If I build small, then after one year, I can review how good business was, and decide
whether to expand. If I build large, there is no further option to enlarge.”
Do not concern yourself with probabilities or payoff values. Simply draw the tree that illustrates the
manager’s decision alternatives and the chance events that go along with them. Use standard symbols for
decision tree construction, and label all parts of your diagram carefully. To simplify, assume that business
in the first year, and in the second, can be only “good” or “bad.”
14) Miles is considering buying a new pickup truck for his lawn service firm. The economy in town seems
to be growing, and he is wondering whether he should opt for a subcompact, compact, or full–size pickup
truck. The smaller truck would have better fuel economy, but would sacrifice capacity and some
durability. A friend at the Bureau of Economic Research told him that there is a 30% chance of lower gas
prices in his area this year, a 20% chance of higher gas prices, and a 50% chance that gas prices will stay
roughly unchanged. Based on this information, Miles has developed a decision table that indicates the
profit amount he would end up with after a year for each combination of truck and gas prices. Develop a
decision tree for this situation and indicate which type of truck he should select.
States of Nature
Alternatives
Lower gas prices
Higher gas prices
Probability
.3
.2
Subcompact
16,000
21,000
Compact
15,000
22,000
Full size
24,000
6,000
15) Earl Shell owns his own Sno-Cone business and lives 30 miles from a beach resort. The sale of Sno-
Cones is highly dependent upon his location and upon the weather. At the resort, he will profit $120 per
day in fair weather, $10 per day in bad weather. At home, he will profit $70 in fair weather, $55 in bad
weather. Assume that on any particular day, the weather service suggests a 40% chance of foul weather.
a. Construct Earl’s decision tree.
b. What decision is recommended by the expected value criterion?
16) Bratt’s Bed and Breakfast, in a small historic New England town, must decide how to subdivide
(remodel) the large old home that will become an inn. There are three alternatives: Option A would
modernize all baths and combine rooms, leaving the inn with four suites, each suitable for two to four
adults. Option B would modernize only the second floor; the results would be six suites, four for two to
four adults, and two for two adults only. Option C (the status quo option) leaves all walls intact. In this
case, there are eight rooms available, but only two are suitable for four adults, and four rooms will not
have private baths. Below are the details of profit and demand patterns that will accompany each option.
Which option has the highest expected value?
Annual profit under various demand
patterns
Capacity
p
Average
p
A (Modernize all)
$90,000
.5
$25,000
.5
B (Modernize 2nd)
$80,000
.4
$70,000
.6
C (Status Quo)
$60,000
.3
$55,000
.7
17) A do-it-yourself homeowner is installing a new toilet. While installing the toilet he must decide on
what kind of connecting pipe he will install to the water supply. There are two available options, one that
has a shut-off valve in case of a leak and a cheaper one without the shut-off valve. Suppose that the shut-
off valve pipe costs an extra ten dollars and that the homeowner must buy one of the two.
a. Draw a decision tree for this scenario, labeling the cost of a leak as X and the chance of a leak as P.
b. If the chance of a leak causing household damage is 1%, at what $ amount of household damage is
the owner neutral on which pipe to buy?
c. If the cost of a leak would be $10,000 what is the maximum % chance to leak at which the homeowner
would prefer to buy the cheaper pipe?
d. If the cost of a leak is $1,000 and the chance to flood .1% which pipe should the homeowner buy?
18) A poker player is considering three different options after his opponent bet 200 before him. If the
player folds, he will lose instantly. If the player calls, he figures he will win half the time. If he raises, he
figures that the opposing player will not re-raise him, but rather will either call or fold. He figures the
opposing player will call only ¼ of the time, folding the other ¾ of the time. If the opposing player calls
his raise, he figures he will never win. The pot size is 1,000 (including the opposing player’s bet).
a. Draw a decision tree for this scenario including the information provided in part b.
b. Suppose that the player is thinking of raising to $400 (he will put in 200 to match the opponent’s bet
and another 200 as a raise, his opponent would then have to put in 200 more to call the raise). Is this the
best option or should he instead call or fold?
c. At what raise size is the player’s EMV of a raise equivalent to simply calling?