Decision theory Problems

INDUSTRIAL ENGINEERING DEPARTMENT
Introduction to Operations Research III
Decision Theory

1. The MBA Movie Studio is trying to decide to distribute its new movie “Claws”. The movie
has the potential of being a great financial success (a “smash”), but the executives are not
sure because the subject is controversial. And they have seen some films heralded as
“smashes” become “flops” with disastrous financial consequences.

The decision facing MBA is whether or not to put out the movie “Claws” on a limited first
run basis. This means that the movie will show only in a few select theaters during the first
six months. After six months it will be released generally. If the movie turns out to be a
success, this is clearly the best approach because the studio makes considerable profit from
these theaters.

The other alternative is to release the film for wide distribution immediately. The profits for
the two alternatives are given in the table below, classified in terms of whether the film is a
“smash”, or “medium” success, or a “flop”.
Profits from Film “Claws”
Level of Success Probability Limited Initial Widespread Release
Release (in millions) (in millions)
Smash 0.3 22 12
Medium 0.4 8 8
Flop 0.3 -10 -2

There is considerable discussion in MBA about the potential of Claws. Management has
finally agreed on the probabilities shown in the table. But which decision to make is still not
clear. One possibility is to have a few sneak previews of the movie and get the audience’s
opinions. The cost of such a process would be about Php 50,000. And several executives in
the company feel it would be money wasted, since sneak preview audience tends to rate a
movie as good or outstanding even when it later tuns out to be flop. To support this, the
following table was produced, describing the company’s past experience with sneak preview
audience reactions.
Sneak Preview Audience Reaction
Audience Smash Medium Flop Total
Rating
Outstanding 9 12 3 24
Good 1 6 5 12
Poor 0 2 2 4
Total 10 20 10 40
a. Draw the decision tree for this problem.
b. Calculate the posterior probabilities for “smash”, “medium”, and “flop” given the various
audience reactions.
c. Assume that MBA is willing to base its decision on Expected Monetary Value. What
decision should the MBA Movie Studio make about the movie “Claws”?

2. The Tarheel Manufacturing Company must decide whether to build a large plant or a small
one to process a new product with an expected life of 10 years. Demand may be high during
the first 2 years, but if many users find the product unsatisfactory, demand will be low for the
remaining 8 years. High demand during the first 2 years may indicate high demand for the

next 8 years. If demand is high during the first 2 years and company does not expand within
the first 2 years, competitive products will be introduced, thus lowering the benefits.

If the company builds a large processing plant, it must keep it for 10 years. If its builds the
small plant, the plant can be expanded in 2 years if demand is high, or the company can stay
in the small plant while making smaller benefits on the small volume of sales. Estimates of
demand are these:

Probability Probability
High demand (first 2 years) 0.5 High demand during first 2 0.6
followed by high demand years
(next 8 years)
High demand (first 2 years 0.1
followed by low demand
(next 8 years)
Low demand (first 2 years) 0.4 Low demand during first 2 0.4
followed by continuing low years
demand (next 8 years)
Low demand (first 2 years) 0
followed by high demand
(next 8 years)

Financial costs and profits are as follows:
♦ A large plant with high demand would yield Php 1 million annually in profits.
♦ A large plant with low demand would yield Php 200,000 annually because of production
inefficiencies.
♦ A small plant, not expanded, with a low demand would yield annual profits of Php
250,000 for 10 years.
♦ A small plant during a 2-year period of high demand would yield Php 450,000 annually;
if high demand continued and if the plant was not expanded, this would drop to Php
300,000 annually for the next 8 years as a result of competition.
♦ A small plant which was expanded after 2 years would yield Php 100,000 annually for 8
years if low demand occurred during that period.
♦ A large plant would cost Php 5 million to build and put into operation.
♦ A small plant would cost Php 1,500,000 to build and put into operation.
♦ Expanding a small plant after 2 years would cost Php 2,500,000.

Under the conditions stated and with the information furnished, analyze the alternatives and
choose the best decision.

3. A machine in a group of 50 machines is serviced when it breaks down. At the end of T
period, preventive maintenance is performed by servicing all 50 machines. The cost of
repairing a broken machine is Php 1000 while Php 100 is the preventive maintenance cost per
machine. Determine the optimum T that minimizes the total cost per period.

Probability Distribution of Breakdown before T
0.03, t =1

P(t ) =  Pt −1 + 0.01, t = 2 10
0.125, t = 11,12


4. An automatic machine produces α (thousands of) units of a certain product per day. As α
increases, the proportion of defective p goes up. The probability density function of p in
terms of α is given by

αp α −1 ,
 0 ≤ p ≤1
f ( p) = 
0,
 otherwise

Each defective item incurs a loss of Php 50. A good item produces a profit of Php 5.
Determine the optimal value of α.

5. A doctor must diagnose the condition of one of her patients. She is certain that the condition
is not life threatening, so there is not risk to the patient’s life. However, in diagnosing the
condition the doctor would like to minimize the cost to the patient. There are three tests she
could conduct on her patient. The doctor has narrowed the range of possibilities down to
three disease conditions, and indicates her professional judgmental probability assessments of
the three states as 40 percent, 25 percent, and 35 percent that the disease is type 1, type 2, and
type 3, respectively. The cost (in Php) to the patient for diagnosis and treatment is given in
the table of possibilities shown in the table below depending on the test employed. Answer
the following by dealing directly with costs (the problem can also be solved by constructing a
table of negative payoffs):
a. Determine the Bayes decision (the test which minimizes the expected cost).
b. Compute the EVPI.
c. Suppose the doctor first administers a special blood test that could be used to refine her
probability estimates. The cost of the blood test is Php 50 and it would indicate one of
two results – O1 or O2. The likelihood probabilities are given in the following table:
Disease Condition
S=1 S=2 S=3
O1 0.80 0.05 0.40 P(O1/S), i = 1, 2, 3
O2 0.20 0.95 0.60 P(O2/S), i = 1, 2, 3
Should the doctor administer the blood test prior to selecting one of the three major tests?
Cost of Diagnosis and Treatment
Test 1 2 3
1 500 300 400
2 600 500 300
3 300 550 450

6. A large mill is faced with the problem of extending Php 100,000 credit to a new customer, a
dress manufacturer. The mill classifies typical companies into the categories: poor risk,
average risk, and good risk. Their experience indicates that 20 percent of similar companies
are poor risks, 50 percent are average risks, and 30 percent are good risks .If credit is
extended, the expected profit for poor risks is- Php 15,000, for average risks Php 10,000, and
for good risks Php 20,000. If credit is not extended, the dress manufacturers will turn to

another mill. The mill is able to consult a credit rating organization for a fee of Php 2,000.
Their experience with this credit –rating company is given by

Credit company Actual credit rating %
evaluation Poor Average Good
Poor 50 40 20
Average 40 50 40
Good 10 10 40

a) What is the Bayes’ action, assuming the credit rating company is not used?
b) How much money can be paid for “ perfect information”?
c) What is the optimal expected loss if the credit rating company data is used? Does it pay
to utilize these data?
d) What is the Bayes’ action if the credit-rating company determines the dress manufacturer
to be a poor risk?

7. The Breezy Breakfast Foods Company is considering marketing a new breakfast cereal. If the
new cereal is successful, it will mean a Php 10 million profit (present value) over the life of
the product. If unsuccessful, a Php 2 million, loss on investment will be incurred.
Management currently feels there is a 50-50 chance that the product will be successful.

Two market research firms have approached Breezy with proposals to obtain more
information. Attitude Surveys collects data on consumer attitudes with respect to specific
characteristics of a product, such as sweetness, caloric content, nutritive value, etc., and
produces a forecast of “success” or “fail”. Of the studies this company has performed on
similar products recently, their experience has been as follows:

Attitude Surveys Experience
Actual outcome
Forecast Success Failure
Success 20 5
Failure 5 20

A second company, Market Competition Inc. ; performs analysis in an entirely different,
independent manner. This company performs extensive analysis on competitive products, and
produce a recommendation of “success” or ”fail” based on the anticipated amount and
quality of competitive products. Their experience with 50 studies has been as follows:

Market Competition Experience
Actual outcome
Forecast Success Failure
Success 22 3
Failure 0 25

Attitude Surveys charges Php 100,000 per survey, while Market competition charges Php
150,000.

a. Consider only Attitude Surveys. Use a decision tree to decide whether or not Breezy
should purchase this survey.

b. Consider only Market Competition, Inc. Use a decision tree to decide whether or not
Breezy should purchase this survey.

8. Inventory Management. Blumberg’s Department Store will hold a one-month suit sale. The
suits can be purchased in lots of 25 each, an the wholesale post per suit is a function of the
number of suites ordered, as shown below:

Number of suits ordered 25 50 75 100
Cost per suit (Php) 800 750 700 650

Each suit left over at the end of the month will be sold at the clearance sale for half the retail
sales price of Php 1400. If a shopping arises during the month, nothing will be done to
replenish inventory, The possible sales levels are 25, 50, 75, and 100, having probabilities of
0.20, 0.30, 0.40 and 0.10, respectively.

a. Construct a payoff table for this decision problem.
b. Construct a decision tree to represent the problem.
c. Fold back the decision tree to determine the optical inventory ordering the decision.
d. Marketing Analysis. Suppose Blumberg’s could purchase the services of a marketing
consultant who would conduct a survey that could be summarized by one of two
outcomes: O1 (a favorable market exists for the sale) or O 2 (an unfavorable market
exists). The likelihoods, estimated by Blumberg’s from past experience with this
consultant, are given in Table 2. What is the maximum amount Blumberg’s should pay
for the consultant’s services? (Hint: conduct a preposterior analysis).

Table 2
Sales (states of nature)
P(0/S) S = 25 S = 50 S = 75 S = 100
O1 0.05 0.50 0.70 0.85
O2 0.95 0.50 0.30 0.15

9. The Profit & and Gambit Company has a major product that has been losing money recently
because of declining sales. In fact, during the current quarter of the year sales will be 4
million units below the break-even point. Since the marginal revenue for each unit sold
exceeds the marginal cost by Php 1. This amount to a loss of Php 4 million for shutting
down. The other alternative is to undertake an intensive advertising campaign to increase
sales and then abandon the product (at the cost of Php 4 million) only if the campaign is not
sufficiently successful. Tentative plans for this advertising campaign have been developed
and analyzed. It would extend over the next three quarters (subject to early cancellation), and
the cost would be Php 6 million in each of the three quarters. It is estimated that the increase
in sales would be approximately 3 million units in the first quarter, another 2 million units in
the second quarter and another 1 million units in the third quarter. However, because of a
number of unpredictable market variables, there is considerable uncertainty as to what impact
the advertising actually would have, and careful analysis indicates that the estimate for each
quarter could turn out to be off by as much as 2 million units in either direction. (To quantify
this uncertainty, assume that the additional increase in sales in the three quarters are
independent of random variables having a uniform distribution with a range from 1 to 5
million, from 0 to 4 million, and from –1 to 3 million, respectively). If the actual increases
are too small, the advertising campaign can be discontinued and the product abandoned at the
end of either of the next 2 quarters.

If the intensive advertising campaign were to be initiated and continued to its competition, it
is estimated that the sales for some time thereafter would continue to be at about the same
level as in the third (last) quarter of the campaign. Therefore, if the sales in that quarter still
are below the break-even point, the product would be abandoned. Otherwise, it is estimated
that the expected discounted profit thereafter would be Php 8 for each unit sold over the
break-even point in the third quarter.

Determine the optimal policy maximizing expected profit.

10. An oil company has some land that is purported to contain oil. The company classifies such
land into four categories by the total number of barrels that are expected to be obtained from
the well, i.e. a 500,000 – barrel well, a 200,000 barrel well, a 50,000- barrel well, or a dry
well. The company is faced with deciding whether to drill for oil, to unconditionally lease the
land, or to conditionally lease the land at a rate depending upon the oil strike. The cost of
drilling a producing well is Php 100,000, and the cost of drilling a dry well is Php 75,000. For
producing wells the profit per barrel of oil is Php 1.50 (after deducting all production costs).
Under the unconditional lease agreement, the company receives Php 45,000 for the land,
whereas under the conditional lease arrangement the company receives 50 cents for each
barrel of oil extracted, provided the land yields a 200,000 – or 500,000- barrel strike;
otherwise, it receives nothing. The possible profits for the oil company are shown below:

a. What should the company’s decision be? Why?
b. If the company has had some experience with well in similar geographical areas and has
concluded that about 10% of the strikes are 500,000 barrel wells, 15% are 200,000 barrel
wells, 25% are 50,000 barrel wells, and 50% are dry wells, what will be the decision
maker’s decision? Why? (Please use different methods to justify you answer.)
c. The decision-maker has the choice of not utilizing seismic soundings or utilizing seismic
sounding which would mean a research study which would cost Php 12,000. Seismic
soundings may reveal information on four possible seismic classification denoted as
follows:

1. there is definitely a closed geological structure to the site ( a very favorable
outcome if the presence of oil is desired);
2. there is probably a closed structure to the site;
3. there is a non-closed structure( a relatively unfavorable report);
4. there is no structure to the site an unfavorable condition).
Based upon past examination of similar geological areas (100 such examinations), the data
below are obtained:

State of 500,000- barrel 200,000- barrel 50,000-barrel well Dry well
seismic well well
nature class
f Prob. f Prob. f Prob. f Prob.
1 7 7/12 9 9/16 11 11/24 9 9/48
2 4 4/12 3 3/16 6 6/24 13 13/48
3 1 1/12 2 2/16 3 3/24 15 15/48
4 0 0/12 2 2/16 4 4/24 11 11/48

Use decision tree to come up with your decision analysis of the problem.

CASE I

A manufacturer produces items having a probability p of being defective. These items are
formed into lots of 150. Past experience indicates that p is either 0.05 or 0.25, and furthermore, in
80 percent of the lots produced p equals 0.05 (and in 20 percent of the lots p equals 0.25). These
items are then used in an assembly, and ultimately, their quality is determined before the final
assembly leaves the plant. The manufacturer can initially screen each item in a lot at a cost of P15
per item, replacing defective items found, or use the items directly without screening. If the latter
action is chosen, the cost per lot can be calculated as:
p = 0.05 p = 0.25
Screen 2,250 2,250
Do not screen 750 3,750

Because screening requires scheduling of inspectors and equipment, the decision to screen or not
screen must be made 2 days before the potential screening takes place. However, one item can be
taken from the lot and sent to a laboratory, and its quality (defective or non-defective) can be
reported before the screen, no-screen decision must be made. The cost of this initial inspection is
P125.

a) What is the Bayes’ action without looking at the single item?
b) How much money can be paid for “perfect information”
c) What is the optimal expected cost if the quality of items is determined before the screen, no-
screen decision is made?
d) What is the Bayes’ action if the quality of one of the items is determined and found to be
defective?

CASE II

In a manufacturing process, lot having 8,10, 12, or 14% defectives are produced
according to the respective probabilities 0.4, 0.3, 0.25, and 0.05. Three customers have contracts
to receive lots from the manufacturer. The contracts specify that percentages of defective in lots
shipped to customers A, B, C should not exceed 8, 12, 14, respectively. If a lot has a higher
percentage of defectives than stipulated, a penalty of Php 100 per percentage point is incurred. On
the other hand, supplying better quality than required costs the manufacturer Php 50 per
percentage point. A sample of rise n = 20 is inspected before each lot is shipped lot to customers.
Suppose that 4 defectives are found in the sample. What would be optimal decision of the firm?

CASE III

A new type of airplane is to be purchased by the Air force, and the number of spare
engines accompanying the order must be determined. The Air Force must order these spare
engines in bathes of 5 and can only choose among 15, 20, or 25 spares. The supplier of these
engines has two plants, and the Air Force must make its decision prior to knowing which plant
will be used. From past experience it is known that the number of spare engines required when
production takes place at plant A is approximated by a Poisson distribution with parameter θ =
21, whereas the number of spare engines when production takes place at plant B is approximated
by Poisson distribution with parameter θ = 24. The cost of a spare engine purchased now is Php
400,000, whereas the cost and interest are to be neglected. Spares must always be supplied if they
are demanded, and unused engines will be scrapped when the airplanes become obsolete.

The Air Force know from past experience that 2/3 of all types of airplane engines are
produced in plant A and only 1/3 in plant B. Furthermore, it is known that a similar type of
engine was produced for an earlier version for the current airplane under consideration. The order
size for this earlier type was the same as for the current model. Furthermore, its non-obsolete life
is identical with that planned for the present version. The engine for the current order will be
produced in the same plant as the previous model, although the Air Force is not aware of which
of the two plant this is. The reason for this lack of knowledge is due to the haste in which the
spare engine decision must be made. The Air Force has access to the data which the spare engine
decision must be made. The Air Force has access to the data on the number of spares actually
required for the older version (which had a Poisson distribution), but it does not have time
determine the production location.

a. What is the Bayes’ action, assuming that the information on the old airplane model is
unavailable?
b. How much money can be paid for “perfect information”?
c. Assuming that cost of data on the old airplane model is free and 30 spares were required,
determine the Bayes’ action.

CASE IV

Carlo Steerio is a chain of retail outlets specializing in sound equipment. Because of its
high volume, Charles stocks its inventory of stereo pickup cartridges by ordering lots of 100 units
from various manufacturers. Carlo Hagano, the owner, decides randomly whether to accept or
reject a particular lot from a supplier, KICO. Each cartridge in a rejected lot is thoroughly
inspected by Carlo, and cartridges that are found to be defective are replaced by the supplier. An
accepted lot is parceled without inspection to the retail stores for sale to customers, who are relied
on to find the defective cartridge, which Charles replaces without charge from its inventory.
KICO will not give Carlo credit for cartridges that have been used by retail customers, even if
they were originally defective.

One of Carlo’s employees suggested that the company sample incoming lots, using the
information thereby obtained as a basis of deciding whether a lot should be accepted or rejected.
Carlo is skeptical about the adventures of the sampling, since the test costs Php 300 per cartridge
tested.

Detailed records of KICO cartridge received by Carlo have been maintained, and the
frequencies of lot proportions defective have been found. These frequencies serve as estimates of
the following prior for values of p of .10, .20 and .30

Possible lot Proportion Defective p Prior Probability
0.10 0.40
0.20 0.30
0.30 0.30
Total 1.00

The cartridges are bought at P150 each by Carlo Steerio from KICO and sold for Php 20,000 per
lot customers.

Carlo calls on you to evaluate his employee’s suggestion but tells you that if sampling is
beneficial, he is noting to sample up to 2 only.

What would your recommendation be?

CASE V
Artex Computers is going to purchase 10,000 units of a certain part that is to be
assembled into the Artex products. The order is to be placed with the lowest bidder with the
10,000 units to be delivered at a rate of 1,000 per month over the next 10 months.

The Frank Machine Shop (FMS) is considering bidding on this order. Two factors are
puzzling Mr. Frank in his attempt to fix a bid price. The first factor deals with FMS’s chances of
winning the bid. Mr. Frank finally decides to consider only two bids - either Php 12 per unit or
Php 13 per unit. He estimates that the chances are two thirds of winning the former price and one
third of winning the latter price.

The second factor involved in the decision is the FMS unit manufacturing cost. Two
production processes are available. The first, process A, is known to cost Php 10 per unit. The
second, process B is one that FMS has not used before. The chief foreman says there is a one-
fourth chance that the per unit cost will be P9; a one-half chance that the cost will be P10; and a
one-fourth chance that the cost will be P11, if process B is used.

The chief foreman has suggested that he conduct an experiment with the new process (B).
He could produce 10 or 15 units; and from the experience gained, he believes he could estimate
unit cost “ pretty well”. The cost of this test would be P500. Then asked to be more specific about
how accurate his estimate of cost would be, the chief foreman provided the following table.

Chance of Various Estimated Cost
Foreman’s Actual per unit cost
estimated cost Php 9 Php 10 Php 11
Php 9 0.8 0.1 0.1
Php 10 0.1 0.8 0.1
Php 11 0.1 0.1 0.8
1 1 1
Mr. Frank is not sure that this information is at all relevant to his problem. The controller
has argued that the test suggested by the foreman may be valuable but should be performed after
the bid is awarded. Otherwise, he argues, the firm may spend Php 500 and not win the contract.
The foreman feels that the test should be done before the bid, since it may influence FMS’s bid
price.

Draw the complete Decision tree diagram. What specific sequence of actions should be
FMS take?

CASE VI

Two batches of a product have had their labels torn off in error. One batch was produced
on Machine A. The proportion of defectives of A is given by the following density function.

80,000 p
f ( PA ) = , 0 < p < 0.015
9
= 0, otherwise

The other was produced by machine B whose density function given by

f ( PA ) = 10,125 p 2 , 0 < p < 1 / 15
= 0, otherwise

The output of Machine B is sol a “ as is” to a user who incurs very little cost when a defective
unit is found. However, Machine A output is sold to a user who charges the manufacturer P5 for
each defective unit, since it is installed in a complex piece of machinery. There are 100 units in
each batch. The manufacturer must find out which batch is which, as he has an urgent order for a
batch of Machine A output.

A statistician has suggested has suggested taking a sample of 20 units from one batch. If there are
one or more defectives in that batch, label it “Machine B output” while if there are zero
defectives, label it “Machine A output”; and label the other batch in the opposite manner. Then
ship the batch labeled
“ Machine A output.”

a. Compute the conditional probabilities of misleading the batches.
b. Assuming that the two machines have been used with the same frequency (based on historical
data), compute the unconditional expected cost of the statisticians’s suggestion.
c. Suppose taking a sample of 20 costs P 2, but a complete batch of 100 units would cost only P
10. Based on your results from (b), should the manufacturer take 100% sample of one of the
batches?

Decision theory Problems

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