Chapter Outline
Applying Decision Analysis Tools
Structuring Decision Problems
Selecting Decision Alternatives
One-Time Decisions Without Event
Probabilities
Repeated Decisions With Event
Probabilities
Expected Value of Perfect
Information
Decision Trees
OM Spotlight: How Computers
Play Chess
OM Spotlight: Collegiate Athletic
Drug Testing
SUPPLEMENTARY
CHAPTER E
Solved Problems
Key Terms and Concepts
Questions for Review and
Discussion
Problems and Activities
Cases
Trendy’s Pies
Service Guarantee Decisions
for McCord Hotels
Endnotes
Decision Analysis
Learning Objectives
• To identify characteristics of management decisions where decision
analysis techniques are used and to define the elements of a
decision problem.
• To evaluate risk in making decisions and apply decision criteria to
select an appropriate decision alternative.
• To construct simple decision trees and use them to select optimal
expected value decisions.
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• “What do you think we should do? We’re down by 10 with 5 minutes left—
plenty of time to get the ball back,” pondered Ken Kendall, head coach of
West High in talking to offensive coach Craig Russell. West was facing fourth
down and short yardage for another first down from their opponent’s 9-yard
line. “Should we try for the first down or go for the field goal?” Craig noted
that statistically a run is better than a field goal attempt inside the 10-yard line.
Ken wasn’t so sure, trying to weigh the risk of not getting the first down or a
touchdown instead of an almost sure field goal.
• Electric utilities face decisions that can have important impacts on the environment. The impacts stem from the by-products of combustion and other
chemicals, equipment, and processes that utilities use to produce electricity.
For example, utilities use large boilers to boil water and make steam to generate electricity. The cleaning process results in a waste solution that may be
hazardous. Whether or not the waste stream will be hazardous is uncertain,
as are the costs and effects of the various management strategies. Several
courses of action—choice of cleaning agent, whether or not to include a prerinse stage, treatment and disposal method, and cleaning frequency—are available. Using techniques of decision analysis, the consulting firm Decision Focus
Incorporated developed a strategy that would save a utility $119,000 for one
boiler over a 20-year horizon.1
Decision analysis is the formal
study of how people make
decisions, particularly when
faced with uncertain
information, as well as a
collection of techniques to
support the analysis of
decision problems.
is the formal study of how people make decisions, particularly when
faced with uncertain information, as well as a collection of techniques to support
the analysis of decision problems. For example, the manufacturer of a new style or
line of seasonal clothing would like to manufacture large quantities of the product
if consumer acceptance and, consequently, demand for the product are going to be
high. Unfortunately, the seasonal clothing items require the manufacturer to make
a production-quantity decision before the actual demand is known. Most decisions
that we face in business and in our personal lives require a choice in the face of an
uncertain future.
Decision analysis has many applications in product selection, facility capacity
expansion and location, inventory analysis, technology and process selection, and
other areas of operations management. The two opening episodes are some examples. In fact, Virgil Carter, a former NFL quarterback, and Robert Machol applied
decision analysis to evaluate football strategies. They found, for example, that the
expected value of having the ball with first down and 10 yards to go varies by field
position. If the ball is close to one’s own goal line, then the team’s expected scoring
value is 1.64, indicating that their opponent is more likely to score as a result of
getting the ball back in good field position. As field position moves closer to the
opponent’s goal line, the expected value becomes positive and increases. A further
analysis of field goal attempts showed that inside the 30-yard line, the run is preferred to the field goal attempt if there are 1 or 2 yards to go, and possibly with
3. Inside the 10-yard line, the run is preferred to the field goal attempt with up to
5 yards to go. These results were contrary to practice, but many coaches continued to employ the field goal far more than the analysis indicated.2
Decision analysis
Supplementary Chapter E: Decision Analysis
APPLYING DECISION ANALYSIS TOOLS
Decision analysis tools should not be used in every decision situation. Characteristics
of management decisions where decision analysis techniques apply are summarized
as follows:3
1. They must be important. Decision analysis techniques would not be appropriate for minor decisions where the consequences of a mistake are so small that
it is not worth our time to study the situation carefully. The consequences of
many decisions, such as building a major facility, are not felt immediately but
may cover a long time period.
2. They are probably unique. Decisions that recur can be programmed and then
delegated. But the ones that are unusual and perhaps occur only one time cannot be handled this way.
3. They allow some time for study. For example, decision analysis techniques
would not be useful in making a decision in the emergency room or when a jet
fighter flames out during takeoff.
4. They are complex. Practical decision problems involve multiple objectives, requiring the evaluation of trade-offs among the objectives. For example, in evaluating routes for proposed pipelines, a decision maker would want to minimize
environmental impact, minimize health and safety hazards, maximize economic
benefit, and maximize social impact. Decisions involve many intangibles, such
as the goodwill of a client, employee morale, and governmental regulations, and
may involve several stakeholders. For instance, to build a plant in a new area,
corporate management may require approval from stockholders, regulatory
agencies, community zoning boards, and perhaps even the courts. Finally, most
decisions are closely allied to other decisions. Choices today affect both the
alternatives available in the future and the desirability of those alternatives.
Thus, a sequence of decisions must often be made.
5. They involve uncertainty and risk. Uncertainty refers to not knowing what will
happen in the future. An advertising campaign may fail, a reservoir may break,
or a new product may be a complete failure. Uncertainty is further complicated
when little or no data are available, or some data are very expensive or timeconsuming to obtain. Faced with such uncertainties, different people view the
same set of information in different ways. Risk is the uncertainty associated with
an undesirable outcome, such as financial loss. To appreciate the importance of
risk, consider the fact that it takes hundreds of millions of dollars and about
10 years for a pharmaceutical company to bring a drug to market. Once there,
seven of ten products fail to return the company’s cost of capital. Decisions involving capital investment and continuation of research over the long development cycle do not lend themselves to traditional financial analysis.4
Structuring Decision Problems
To illustrate the process of defining a decision problem, we present an example of a
medium-size producer of industrial chemical products, Commonwealth Chemicals
Company, that is facing a decision about capacity expansion. The company has
recently developed a new synthetic industrial lubricant that will increase tool life
for machining operations in metal-fabrication industries. A new factory would be
necessary to produce the lubricant on a large scale, but expanding the existing
facilities would allow production on a smaller scale.
Managers are uncertain which decision to choose. Clearly, the best decision
depends on future demand. If the demand for the product is high, the expansion
E3
Learning Objective
To identify characteristics of
management decisions where
decision analysis techniques are
used and to define the elements
of a decision problem.
Uncertainty refers to not
knowing what will happen in
the future.
Risk is the uncertainty
associated with an
undesirable outcome, such as
financial loss.
E4
Decision alternatives represent
the choices that a decision
maker can make.
Supplementary Chapter E: Decision Analysis
alternative will not provide enough capacity to meet all the demand and profits will
be lost. If demand is low, and a new factory is built, the excess capacity will substantially reduce the return on investment. With an unstable economy, it is difficult to predict actual demand for the product.
The first step in structuring a decision problem is to define the decision alternatives. Decision alternatives represent the choices that a decision maker can make.
In this case, the alternatives are whether to expand the existing plant or to build a
new factory. Let
d1 decision to expand the existing plant
d2 decision to build a new plant
Events represent the future
outcomes that can occur after
a decision is made and that
are not under the control of
the decision maker.
The second step is to define the events that might occur after a decision is made.
represent the future outcomes that can occur after a decision is made and
that are not under the control of the decision maker. For each combination of
production-volume decision and subsequent event, a payoff can be computed. For
instance, if the manufacturer decides to produce 10,000 units, but demand is low,
the manufacturer will incur the cost of producing the 10,000 units but will receive
revenue for sales of only 5,000; the remaining units will have to be disposed of at
a loss. On the other hand, if sales are medium or high, all 10,000 units will be sold,
and the net profit can be computed. The payoff would be the net profit.
For instance, in deciding to expand an existing plant or build a new one, Commonwealth Chemicals needs to consider the future demand for the product. Different possible levels of demand represent the events. Demand might be expressed
quantitatively in sales units or dollars. In this example, events might be designated
as “high demand,” “medium demand,” and “low demand.” Alternatively, they
might be quantified as “demand estimated as 15,000 units,” “demand estimated as
10,000 units,” and “demand estimated as 5,000 units.” If you are planning a spring
break vacation to Florida in January, you might define events as the weather that
you might encounter. Uncertain weather-related outcomes might be defined qualitatively, for example, sunny and warm, sunny and cold, rainy and warm, or rainy
and cold. For the Commonwealth Chemicals decision problem, we will define the
events as
Events
s1 low product demand
s2 high product demand
A numerical value associated
with a decision coupled with
some event is called a payoff.
Next, we need well-defined decision criteria on which to evaluate potential options. Decision criteria might be net profit, customer service, cost, social benefits,
or any other measure of output that may be appropriate for the particular situation being analyzed. A numerical value associated with a decision coupled with
some event is called a payoff. Using the best information available, the managers of
Commonwealth Chemicals have estimated the payoffs, expressed as profits, shown
in Exhibit E.1. A table of this form is referred to as a payoff table. The notation
we use for the entries in the payoff table is V(di , sj), which denotes the payoff, V,
associated with decision alternative di and event sj. Using this notation, we see that
V(d2 , s1) $100,000.
Exhibit E.1
Payoff Table for Commonwealth
Chemicals
Possible Future Events
Decision Alternative
Expand existing plant (d1)
Build new plant (d2)
Low Product Demand (s1)
High Product Demand (s2)
$200,000
$100,000
$300,000
$450,000
Supplementary Chapter E: Decision Analysis
E5
In many decision problems, the probabilities of events can be estimated, either
from historical data or managerial judgment. Knowing the likelihood of the occurrence of events helps to assess risk when making a decision. In some cases, however, event probabilities may not be available or appropriate to try to assess. We
will provide examples of both situations in the following sections.
In summary, the elements of a decision problem are (1) decision alternatives,
(2) events, (3) estimated payoffs for each combination of decision alternatives and
events, and possibly (4) probabilities of the events.
SELECTING DECISION ALTERNATIVES
Making decisions with uncertain future consequences is often quite frustrating and
a source of anxiety for individuals and managers alike. We run the risk that any
decision we choose may result in undesirable consequences once we see what the
future holds in store. There are two principal ways of viewing a decision strategy,
and these depend on the frequency with which the decision will be made. For onetime decisions, managers must take into account the risk associated with making
the wrong decision. However, for decisions that are repeated over and over, managers can choose decisions based on the expected payoffs that might occur.
One-Time Decisions Without Event Probabilities
The Commonwealth Chemicals decision is clearly a one-time decision. So how
should the choice be made? Different criteria can be used to reflect different attitudes toward risk, and they may result in different decision recommendations. For
a problem in which the payoff is profit, as it is in the Commonwealth Chemicals
problem, three common criteria are
1. Maximax—choose the decision that will maximize the maximum possible profit
among all events. This is an aggressive, or risk-taking, approach.
2. Maximin—choose the decision that will maximize the minimum possible profit
among all events. This is a conservative, or risk-averse, approach.
3. Minimax regret—choose the decision that will minimize the maximum opportunity loss associated with the events. Opportunity loss represents the regret, or
ill-feeling, that people often have after making a nonoptimal decision (“I should
have bought that stock years ago . . .”). This approach is neither aggressive nor
conservative, but focuses on not erring too much in either direction.
We will apply these criteria for the Commonwealth Chemicals problem. For the
maximax criterion, we see that if d1 is selected, the maximum payoff is $300,000,
and it occurs for s2. If d2 is selected, the maximum payoff is $450,000, also for s2.
The decision maker should choose d2, build a new plant, since it results in the
largest possible payoff.
For the maximin criterion, we see that if d1 is chosen, the minimum payoff is
$200,000, whereas if d2 is selected, the minimum payoff is $100,000. Thus, to maximize the minimum payoff, the decision maker should choose d1, expand the existing plant.
To apply the minimax-regret criterion, we must first construct a regret or
opportunity-loss matrix. The opportunity loss associated with a particular decision,
di, and state of nature, sj, is the difference between the best payoff that the decision maker can receive by making the optimal decision d* corresponding to sj,
V(d*, sj), and the payoff for choosing any arbitrary decision di and having sj occur,
V(di , sj). For example, if we know that s1 will occur, the best decision is to choose
Learning Objective
To evaluate risk in making
decisions and apply decision
criteria to select an appropriate
decision alternative.
E6
Supplementary Chapter E: Decision Analysis
d* d1 and receive a payoff of $200,000; the opportunity loss will be zero. If we
choose d2, we will receive only $100,000 and will lose the opportunity to receive
$200,000 $100,000 $100,000. Similarly, if we know that s2 will occur, the
best decision is d* d2; an opportunity loss of $450,000 $300,000 $150,000
will occur if we choose d1. Exhibit E.2 shows the complete opportunity-loss matrix
for this situation. We see that the smallest maximum opportunity loss occurs for
d2, so using this criterion, Commonwealth should build the new plant.
We see that different criteria can result in different decisions; which to use is
purely a judgment call on the part of the decision maker and reflects the person’s
values and attitudes toward risk.
For problems in which the payoff is cost, the criteria change somewhat. The
aggressive decision criterion is minimin—minimize the minimum possible payoff
over all events. The conservative decision is minimax—minimize the maximum possible payoff over all events. Finally, the minimax-regret criterion does not change,
since opportunity loss is always a cost. However, care is needed in computing the
opportunity loss correctly. It is still the difference between the best possible payoff
(received by making the optimal decision) and the payoff of any other decision. The
only difference when the output measure is cost is that the “best” payoff is the lowest cost, not the highest profit. The difference must be viewed as an absolute value,
that is, the savings in cost, since it does not make sense for opportunity losses to
be negative.
Repeated Decisions With Event Probabilities
The expected value approach
is to select the decision
alternative with the best
expected payoff.
If an individual or business faces the same decision problem repeatedly, then over
the long run, the decision can be made based on expected value. The expected value
approach is to select the decision alternative with the best expected payoff. The
expected value criterion requires probability estimates for the events. In many situations, good probability estimates can be developed from historical data or judgmentally. Let
P(sj) probability of occurrence for event sj
N number of events
Since one and only one of the N states of nature can occur, the associated probabilities must satisfy these two conditions:
P(sj) 0 for all j
P(sj) P(s1) P(s2) . . . P(sN) 1
The expected value for decision alternative di is given by
EV(di) jP(sj)V(di, sj)
(E.1)
The EV criterion is used in revenue management applications (see Chapter 10).
Most airlines, for example, offer discount fares for advanced purchase. Assume that
only two fares are available: full and discount. The airline must make the decision
of whether or not to accept the next request for a discount seat. If it accepts the
discount request, the revenue it earns is the discount fare. If it rejects the discount
request, two outcomes are possible. First, the seat may remain empty and the air-
Exhibit E.2
Opportunity-Loss Matrix for
Commonwealth Chemicals
Decision
Expand existing plant (d1)
Build new plant (d2)
Low Product
Demand (s1)
High Product
Demand (s2)
Maximum
Opportunity Loss
0
$100,000
$150,000
0
$150,000
$100,000
Supplementary Chapter E: Decision Analysis
E7
line will not realize additional revenue. Alternatively, the remaining seat may be
filled by a full-fare passenger, either because full-fare passenger demand is sufficient to fill the seats or because discount-fare passengers choose to pay full fare
when told the discount fare is not available.
This decision situation is illustrated by an example in Exhibit E.3. Suppose that
a full-fare ticket is $560 and the discount fare is $400. The decision depends on
the probability, p, of getting a full-fare request when a discount request is rejected.
The expected value of rejecting the discount seat request is p times the full-fare
value. Thus, if p .75, the expected value of rejecting the discount request is
.25(0) .75 ($560) $420. Since this is higher than the discount fare, the discount request should be rejected. Since an airline makes hundreds or thousands
of such decisions each day, the expected value criterion is appropriate.
Expected Value of Perfect Information
By perfect information, we mean knowing in advance what state of nature will occur. Although we never have perfect information in practice, it is worth knowing
how much we could improve the value of our decision if we had such information.
This is called the expected value of perfect information, or EVPI, which is the difference
between the expected payoff under perfect information and the expected payoff of
the optimal decision without perfect information. We compute EVPI by asking the
following question: If each event occurs, what would be the best decision and payoff? Then we weight these payoffs by the probabilities associated with the events
to obtain the expected payoff under perfect information.
Suppose the airline somehow knew in advance that it could not sell the full-fare
ticket to a particular customer (perhaps based on demographic profiles and analysis of past behavior). Then clearly the best decision would be to accept the discount
request and receive revenue of $400. On the other hand, if it knows that it can sell
the full-fare ticket, then obviously it should reject the request and receive $560.
However, on average, we know that only 75 percent of customers will buy the fullfare ticket if the request is rejected and 25 percent will not. So the expected value
of having perfect information would be
Expected Value of Perfect
Information, or EVPI, which
is the difference between the
expected payoff under perfect
information and the expected
payoff of the optimal decision
without perfect information.
(.75)(560) (.25)(400) $520
Recall that without the perfect information, the best decision is to always choose
d1, which has an expected value of $420. By having perfect information about what
a particular customer might do, we see that the value of the expected payoff can
be increased by
$520 420 $100
This difference is the expected value of perfect information (EVPI), and it represents the maximum amount the company should be willing to pay for any information about the events, no matter how good it is. In this case, we might interpret
it as the maximum incentive that the airline might give to a customer who is unwilling to purchase the full-fare ticket.
Exhibit E.3
Events
Decision
Reject request
Accept request
Probability of event
Sell
Full-Fare Ticket
Do Not Sell
Full-Fare Ticket
$560
$400
.75
$0
$400
.25
Airline Discount-Fare Request
Decision
Expected Value
$560 .75 $420
$400
E8
Learning Objective
To construct simple decision
trees and use them to select
optimal expected value
decisions.
A decision tree is a graphical
schematic of the logical order
with which decisions are
made and events occur.
Nodes refer to the
intersections, or junction
points, of the tree.
Arcs are the connectors
between the nodes.
Supplementary Chapter E: Decision Analysis
DECISION TREES
Decision problems can be depicted graphically using a decision tree. A decision tree
is a graphical schematic of the logical order with which decisions are made and
events occur. In the terminology associated with decision trees, nodes refer to the
intersections, or junction points, of the tree. Arcs are the connectors between the
nodes. Arcs are sometimes called branches. When the branches leaving a given node
are decision branches, we refer to the node as a decision node. Decision nodes are
usually denoted by squares. Similarly, when the branches leaving a given node are
event branches, we refer to the node as an event node. Event nodes are denoted by
circles. The number at each endpoint of the tree represents the payoff associated
with a particular chain of events.
Exhibit E.4 is a decision tree of the airline fare request decision. Note that the
tree shows the natural, or logical, progression of the decision-making process. First,
the firm must make its decision (d1 or d2); then, once the decision is implemented,
an event (s1 or s2) occurs. Note that in this case, if the request is accepted it does
not matter if the airline could have sold the full fare or not, so we do not have to
include event branches for this decision. The OM Spotlight: How Computers Play
Chess provides another interesting application of decision trees.
Exhibit E.4
Airline Discount-Fare Request
Decision
Decision
Event
Sell full fare
P 0.75
Payoff
$560
Reject request
Do not sell
1 p .25
Accept request
$400
Exhibit E.5
How Computers Play Chess
$0
Supplementary Chapter E: Decision Analysis
E9
OM SPOTLIGHT
How Computers Play Chess5
Humans play chess by learning the
nuances of the game, reading books
and learning patterns of play, and ultimately develop strategies and tactics about how they will play. When computers play chess
they do none of this. In fact, the computer is calculating
strategies through sets of formulas based in full or in part
on decision trees. Exhibit E.5 shows a decision tree for the
start of a chess game.
Assume there are 20 possible moves to begin the game
for the white chess pieces. The player chooses from those 20
moves and makes the move. Then the player with the black
pieces must choose among 20 possible moves. When black
moves, white must respond, and so on. For just a sequence
of three moves as shown in the figure, there are 20 20
20 8,000 possible combinations. If you were to carry
these computations out for a typical chess game, you end up
with at least 10120 moves, give or take a few. No computer
or computers are ever going to be able to calculate the entire tree, so complex algorithms are used to prune the tree
branches and only evaluate the most promising branches
of the decision tree. So what some people think is intelligence is completely mechanical and involves no thought
whatsoever—hence the name artificial intelligence!
Decision trees are useful for more complex business decisions. For example, a
nationwide restaurant franchise that frequently introduces new products might
develop the decision tree shown in Exhibit E.6 to help make a decision on how
to best market the products. Even if the tree is not used analytically to evaluate
expected payoffs, it can be of substantial benefit in helping decision makers to logically determine what decisions need to be made and how to react to external forces
such as competitor strategies or economic changes beyond their control.
Expected value calculations can be made directly on the tree to arrive at the best
decision strategy. Working backward through the decision tree, we first compute
Exhibit E.6
New Product Introduction
Decision Tree
E10
Supplementary Chapter E: Decision Analysis
the expected monetary value of each event node by weighting the possible payoffs
by their chances of occurrence. In the airline fare example, the expected value for
the event node corresponding to the reject request decision is
EV .75 (560) .25 (0) $420
shown in the box in Exhibit E.7. We continue backward through the tree to the
decision node. At this point we compare the expected value for rejecting the request with the value of the branch associated with accepting the request. Because
the value for rejecting the request is higher, it corresponds to the best decision.
Many unique uses of decision trees exist, such as the one described in the OM
Spotlight: Collegiate Athletic Drug Testing.
Exhibit E.7
Calculation of Optimal Decision
Strategy
Decision
Event
Payoff
Sell full fare
p 0.75
$560
Reject request
$420
Do not sell
$420
1 p .25
$0
Accept request
$400
OM SPOTLIGHT
Collegiate Athletic Drug Testing6
The athletic board of Santa Clara University had to decide whether to
recommend implementing a drugtesting program for intercollegiate
athletes. One of the board members, who was a management science professor, developed a simple decision model
to address the question of whether or not to test a single
individual for the presence of drugs. The model focused on
the key issue of the reliability of the testing procedures, consequences of testing errors, and the benefits of identifying
a drug user compared with the costs of false accusations
and nonidentification of users.
Exhibit E.8 shows the decision tree developed for testing
an individual for drug use. The two main alternatives are “test”
or “don’t test.” The model evaluates the expected cost of
testing for drug use compared with that of not testing. If testing is chosen, the test is given and the result, positive or negative, is observed. If the result is positive, action is taken.
Since not all those who test positively are actually users, there
is some chance of a false accusation, which costs an amount
C1. If the result is negative, then some drug users are not
identified, which costs C2. Nonusers who test negatively
might be expected to experience some cost, C3, perhaps
based on invasion of privacy. Following the lower path of the
tree, if the alternative “don’t test” were selected, the expected cost is just the cost of an unidentified user, C2, multiplied by the prior probability that an individual is a drug user.
The model’s results surprised many board members. For
instance, the model showed that if a test that is 95 percent
reliable is applied to a population of 5 percent drug users,
only 50 percent of all those who tested positively will actually be drug users. Most board members had read about the
reliability of drug tests in various publications and agreed that
95 percent reliability was a representative value. As a result,
the board concluded that a false accusation was more serious than not identifying drug users and rejected the proposal. The university administration later accepted this
recommendation.
Supplementary Chapter E: Decision Analysis
E11
Exhibit E.8
Decision Tree for Drug-Use
Testing
SOLVED PROBLEMS
SOLVED PROBLEM #1
Maling Manufacturing needs to purchase a new piece
of machining equipment. The two choices are a conventional (labor-intensive) machine and an automated
(computer-controlled) machine. Profitability will depend on demand volume. The following data provide
an estimate of profits over the next three years.
Solution:
Decision
Maximum Profit
Conventional (d1)
$21,000
Automated (d2)
$35,000
Maximax decision d2
Decision
Demand Volume
Low (s1)
High (s2)
Opportunity-Loss Matrix
Decision
Low
Conventional machine (d1)
Automated machine (d2)
$15,000
$ 9,000
Conventional (d1)
Automated (d2)
$21,000
$35,000
0
$6,000
Minimum Profit
$15,000
$ 9,000
Maximin decision d1
High
Maximum
$14,000
$14,000
0
$ 6,000
Minimax-regret decision d2
What decisions would be indicated by maximax, maximin, and minimax-regret criteria?
SOLVED PROBLEM #2
Martin’s Service Station is considering investing in a
heavy-duty snowplow this fall. Martin has analyzed the
situation carefully and feels that this would be a very
profitable investment if the snowfall is heavy, somewhat
profitable if the snowfall is moderate, and would result
in a loss if the snowfall is light. Specifically, Martin forecasts a profit of $7,000 if snowfall is heavy and $2,000
if it is moderate, and a $9,000 loss if it is light. From
the Weather Bureau’s long-range forecast, Martin estimates that P(heavy snowfall) .4, P(moderate snowfall) .3, and P(light snowfall) .3.
a. Prepare a decision tree for Martin’s problem.
b. Using the EV criterion, would you recommend that
Martin invest in the snowplow?
c. Discuss the value of using EV for this situation.
Solution:
a. Exhibit E.9 is the decision tree and the variables are
defined as follows: d1 invest, d2 do not invest,
s1 heavy, s2 moderate, s3 light, P(s1) .4,
P(s2) .3, and P(s3) .3.
b. Recommended decision: d1 (invest) since EV (d1) .4(7,000) .3(2,000) .3(29,000) $700 and EV
(d2) 0.
c. Although the decision tree helps structure the problem, the fact remains that this is a one-time decision.
The expected value criterion does not incorporate
risk into the decision.
E12
Supplementary Chapter E: Decision Analysis
Exhibit E.9
Decision Tree for Solved
Problem 2
KEY TERMS AND CONCEPTS
Arcs/branches
Decision alternatives
Decision analysis
Decision trees
Events
Expected value
Expected value approach
Expected value of perfect information (EVPI)
Five decision analysis characteristics
Maximax criterion
Maximin criterion
Minimax regret criterion
Nodes
Opportunity-loss matrix
Payoff matrix
Regret
Risk
Three decision analysis elements
Uncertainty
QUESTIONS FOR REVIEW AND DISCUSSION
1. Describe two situations where decision analysis
could be applied in operations to support management decision-making.
2. What are five characteristics of management decisions for which decision analysis techniques should
be used? Do all five characteristics have to exist to
apply decision analysis techniques?
3. Define the four major elements of a decision problem.
4. Explain the difference between uncertainty and risk.
Provide some examples from OM of each of these
concepts.
5. What information is provided in a payoff table?
Supplementary Chapter E: Decision Analysis
E13
6. How do managers determine the probabilities for
expected value decision-making?
9. Explain the concept of regret, or opportunity loss,
in decision analysis.
7. Describe how the following criteria are applied to a
decision problem in which the objective is maximization:
10. In what situations would the expected monetary
value criterion be useful? In what situations would
it not be useful?
a. maximax
b. maximin
c. minimax regret
11. What does the expected value of perfect information provide to a decision maker?
12. Explain the structure and purpose of a decision tree.
8. How do the criteria in Question 7 change if the objective is minimization?
PROBLEMS AND ACTIVITIES
1. Suppose a decision maker faced with four decision
alternatives and four states of nature develops the
profit-payoff table shown as follows:
Decision
s1
d1
d2
d3
d4
14
11
9
8
State of Nature
s2
s3
9
10
10
10
10
8
10
11
s4
5
7
11
13
a. If the decision maker knows nothing about the
chances or probability of occurrence of the four
states of nature, what decision would be indicated by the maximax, maximin, and minimaxregret criteria?
b. Which decision criterion do you prefer? Explain.
Should the decision maker establish the most appropriate decision criterion before analyzing the
problem? Explain.
c. Assume the payoff table provides cost, rather
than profit, payoffs. What is the recommended
decision using the optimistic, conservative, and
minimax-regret decision criteria?
2. Suppose the decision maker in Problem 1 obtains
information that enables these probability estimates
to be made: P(s1) .5, P(s2) .2, P(s3) .2, P(s4)
.1.
a. Use the expected value (EV) criterion to determine the optimal decision.
b. Now assuming the entries in the payoff table
are costs, use the EV criterion to determine the
minimum-cost solution.
3. Southland Corporation’s decision to produce a new
line of recreational products has resulted in the need
to construct either a small plant or a large plant.
The decision as to which size to select depends on
the marketplace reaction to the new product line.
To conduct an analysis, marketing managers have
decided to view the possible long-run demand as
low, medium, or high. The payoff table gives the
projected profits in millions of dollars as follows:
Decision
Low
Small plant
Large plant
$150
50
Long-Run Demand
Medium
High
$200
200
$200
500
a. Construct a decision tree for this problem and
determine the best decisions using the maximax,
maximin, and minimax-regret decision criteria.
b. Assume that the best estimate of the probability
of low long-run demand is .20, of medium longrun demand is .15, and of high long-run demand
is .65. What is the best decision using the EV
criterion?
4. Milford Trucking, located in Chicago, has requests
to haul two shipments, one to St. Louis and one to
Detroit. Because of a scheduling problem, Milford
will be able to accept only one of these assignments.
The St. Louis customer has guaranteed a return shipment, and the Detroit customer has not. Thus, if
Milford accepts the Detroit shipment and cannot
find a Detroit-to-Chicago return shipment, the truck
will return to Chicago empty. The payoff table for
profit is shown as follows:
Destination
Return Shipment
from Detroit (s1)
No Return Shipment
from Detroit (s1)
St. Louis (d1)
Detroit (d2)
$2,000
2,500
$2,000
1,000
E14
Supplementary Chapter E: Decision Analysis
If the probability of a Detroit return shipment is .4,
what should Milford do?
5. McHuffter Condominiums, Inc., of Pensacola,
Florida, recently purchased land near the Gulf of
Mexico and is attempting to determine the size of
the condominium development it should build there.
Three sizes of developments are being considered:
small, d1, medium, d2, and large, d3. At the same
time an uncertain economy makes it difficult to ascertain the demand for the new condominiums.
McHuffter’s managers realize that a large development followed by a low demand could be very costly
to the company. However, if McHuffter makes a
conservative, small-development decision and then
finds high demand, the firm’s profits will be lower
than they might have been. With the three levels of
demand—low, medium, and high—McHuffter’s
managers prepared the payoff table as follows:
Decision
Small
Medium
Large
Demand (in Thousands of Dollars)
Low
Medium
High
$400
100
300
$400
600
300
$400
600
900
If P(low) .20, P(medium) .35, and P(high) .45, what decision is recommended by the EV criterion?
6. Construct a decision tree for the McHuffter Condominiums problem (Problem 5). What is the expected
value at each state-of-nature node? What is the optimal decision?
7. Construct a decision tree for Solved Problem 1. Suppose the probabilities of low- and high-demand volume are estimated to be .7 and .3, respectively. What
decision would you recommend?
8. Refer again to the investment problem faced by Martin’s Service Station (Solved Problem 2). Martin can
purchase a blade to attach to his service truck that
can be used to plow driveways and parking lots.
Since this truck would also need to be available to
start cars and do other tasks, Martin would not be
able to generate as much revenue plowing snow if
he elects this alternative, but he would keep his loss
smaller if there is light snowfall. Under this alternative Martin forecasts a profit of $3,500 if snowfall is heavy, $1,000 if it is moderate, and a loss of
$1,500 if snowfall is light.
a. Prepare a new decision tree showing all three alternatives.
b. Using the EV approach, what is the optimal
decision?
9. The Gorman Manufacturing Company must decide
whether to purchase a component part from a supplier or to manufacture the component at its own
plant. If demand is high, it would be to Gorman’s
advantage to manufacture the component. If demand is low, however, Gorman’s unit manufacturing cost will be high because of underutilization of
equipment. The projected profit in thousands of
dollars for Gorman’s make-or-buy decision is as
follows:
Decision
Low
Demand
Medium
High
Manufacture component
Purchase component
$220
210
$40
45
$100
70
The states of nature have these probabilities: P(low
demand) .35, P(medium demand) .35, and
P(high demand) .30. Use a decision tree to recommend a decision.
10. A firm produces a perishable food product at a cost
of $10 per case. The product sells for $15 per case.
For planning purposes, the company is considering
possible demands of 100, 200, and 300 cases. If the
demand is less than production, the excess production is discarded. If demand is more than production, the firm, in an attempt to maintain a good
service image, will satisfy the excess demand with a
special production run at a cost of $18 per case. The
product, however, always sell at $15 per case.
a. Set up the payoff table for this problem.
b. If P(100) .2, P(200) .2, and P(300) .6,
should the company produce 100, 200, or 300
cases?
11. Sealcoat, Inc. has a contract with one of its customers to supply a unique liquid chemical product
used in the manufacture of a lubricant for airplane
engines. Because of the chemical process Sealcoat
uses, batch sizes for the product must be 1,000
pounds. The customer has agreed to adjust manufacturing to the full-batch quantities and will order
either one, two, or three batches every three months.
Since production includes a one-month aging
process, Sealcoat must make its production (how
much to make) decision before the customer places
an order. Thus the product demand alternatives
are 1,000, 2,000, and 3,000 pounds, but the exact
demand is unknown.
Sealcoat’s manufacturing costs are $150 per
pound, and the product sells at the fixed contract
price of $200 per pound. If the customer orders
more than Sealcoat has produced, Sealcoat has
agreed to absorb the added cost of filling the order
Supplementary Chapter E: Decision Analysis
E15
by purchasing a higher-quality substitute product
from another chemical firm. The substitute product,
including transportation expenses, will cost Sealcoat
$240 per pound. Since the product cannot be stored
more than two months without spoilage, Sealcoat
cannot inventory excess production until the customer’s next three-month order. Therefore, if the
customer’s current order is less than Sealcoat has
produced, the excess production will be reprocessed
and will then be valued at $50 per pound.
The decision in this problem is: How much
should Sealcoat produce given the costs and the possible demands of 1,000, 2,000, and 3,000 pounds?
From historical data and analysis of the customer’s
future demands, Sealcoat has developed the probability distribution for demand as follows.
Demand
Probability
1,000
2,000
3,000
.3
.5
.2
a. Develop a payoff table for the problem.
b. How many batches should Sealcoat produce
every three months?
12. A quality control procedure involves 100-percent
inspection of parts received from a supplier. Historical records show the observed defect rates as
follows.
Percent Defective
Probability
0
1
2
3
.15
.25
.40
.20
The cost to inspect 100 percent of the parts received
is $250 for each shipment of 500 parts. If the shipment is not 100-percent inspected, defective parts
will cause rework problems later in the production
process. The rework cost is $25 per each defective
part.
a. Complete the payoff table shown here, in which
entries represent the total cost of inspection and
reworking.
Decision
100% inspection
No inspection
0
$250
?
Percent Defective
1
2
$250
?
$250
?
3
$250
?
b. The plant manager is considering eliminating the
inspection process to save the $250 inspection
cost per shipment. Do you support this action?
Use EV to justify your answer.
c. Show the decision tree for this problem.
13. The R&D manager of the Beck Company is trying
to decide whether or not to fund a project to develop a new lubricant. It is assumed that the project
will be a major technical success, a minor technical
success, or a failure. The company estimates the
value of a major technical success as $150,000, since
the lubricant could be used in a number of products
the company is making. If the project is a minor
technical success, its value is estimated as $10,000,
since Beck feels the knowledge gained will benefit
some other ongoing projects. If the project is a failure, it will cost the company $100,000. Based on
the opinion of the scientists involved and the manager’s own subjective assessment, the following
probabilities are assigned:
P(major success) .15
P(minor success) .45
P(failure) .40
a. According to the EV criterion, should the project be funded?
b. Suppose a group of expert scientists from a research institute could be hired as consultants to
study the project and make a recommendation.
If this study would cost $30,000, should the
Beck Company hire the consultants?
14. Consider again the problem faced by the Beck Company R&D manager (Problem 13). Suppose an experiment can be conducted to shed some light on
the technical feasibility of the project. There are
three possible outcomes of the experiment:
I1 prototype lubricant works well at all
temperatures
I2 prototype lubricant works well only at
temperatures above 10°F
I3 prototype lubricant does not work well at
any temperature
How would the decision tree be modified to include
this information?
15. Explain how decision analysis techniques can be implemented on spreadsheets. Design spreadsheets for
the examples in this chapter.
E16
Supplementary Chapter E: Decision Analysis
CASES
TRENDY’S PIES
Trendy’s is a national chain specializing in selling pies,
either whole or by the slice, from small facilities with
drive-through capabilities. Trendy’s corporate kitchen
staff has developed a new type of pie and needs to make
a decision on whether to introduce it nationally across
the chain or to try a regional test market first. Tina
Trendy, the franchise founder, and her staff sketched
out the decision tree described earlier in the chapter in
Exhibit E.6.
Based on various research reports and industry
knowledge and judgment, Trendy and her staff came up
with the following financial estimates and risk probabilities. If they decide to roll the product out nationally,
they would incur costs of $200,000. A high consumer
response would result in expected revenues of $700,000,
with a .6 probability; whereas a low consumer response
would result in only $150,000 of revenue, with a .4
probability. If they first introduce the product in a regional test market, they would incur $30,000 in costs
and expect a 70 percent chance of a high regional response and a 30 percent chance of a low regional response. Regardless of the outcome, they still have to
make a decision whether to remain regional with the
product (thereby avoiding potential risks of national
failure), market the product nationally, or drop the idea.
If the regional response is high, they anticipate that re-
maining regional would result in revenues of $200,000;
remaining regional with a low regional response would
result in revenues of only $100,000. If they decide to
market nationally after a high regional test market response, Trendy estimates that there is a .9 probability of
a high national response that would result in revenues
of $700,000 and a .1 probability of a low national response with revenues of $150,000. If they market nationally after a low regional test market response, the
probability of a high national response is only .05; the
probability of a low national response would be .95
(revenue estimates would remain the same).
a. Use these cost, revenue, and probability estimates
along with the decision tree to identify the best
decision strategy for Trendy’s Pies.
b. Suppose that Trendy is concerned about her
probability estimates of the consumer response
to the regional test market. Although her estimates are .7 for a high response and .3 for a low
response, she is not very confident of these values. Determine how the decision strategy would
change if the probability of a high response
varies from .1 to .9 in increments of .1. How
sensitive is the best strategy in part a to this probability assumption?
SERVICE GUARANTEE DECISIONS FOR MCCORD HOTELS
McCord Hotels is a small chain of 25 hotels located in
four states—Indiana, Kentucky, Ohio, and West Virginia. The hotel prides itself on superior customer service and is well known in the region. Gregory Hamlet,
the chief executive officer of McCord Hotels, is considering whether to offer a service guarantee at all McCord hotels. If he decides to offer a service guarantee
due to service upsets and mistakes, he will also have to
decide whether to incur the cost of service-recovery
training. This type of training develops a list of the top
20 types of service upsets and trains all employees what
their response should be. The service-recovery training
program includes studying training manuals, in-class
exercises and videos, and out-of-class reading for all
employees. The probabilities for states of nature are
shown in Exhibit E.10.
Revenue-producing room-nights for all 25 hotels for
each payoff scenario are shown in Exhibit E.11. The payoff matrix is stated in total annual room-nights because
the hotel reservation system and pricing policies are very
consistent in all 25 hotel properties. A room-night is defined as one customer staying one night in a hotel room.
Therefore, a room-night generates revenue; if the room
is empty, it generates zero revenue. Hotel rooms represent perishable service capacity that is time-dependent.
Service guarantees were described in Chapter 6. If McCord Hotels does not adopt a service guarantee, it is
Exhibit E.10
Probabilities for McCord Hotels Service Guarantee Decision
Variable/State of Nature
Adopt service guarantees (ASG)
No service guarantees (NSG)
Do service recovery training (DST)
No service recovery training (NST)
High Demand (HD)
Low Demand (LD)
Probabilities
—
—
.67
.33
.60
.40
Supplementary Chapter E: Decision Analysis
E17
Exhibit E.11 Room-Night Payoff Matrix for McCord Hotels
Decision Tree Branches
ASG—DST—HD
ASG—DST—LD
ASG—NST—HD
ASG—NST—LD
NSG
Incremental Annual
Room-Night Payoffs
75,000
20,000
50,000
15,000
2,500
expected to lose 12,500 room-nights annually for its 25
hotels because some competing hotels are offering such
guarantees.
a. Construct a decision tree to help make this decision.
Using expected value, what decision does it support?
Explain.
b. If the average room-night was valued at $80 of revenue, total annual service-guarantee training cost for
all 25 hotels at a cost of $312,500, new marketing
and advertising materials at a cost of $200,000 annually, and the economic loss from service upset payouts were estimated to average less than $150,000
per year, evaluate the economics of the situation.
What other costs might be considered?
c. What are your final recommendations to Hamlet?
ENDNOTES
1
Balson, William E., Welsh, Justin L., and Wilson, Donald S., “Using Decision Analysis and Risk Analysis to Manage Utility Environmental
Risk,” Interfaces 22, no. 6, November–December 1992, pp. 126–139.
2 Carter, Virgil, and Machol, Robert E., “Optimal Strategies on Fourth Down,” Management Science 24, no. 16, December 1978, pp.
1758–1762.
3 Baird, Bruce F., Managerial Decisions Under Uncertainty, New York: John Wiley & Sons, 1989, p. 6; and Keeney, Ralph L., “Decision
Analysis: An Overview,” Operations Research 30, no. 5, September–October 1982, pp. 803–838.
4 Nichols, Nancy A., “Scientific Management at Merck: An Interview with CFO Judy Lewent,” Harvard Business Review January–February
1994, pp. 89–99.
5 Brain, M., “How Chess Computers Work,” http://www.ibs.howstuffworks.com/ibs//chess1.htm, October 20, 2004.
6 Adapted from: Feinstein, Charles D., “Deciding Whether to Test Student Athletes for Drug Use,” Interfaces 20, no. 3, May–June 1990, pp.
80–87.