Decision Analysis
Example 1: Mass. Bay Production (MBP) is planning a new manufacturing facility for a
new product. MBP is considering three plant sizes, small, medium, and large. The
demand for the product is not fully known, but MBP assumes two possibilities, 1. High
demand, and 2. Low demand. The profits (payoffs) associated with each plant size and
demand level is given in the table below.
Decision
Plant Size
State of Nature
High Demand (S1)
$200 K
Low Demand (S2)
$-20 K
$150 K
$ 20 K
$100 K
$ 60 K
Large (d1)
Medium (d2)
Small (d3)
1.
2.
3.
4.
Analyze this decision using the maximax (optimistic) approach.
Analyze this decision using the maximin (conservative) approach.
Analyze this decision using the minimax regret criterion.1
Now assume the decision makers have probability information about the states of
nature. Assume that P(S1)=.3, and P(S2)=.7. Analyze the problem using the expected
value criterion.2
5. How much would you be willing to pay in this example for perfect information about
the actual demand level? (EVPI)
6. Compute the expected opportunity loss (EOL) for this problem. Compare EOL and
EVPI.
D.W. Bunn discusses the regret criterion as follows. “The minimax regret criterion often has considerable
appeal, particularly wherever decision makers tend to be evaluated with hindsight. Of course, hindsight is
an exact science, and our actions are sometimes unfairly compared critically with what might have been
done. Many organizations seem implicitly to review and reward their employees in this way.” Bunn, D.
W., Applied Decision Analysis.
2
Note that that P(S1) and P(S2) are complements, so that that P(S1)+P(S2)=1.0.
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Decision Analysis
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Decision Analysis
7. Analyze the same problem using a decision tree.
Large
P(S1) =.3
$200K
P(S2)=.7
$-20K
P(S1) =.3
$150 K
P(S2)=.7
$20 K
Medium
Small
P(S1) =.3
$100 K
P(S2)=.7
$60
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Decision Analysis
Now suppose that MBP has the option of doing market research to get a better estimate of
the likely level of demand. Market Research Inc. (MRI) has done considerable research
in this area and established a documented track record for forecasting demand. Their
accuracy is stated in terms of probabilities, conditional probabilities, to be exact.
Let F be the event: MRI forecasts high demand (i.e., MRI forecasts S1)
Let U be the event: MRI forecasts low demand (i.e., MRI forecasts S2)
The conditional probabilities, which quantify MRI’s accuracy, would be:
P ( F S1 )
and
P (U S 2 )
Suppose that
P( F S1 )  .80
and
P(U S 2 )  .75
This would say that 80% of the time when demand is high, MRI forecasts high demand.
In addition, 75% of the time when the demand is low, MRI forecasts low demand. In the
calculations, which follow, however, we will need to reverse these conditional
probabilities. That is, we will need to know:
P ( S1 F )
and
P(S 2 U )
These are the probabilities that a certain event will occur, given a forecast. To calculate
these probabilities we will need Bayes Law. If we have two events, A and B, (which we
assume are not complements) and we know: P( B), P( A B), and P( A B) . Then we can
find P( B A) using Bayes Law:
P( B A) 
P( A B) P( B)
(1)
P( A B) P( B)  P( A B) P( B)
In this equation, P(B) is called the prior probability of B and P( B A) is called the
posterior, or sometimes the revised probability of B. The idea here is that we have some
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Decision Analysis
initial estimate of P(B) , and then we get some additional information about whether A
happens or not, and then we use Bayes to compute this revised probability of B.
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Decision Analysis
Bayes Law can also be computed using a tabular approach as in the tables below.
Bayes Law Using a Tabular Approach (finding posteriors for F given)
States of Prior
Conditional
Joint Probabilities
Nature
Probabilities Probabilities P( F  S )  P( F S j ) P(S j )
Sj
P( S j )
P( F S j )
Posterior
Probabilities
P(S j F )
S1
.30
.80
(.80)(.30)=.24
S2
.70
.25
(.25)(.70)=.175
.175
P( F )  .24  .175  .415
Note: The two
numbers above are
complements
Note: The two
numbers above
are
complements
Bayes Law Using a Tabular Approach (finding posteriors for U given)
States of Prior
Conditional
Joint Probabilities
Nature
Probabilities Probabilities P(U  S )  P(U S j ) P(S j )
Sj
P( S j )
P(U S j )
.24
.415
.415
 .578
 .422
Posterior
Probabilities
P(S j U )
S1
.30
.20
(.20)(.30)=.06
S2
.70
.75
(.75)(.70)=.525
.525
P(U )  .06  .525  .585
Note: The two
numbers above are
complements
Note: The two
numbers above
are
complements
.06
.585
.585
 .103
 .897
Now, using Bayes Law, we can construct a new decision tree, which will give us a
decision strategy: Should we pay MRI for the market research? If we do not do the
market research, what should our decision be? If we do the market research and get an
indication of high demand, what should our decision be? If we get an indication of low
demand, what should our decision be? We will use a decision tree as shown below to
determine this strategy.
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Decision Analysis
7a. Finish the decision tree below and state the best strategy for MBP.
P(S1 |F) = .578
P(S2|F)= .422
P(S1 |F) = .578
Large
$200K
$-20K
$150K
Medium
Favorable
Forecast
P(S2|F)= .422
P(S1 |F) = .578
Small
P(S2|F)= .422
Unfavorable
Forecast
Large
Medium
Small
b. How much should MBP be willing to pay for MRI’s market research? (EVSI).
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$20K
$100K
$60K
Decision Analysis
Example 2: The LaserLens Company (LLC) is considering introducing a new product,
which to some extent will replace an existing product. LLC is unsure about whether to
do this because the financial results depend upon the state of the economy. The payoff
table below gives the profits in K$ for each decision and each economic state.
Decision
State of Nature
Strong Economy (S1)
$140K
Weak Economy (S2)
$-12 K
$ 25 K
$ 35 K
Introduce New Product (d1)
Keep Old Product (d2)
1.
2.
3.
4.
Analyze this decision using the maximax (optimistic) approach.
Analyze this decision using the maximin (conservative) approach.
Analyze this decision using the minimax regret criterion.
Now assume the decision makers have probability information about the states of
nature. Assume that P(S1)=.4. Analyze the problem using the expected value
criterion.
5. How much would you be willing to pay in this example for perfect information
about the actual state of the economy? (EVPI)
6. Compute the expected opportunity loss (EOL) for this problem. Compare EOL
and EVPI.
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Decision Analysis
Now suppose that LLC has the option of contracting with an economic forecasting firm
to get a better estimate of the future state of the economy. Economics Research Inc.
(ERI) is the forecasting firm being considered. After investigating ERI’s forecasting
record, it is found that in the past, 64% of the time when the economy was strong, ERI
predicted a strong economy. Also, 95% of the time when the economy was weak, ERI
predicted a weak economy.
Bayes Law Using a Tabular Approach (finding posteriors)
States of Prior
Conditional
Joint Probabilities
Nature
Probabilities Probabilities P( F  S )  P( F S j ) P(S j )
Sj
P( S j )
P( F S j )
Posterior
Probabilities
P(S j F )
Bayes Law Using a Tabular Approach (finding posteriors)
States of Prior
Conditional
Joint Probabilities
Nature
Probabilities Probabilities P(U  S )  P(U S j ) P(S j )
Sj
P( S j )
P(U S j )
Posterior
Probabilities
P(S j U )
7a. Determine LLC’s best decision strategy. Should they hire ERI or go ahead without
additional information? If they buy the economic forecast, what should their subsequent
decision strategy be?
7b. Determine how much LLC should be willing to pay (maximum) to ERI for an
economic forecast.
7c. What is the efficiency of the information provided by ERI?
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Decision Analysis
F o re ca s t
o f S tron g
E c on om y
F o re ca s t
of w eak
E c on om y
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Decision Analysis
Decision Making with Cost Data
Consider the following payoff table, which gives three decisions and their costs under
each state of nature. The company’s objective is to minimize cost.
Decision
d1
d2
d3
1.
2.
3.
4.
5.
State of Nature
S2
40 K$
110 K$
75 K$
S1
100 K$
30 K$
60 K$
S3
100 K$
110 K$
120 K$
Apply the optimistic (minimin cost) criterion.
Apply the conservative (minimax cost) criterion.
Apply the minimax regret criterion.
Assume that P(S1)=.40 and P(S2)=.20 Apply the expected value criterion.
Compute EVPI.
6. Compute EOL.
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