Chapter 8
Decision Analysis
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Decision Analysis

A method for determining optimal strategies
when faced with several decision alternatives
and an uncertain pattern of future events.
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The Decision Analysis Approach
Identify the decision alternatives - di
 Identify possible future events - sj

 mutually
exclusive - only one state can occur
 exhaustive - one of the states must occur
Determine the payoff associated with each
decision and each state of nature - Vij
 Apply a decision criterion

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Types of Decision Making Situations

Decision making under certainty
 state
of nature is known
 decision is to choose the alternative with the best
payoff
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Types of Decision Making Situations

Decision making under uncertainty
 The
decision maker is unable or unwilling to
estimate probabilities
 Apply a common sense criterion
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Decision Making Under Uncertainty

Maximax Criterion (for profits) - optimistic
 list
maximum payoff for each alternative
 choose alternative with the largest maximum
payoff
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Decision Making Under Uncertainty

Maximin Criterion (for profits) - pessimistic
 list
minimum payoff for each alternative
 choose alternative with the largest minimum
payoff
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Decision Making Under Uncertainty

Minimax Regret Criterion
 calculate
the regret for each alternative and each
state
 list the maximum regret for each alternative
 choose the alternative with the smallest maximum
regret
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Decision Making Under Uncertainty

Minimax Regret Criterion
 Regret
- amount of loss due to making an
incorrect decision - opportunity cost
Rij | V
* j
 Vij |
Where V * j is the
best result for
th
the j state of nature
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Types of Decision Making Situations
Decision making under risk
 Expected Value Criterion

 compute
expected value for each decision
alternative
 select alternative with “best” expected value
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Computing Expected Value

Let:
 P(sj)=probability of

occurrence for state sj
and
 N=the
total number of states
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Computing Expected Value

Since the states are mutually exclusive and
exhaustive
N
 P( s )  P( s )  P( s )      P( s )  1
j
1
2
N
j 1
and
P( sj )  0 for all j
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Types of Decision Making Situations

Then the expected value of any decision di is
N
EV (di )   P(sj )Vij
j 1
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Decision Trees
A graphical representation of a decision
situation
 Most useful for sequential decisions

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P(S1) = .3
Large
$200K
2
Medium
P(S2) = .7
$-20K
P(S1) = .3
$150K
P(S2) = .7
$20K
3
1
Small
P(S1) = .3
$100K
4
$60K
P(S2) = .7
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EV2 = 46
Large
$200K
2
EV3 = 59
Medium
1
P(S1) = .3
P(S2) = .7
$-20K
P(S1) = .3
$150K
P(S2) = .7
$20K
3
Small
EV4 = 72
P(S1) = .3
$100K
4
$60K
P(S2) = .7
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Decision Making Under Risk:
Another Criterion

Expected Regret Criterion
 Compute
the regret table
 Compute the expected regret for each alternative
 Choose the alternative with the smallest expected
regret

The expected regret criterion will always yield
the same decision as the expected value
criterion.
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Expected Regret Criterion
The expected regret for the preferred decision
is equal to the Expected Value of Perfect
Information - EVPI
 EVPI is the expected value of knowing which
state will occur.

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EVPI – Alternative to Expected Regret
EVPI – Expected Value of Perfect Information
 EVwPI – Expected Value with Perfect
Information about the States of Nature
 EVwoPI – Expected Value without Perfect
Information about the States of Nature
 EVPI=|EVwPI-EVwoPI|

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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
State of Nature
Plant Size
High Demand (S1)
Low Demand (S2)
Large (d1)
$200 K
$-20 K
Medium (d2)
$150 K
$ 20 K
Small (d3)
$100 K
$ 60 K
1.Analyze this decision using the maximax (optimistic) approach.
2.Analyze this decision using the maximin (conservative) approach.
3.Analyze this decision using the minimax regret criterion.[1]
4.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.
[1] 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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Bayes Law
P( A | B) P( B)
P( B | A) 
P( A | B) P( B)  P( A | B ) P( B )
P( A)  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 initial
estimate of P(B) , and then we get some additional
information about whether A happens or not, and then we
use Bayes Law to compute this revised probability of B.
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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 )
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Blank page for work
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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
Posterior
Nature
Probabilities Probabilities
P( F  S )  P( F S j ) P(S j ) Probabilities
Sj
P(S j )
P( F S j )
P(S j F )
.24
 .578
.30
.80
(.80)(.30)=.24
.415
S1
S2
.70
.25
(.25)(.70)=.175
P( F )  .24  .175  .415
Note: The two
numbers above
are
complements
.175
.415
 .422
Note: The two
numbers above are
complements
Bayes Law Using a Tabular Approach (finding posteriors for U given)
States of
Nature
Sj
S1
S2
Prior
Conditional Joint Probabilities
Probabilities Probabilities P(U  S )  P(U S j ) P(S j )
P(S j )
P(U S )
j
.30
.70
Note: The two
numbers above
are
complements
.20
.75
(.20)(.30)=.06
(.75)(.70)=.525
P(U )  .06  .525  .585
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Posterior
Probabilities
P(S j U )
.06
.525
.585
.585
 .103
 .897
Note: The two
numbers above are
complements
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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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EV4= $107.16K
P(S1|F)= .578
Large
EV2= 107.16
2
Favorable
Forecast
4
EV5= $95.14K
Medium
5
EV6= $83.12K
Small
6
P(F)= .415
1
EV7= $2.66K
P(S2|F)=.422
P(S1|F)= .578
$150K
$20K
$100K
$60K
$200K
P(S2|U)=.897
$-20K
EV8= $33.39K P(S |U)= .103
1
$150K
7
P(U)= .585
Medium
3
EV3= 64.12
$-20K
P(S1|U)= .103
Large
Do
Survey
P(S1|F)= .578
P(S2|F)=.422
EV1= $81.98K
Unfavorable
Forecast
P(S2|F)=.422
$200K
8
EV9= $64.12K
Small
P(S2|U)=.897
$20K
P(S1|U)=.103
$100K
9
P(S2|U)=.897
$60K
Don’t do
Survey
$72K
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Expected Value of Sample Information
– EVSI
EVSI – Expected Value of Sample Information
 EVwSI – Expected Value with Sample
Information about the States of Nature
 EVwoSI – Expected Value without Sample
Information about the States of Nature
 EVSI=|EVwSI-EVwoSI|

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Efficiency of Sample Information – E

Perfect Information has an efficiency rating of 100%, the
efficiency rating E for sample information is computed as
follows:
EVSI
E
 100
EVPI

Note: Low efficiency ratings for sample information might
lead the decision maker to look for other types of information
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Example 2: The LaserLens Company (LLC) is considering introducing a new product, which to some extent will replace an existing
product. LLC is unsure about whether to do this because the financial results depend upon the state of the economy. The payoff table
below gives the profits in K$ for each decision and each economic state.
Decision
State of Nature
Strong Economy (S1)
Weak Economy (S2)
Introduce New Product (d1)
$140K
$-12 K
Keep Old Product (d2)
$ 25 K
$ 35 K
1.Analyze this decision using the maximax (optimistic) approach.
2.Analyze this decision using the maximin (conservative) approach.
3.Analyze this decision using the minimax regret criterion.
4.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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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 Nature
Sj
Prior Probabilities
P(S j )
Conditional
Probabilities
P( F S j )
Joint Probabilities
Posterior Probabilities
P( F  S )  P( F S j ) P(S j ) P ( S j F )
Bayes Law Using a Tabular Approach (finding posteriors)
States of Nature
Sj
Prior Probabilities
P(S j )
Conditional
Probabilities
P(U S j )
Joint Probabilities
Posterior Probabilities
P(U  S )  P(U S j ) P(S j ) P ( S j U )
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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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EV4= $124.04K
P(S1|F)= .895
4
d1
Favorable
Forecast
P(S2|F)=.105
EV5= $26.05K
P(S1|F)= .895
5
P(F)= .286
$35K
EV6= $18.70K
EV1= $59.02
P(U)= .714
$25K
P(S2|F)=.105
1
Hire
ERI
$-12K
2
d2
Unfavorable
Forecast
$140K
d1
6
P(S1|U)= .202
P(S2|U)=.798
$140K
$-12K
3
EV7= $32.98K
d2
P(S1|U)= .202
7
$25K
$35K
P(S2|U)=.798
Don’t hire
ERI
$48.8K
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Decision Making with Cost Data
Consider the following payoff table, which gives three decisions and their costs under each state of nature. The company’s objective is to
minimize cost.
State of Nature
Decision
S1
S2
S3
d1
100 K$
40 K$
100 K$
d2
30 K$
110 K$
110 K$
d3
60 K$
75 K$
120 K$
1.
2.
3.
4.
5.
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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