Decision Analysis

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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

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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
• Nthe total number of states

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Computing Expected Value

• Since the states are mutually exclusive and
  exhaustive

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Types of Decision Making Situations

• Then the expected value of any decision di is

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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
200K
2
Large
-20K
P(S2) .7
150K
P(S1) .3
Medium
3
1
20K
P(S2) .7
Small
P(S1) .3
100K
4
60K
P(S2) .7
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EV2 46
P(S1) .3
200K
2
Large
-20K
P(S2) .7
EV3 59
150K
P(S1) .3
Medium
1
3
20K
P(S2) .7
Small
P(S1) .3
EV4 72
100K
4
60K
P(S2) .7
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Decision Making Under RiskAnother 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
• EVPIEVwPI-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.

• 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
• How much would you be willing to pay in this
  example for perfect information about the actual
  demand level? (EVPI)
• 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

• In this equation, P(B) is called the prior
  probability of B and P(BA) 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 MRIs accuracy,
would be
Suppose that
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
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Bayes Law can also be computed using a tabular
approach as in the tables below.
States of Nature
Joint Probabilities
Prior Probabilities
Conditional Probabilities
Posterior Probabilities
Posterior Probabilities
Prior Probabilities
Conditional Probabilities
States of Nature
Joint Probabilities
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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
200K
P(S1F) .578
4
Large
-20K
P(S2F).422
EV2 107.16
EV5 95.14K
150K
P(S1F) .578
Medium
5
2
20K
P(S2F).422
Favorable Forecast
EV6 83.12K
100K
P(S1F) .578
Small
6
60K
P(S2F).422
P(F) .415
EV7 2.66K
1
EV1 81.98K
Large
7
P(U) .585
Unfavorable Forecast
EV8 33.39K
Medium
8
3
Do Survey
EV3 64.12
EV9 64.12K
Small
9
Dont 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
• EVSIEVwSI-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
• 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.

• 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.
• How much would you be willing to pay in this
  example for perfect information about the actual
  state of the economy? (EVPI)
• 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 ERIs 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.
Prior Probabilities
States of Nature
Conditional Probabilities
Joint Probabilities
Posterior Probabilities
States of Nature
Conditional Probabilities
Joint Probabilities
Posterior Probabilities
Prior Probabilities
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7a. Determine LLCs 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
140K
P(S1F) .895
4
d1
-12K
P(S2F).105
2
Favorable Forecast
EV5 26.05K
P(S1F) .895
25K
d2
5
35K
P(F) .286
P(S2F).105
1
EV6 18.70K
EV1 59.02
d1
6
P(U) .714
Unfavorable Forecast
3
Hire ERI
EV7 32.98K
d2
7
Dont 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 companys objective is to minimize
cost.
1. Apply the optimistic (minimin cost)
criterion. 2. Apply the conservative (minimax
cost) criterion. 3. Apply the minimax regret
criterion. 4. Assume that P(S1).40 and
P(S2).20 Apply the expected value
criterion. 5. Compute EVPI. 6. Compute EOL.