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Chapter 8 Decision Analysis Decision Analysis A method for determining optimal strategies when faced with several decision alternatives and an uncertain pattern of future events. The Decision Analysis Approach Identify the decision alternatives - d i Identify possible future events - s j

Chapter 8

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Decision Analysis

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- A method for determining optimal strategies when faced with several decision alternatives and an uncertain pattern of future events.

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- 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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- Decision making under certainty
- state of nature is known
- decision is to choose the alternative with the best payoff

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- Decision making under uncertainty
- The decision maker is unable or unwilling to estimate probabilities
- Apply a common sense criterion

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- Maximax Criterion (for profits) - optimistic
- list maximum payoff for each alternative
- choose alternative with the largest maximum payoff

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- Maximin Criterion (for profits) - pessimistic
- list minimum payoff for each alternative
- choose alternative with the largest minimum payoff

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- 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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- Minimax Regret Criterion
- Regret - amount of loss due to making an incorrect decision - opportunity cost

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- Decision making under risk
- Expected Value Criterion
- compute expected value for each decision alternative
- select alternative with “best” expected value

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- Let:
- P(sj)=probability of occurrence for state sj

- and
- N=the total number of states

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- Since the states are mutually exclusive and exhaustive

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- Then the expected value of any decision di is

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- A graphical representation of a decision situation
- Most useful for sequential decisions

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$200K

P(S1) = .3

2

Large

$-20K

P(S2) = .7

$150K

P(S1) = .3

Medium

1

3

$ 20K

P(S2) = .7

Small

$100K

P(S1) = .3

4

$ 60K

P(S2) = .7

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EV2 = 46

$200K

P(S1) = .3

2

Large

$-20K

P(S2) = .7

EV3 = 59

$150K

P(S1) = .3

Medium

1

3

P(S2) = .7

$ 20K

Small

EV4 = 72

$100K

P(S1) = .3

4

$ 60K

P(S2) = .7

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- 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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- 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 – 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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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.
- 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.[1]
- 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] Note that that P(S1) and P(S2) are complements, so that that P(S1)+P(S2)=1.0.

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- 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 Fbe the event: MRI forecasts high demand (i.e., MRI forecasts S1)

Let Ube the event: MRI forecasts low demand (i.e., MRI forecasts S2)

The conditional probabilities, which quantify MRI’s accuracy, would be:

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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States of Nature

Prior Probabilities

Conditional Probabilities

Joint Probabilities

Posterior Probabilities

Posterior Probabilities

States of Nature

Prior Probabilities

Conditional Probabilities

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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$200K

P(S1|U)= .103

$-20K

P(S2|U)=.897

P(S1|U)= .103

$150K

$20K

P(S2|U)=.897

$100K

P(S1|U)=.103

$60K

P(S2|U)=.897

EV4= $107.16K

$200K

P(S1|F)= .578

4

Large

$-20K

P(S2|F)=.422

EV2= 107.16

EV5= $95.14K

$150K

P(S1|F)= .578

Medium

5

2

$20K

Favorable Forecast

P(S2|F)=.422

EV6= $83.12K

$100K

P(S1|F)= .578

Small

6

$60K

P(F)= .415

P(S2|F)=.422

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

Don’t do

Survey

$72K

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- 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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- 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 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.

States of Nature

Prior Probabilities

Conditional Probabilities

Joint Probabilities

Posterior Probabilities

States of Nature

Prior Probabilities

Conditional Probabilities

Joint Probabilities

Posterior Probabilities

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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

$140K

P(S1|F)= .895

4

d1

$-12K

P(S2|F)=.105

2

Favorable Forecast

EV5= $26.05K

P(S1|F)= .895

$ 25K

d2

5

$ 35K

P(F)= .286

P(S2|F)=.105

1

EV6= $18.70K

$140K

EV1= $59.02

P(S1|U)= .202

d1

6

$ -12K

P(U)= .714

P(S2|U)=.798

Unfavorable Forecast

3

Hire ERI

EV7= $32.98K

P(S1|U)= .202

$ 25K

d2

7

$ 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.

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.

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