mt 2351 chapter 8 decision analysis. mt 2352 decision analysis a method for determining optimal...

MT 235 1

Chapter 8

Decision Analysis

MT 235 2

Decision Analysis A method for determining optimal strategies

when faced with several decision alternatives and an uncertain pattern of future events.

MT 235 3

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

MT 235 4

Types of Decision Making Situations

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

payoff

MT 235 5


Decision making under uncertainty The decision maker is unable or unwilling to

estimate probabilities Apply a common sense criterion

MT 235 6

Decision Making Under Uncertainty

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

payoff

MT 235 7

MT 235 8


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

payoff

MT 235 9

MT 235 10


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

MT 235 11


Minimax Regret Criterion Regret - amount of loss due to making an

incorrect decision - opportunity cost

|| * ijjij VVR

nature of state j the

for result best

theis jV Where

th

*

MT 235 12

MT 235 13


Decision making under risk Expected Value Criterion

compute expected value for each decision alternative

select alternative with “best” expected value

MT 235 14

Computing Expected Value Let:

P(sj)=probability of occurrence for state sj

and N=the total number of states

MT 235 15

Computing Expected Value

Since the states are mutually exclusive and exhaustive

jsP

sPsPsPsP

j

N

j

Nj

allfor 0)(

and

1)()()()(1

21

MT 235 16


Then the expected value of any decision di is

ij

N

j

ji VsPdEV )()(1

MT 235 17

MT 235 18

Decision Trees A graphical representation of a decision

situation Most useful for sequential decisions

MT 235 19

$200K

$-20K

$150K

$20K

$100K

$60K

Large

Medium

Small

P(S1) = .3

P(S2) = .7

P(S1) = .3

P(S2) = .7

P(S1) = .3

P(S2) = .7

1

2

4

3

MT 235 20

$200K

$-20K

$150K

$20K

$100K

$60K

Large

Medium

Small

P(S1) = .3

P(S2) = .7

P(S1) = .3

P(S2) = .7

P(S1) = .3

P(S2) = .7

1

2

4

3

EV2 = 46

EV3 = 59

EV4 = 72

MT 235 21

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.

MT 235 22

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.

MT 235 23

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|

MT 235 24

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.

MT 235 25

MT 235 26

MT 235 27

Bayes Law

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.

)()|()()|()(

)()|()()|(

)()|()|(

BPBAPBPBAPAP

BPBAPBPBAP

BPBAPABP

MT 235 28

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:

)(

)(

2

1

SUP

and

SFP

Suppose that

75.)(

80.)(

2

1

SUP

and

SFP

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:

)(

)(

2

1

USP

and

FSP

MT 235 29

Blank page for work

MT 235 30

Bayes Law can also be computed using a tabular approach as in the tables below.

States of Nature

jS

Prior Probabilities

)( jSP

Conditional Probabilities

)( jSFP

Joint Probabilities)()()( jj SPSFPSFP

Posterior Probabilities

)( FSP j

1S578.415.

24.

2S

422.415.175.

415.175.24.)( FP

Bayes Law Using a Tabular Approach (finding posteriors for F given)

.30 .80 (.80)(.30)=.24

.70 .25 (.25)(.70)=.175

Note: The two numbers above are complements


States of Nature

jS

Prior Probabilities

)( jSP

Conditional Probabilities

)( jSUP

Joint Probabilities)()()( jj SPSUPSUP


)( USP j

1S 103.585.06.

2S

897.585.525.

585.525.06.)( UP

Bayes Law Using a Tabular Approach (finding posteriors for U given)

.30 .20 (.20)(.30)=.06

.70 .75 (.75)(.70)=.525



MT 235 31

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.

MT 235 32

$200K

$-20K

$150K

$20K

$100K

$60K

$200K

$-20K

$150K

$20K

$100K

$60K

P(S1|F)= .578

P(S2|F)=.422

P(S1|F)= .578

P(S1|F)= .578

P(S2|F)=.422

P(S2|F)=.422

P(S1|U)= .103

P(S1|U)= .103

P(S1|U)=.103

P(S2|U)=.897

P(S2|U)=.897

P(S2|U)=.897

Large

Medium

Small

Large

Medium

Small

Favorable Forecast

Unfavorable Forecast

EV2= 107.16

EV3= 64.12

EV4= $107.16K

EV5= $95.14K

EV6= $83.12K

EV7= $2.66K

EV8= $33.39K

EV9= $64.12K

EV1= $81.98K

P(U)= .585

P(F)= .415

1

2

3

4

5

6

7

8

9

Do Survey

$72KDon’t do

Survey

MT 235 33

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|

MT 235 34

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

100EVPI

EVSIE

MT 235 35

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(S 1)=.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.

MT 235 36

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

jS

Prior Probabilities

)( jSPConditional Probabilities

)( jSFP

Joint Probabilities

)()()( jj SPSFPSFP


)( FSP j

Bayes Law Using a Tabular Approach (finding posteriors)

States of Nature

jS

Prior Probabilities

)( jSPConditional Probabilities

)( jSUP

Joint Probabilities

)()()( jj SPSUPSUP


)( USP j

Bayes Law Using a Tabular Approach (finding posteriors)

MT 235 37

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?

MT 235 38

$140K

$-12K

$25K

$35K

d1

d2

d1

d2

Favorable Forecast

Unfavorable Forecast

EV4= $124.04K

EV5= $26.05K

EV6= $18.70K

EV7= $32.98K

EV1= $59.02

P(U)= .714

P(F)= .286

1

2

3

4

5

6

7

Hire ERI

$48.8KDon’t hire

ERI

$140K

$-12K

$25K

$35K

P(S1|F)= .895

P(S2|F)=.105

P(S1|F)= .895

P(S2|F)=.105

P(S1|U)= .202

P(S1|U)= .202

P(S2|U)=.798

P(S2|U)=.798

MT 235 39

Decision Making with Cost DataConsider 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. 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.

mt 2351 chapter 8 decision analysis. mt 2352 decision analysis a method for determining optimal...

Documents