Chapter 8 Decision Analysis MT 235

Decision Analysis A method for determining optimal strategies when faced with several decision alternatives and an uncertain pattern of future events. MT 235

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

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

Types of Decision Making Situations Decision making under uncertainty The decision maker is unable or unwilling to estimate probabilities Apply a common sense criterion MT 235

Decision Making Under Uncertainty Maximax Criterion (for profits) - optimistic list maximum payoff for each alternative choose alternative with the largest maximum payoff MT 235

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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 MT 235

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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 MT 235

Decision Making Under Uncertainty Minimax Regret Criterion Regret - amount of loss due to making an incorrect decision - opportunity cost MT 235

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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 MT 235

Computing Expected Value Let: P(sj)=probability of occurrence for state sj and N=the total number of states MT 235

Computing Expected Value Since the states are mutually exclusive and exhaustive MT 235

Types of Decision Making Situations Then the expected value of any decision di is MT 235

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Decision Trees A graphical representation of a decision situation Most useful for sequential decisions MT 235

$200K 2 $-20K $150K 3 1 $20K $100K 4 $60K P(S1) = .3 Large P(S2) = .7 Medium 3 1 $20K P(S2) = .7 Small P(S1) = .3 $100K 4 $60K P(S2) = .7 MT 235

$200K 2 $-20K $150K 1 3 $20K $100K 4 $60K EV2 = 46 P(S1) = .3 Large Medium 1 3 $20K P(S2) = .7 Small EV4 = 72 P(S1) = .3 $100K 4 $60K P(S2) = .7 MT 235

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

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

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

Analyze this decision using the maximax (optimistic) approach. 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 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. MT 235

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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. MT 235

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: 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: MT 235

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Bayes Law Using a Tabular Approach (finding posteriors for F given) 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) .30 .80 (.80)(.30)=.24 .70 .25 (.25)(.70)=.175 Note: The two numbers above are complements States of Nature Prior Probabilities Conditional Probabilities Joint Probabilities Posterior Probabilities Bayes Law Using a Tabular Approach (finding posteriors for U given) .30 .20 (.20)(.30)=.06 .70 .75 (.75)(.70)=.525 Note: The two numbers above are complements Posterior Probabilities States of Nature Prior Probabilities Conditional Probabilities Joint Probabilities MT 235

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

$200K 4 Large $-20K $150K Medium 5 2 $20K Favorable Forecast $100K 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 $200K $-20K $150K $20K $100K $60K EV1= $81.98K P(S1|U)= .103 P(S1|U)=.103 P(S2|U)=.897 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 MT 235

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

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 MT 235

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 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. MT 235

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 Prior Probabilities Conditional Probabilities Joint Probabilities Posterior Probabilities Bayes Law Using a Tabular Approach (finding posteriors) States of Nature Prior Probabilities Conditional Probabilities Joint Probabilities Posterior Probabilities MT 235

7a. Determine LLC’s best decision strategy 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

$140K 4 d1 $-12K 2 Favorable Forecast $25K d2 5 $35K 1 $140K $-12K 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 $-12K $25K $35K EV1= $59.02 P(S1|U)= .202 P(S2|U)=.798 d1 6 P(U)= .714 Unfavorable Forecast 3 Hire ERI EV7= $32.98K d2 7 Don’t hire ERI $48.8K MT 235

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$ d2 30 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 235