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MT 2351 Chapter 8 Decision Analysis

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MT 2352 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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MT 2353 The Decision Analysis Approach Identify the decision alternatives - d i Identify possible future events - s j mmutually exclusive - only one state can occur eexhaustive - one of the states must occur Determine the payoff associated with each decision and each state of nature - V ij Apply a decision criterion

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MT 2354 Types of Decision Making Situations Decision making under certainty sstate of nature is known ddecision is to choose the alternative with the best payoff

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MT 2355 Types of Decision Making Situations Decision making under uncertainty TThe decision maker is unable or unwilling to estimate probabilities AApply a common sense criterion

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MT 2356 Decision Making Under Uncertainty Maximax Criterion (for profits) - optimistic llist maximum payoff for each alternative cchoose alternative with the largest maximum payoff

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MT 2357

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8 Decision Making Under Uncertainty Maximin Criterion (for profits) - pessimistic llist minimum payoff for each alternative cchoose alternative with the largest minimum payoff

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MT 2359

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10 Decision Making Under Uncertainty Minimax Regret Criterion ccalculate the regret for each alternative and each state llist the maximum regret for each alternative cchoose the alternative with the smallest maximum regret

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

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MT 23512

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MT Types of Decision Making Situations Decision making under risk Expected Value Criterion ccompute expected value for each decision alternative sselect alternative with “best” expected value

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MT Computing Expected Value Let: PP(s j )=probability of occurrence for state s j and NN=the total number of states

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

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MT Types of Decision Making Situations Then the expected value of any decision d i is

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MT 23517

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

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

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MT $200K $-20K $150K $20K $100K $60K Large Medium Small P(S 1 ) =.3 P(S 2 ) =.7 P(S 1 ) =.3 P(S 2 ) =.7 P(S 1 ) =.3 P(S 2 ) = EV 2 = 46 EV 3 = 59 EV 4 = 72

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MT Decision Making Under Risk: Another Criterion Expected Regret Criterion CCompute the regret table CCompute the expected regret for each alternative CChoose 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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MT 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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MT 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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MT 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. DecisionState of Nature Plant SizeHigh Demand (S 1 )Low Demand (S 2 ) Large (d 1 )$200 K$-20 K Medium (d 2 )$150 K$ 20 K Small (d 3 )$100 K$ 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][1] 4.Now assume the decision makers have probability information about the states of nature. Assume that P(S 1 )=.3, and P(S 2 )=.7. Analyze the problem using the expected value criterion.[2][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][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][2] Note that that P(S 1 ) and P(S 2 ) are complements, so that that P(S 1 )+P(S 2 )=1.0.

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MT 23525

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MT 23526

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MT 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.

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MT 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 S 1 ) Let U be the event: MRI forecasts low demand (i.e., MRI forecasts S 2 ) 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:

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MT Blank page for work

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MT Bayes Law can also be computed using a tabular approach as in the tables below. States of Nature Prior Probabilities Conditional Probabilities Joint Probabilities Posterior Probabilities Bayes Law Using a Tabular Approach (finding posteriors for F given).30.80(.80)(.30)= (.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)= (.75)(.70)=.525 Note: The two numbers above are complements

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MT 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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MT $200K $-20K $150K $20K $100K $60K $200K $-20K $150K $20K $100K $60K P(S 1 |F)=.578 P(S 2 |F)=.422 P(S 1 |F)=.578 P(S 2 |F)=.422 P(S 1 |U)=.103 P(S 2 |U)=.897 Large Medium Small Large Medium Small Favorable Forecast Unfavorable Forecast EV 2 = EV 3 = EV 4 = $107.16K EV 5 = $95.14K EV 6 = $83.12K EV 7 = $2.66K EV 8 = $33.39K EV 9 = $64.12K EV 1 = $81.98K P(U)=.585 P(F)= Do Survey $72K Don’t do Survey

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MT 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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MT 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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MT 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. DecisionState of Nature Strong Economy (S 1 )Weak Economy (S 2 ) Introduce New Product (d 1 )$140K$-12 K Keep Old Product (d 2 )$ 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.

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

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MT a. 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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MT $140K $-12K $25K $35K d1d1 d2d2 d1d1 d2d2 Favorable Forecast Unfavorable Forecast EV 4 = $124.04K EV 5 = $26.05K EV 6 = $18.70K EV 7 = $32.98K EV 1 = $59.02 P(U)=.714 P(F)= Hire ERI $48.8K Don’t hire ERI $140K $-12K $25K $35K P(S 1 |F)=.895 P(S 2 |F)=.105 P(S 1 |F)=.895 P(S 2 |F)=.105 P(S 1 |U)=.202 P(S 2 |U)=.798

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MT 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 DecisionS1S1 S2S2 S3S3 d1d1 100 K$40 K$100 K$ d2d2 30 K$110 K$ d3d3 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(S 1 )=.40 and P(S 2 )=.20 Apply the expected value criterion. 5. Compute EVPI. 6. Compute EOL.

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