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Decision Analysis Managers often must make decisions in environments that are fraught with uncertainty. Some Examples A manufacturer introducing a new product into the marketplace What will be the reaction of potential customers? How much should be produced? Should the product be test-marketed? How much advertising is needed? A financial firm investing in securities Which are the market sectors and individual securities with the best prospects? Where is the economy headed? How about interest rates? How should these factors affect the investment decisions?

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Decision Analysis Managers often must make decisions in environments that are fraught with uncertainty. Some Examples A government contractor bidding on a new contract. What will be the actual costs of the project? Which other companies might be bidding? What are their likely bids? An agricultural firm selecting the mix of crops and livestock for the season. What will be the weather conditions? Where are prices headed? What will costs be? An oil company deciding whether to drill for oil in a particular location. How likely is there to be oil in that location? How much? How deep will they need to drill? Should geologists investigate the site further before drilling?

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**The Goferbroke Company Problem**

The Goferbroke Company develops oil wells in unproven territory. A consulting geologist has reported that there is a one-in-four chance of oil on a particular tract of land. Drilling for oil on this tract would require an investment of about $100,000. If the tract contains oil, it is estimated that the net revenue generated would be approximately $800,000. Another oil company has offered to purchase the tract of land for $90,000. Question: Should Goferbroke drill for oil or sell the tract?

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**Prospective Profits Profit Status of Land Oil Dry Alternative**

Drill for oil $700,000 –$100,000 Sell the land 90,000 Chance of status 1 in 4 3 in 4 Table 9.1 Prospective profits for the Goferbroke Company.

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**Decision Analysis Terminology**

The decision maker is the individual or group responsible for making the decision. The alternatives are the options for the decision to be made. The outcome is affected by random factors outside the control of the decision maker. These random factors determine the situation that will be found when the decision is executed. Each of these possible situations is referred to as a possible state of nature. The decision maker generally will have some information about the relative likelihood of the possible states of nature. These are referred to as the prior probabilities. Each combination of a decision alternative and a state of nature results in some outcome. The payoff is a quantitative measure of the value to the decision maker of the outcome. It is often the monetary value.

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**Prior Probabilities State of Nature Prior Probability**

The tract of land contains oil 0.25 The tract of land is dry (no oil) 0.75 Table 9.2 Prior probabilities for the first Goferbroke Co. problem.

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**Payoff Table (Profit in $Thousands)**

State of Nature Alternative Oil Dry Drill for oil 700 –100 Sell the land 90 Prior probability 0.25 0.75 Table 9.3 Payoff table (profit in $thousands) for the first Goferbroke Co. problem.

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The Maximax Criterion The maximax criterion is the decision criterion for the eternal optimist. It focuses only on the best that can happen. Procedure: Identify the maximum payoff from any state of nature for each alternative. Find the maximum of these maximum payoffs and choose this alternative. State of Nature Alternative Oil Dry Maximum in Row Drill for oil 700 –100 700 Maximax Sell the land 90 Table 9.4 Application of the maximax criterion to the first Goferbroke Co. problem.

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The Maximin Criterion The maximin criterion is the decision criterion for the total pessimist. It focuses only on the worst that can happen. Procedure: Identify the minimum payoff from any state of nature for each alternative. Find the maximum of these minimum payoffs and choose this alternative. State of Nature Alternative Oil Dry Minimum in Row Drill for oil 700 –100 Sell the land 90 90 Maximin Table 9.5 Application of the maximin criterion to the first Goferbroke Co. problem.

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**The Maximum Likelihood Criterion**

The maximum likelihood criterion focuses on the most likely state of nature. Procedure: Identify the state of nature with the largest prior probability Choose the decision alternative that has the largest payoff for this state of nature. State of Nature Alternative Oil Dry Drill for oil 700 –100 Sell the land 90 90 Step 2: Maximum Prior probability 0.25 0.75 Step 1: Maximum Table 9.6 Application of the maximum likelihood criterion to the first Goferbroke Co. problem.

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Bayes’ Decision Rule Bayes’ decision rule directly uses the prior probabilities. Procedure: For each decision alternative, calculate the weighted average of its payoff by multiplying each payoff by the prior probability and summing these products. This is the expected payoff (EP). Choose the decision alternative that has the largest expected payoff. Figure 9.1 This spreadsheet shows the application of Bayes’ decision rule to the first Goferbroke Co. problem, where a comparison of the expected payoffs in cells F5:F6 indicates that the Drill alternative should be chosen because it has the largest expected payoff.

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**Bayes’ Decision Rule Features of Bayes’ Decision Rule**

It accounts for all the states of nature and their probabilities. The expected payoff can be interpreted as what the average payoff would become if the same situation were repeated many times. Therefore, on average, repeatedly applying Bayes’ decision rule to make decisions will lead to larger payoffs in the long run than any other criterion. Criticisms of Bayes’ Decision Rule There usually is considerable uncertainty involved in assigning values to the prior probabilities. Prior probabilities inherently are at least largely subjective in nature, whereas sound decision making should be based on objective data and procedures. It ignores typical aversion to risk. By focusing on average outcomes, expected (monetary) payoffs ignore the effect that the amount of variability in the possible outcomes should have on decision making.

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Decision Trees A decision tree can apply Bayes’ decision rule while displaying and analyzing the problem graphically. A decision tree consists of nodes and branches. A decision node, represented by a square, indicates a decision to be made. The branches represent the possible decisions. An event node, represented by a circle, indicates a random event. The branches represent the possible outcomes of the random event.

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**Decision Tree for Goferbroke**

Figure 9.2 The decision tree for the first Goferbroke Co. problem as presented in Table 9.3.

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Using TreePlan TreePlan, an Excel add-in developed by Professor Michael Middleton, can be used to construct and analyze decision trees on a spreadsheet. Choose Decision Tree under the Tools menu. Click on New Tree, and it will draw a default tree with a single decision node and two branches, as shown below. The labels in D2 and D7 (originally Decision 1 and Decision 2) can be replaced by more descriptive names (e.g., Drill and Sell). Figure 9.3 The default decision tree created by TreePlan by selecting Decision Tree from the Tools menu, clicking on New Tree, and then entering Drill and Sell labels for the two decision alternatives.

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Using TreePlan To replace a node (such as the terminal node of the drill branch in F3) by a different type of node (e.g., an event node), click on the cell containing the node, choose Decision Tree again from the Tools menu, and select “Change to event node”. Figure 9.4 The TreePlan dialogue box that is used for making various kinds of changes in the decision tree.

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**Using TreePlan Enter the correct probabilities in H1 and H6.**

Enter the partial payoffs for each decision and event in D6, D14, H4, and H9. Figure 9.5 The decision tree constructed and solved by TreePlan for the first Goferbroke Co. problem as presented in Table 9.3, where the 1 in cell B9 indicates that the top branch (the Drill alternative) should be chosen.

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TreePlan Results The numbers inside each decision node indicate which branch should be chosen (assuming the branches are numbered consecutively from top to bottom). The numbers to the right of each terminal node is the payoff if that node is reached. The number 100 in cells A10 and E6 is the expected payoff at those stages in the process. Figure 9.5 The decision tree constructed and solved by TreePlan for the first Goferbroke Co. problem as presented in Table 9.3, where the 1 in cell B9 indicates that the top branch (the Drill alternative) should be chosen.

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**Consolidate the Data and Results**

Figure 9.6 In preparation for performing sensitivity analysis on the first Goferbroke Co. problem, the data and results have been consolidated on the spreadsheet below the decision tree.

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**Sensitivity Analysis: Prior Probability of Oil = 0.15**

Figure 9.7 Performing sensitivity analysis for the first Goferbroke Co. problem by trying a value of 0.15 for the prior probability of oil.

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**Sensitivity Analysis: Prior Probability of Oil = 0.35**

Figure 9.7 Performing sensitivity analysis for the first Goferbroke Co. problem by trying a value of 0.35 for the prior probability of oil.

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**Using Data Tables to Do Sensitivity Analysis**

Figure 9.8 Expansion of the spreadsheet in Figure 9.6 to prepare for generating a data table, where the choice of E22 for the column input cell in the Table dialogue box indicates that this is the data cell that is being changed in the first column of the data table.

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**Data Table Results The Effect of Changing the Prior Probability of Oil**

Figure 9.9 After the preparation in Figure 9.8, clicking OK generates this data table that shows the optimal action and expected payoff for various trial values of the prior probability of oil.

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**Using Utilities to Better Reflect the Values of Payoffs**

Thus far, when applying Bayes’ decision rule, we have assumed that the expected payoff in monetary terms is the appropriate measure. In many situations, this is inappropriate. Suppose an individual is offered the following choice: Accept a chance of winning $100,000. Receive $40,000 with certainty. Many would pick $40,000, even though the expected payoff on the chance of winning $100,000 is $50,000. This is because of risk aversion. A utility function for money is a way of transforming monetary values to an appropriate scale that reflects a decision maker’s preferences (e.g., aversion to risk).

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**A Typical Utility Function for Money**

Figure A typical utility function for money, where U(M) is the utility of obtaining an amount of money M.

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**Shape of Utility Functions**

Figure The shape of the utility function for (a) risk-averse, (b) risk-seeking, and (c) risk-neutral individuals.

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Utility Functions When a utility function for money is incorporated into a decision analysis approach, it must be constructed to fit the current preferences and values of the decision maker. Fundamental Property: Under the assumptions of utility theory, the decision maker’s utility function for money has the property that the decision maker is indifferent between two alternatives if the two alternatives have the same expected utility. When the decision maker’s utility function for money is used, Bayes’ decision rule replaces monetary payoffs by the corresponding utilities. The optimal decision (or series of decisions) is the one that maximizes the expected utility.

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**Illustration of Fundamental Property**

By the fundamental property, a decision maker with the utility function below-right will be indifferent between each of the three pairs of alternatives below-left. 25% chance of $100,000 $10,000 for sure Both have E(Utility) = 0.25. 50% chance of $100,000 $30,000 for sure Both have E(Utility) = 0.5. 75% chance of $100,000 $60,000 for sure Both have E(Utility) = 0.75.

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**The Equivalent Lottery Method**

Determine the largest potential payoff, M=Maximum. Assign U(Maximum) = 1. Determine the smallest potential payoff, M=Minimum. Assign U(Minimum) = 0. To determine the utility of another potential payoff M, consider the two aleternatives: A1: Obtain a payoff of Maximum with probability p. Obtain a payoff of Minimum with probability 1–p. A2: Definitely obtain a payoff of M. Question to the decision maker: What value of p makes you indifferent? Then, U(M) = p.

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**Generating the Utility Function for Max Flyer**

The possible monetary payoffs in the Goferbroke Co. problem are –130, –100, 0, 60, 90, 670, and 700 (all in $thousands). Set U(Maximum) = U(700) = 1. Set U(Minimum) = U(–130) = 0. To find U(M), use the equivalent lottery method. For example, for M=90, consider the two alternatives: A1: Obtain a payoff of 700 with probability p Obtain a payoff of –130 with probability 1–p. A2: Definitely obtain a payoff of 90 If Max chooses a point of indifference of p = 1/3, then U(90) = 1/3.

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**Max’s Utility Function for Money**

Figure Max’s utility function for money as the owner of Goferbroke Co.

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**Utilities for the Goferbroke Co. Problem**

Monetary Payoff, M Utility, U(M) –130 0.00 –100 0.05 60 0.30 90 0.33 670 0.97 700 1.00 Table 9.9 Utilities for the Goferbroke Co. problem.

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**Decisions Under Certainty**

State of nature is certain (one state). Select decision that yields highest return (e.g., linear programming, integer programming). Examples: Product mix Diet problem Distribution Scheduling Slides 9.61–9.76 are based upon a lecture from the MBA core course in Management Science at the University of Washington (as taught by one of the authors). It covers the basics of decision analysis, including decision trees and Bayes’ Criterion.

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**Decisions Under Uncertainty (or Risk)**

State of nature is uncertain (several possible states) Examples Drilling for oil Uncertainty: Oil found? How much? How deep? Selling Price? Decision: Drill or not? Developing a new product Uncertainty: R&D Cost, demand, etc. Decisions: Design, quantity, produce or not? Newsvendor problem Uncertainty: Demand Decision: Stocking levels Producing a movie Uncertainty: Cost, gross, etc. Decisions: Develop? Arnold or Keanu?

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Oil Drilling Problem Consider the problem faced by an oil company that is trying to decide whether to drill an exploratory oil well on a given site. Drilling costs $200,000. If oil is found, it is worth $800,000. If the well is dry, it is worth nothing. State of Nature Decision Wet Dry Drill 600 –200 Do not drill

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**Decision Criteria Which decision is best? “Optimist” “Pessimist”**

State of Nature Decision Wet Dry Drill 600 –200 Do not drill Which decision is best? “Optimist” “Pessimist” “Second–Guesser” “Joe Average” “Optimist” discuss Maximax Max(Drill) = 600 Max(Do not drill) = 0 Maximax = 600 with Drill “Pessimist” discuss Maximin Min(Drill) = –200 Min(Do not drill) = 0 Maximin = 0 with Do not drill “Second-Guesser” discuss Minimax Regret Max Regret (Drill) = 200 Max Regret (Do not drill) = 600 Minimax Regret = 200 with Drill “Joe Average” discuss Equally Likely Average (Drill) = 200 Average (Do not drill) = 0 Maximum = 200 with Drill

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Bayes’ Decision Rule Suppose that the oil company estimates that the probability that the site is “Wet” is 40%. State of Nature Decision Wet Dry Drill 600 –200 Do not drill Prior Probability 0.4 0.6 Expected value of payoff (Drill) = (0.4)(600) + (0.6)(–200) = 120 Expected value of payoff (Do not drill) = (0.4)(0) + (0.6)(0) = 0 Bayes’ Decision Rule: Choose the decision that maximizes the expected payoff (Drill).

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**Features of Bayes’ Decision Rule**

Accounts not only for the set of outcomes, but also their probabilities. Represents the average monetary outcome if the situation were repeated indefinitely. Can handle complicated situations involving multiple related risks.

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