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Chapter 12 - Decision Analysis 1 Chapter 12 Decision Analysis Introduction to Management Science 8th Edition by Bernard W. Taylor III

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Chapter 12 - Decision Analysis 2 Components of Decision Making Decision Making without Probabilities Decision Making with Probabilities Decision Analysis with Additional Information Utility Chapter Topics

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Chapter 12 - Decision Analysis 3 Table 12.1 Payoff Table A state of nature is an actual event that may occur in the future. A payoff table is a means of organizing a decision situation, presenting the payoffs from different decisions given the various states of nature. Decision Analysis Components of Decision Making

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Chapter 12 - Decision Analysis 4 Decision situation: Decision-Making Criteria: maximax, maximin, minimax, minimax regret, Hurwicz, and equal likelihood Table 12.2 Payoff Table for the Real Estate Investments Decision Analysis Decision Making without Probabilities

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Chapter 12 - Decision Analysis 5 Table 12.3 Payoff Table Illustrating a Maximax Decision In the maximax criterion the decision maker selects the decision that will result in the maximum of maximum payoffs; an optimistic criterion. Decision Making without Probabilities Maximax Criterion

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Chapter 12 - Decision Analysis 6 Table 12.4 Payoff Table Illustrating a Maximin Decision In the maximin criterion the decision maker selects the decision that will reflect the maximum of the minimum payoffs; a pessimistic criterion. Decision Making without Probabilities Maximin Criterion

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Chapter 12 - Decision Analysis 7 Table 12.6 Regret Table Illustrating the Minimax Regret Decision Regret is the difference between the payoff from the best decision and all other decision payoffs. The decision maker attempts to avoid regret by selecting the decision alternative that minimizes the maximum regret. Decision Making without Probabilities Minimax Regret Criterion

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Chapter 12 - Decision Analysis 8 The Hurwicz criterion is a compromise between the maximax and maximin criterion. A coefficient of optimism, , is a measure of the decision maker’s optimism. The Hurwicz criterion multiplies the best payoff by and the worst payoff by 1- ., for each decision, and the best result is selected. Decision Values Apartment building $50,000(.4) + 30,000(.6) = 38,000 Office building $100,000(.4) - 40,000(.6) = 16,000 Warehouse $30,000(.4) + 10,000(.6) = 18,000 Decision Making without Probabilities Hurwicz Criterion

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Chapter 12 - Decision Analysis 9 The equal likelihood ( or Laplace) criterion multiplies the decision payoff for each state of nature by an equal weight, thus assuming that the states of nature are equally likely to occur. Decision Values Apartment building $50,000(.5) + 30,000(.5) = 40,000 Office building $100,000(.5) - 40,000(.5) = 30,000 Warehouse $30,000(.5) + 10,000(.5) = 20,000 Decision Making without Probabilities Equal Likelihood Criterion

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Chapter 12 - Decision Analysis 10 A dominant decision is one that has a better payoff than another decision under each state of nature. The appropriate criterion is dependent on the “risk” personality and philosophy of the decision maker. Criterion Decision (Purchase) MaximaxOffice building MaximinApartment building Minimax regretApartment building HurwiczApartment building Equal likelihoodApartment building Decision Making without Probabilities Summary of Criteria Results

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Chapter 12 - Decision Analysis 11 Exhibit 12.1 Decision Making without Probabilities Solution with QM for Windows (1 of 3)

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Chapter 12 - Decision Analysis 12 Exhibit 12.2 Decision Making without Probabilities Solution with QM for Windows (2 of 3)

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Chapter 12 - Decision Analysis 13 Exhibit 12.3 Decision Making without Probabilities Solution with QM for Windows (3 of 3)

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Chapter 12 - Decision Analysis 14 Expected value is computed by multiplying each decision outcome under each state of nature by the probability of its occurrence. EV(Apartment) = $50,000(.6) + 30,000(.4) = 42,000 EV(Office) = $100,000(.6) - 40,000(.4) = 44,000 EV(Warehouse) = $30,000(.6) + 10,000(.4) = 22,000 Table 12.7 Payoff table with Probabilities for States of Nature Decision Making with Probabilities Expected Value

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Chapter 12 - Decision Analysis 15 The expected opportunity loss is the expected value of the regret for each decision. The expected value and expected opportunity loss criterion result in the same decision. EOL(Apartment) = $50,000(.6) + 0(.4) = 30,000 EOL(Office) = $0(.6) + 70,000(.4) = 28,000 EOL(Warehouse) = $70,000(.6) + 20,000(.4) = 50,000 Table 12.8 Regret (Opportunity Loss) Table with Probabilities for States of Nature Decision Making with Probabilities Expected Opportunity Loss

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Chapter 12 - Decision Analysis 16 Exhibit 12.4 Expected Value Problems Solution with QM for Windows

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Chapter 12 - Decision Analysis 17 Exhibit 12.5 Expected Value Problems Solution with Excel and Excel QM (1 of 2)

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Chapter 12 - Decision Analysis 18 Exhibit 12.6 Expected Value Problems Solution with Excel and Excel QM (2 of 2)

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Chapter 12 - Decision Analysis 19 The expected value of perfect information (EVPI) is the maximum amount a decision maker would pay for additional information. EVPI equals the expected value given perfect information minus the expected value without perfect information. EVPI equals the expected opportunity loss (EOL) for the best decision. Decision Making with Probabilities Expected Value of Perfect Information

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Chapter 12 - Decision Analysis 20 Table 12.9 Payoff Table with Decisions, Given Perfect Information Decision Making with Probabilities EVPI Example (1 of 2)

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Chapter 12 - Decision Analysis 21 Decision with perfect information: $100,000(.60) + 30,000(.40) = $72,000 Decision without perfect information: EV(office) = $100,000(.60) - 40,000(.40) = $44,000 EVPI = $72,000 - 44,000 = $28,000 EOL(office) = $0(.60) + 70,000(.4) = $28,000 Decision Making with Probabilities EVPI Example (2 of 2)

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Chapter 12 - Decision Analysis 22 Exhibit 12.7 Decision Making with Probabilities EVPI with QM for Windows

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Chapter 12 - Decision Analysis 23 A decision tree is a diagram consisting of decision nodes (represented as squares), probability nodes (circles), and decision alternatives (branches). Table 12.10 Payoff Table for Real Estate Investment Example Decision Making with Probabilities Decision Trees (1 of 4)

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Chapter 12 - Decision Analysis 24 Figure 12.1 Decision Tree for Real Estate Investment Example Decision Making with Probabilities Decision Trees (2 of 4)

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Chapter 12 - Decision Analysis 25 The expected value is computed at each probability node: EV(node 2) =.60($50,000) +.40(30,000) = $42,000 EV(node 3) =.60($100,000) +.40(-40,000) = $44,000 EV(node 4) =.60($30,000) +.40(10,000) = $22,000 Branches with the greatest expected value are selected. Decision Making with Probabilities Decision Trees (3 of 4)

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Chapter 12 - Decision Analysis 26 Figure 12.2 Decision Tree with Expected Value at Probability Nodes Decision Making with Probabilities Decision Trees (4 of 4)

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Chapter 12 - Decision Analysis 27 Exhibit 12.8 Decision Making with Probabilities Decision Trees with QM for Windows

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Chapter 12 - Decision Analysis 28 Exhibit 12.9 Decision Making with Probabilities Decision Trees with Excel and TreePlan (1 of 4)

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Chapter 12 - Decision Analysis 29 Exhibit 12.10 Decision Making with Probabilities Decision Trees with Excel and TreePlan (2 of 4)

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Chapter 12 - Decision Analysis 30 Exhibit 12.11 Decision Making with Probabilities Decision Trees with Excel and TreePlan (3 of 4)

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Chapter 12 - Decision Analysis 31 Exhibit 12.12 Decision Making with Probabilities Decision Trees with Excel and TreePlan (4 of 4)

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Chapter 12 - Decision Analysis 32 Decision Making with Probabilities Sequential Decision Trees (1 of 4) A sequential decision tree is used to illustrate a situation requiring a series of decisions. Used where a payoff table, limited to a single decision, cannot be used. Real estate investment example modified to encompass a ten-year period in which several decisions must be made:

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Chapter 12 - Decision Analysis 33 Figure 12.3 Sequential Decision Tree Decision Making with Probabilities Sequential Decision Trees (2 of 4)

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Chapter 12 - Decision Analysis 34 Decision Making with Probabilities Sequential Decision Trees (3 of 4) Decision is to purchase land; highest net expected value ($1,160,000). Payoff of the decision is $1,160,000.

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Chapter 12 - Decision Analysis 35 Figure 12.4 Sequential Decision Tree with Nodal Expected Values Decision Making with Probabilities Sequential Decision Trees (4 of 4)

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Chapter 12 - Decision Analysis 36 Exhibit 12.13 Sequential Decision Tree Analysis Solution with QM for Windows

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Chapter 12 - Decision Analysis 37 Exhibit 12.14 Sequential Decision Tree Analysis Solution with Excel and TreePlan

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Chapter 12 - Decision Analysis 38 Bayesian analysis uses additional information to alter the marginal probability of the occurrence of an event. In real estate investment example, using expected value criterion, best decision was to purchase office building with expected value of $444,000, and EVPI of $28,000. Table 12.11 Payoff Table for the Real Estate Investment Example Decision Analysis with Additional Information Bayesian Analysis (1 of 3)

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Chapter 12 - Decision Analysis 39 A conditional probability is the probability that an event will occur given that another event has already occurred. Economic analyst provides additional information for real estate investment decision, forming conditional probabilities: g = good economic conditions p = poor economic conditions P = positive economic report N = negative economic report P(P g) =.80P(N G) =.20 P(P p) =.10P(N p) =.90 Decision Analysis with Additional Information Bayesian Analysis (2 of 3)

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Chapter 12 - Decision Analysis 40 A posteria probability is the altered marginal probability of an event based on additional information. Prior probabilities for good or poor economic conditions in real estate decision: P(g) =.60; P(p) =.40 Posteria probabilities by Bayes’ rule: (g P) = P(P G)P(g)/[P(P g)P(g) + P(P p)P(p)] = (.80)(.60)/[(.80)(.60) + (.10)(.40)] =.923 Posteria (revised) probabilities for decision: P(g N) =.250P(p P) =.077P(p N) =.750 Decision Analysis with Additional Information Bayesian Analysis (3 of 3)

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Chapter 12 - Decision Analysis 41 Decision Analysis with Additional Information Decision Trees with Posterior Probabilities (1 of 4) Decision tree with posterior probabilities differ from earlier versions in that: Two new branches at beginning of tree represent report outcomes. Probabilities of each state of nature are posterior probabilities from Bayes’ rule.

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Chapter 12 - Decision Analysis 42 Figure 12.5 Decision Tree with Posterior Probabilities Decision Analysis with Additional Information Decision Trees with Posterior Probabilities (2 of 4)

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Chapter 12 - Decision Analysis 43 Decision Analysis with Additional Information Decision Trees with Posterior Probabilities (3 of 4) EV (apartment building) = $50,000(.923) + 30,000(.077) = $48,460 EV (strategy) = $89,220(.52) + 35,000(.48) = $63,194

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Chapter 12 - Decision Analysis 44 Figure 12.6 Decision Tree Analysis Decision Analysis with Additional Information Decision Trees with Posterior Probabilities (4 of 4)

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Chapter 12 - Decision Analysis 45 Table 12.12 Computation of Posterior Probabilities Decision Analysis with Additional Information Computing Posterior Probabilities with Tables

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Chapter 12 - Decision Analysis 46 The expected value of sample information (EVSI) is the difference between the expected value with and without information: For example problem, EVSI = $63,194 - 44,000 = $19,194 The efficiency of sample information is the ratio of the expected value of sample information to the expected value of perfect information: efficiency = EVSI /EVPI = $19,194/ 28,000 =.68 Decision Analysis with Additional Information Expected Value of Sample Information

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Chapter 12 - Decision Analysis 47 Table 12.13 Payoff Table for Auto Insurance Example Decision Analysis with Additional Information Utility (1 of 2)

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Chapter 12 - Decision Analysis 48 Expected Cost (insurance) =.992($500) +.008(500) = $500 Expected Cost (no insurance) =.992($0) +.008(10,000) = $80 Decision should be do not purchase insurance, but people almost always do purchase insurance. Utility is a measure of personal satisfaction derived from money. Utiles are units of subjective measures of utility. Risk averters forgo a high expected value to avoid a low- probability disaster. Risk takers take a chance for a bonanza on a very low- probability event in lieu of a sure thing. Decision Analysis with Additional Information Utility (2 of 2)

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Chapter 12 - Decision Analysis 49 Decision Analysis Example Problem Solution (1 of 9)

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Chapter 12 - Decision Analysis 50 Decision Analysis Example Problem Solution (2 of 9) a.Determine the best decision without probabilities using the 5 criteria of the chapter. b.Determine best decision with probabilities assuming.70 probability of good conditions,.30 of poor conditions. Use expected value and expected opportunity loss criteria. c.Compute expected value of perfect information. d.Develop a decision tree with expected value at the nodes. e.Given following, P(P g) =.70, P(N g) =.30, P(P p) = 20, P(N p) =.80, determine posteria probabilities using Bayes’ rule. f.Perform a decision tree analysis using the posterior probability obtained in part e.

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Chapter 12 - Decision Analysis 51 Step 1 (part a): Determine decisions without probabilities. Maximax Decision: Maintain status quo DecisionsMaximum Payoffs Expand $800,000 Status quo1,300,000 (maximum) Sell 320,000 Maximin Decision: Expand DecisionsMinimum Payoffs Expand$500,000 (maximum) Status quo -150,000 Sell 320,000 Decision Analysis Example Problem Solution (3 of 9)

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Chapter 12 - Decision Analysis 52 Minimax Regret Decision: Expand DecisionsMaximum Regrets Expand$500,000 (minimum) Status quo 650,000 Sell 980,000 Hurwicz ( =.3) Decision: Expand Expand $800,000(.3) + 500,000(.7) = $590,000 Status quo$1,300,000(.3) - 150,000(.7) = $285,000 Sell $320,000(.3) + 320,000(.7) = $320,000 Decision Analysis Example Problem Solution (4 of 9)

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Chapter 12 - Decision Analysis 53 Equal Likelihood Decision: Expand Expand $800,000(.5) + 500,000(.5) = $650,000 Status quo $1,300,000(.5) - 150,000(.5) = $575,000 Sell $320,000(.5) + 320,000(.5) = $320,000 Step 2 (part b): Determine Decisions with EV and EOL. Expected value decision: Maintain status quo Expand $800,000(.7) + 500,000(.3) = $710,000 Status quo $1,300,000(.7) - 150,000(.3) = $865,000 Sell $320,000(.7) + 320,000(.3) = $320,000 Decision Analysis Example Problem Solution (5 of 9)

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Chapter 12 - Decision Analysis 54 Expected opportunity loss decision: Maintain status quo Expand $500,000(.7) + 0(.3) = $350,000 Status quo 0(.7) + 650,000(.3) = $195,000 Sell $980,000(.7) + 180,000(.3) = $740,000 Step 3 (part c): Compute EVPI. EV given perfect information = 1,300,000(.7) + 500,000(.3) = $1,060,000 EV without perfect information = $1,300,000(.7) - 150,000(.3) = $865,000 EVPI = $1.060,000 - 865,000 = $195,000 Decision Analysis Example Problem Solution (6 of 9)

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Chapter 12 - Decision Analysis 55 Step 4 (part d): Develop a decision tree. Decision Analysis Example Problem Solution (7 of 9)

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Chapter 12 - Decision Analysis 56 Step 5 (part e): Determine posterior probabilities. P(g P) = P(P G)P(g)/[P(P g)P(g) + P(P p)P(p)] = (.70)(.70)/[(.70)(.70) + (.20)(.30)] =.891 P(p P) =.109 P(g N) = P(N G)P(g)/[P(N g)P(g) + P(N p)P(p)] = (.30)(.70)/[(.30)(.70) + (.80)(.30)] =.467 P(p N) =.533 Decision Analysis Example Problem Solution (8 of 9)

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Chapter 12 - Decision Analysis 57 Step 6 (part f): Decision tree analysis. Decision Analysis Example Problem Solution (9 of 9)

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