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Chapter 3 - Decision Analysis 1 Chapter 3 Decision Analysis Introduction to Management Science 8th Edition by Bernard W. Taylor III

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Chapter 3 - 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 3 - Decision Analysis 3 Table 3.1 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 3 - Decision Analysis 4 Decision situation: Decision-Making Criteria: maximax, maximin, minimax (minimal regret), Hurwicz, and equal likelihood Table 3.2 Payoff Table for the Real Estate Investments Decision Analysis Decision Making without Probabilities

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Chapter 3 - Decision Analysis 5 Table 3.3 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 3 - Decision Analysis 6 Table 3.4 Payoff Table Illustrating a Maximin Decision In the maximin criterion the decision maker selects the decision that will reflect the maximum of the minimum (best of the worst-case) payoffs; a pessimistic criterion. Decision Making without Probabilities Maximin Criterion conservative

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Chapter 3 - Decision Analysis 7 Table 3.6 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 Maximal regrets $ 50,000 $ 70,000 $ 70,000 Maximal regrets $ 50,000 $ 70,000 $ 70,000 Highest payoff $100,000 - $50,000

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Chapter 3 - Decision Analysis 8 The Hurwicz criterion is a compromise between the maximax (optimist) and maximin (conservative) 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 = 0.4

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Chapter 3 - 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. For 2 states of nature, the =.5 case of the Hurwicz method In general, it is essentially different ! 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 3 - 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 3 - Decision Analysis 11 Exhibit 3.1 Decision Making without Probabilities Solution with QM for Windows (1 of 3)

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Chapter 3 - Decision Analysis 12 Exhibit 3.2 Decision Making without Probabilities Solution with QM for Windows (2 of 3)

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Chapter 3 - Decision Analysis 13 Exhibit 3.3 Decision Making without Probabilities Solution with QM for Windows (3 of 3)

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Chapter 3 - 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 3.7 Payoff table with Probabilities for States of Nature Decision Making with Probabilities Expected Value

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Chapter 3 - 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 3.8 Regret (Opportunity Loss) Table with Probabilities for States of Nature Decision Making with Probabilities Expected Opportunity Loss

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Chapter 3 - Decision Analysis 16 Exhibit 3.4 Expected Value Problems Solution with QM for Windows

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Chapter 3 - Decision Analysis 17 Exhibit 3.5 Expected Value Problems Solution with Excel and Excel QM (1 of 2)

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Chapter 3 - Decision Analysis 18 The expected value of perfect information (EVPI) is the maximum amount a decision maker should pay for additional information. EVPI equals the expected value (with) given perfect information (insider information, genie) minus the expected value calculated 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 3 - Decision Analysis 19 Table 3.9 Payoff Table with Decisions, Given Perfect Information Decision Making with Probabilities EVPI Example (1 of 2)

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Chapter 3 - Decision Analysis 20 Decision with perfect (insider/genie) 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) EV $42,000 $44,000 $22,000 The “genie pick”

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Chapter 3 - Decision Analysis 21 Exhibit 3.6 Expected Value Problems Solution with Excel and Excel QM (2 of 2) $100,000*0.6+$30,000*0.4 = $72,000

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Chapter 3 - Decision Analysis 22 Exhibit 3.7 Decision Making with Probabilities EVPI with QM for Windows

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Chapter 3 - Decision Analysis 23 A decision tree is a diagram consisting of decision nodes (represented as squares), probability nodes (circles), and decision alternatives (branches). Table 3.10 Payoff Table for Real Estate Investment Example Decision Making with Probabilities Decision Trees (1 of 4)

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Chapter 3 - Decision Analysis 24 Figure 3.1 Decision Tree for Real Estate Investment Example Decision Making with Probabilities Decision Trees (2 of 4) controllable uncontrollable

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Chapter 3 - Decision Analysis 25 The expected value is computed at each probability (uncontrollable) 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 populating the decision tree from right to left. The branch(es) with the greatest expected value are then selected, starting from the left and progressing to the right. Decision Making with Probabilities Decision Trees (3 of 4)

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Chapter 3 - Decision Analysis 26 Figure 3.2 Decision Tree with Expected Value at Probability Nodes Decision Making with Probabilities Decision Trees (4 of 4)

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Chapter 3 - Decision Analysis 27 Exhibit 3.8 Decision Making with Probabilities Decision Trees with QM for Windows

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Chapter 3 - Decision Analysis 28 Exhibit 3.9 Decision Making with Probabilities Decision Trees with Excel and TreePlan (1 of 4)

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Chapter 3 - Decision Analysis 29 Exhibit 3.10 Decision Making with Probabilities Decision Trees with Excel and TreePlan (2 of 4)

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Chapter 3 - Decision Analysis 30 Exhibit 3.11 Decision Making with Probabilities Decision Trees with Excel and TreePlan (3 of 4)

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Chapter 3 - Decision Analysis 31 Exhibit 3.12 Decision Making with Probabilities Decision Trees with Excel and TreePlan (4 of 4)

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Chapter 3 - 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 (a sequence) of decisions. It is often chronological, and always logical in order. 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 3 - Decision Analysis 33 Figure 3.3 Sequential Decision Tree Decision Making with Probabilities Sequential Decision Trees (2 of 4) The decision to be made at [1] logically depends on the decisions (to be) made at [4] and [5].

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Chapter 3 - Decision Analysis 34 Figure 3.4 Sequential Decision Tree with Nodal Expected Values Decision Making with Probabilities Sequential Decision Trees (3 of 4)

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Chapter 3 - Decision Analysis 35 Decision Making with Probabilities Sequential Decision Trees (4 of 4) Decision is to purchase land; highest net expected value ($1,160,000, at node [1] ). Payoff of the decision is $1,160,000. (That’s the payoff that this decision is expected to yield.)

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Chapter 3 - Decision Analysis 36 Exhibit 3.13 Sequential Decision Tree Analysis Solution with QM for Windows

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Chapter 3 - Decision Analysis 37 Exhibit 3.14 Sequential Decision Tree Analysis Solution with Excel and TreePlan

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Chapter 3 - 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 $44,000, and EVPI of $28,000. Table 3.11 Payoff Table for the Real Estate Investment Example Decision Analysis with Additional Information Bayesian Analysis (1 of 3)

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Chapter 3 - 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) as before… new info new, given

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Chapter 3 - Decision Analysis 40 A posterior 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 Posterior probabilities by Bayes’ rule: P(g P) = P(P g)P(g)/[P(P g)P(g) + P(P p)P(p)] = (.80)(.60)/[(.80)(.60) + (.10)(.40)] =.923 Posterior (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 3 - 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 (prior probabilities) in that: Two (or more) new branches at beginning of tree represent report/survey… outcomes. Probabilities of each state of nature, thereafter, are posterior probabilities from Bayes’ rule. Bayes’ rule can be simplified, since P(A|B)P(B)=P(AB) is the joint prob., and i P(AB i )=P(A) is the marginal prob. So: P(B k |A)=P(A|B k )P(B k )/[ i P(A|B i )P(B i )] = P(AB k )/P(A), much quicker, if the joint and marginal prob’s are known.

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Chapter 3 - Decision Analysis 42 Figure 3.5 Decision Tree with Posterior Probabilities Decision Analysis with Additional Information Decision Trees with Posterior Probabilities (2 of 4) P(P|g)=.80 P(N|g)=.20 P(P|p)=.10 P(N|p)=.90 P(g)=.60 P(p)=.40 P(g|P)=.923 P(p|P)=.077 P(g|N)=.250 P(p|N)=.750

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Chapter 3 - 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 (office building) = $100,000(.923) – 40,000(.077) = $89,220 EV (warehouse) = $30,000(.923) + 10,000(.077) = $28,460 Then do the same with the “Negative report” probabilities. So, finally: EV (whole strategy) = $89,220(.52) + 35,000(.48) = $63,194 “Positive report” “Negative report”

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Chapter 3 - Decision Analysis 44 Figure 3.6 Decision Tree Analysis Decision Analysis with Additional Information Decision Trees with Posterior Probabilities (4 of 4)

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Chapter 3 - Decision Analysis 45 Table 3.12 Computation of Posterior Probabilities Decision Analysis with Additional Information Computing Posterior Probabilities with Tables Indeed, this equals [ P(P|g)P(g)+P(P|p)P(p) ] = P(P&g) + P(P&p) = P(P).

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Chapter 3 - 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 3 - Decision Analysis 47 Table 3.13 Payoff Table for Auto Insurance Example Decision Analysis with Additional Information Utility (1 of 2) Cost

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Chapter 3 - 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 (evaders) 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 3 - Decision Analysis 49 Decision Analysis Example Problem Solution (1 of 9) Decisions Good Foreign Competitive Conditions Poor Foreign Competitive Conditions Expand$800,000$500,000 Maintain Status Quo$1,300,00–$150,000 Sell Now$320,000 States of Nature

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Chapter 3 - 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 posterior probabilities using Bayes’ rule. f.Perform a decision tree analysis using the posterior probability obtained in part e.

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Chapter 3 - Decision Analysis 51 Step 1 (part a): Determine decisions without probabilities. Maximax (Optimist) Decision: Maintain status quo Decisionsmaximum Payoffs Expand $800,000 Status quo1,300,000 (Maximum) Sell 320,000 Maximin (Conservative) 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 3 - Decision Analysis 52 Minimax (Optimal) 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 3 - Decision Analysis 53 Equal Likelihood (Laplace) 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 3 - 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 3 - Decision Analysis 55 Step 4 (part d): Develop a decision tree. Decision Analysis Example Problem Solution (7 of 9)

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Chapter 3 - 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 3 - Decision Analysis 57 Step 6 (part f): Decision tree analysis. Decision Analysis Example Problem Solution (9 of 9) Without the report, maintain status quo, based on the expected payoff value $865,000. With the report, the payoff may be expected to be even $1,141,950. Thus, the opportunity loss is $1,141,950 – $865,000 = $276,950. Therefore, no more than $276,950 should be paid to obtain such a report. (EVPI)

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Chapter 3 - Decision Analysis 58

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