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

Chapter Topics Components of Decision Making Decision Making without Probabilities Decision Making with Probabilities Decision Analysis with Additional Information Utility Chapter 3 - Decision Analysis

Components of Decision Making Decision Analysis Components of Decision Making 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. Table 3.1 Payoff Table Chapter 3 - Decision Analysis

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

Payoff Table Illustrating a Maximax Decision Decision Making without Probabilities Maximax Criterion In the maximax criterion the decision maker selects the decision that will result in the maximum of maximum payoffs; an optimistic criterion. Table 3.3 Payoff Table Illustrating a Maximax Decision Chapter 3 - Decision Analysis

Payoff Table Illustrating a Maximin Decision Decision Making without Probabilities Maximin Criterion 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. conservative Table 3.4 Payoff Table Illustrating a Maximin Decision Chapter 3 - Decision Analysis

Regret Table Illustrating the Minimax Regret Decision Decision Making without Probabilities Minimax Regret Criterion 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. Highest payoff Maximal regrets $ 50,000 $ 70,000 $ 70,000 $100,000 - $50,000 Table 3.6 Regret Table Illustrating the Minimax Regret Decision Chapter 3 - Decision Analysis

Decision Making without Probabilities Hurwicz Criterion 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  = 0.4 Chapter 3 - Decision Analysis

Decision Making without Probabilities Equal Likelihood Criterion 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 Chapter 3 - Decision Analysis

Decision Making without Probabilities Summary of Criteria Results 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) Maximax Office building Maximin Apartment building Minimax regret Apartment building Hurwicz Apartment building Equal likelihood Apartment building Chapter 3 - Decision Analysis

Decision Making without Probabilities Solution with QM for Windows (1 of 3) Exhibit 3.1 Chapter 3 - Decision Analysis

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

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

Decision Making with Probabilities Expected Value 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 Chapter 3 - Decision Analysis

Decision Making with Probabilities Expected Opportunity Loss 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 Chapter 3 - Decision Analysis

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

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

Decision Making with Probabilities Expected Value of Perfect Information 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. Chapter 3 - Decision Analysis

Payoff Table with Decisions, Given Perfect Information Decision Making with Probabilities EVPI Example (1 of 2) Table 3.9 Payoff Table with Decisions, Given Perfect Information Chapter 3 - Decision Analysis

Decision Making with Probabilities EVPI Example (2 of 2) 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 The “genie pick” EV $42,000 $44,000 $22,000 Chapter 3 - Decision Analysis

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

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

Payoff Table for Real Estate Investment Example Decision Making with Probabilities Decision Trees (1 of 4) 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 Chapter 3 - Decision Analysis

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

Decision Making with Probabilities Decision Trees (3 of 4) 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. Chapter 3 - Decision Analysis

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

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

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

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

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

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

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: Chapter 3 - Decision Analysis

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]. Figure 3.3 Sequential Decision Tree Chapter 3 - Decision Analysis

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

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.) Chapter 3 - Decision Analysis

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

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

Payoff Table for the Real Estate Investment Example Decision Analysis with Additional Information Bayesian Analysis (1 of 3) 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 Chapter 3 - Decision Analysis

Decision Analysis with Additional Information Bayesian Analysis (2 of 3) 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) = .80 P(Ng) = .20 P(Pp) = .10 P(Np) = .90 as before… new info new, given Chapter 3 - Decision Analysis

Decision Analysis with Additional Information Bayesian Analysis (3 of 3) 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) = .250 P(pP) = .077 P(pN) = .750 Chapter 3 - Decision Analysis

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 iP(ABi)=P(A) is the marginal prob. So: P(Bk|A)=P(A|Bk)P(Bk)/[iP(A|Bi)P(Bi)] = P(ABk)/P(A), much quicker, if the joint and marginal prob’s are known. Chapter 3 - Decision Analysis

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 Figure 3.5 Decision Tree with Posterior Probabilities Chapter 3 - Decision Analysis

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” Chapter 3 - Decision Analysis

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

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) . Table 3.12 Computation of Posterior Probabilities Chapter 3 - Decision Analysis

Decision Analysis with Additional Information Expected Value of Sample Information 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 Chapter 3 - Decision Analysis

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

Decision Analysis with Additional Information Utility (2 of 2) 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. Chapter 3 - Decision Analysis

Example Problem Solution (1 of 9) Decision Analysis Example Problem Solution (1 of 9) States of Nature 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 Chapter 3 - Decision Analysis

Example Problem Solution (2 of 9) Decision Analysis Example Problem Solution (2 of 9) Determine the best decision without probabilities using the 5 criteria of the chapter. Determine best decision with probabilities assuming .70 probability of good conditions, .30 of poor conditions. Use expected value and expected opportunity loss criteria. Compute expected value of perfect information. Develop a decision tree with expected value at the nodes. Given following, P(Pg) = .70, P(Ng) = .30, P(Pp) = .20, P(Np) = .80, determine posterior probabilities using Bayes’ rule. Perform a decision tree analysis using the posterior probability obtained in part e. Chapter 3 - Decision Analysis

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

Example Problem Solution (4 of 9) Decision Analysis Example Problem Solution (4 of 9) Minimax (Optimal) Regret Decision: Expand Decisions maximum 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 Chapter 3 - Decision Analysis

Example Problem Solution (5 of 9) Decision Analysis Example Problem Solution (5 of 9) 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 Chapter 3 - Decision Analysis

Example Problem Solution (6 of 9) Decision Analysis Example Problem Solution (6 of 9) 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 Chapter 3 - Decision Analysis

Example Problem Solution (7 of 9) Decision Analysis Example Problem Solution (7 of 9) Step 4 (part d): Develop a decision tree. Chapter 3 - Decision Analysis

Example Problem Solution (8 of 9) Decision Analysis Example Problem Solution (8 of 9) 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 Chapter 3 - Decision Analysis

Example Problem Solution (9 of 9) Decision Analysis Example Problem Solution (9 of 9) Step 6 (part f): Decision tree analysis. 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) Chapter 3 - Decision Analysis

Chapter 3 - Decision Analysis