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Chapter 3 Decision Analysis

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kasimir-buckley
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Introduction to Management Science 8th Edition by Bernard W. Taylor III. Chapter 3 Decision Analysis. Chapter Topics. Components of Decision Making Decision Making without Probabilities Decision Making with Probabilities Decision Analysis with Additional Information Utility.

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

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

  3. 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

  4. 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

  5. Decision Making without Probabilities Maximax Criterion • In themaximaxcriterion 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

  6. conservative 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. Table 3.4 Payoff Table Illustrating a Maximin Decision Chapter 3 - Decision Analysis

  7. Maximalregrets $ 50,000$ 70,000$ 70,000 Highestpayoff $100,000- $50,000 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. Table 3.6 Regret Table Illustrating the Minimax Regret Decision Chapter 3 - Decision Analysis

  8. Decision Making without Probabilities Hurwicz Criterion • TheHurwiczcriterion is a compromise between the maximax (optimist) and maximin (conservative) criterion. • A coefficient of optimism, , is a measureof 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. • DecisionValues • 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

  9. 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 methodIn general, it is essentially different ! • DecisionValues • 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

  10. Decision Making without Probabilities Summary of Criteria Results • A dominantdecision 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. • CriterionDecision (Purchase) • Maximax Office building • Maximin Apartment building • Minimax regret Apartment building • Hurwicz Apartment building • Equal likelihood Apartment building Chapter 3 - Decision Analysis

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

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

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

  14. 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

  15. 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

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

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

  18. 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

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

  20. The“genie pick” EV $42,000 $44,000 $22,000 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 Chapter 3 - Decision Analysis

  21. 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

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

  23. 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

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

  25. 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

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

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

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

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

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

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

  33. Decision Making with Probabilities Sequential Decision Trees (2 of 4) The decision to bemade at [1] logicallydepends on the decisions(to be) made at [4] and [5]. Figure 3.3 Sequential Decision Tree Chapter 3 - Decision Analysis

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

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

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

  38. 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

  39. as before… new info new, given 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 Chapter 3 - Decision Analysis

  40. 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

  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 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

  42. 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 Decision Analysis with Additional Information Decision Trees with Posterior Probabilities (2 of 4) Figure 3.5 Decision Tree with Posterior Probabilities Chapter 3 - Decision Analysis

  43. “Positive report” “Negative report” 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 Chapter 3 - Decision Analysis

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

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

  46. 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

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

  48. 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. • Utilityis 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

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

  50. 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