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

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

3. 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 12.1
Payoff Table
Chapter 12 - Decision Analysis

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

5. 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 12.3
Payoff Table Illustrating a Maximax Decision
Chapter 12 - Decision Analysis

6. 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 payoffs; a pessimistic criterion.
Table 12.4
Payoff Table Illustrating a Maximin Decision
Chapter 12 - Decision Analysis

7. 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.
Table 12.6
Regret Table Illustrating the Minimax Regret Decision
Chapter 12 - Decision Analysis

8. Decision Making without Probabilities Hurwicz Criterion
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
Chapter 12 - 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.
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 12 - Decision Analysis

10. 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 12 - Decision Analysis

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

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

13. Decision Making without Probabilities
Solution with QM for Windows (3 of 3)
Exhibit 12.3
Chapter 12 - 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 12.7
Payoff table with Probabilities for States of Nature
Chapter 12 - 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 12.8
Regret (Opportunity Loss) Table with Probabilities for States of Nature
Chapter 12 - Decision Analysis

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

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

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

19. Decision Making with Probabilities
Expected Value of Perfect Information
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.
Chapter 12 - Decision Analysis

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

21. Decision Making with Probabilities EVPI Example (2 of 2)
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 = $28,000
EOL(office) = $0(.60) + 70,000(.4) = $28,000
Chapter 12 - Decision Analysis

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

23. 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 12.10
Payoff Table for Real Estate Investment Example
Chapter 12 - Decision Analysis

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

25. Decision Making with Probabilities Decision Trees (3 of 4)
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.
Chapter 12 - Decision Analysis

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

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

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

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

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

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

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

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

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

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

38. 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 $444,000, and EVPI of $28,000.
Table 12.11
Payoff Table for the Real Estate Investment Example
Chapter 12 - Decision Analysis

39. 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 12 - Decision Analysis

40. Decision Analysis with Additional Information
Bayesian Analysis (3 of 3)
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) = .250 P(pP) = .077 P(pN) = .750
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.
Chapter 12 - Decision Analysis

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

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

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

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

48. Decision Analysis with Additional Information Utility (2 of 2)
Expected Cost (insurance) = .992($500) (500) = $500
Expected Cost (no insurance) = .992($0) (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.
Chapter 12 - Decision Analysis

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

50. 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 posteria probabilities using Bayes’ rule.
Perform a decision tree analysis using the posterior probability obtained in part e.
Chapter 12 - Decision Analysis

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

52. Example Problem Solution (4 of 9)
Decision Analysis
Example Problem Solution (4 of 9)
Minimax Regret Decision: Expand
Decisions Maximum Regrets
Expand $500,000 (minimum)
Status quo 650,000
Sell ,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 12 - Decision Analysis

53. Example Problem Solution (5 of 9)
Decision Analysis
Example Problem Solution (5 of 9)
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
Chapter 12 - Decision Analysis

54. 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 (.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 = $195,000
Chapter 12 - Decision Analysis

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

56. 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 12 - Decision Analysis

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