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. 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

4. 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

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 3.3
Payoff Table Illustrating a Maximax Decision
Chapter 3 - 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 (best of the worst-case) payoffs; a pessimistic criterion.
conservative
Table 3.4
Payoff Table Illustrating a Maximin Decision
Chapter 3 - 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.
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

8. 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

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 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

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 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. 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

20. 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

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. 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

24. 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

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 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

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. 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

34. 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

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. 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

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
as before…
new info
new, given
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. 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

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”
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. 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

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 3 - Decision Analysis

47. 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

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 (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. 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

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 posterior probabilities using Bayes’ rule.
Perform a decision tree analysis using the posterior probability obtained in part e.
Chapter 3 - 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 (Optimist) Decision: Maintain status quo
Decisions maximum Payoffs
Expand $800,000
Status quo 1,300,000 (Maximum)
Sell ,000
Maximin (Conservative) Decision: Expand
Decisions minimum Payoffs
Expand $500,000 (Maximum)
Status quo -150,000
Sell ,000
Chapter 3 - Decision Analysis

52. 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 ,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

53. 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

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 3 - 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 3 - 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 3 - Decision Analysis

57. 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

58. Chapter 3 - Decision Analysis