1. Decision Analysis
Chapter 15

2. Introduction Business analytics is about making better decisions
Decision analysis can be used to develop an optimal strategy:
When a decision maker is faced with several decision alternatives and an uncertain or risk-filled pattern of future events
For example: The State of North Carolina used decision analysis in evaluating whether to implement a medical screening test to detect metabolic disorders in newborns
A good decision analysis includes careful consideration of risk

3. Introduction
Risk analysis helps to provide the probability information about the favorable as well as the unfavorable outcomes that may occur
Decision analysis considers problems that involve reasonably few decision alternatives and reasonably few possible future events
Topics to be discussed under decision analysis:
Payoff tables and decision trees
Sensitivity analysis
Use of Bayes’ theorem

4. Problem Formulation
Payoff Tables
Decision Trees

5. Problem Formulation
The first step in the decision analysis process is problem formulation:
Create verbal statement of the problem
Identify the decision alternatives:
The uncertain future events, referred to as chance events
The outcomes associated with each combination of decision alternative and chance event outcome

6. Problem Formulation
Example: Construction project of Pittsburgh Development Corporation
PDC commissioned preliminary architectural drawings for three different projects:
One with 30 condominiums
One with 60 condominiums
One with 90 condominiums
The financial success of the project depends on:
The size of the condominium complex
The chance event concerning the demand for the condominiums

7. Problem Formulation
The statement of the PDC decision problem is to select the size of the new luxury condominium project that will lead to the largest profit given the uncertainty concerning the demand for the condominiums
Given the statement of the problem, it is clear that the decision is to select the best size for the condominium complex
PDC has the following three decision alternatives:
d1 = a small complex with 30 condominiums
d2 = a medium complex with 60 condominiums
d3 = a large complex with 90 condominiums

8. Problem Formulation
In decision analysis, the possible outcomes for a chance event are the states of nature
The states of nature are mutually exclusive (no more than one can occur) and collectively exhaustive (at least one must occur)
Thus one and only one of the possible states of nature will occur
For PDC example: The chance event concerning the demand for the condominiums has two states of nature:
s1 = strong demand for the condominiums
s2 = weak demand for the condominiums

9. Problem Formulation Payoff Tables
Table 15.1: Payoff Table for the PDC Condominium Project ($ Millions)
Payoff Tables
Payoff is the outcome resulting from a specific combination of a decision alternative and a state of nature
Payoff table is a table showing payoffs for all combinations of decision alternatives and states of nature

10. Problem Formulation
We will use the notation Vij to denote the payoff associated with decision alternative i and state of nature j
Using Table 15.1, V31 = 20 indicates that a payoff of $20 million occurs if the decision is to build a large complex (d3) and the strong demand state of nature (s1) occurs

11. Problem Formulation Decision Tree
A decision tree provides a graphical representation of the decision- making process
Shows the natural or logical progression that will occur over time
Example: The topmost payoff of 8 indicates that an $8 million profit is anticipated if PDC constructs a small condominium complex (d1) and demand turns out to be strong (s1)

12. Figure 15.1: Decision Tree For The PDC Condominium Project ($ Millions)

13. Problem Formulation The decision tree in Figure 15.1 shows:
Four nodes, numbered 1–4
Nodes are used to represent decisions and chance events
Squares are used to depict decision nodes, circles are used to depict chance nodes
Node 1 is a decision node, and nodes 2, 3, and 4 are chance nodes

14. Problem Formulation
The branches connect the nodes; those leaving the decision node correspond to the decision alternatives
The branches leaving each chance node correspond to the states of nature
The outcomes (payoffs) are shown at the end of the states-of-nature branches

15. Decision Analysis Without Probabilities
Optimistic Approach
Conservative Approach
Minimax Regret Approach

16. Decision Analysis Without Probabilities
Decision analysis without probabilities is appropriate in situations:
In which a simple best-case and worst-case analysis is sufficient
Where the decision maker has little confidence in his or her ability to assess the probabilities

17. Decision Analysis Without Probabilities
Optimistic Approach
The optimistic approach evaluates each decision alternative in terms of the best payoff that can occur
The decision alternative that is recommended is the one that provides the best possible payoff
For minimization problems, this approach leads to choosing the alternative with the smallest payoff

18. Decision Analysis Without Probabilities
Table 15.2: Maximum Payoff For Each PDC Decision Alternative
In the PDC problem, the optimistic approach would lead the decision maker to choose the alternative corresponding to the largest profit

19. Decision Analysis Without Probabilities
Conservative Approach
The conservative approach evaluates each decision alternative in terms of the worst payoff that can occur
The decision alternative recommended is the one that provides the best of the worst possible payoffs
For problems involving minimization (for example, when the output measure is cost), this approach identifies the alternative that will minimize the maximum payoff

20. Decision Analysis Without Probabilities
Table 15.3: Minimum Payoff For Each PDC Decision Alternative
In the PDC problem, the conservative approach would lead the decision maker to choose the alternative that maximizes the minimum possible profit that could be obtained

21. Decision Analysis Without Probabilities
Minimax Regret Approach
Regret is the difference between the payoff associated with a particular decision alternative and the payoff associated with the decision that would yield the most desirable payoff for a given state of nature
Regret is often referred to as opportunity loss
Under the minimax regret approach, one would choose the decision alternative that minimizes the maximum state of regret that could occur over all possible states of nature

22. Decision Analysis Without Probabilities
Using equation (15.1) and the payoffs in Table 15.1, the regret associated with each combination of decision alternative di and state of nature sj is computed
To compute the regret, subtract each entry in a column from the largest entry in the column

23. Decision Analysis Without Probabilities
Table 15.4: Opportunity Loss, or Regret, Table for the PDC Condominium Project ($ Millions)
Table 15.5: Maximum Regret for Each PDC Decision Alternative

24. Decision Analysis Without Probabilities
The next step in applying the minimax regret approach is to list the maximum regret for each decision alternative
For the PDC problem, the alternative to construct the medium condominium complex, with a corresponding maximum regret of $6 million, is the recommended minimax regret decision

25. Decision Analysis with Probabilities
Expected Value Approach
Risk Analysis
Sensitivity Analysis

26. Decision Analysis With Probabilities
Expected Value Approach
The expected value (EV) of a decision alternative is the sum of weighted payoffs for the decision alternative
The weight for a payoff is the probability of the associated state of nature and therefore the probability that the payoff will occur

27. Figure 15.2: PDC Decision Tree with State-of-Nature Branch Probabilities

28. Figure 15.3: Applying the Expected Value Approach Using a Decision Tree for the PDC Condominium Project

29. Decision Analysis With Probabilities
Select the decision branch leading to the chance node with the best expected value
The decision alternative associated with this branch is the recommended decision
In practice, obtaining precise estimates of the probabilities for each state of nature is often impossible, so historical data is preferred to use for estimating the probabilities for the different states of nature

30. Decision Analysis With Probabilities
Risk Analysis
Risk analysis helps the decision maker recognize the difference between the expected value of a decision alternative and the payoff that may actually occur
Decision alternative and a state of nature combine to generate the payoff associated with a decision
Risk profile for a decision alternative shows the possible payoffs along with their associated probabilities

31. Figure 15.4: Risk Profile for the Large Complex Decision Alternative for the PDC Condominium Project

32. Decision Analysis With Probabilities
Sensitivity Analysis
Sensitivity analysis determines how changes in the probabilities for the states of nature or changes in the payoffs affect the recommended decision alternative
In many cases, the probabilities for the states of nature and the payoffs are based on subjective assessments
Sensitivity analysis helps the decision maker understand which of these inputs are critical to the choice of the best decision alternative
If a small change in the value of one of the inputs causes a change in the recommended decision alternative, the solution to the decision analysis problem is sensitive to that particular input

33. Decision Analysis With Probabilities
Example: Suppose that, in the PDC problem, the probability for a strong demand is revised to 0.2 and the probability for a weak demand is revised to 0.8
EV(d1 ) = 0.2 (8) (7) = 7.2
EV(d2 ) = 0.2 (14) (5) = 6.8
EV(d3 ) = 0.2 (20) (29) = 23.2
With these probability assessments, the recommended decision alternative is to construct a small condominium complex (d1), with an expected value of $7.2 million
When the probability of strong demand is large, PDC should build the large complex; when the probability of strong demand is small, PDC should build the small complex

34. Decision Analysis with Sample Information
Expected Value of Sample Information
Expected Value of Perfect Information

35. Decision Analysis with Sample Information
Decision makers have the ability to collect additional information about the states of nature
Additional information is obtained through experiments designed to provide sample information about the states of nature
The preliminary or prior probability assessments for the states of nature that are the best probability values available prior to obtaining additional information
Posterior probabilities are revised probabilities after obtaining additional information

36. Decision Analysis with Sample Information
Example: PDC management is considering a 6-month market research study designed to learn more about potential market acceptance of the PDC condominium project, anticipating two results:
Favorable report: A substantial number of the individuals contacted express interest in purchasing a PDC condominium
Unfavorable report: Very few of the individuals contacted express interest in purchasing a PDC condominium

37. Figure 15.5: The PDC Decision Tree Including the Market Research Study

38. Figure 15.6: The PDC Decision Tree With Branch Probabilities

39. Figure 15.7: PDC Decision Tree after Computing Expected Values at Chance Nodes 6 to 14

40. Figure 15.8: PDC Decision Tree after Choosing Best Decisions at Nodes 3, 4, And 5

41. Figure 15.9: PDC Decision Tree Reduced to Two Decision Branches

42. Decision Analysis with Sample Information
If the market research is favorable, construct the large condominium complex
If the market research is unfavorable, construct the medium condominium complex 

43. Decision Analysis with Sample Information
Expected Value of Sample Information
From Figure 15.9 we can conclude that the difference, – = 1.73, is the expected value of sample information (EVSI)

44. Decision Analysis with Sample Information
Expected Value of Perfect Information
A special case of gaining additional information related to a decision problem is when the sample information provides perfect information on the states of nature

45. Decision Analysis with Sample Information
Table 15.6: Payoff Table for the PDC Condominium Project ($ Millions)
We can state PDC’s optimal decision strategy when the perfect information becomes available as follows:
If s1, select d3 and receive a payoff of $20 million
If s2, select d1 and receive a payoff of $7 million

46. Decision Analysis with Sample Information
The original probabilities for the states of nature:
P(s1) = 0.8 and P(s2) = 0.2
From equation (12.2) the expected value of the decision strategy that uses perfect information is 0.8(20) + 0.2(7) = 17.4 (i.e., expected value with perfect information (EVwPI))
Earlier, we found the expected value approach is decision alternative d3 $14.2 million; this is referred to as the expected value without perfect information (EVwoPI)

47. Decision Analysis with Sample Information
Example for PDF: Expected value of the perfect information (EVPI) is $17.4 – $14.2 = $3.2 million
In general, expected value for perfect information (EVPI) is computed as:

48. Computing Branch Probabilities with Bayes’ Theorem

49. Computing Branch Probabilities with Bayes’ Theorem
Bayes’ theorem can be used to compute branch probabilities for decision trees
The notation | in P(s1|F) and P(s2|F) is read as “given” and indicates a conditional probability because we are interested in the probability of a particular state of nature “conditioned” on the fact that we receive a favorable market report
P(s1|F) and P(s2|F) are referred to as posterior probabilities because they are conditional probabilities based on the outcome of the sample information

50. Figure 15.10: The PDC Decision Tree
F = favorable market research report U = unfavorable market research report S1 = strong demand (state of nature 1) S2 = weak demand (state of nature 2)

51. Computing Branch Probabilities with Bayes’ Theorem
In performing the probability computations, we need to know PDC’s assessment of the probabilities of the two states of nature, P(s1) and P(s2)
We must know the conditional probability of the market research outcomes given each state of nature
To carry out the probability calculations, we need conditional probabilities for all sample outcomes given all states of nature

52. Computing Branch Probabilities with Bayes’ Theorem
In the PDC problem we assume that the following assessments are available for these conditional probabilities:
Bayes’ Theorem restated is:

53. Utility Theory Utility and Decision Analysis Utility Functions
Exponential Utility Function

54. Utility Theory
When monetary value does not necessarily lead to the most preferred decision, expressing the value (or worth) of a consequence in terms of its utility will permit the use of expected utility to identify the most desirable decision alternative
Utility is a measure of the total worth or relative desirability of a particular outcome
Reflects the decision maker’s attitude toward a collection of factors such as profit, loss, and risk

55. Utility Theory
Example of a situation in which utility can help in selecting the best decision alternative:
Swofford Inc. currently has two investment opportunities that require approximately the same cash outlay
The cash requirements necessary prohibit Swofford from making more than one investment at this time
Consequently, three possible decision alternatives may be considered

56. Utility Theory The three decision alternatives are:
d1 = make investment A
d2 = make investment B
d3 = do not invest
The states of nature are:
s1 = real estate prices go up
s2 = real estate prices remain stable
Table 15.7: Payoff Table for Swofford, Inc.

57. Utility Theory Utility and Decision Analysis
A decision maker who would choose a guaranteed payoff over a lottery with a superior expected payoff is a risk avoider
The following steps state in general terms the procedure used to solve the Swofford investment problem:
Step 1. Develop a payoff table using monetary values
Step 2. Identify the best and worst payoff values in the table and assign each a utility, with u(best payoff)> u(worst payoff)

58. U(M) = pU(best payoff) + (1 - p)U(worst payoff)
Utility Theory
Step 3. For every other monetary value m in the original payoff table, do the following to determine its utility:
Define the lottery such that there is a probability p of the best payoff and a probability (1 - p) of the worst payoff
Determine the value of p such that the decision maker is indifferent between a guaranteed payoff of m and the lottery defined in step 3(a)
Calculate the utility of m as follows:
U(M) = pU(best payoff) + (1 - p)U(worst payoff)

59. Utility Theory
Step 4. Convert each monetary value in the payoff table to a utility
Step 5. Apply the expected utility approach to the utility table developed in Step 4 and select the decision alternative with the highest expected utility
We can compute the expected utility (EU) of the utilities in a similar fashion as we computed expected value

60. Utility Theory
Table 15.8: Utility of Monetary Payoffs for Swofford, Inc.
Table 15.9: Utility Table for Swofford, Inc.

61. Utility Theory Utility Functions
Different decision makers may approach risk in terms of their assessment of utility
A risk taker is a decision maker who would choose a lottery over a guaranteed payoff when the expected value of the lottery is inferior to the guaranteed payoff

62. Utility Theory
Analyze the decision problem faced by Swofford from the point of view of a decision maker who would be classified as a risk taker
Compare the conservative point of view of Swofford’s president (a risk avoider) with the behavior of a decision maker who is a risk taker

63. Table 15.11: Payoff Table for Swofford, Inc.
Utility Theory
Table 15.10: Revised Utilities for Swofford, Inc., Assuming a Risk Taker
Table 15.11: Payoff Table for Swofford, Inc.

64. Utility Theory
Using the state-of-nature probabilities P(s1) = 0.3, P(s2) = 0.5, and P(s3) = 0.2, the expected utility for decision alternative is:
EU(d2 ) = 0.3 (10) (1.5 ) (1.0 ) = 3.95
EU(d1 ) = 3.50
EU(d3 ) = 2.50
The analysis recommends investment B, with the highest expected utility of 3.95
Table 15.12: Utility Table of a Risk Taker for Swofford, Inc.

65. Utility Theory
Utility function for a risk avoider shows a diminishing marginal return for money
Utility function for a risk taker shows an increasing marginal return
These values can be plotted on a graph (Figure 15.11) as the utility function for money
Top curve is utility function for risk avoider
Bottom curve is utility function for risk taker
Utility function for a decision maker neutral to risk shows a constant return (middle line)

66. Figure 15.11: Utility Function for Money for Risk-Avoider, Risk-Taker, and Risk-Neutral Decision Makers

67. Utility Theory
The following characteristics are associated with a risk-neutral decision maker:
The utility function can be drawn as a straight line connecting the “best” and the “worst” points
The expected utility approach and the expected value approach applied to monetary payoffs result in the same action

68. Utility Theory Exponential Utility Function
Used as an alternative to assume that the decision maker’s utility is defined when decision maker provides enough indifference values to create a utility function
All the exponential utility functions indicate that the decision maker is risk averse

69. Figure 15.12: Exponential Utility Functions with Different Risk Tolerance (R) Values

70. Utility Theory
The R parameter in equation (15.7) represents the decision maker’s risk tolerance; it controls the shape of the exponential utility function
Larger R values create flatter exponential functions, indicating that the decision maker is less risk averse (closer to risk neutral)
Smaller R values indicate that the decision maker has less risk tolerance (is more risk averse)
Example: If the decision maker is comfortable accepting a gamble with a 50 percent chance of winning $2,000 and a 50 percent chance of losing $1,000, but not with a gamble with a 50 percent chance of winning $3,000 and a 50 percent chance of losing $1,500, then we would use R = $2,000 in equation (12.7)