Cengage_EBA_Chapter12.pptx

Chapter 12
Decision Analysis
1

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.
• E.g. 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.
2

Introduction
• Risk analysis helps to provide the probability information about the
favourable as well as the unfavourable 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
3

Problem Formulation
• The steps in the decision analysis process are as follows:
• 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.
5

Problem Formulation
• Example:
PDC commissioned preliminary architectural drawings for three different
projects: one with 30 condominiums, one with 60 condominiums, and one
with 90 condominiums. The financial success of the project depends on the
size of the condominium complex and the chance event concerning the
demand for the condominiums. 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
6

Problem Formulation
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
7

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.
8

Problem Formulation
• 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
• 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

Table 12.1 - Payoff Table For The PDC
Condominium Project ($ Millions)
11

Problem Formulation
• Example:
We will use the notation Vij to denote the payoff associated
with decision alternative i and state of nature j. Using Table
12.1, 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.
12

Problem Formulation
• Decision Tree
• Provides a graphical representation of the decision-making
process
• It 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).
13

Figure 12.1 - Decision Tree For The PDC Condominium Project ($
Millions)
14

Problem Formulation
• The decision tree in Figure 12.1 shows:
• Four nodes, numbered 1–4.
• Nodes: They are used to represent decisions and chance events.
• Squares are used to depict decision nodes, circles are used to
depict chance nodes.
• Thus, node 1 is a decision node, and nodes 2, 3, and 4 are chance
nodes.
15

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.
16

Decision Analysis Without
Probabilities
17

18
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.

19
• 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.

20
• In the PDC problem,
• the optimistic approach would lead the decision maker to choose the
alternative corresponding to the largest profit.
• for minimization problems, this approach leads to choosing the
alternative with the smallest payoff.

21
Table 12.2 - Maximum Payoff For Each PDC
Decision Alternative

• 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.
22

• 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.
• For problems involving minimization (for example, when the output
measure is cost), this approach identifies the alternative that will
minimize the maximum payoff.
23

24
Table 12.3 - Minimum Payoff For Each PDC Decision
Alternative

• Minimax Regret Approach
• Regret is the difference between the payoff associated with a
particular decision alternative and the payoff associated with the
decision 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.
25

Equation 12.1 - Regret (Opportunity Loss)
26

27
• Using equation (12.1) and the payoffs in Table 12.1, 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.

Table 12.4 - Opportunity Loss, Or Regret, Table For The PDC
Condominium Project ($ Millions)
28

Table 12.5 - Maximum Regret For Each PDC Decision
Alternative
29

30
• 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.

Decision Analysis With
Probabilities
31

• Expected Value Approach
• The expected value 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.
• Figure 12.2 shows the decision tree for the PDC problem with
state-of-nature branch probabilities.
Decision Analysis With Probabilities
32

Equation 12.2 - Expected Value of Decision Alternative Di
33

Figure 12.2 - PDC Decision Tree With State-of-nature
Branch Probabilities
34

Figure 12.3 - Applying The Expected Value Approach Using
A Decision Tree For The PDC Condominium Project
35

• 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.
36

• 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.
37

Figure 12.4 - Risk Profile For The Large Complex Decision
Alternative For The PDC Condominium Project
38

• 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.
39

• So 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.
40

• 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) + 0.8 (7) = 7.2
EV(d2 ) = 0.2 (14) + 0.8 (5) = 6.8
EV(d3 ) = 0.2 (20) + 0.8 (-9) = -3.2
41

42
• With these probability assessments, the recommended decision
alternative is to construct a small condominium complex (d1), with
an expected value of $7.2 million.
• Thus, 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.

Decision Analysis with Sample
Information
43

• 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.
Decision Analysis with Sample Information
44

• Posterior probabilities are revised probabilities after obtaining
additional information.
45

• 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.
46

Figure 12.5 - The PDC Decision Tree Including The
Market Research Study
47

FIGURE 12.6 THE PDC DECISION TREE WITH
BRANCH PROBABILITIES
48

FIGURE 12.7 PDC DECISION TREE AFTER COMPUTING
EXPECTED VALUES AT CHANCE NODES 6–14
49

FIGURE 12.8 PDC DECISION TREE AFTER CHOOSING BEST
DECISIONS AT NODES 3, 4, AND 5
50

FIGURE 12.9 PDC DECISION TREE REDUCED TO TWO
DECISION BRANCHES
51

• If the market research is favorable, construct the large
condominium complex.
• If the market research is unfavorable, construct the medium
condominium complex.
52

• Expected Value of Sample Information
• From Figure 12.9 we can conclude that the difference, 15.93 -
14.20 = 1.73, is the expected value of sample information (EVSI).
53

• 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.
• 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.
54

• If s2, select d1 and receive a payoff of $7 million.
• The original probabilities for the states of nature:
• P(s1) = 0.8 and P(s2) = 0.2.
55

• 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 using the expected value approach is decision alternative d3
$14.2 million is referred to as the expected value without perfect
information (EVwoPI).
• So, Expected value of the perfect information (EVPI) is $17.4 -
$14.2 = $3.2 million.
56

TABLE 12.6 PAYOFF TABLE FOR THE PDC CONDOMINIUM PROJECT
($ MILLIONS)
57

Computing Branch Probabilities
with Bayes’ Theorem
58

• Baye’s Theorem
• 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.
Computing Branch probabilities with Bayes’
Theorem
59

• 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.
Theorem
60

• The steps used to develop this table computations for the PDC
problem based on a favorable market research report (F) are as
follows:
• Step 1. In column 1, enter the states of nature
• Step 2. In column 2, enter the prior probabilities for the states of
nature
• Step 3. In column 3, enter the conditional probabilities of a favorable
market research
Theorem
61

• report (F) given each state of nature
• Step 4. In column 4, compute the joint probabilities by multiplying
the prior probability
• values in column 2 by the corresponding conditional probability
values
• in column 3
• Step 5. Sum the joint probabilities in column 4 to obtain the
probability of a favorable
Theorem
62

• market research report, P(F)
• Step 6. Divide each joint probability in column 4 by P(F) = 0.77 to
obtain the revised
• or posterior probabilities, P(s1 |F) and P(s2 |F)
Theorem
63

TABLE 12.7 BRANCH PROBABILITIES FOR THE PDC CONDOMINIUM
PROJECT BASED
ON A FAVORABLE MARKET RESEARCH REPORT
64

TABLE 12.8 BRANCH PROBABILITIES FOR THE PDC CONDOMINIUM
PROJECT BASED ON AN UNFAVORABLE MARKET RESEARCH
REPORT
65

• 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 Theory
67

• Utility
• 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.
• Example of a situation in which utility can help in selecting the best
decision alternative:
Utility Theory
68

• 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.
Utility Theory
69

TABLE 12.9 PAYOFF TABLE FOR SWOFFORD, INC.
70

• Utility and Decision Analysis
• The following steps state in general terms the procedure used to
solve the Swofford, Inc., 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)
Utility Theory
71

• Step 3. For every other monetary value m in the original payoff table,
do the following to determine its utility:
• a. Define the lottery such that there is a probability p of the best
payoff and a probability (1 - p) of the worst payoff
• b. 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)
• c. Calculate the utility of m as follows:
• U(M) = pU(best payoff) + (1 - p)U(worst payoff)
Utility Theory
72

• 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
Utility Theory
73

TABLE 12.10 UTILITY OF MONETARY PAYOFFS FOR SWOFFORD,
INC.
74

TABLE 12.11 UTILITY TABLE FOR SWOFFORD, INC.
75

• Utility Functions
• We describe how 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.
Utility Theory
76

• We 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.
Utility Theory
77

TABLE 12.12 REVISED UTILITIES FOR SWOFFORD, INC.,
ASSUMING A RISK TAKER
78

TABLE 12.13 PAYOFF TABLE FOR SWOFFORD, INC.
79

• 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) + 0.5 (1.5 ) + 0.2 (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
Utility Theory
80

FIGURE 12.11 UTILITY FUNCTION FOR MONEY FOR RISK-
AVOIDER, RISK- TAKER, AND RISK-NEUTRAL DECISION
MAKERS
81

• Utility function for a risk avoider shows a diminishing marginal
return for money. For example, the increase in utility going from a
monetary value of -$30,000 to $0 is 7.5 - 4.0 = 3.5, whereas the
increase in utility in going from $0 to $30,000 is only 9.5 - 7.5 = 2.0.
• Utility function for a risk taker shows an increasing marginal
return. For example, the increase in utility for the risk taker in
going from -$30,000 to $0 is 2.5 - 1.0 = 1.5, whereas the increase
in utility in going from $0 to $30,000 for the risk taker is 5.0 - 2.5 =
2.5.
Utility Theory
82

• Utility function for a decision maker neutral to risk shows a
constant return.
• 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.
Utility Theory
83

• 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.
Utility Theory
84

FIGURE 12.12 EXPONENTIAL UTILITY FUNCTIONS
WITH DIFFERENT RISK TOLERANCE (R) VALUES
85

• The R parameter in equation (12.7) 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).
Utility Theory
86

• Smaller R values indicate that the decision maker has less risk
tolerance (is more risk averse). For example:
• For instance, 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).
Utility Theory
87

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