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Business Analytics

Decision Analysis

Chapter 15

Introduction (Slide 1 of 2)

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.

Introduction (Slide 2 of 2)

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.

Problem Formulation

Payoff Tables

Decision Trees

Problem Formulation (Slide 1 of 10)

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.

Problem Formulation (Slide 2 of 10)

Illustration: Pittsburgh Development Corporation (PDC):

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.

Problem Formulation (Slide 3 of 10)

Illustration: Pittsburgh Development Corporation (PDC) (cont.):

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:

Problem Formulation (Slide 4 of 10)

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:

Problem Formulation (Slide 5 of 10)

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.

Problem Formulation (Slide 6 of 10)

Payoff Tables (cont.):

Problem Formulation (Slide 7 of 10)

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.

Problem Formulation (Slide 8 of 10)

Figure 15.1: Decision Tree for the PDC Condominium Project ($ Millions)

Problem Formulation (Slide 9 of 10)

Decision Tree (cont.):

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.

Problem Formulation (Slide 10 of 10)

Decision Tree (cont.):

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.

Decision Analysis Without Probabilities

Optimistic Approach

Conservative Approach

Minimax Regret Approach

Decision Analysis without Probabilities (Slide 1 of 10)

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.

Decision Analysis without Probabilities (Slide 2 of 10)

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.

Decision Analysis without Probabilities (Slide 3 of 10)

Optimistic Approach (cont.):

In the PDC problem, the optimistic approach would lead the decision maker to choose the alternative corresponding to the largest profit.

Table 15.2: Maximum Payoff For Each PDC Decision Alternative

Decision Analysis without Probabilities (Slide 4 of 10)

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.

Decision Analysis without Probabilities (Slide 5 of 10)

Conservative Approach (cont.):

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.

Table 15.3: Minimum Payoff For Each PDC Decision Alternative

Decision Analysis without Probabilities (Slide 6 of 10)

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.

Decision Analysis without Probabilities (Slide 7 of 10)

Minimax Regret Approach (cont.):

Decision Analysis without Probabilities (Slide 8 of 10)

Table 15.4: Opportunity Loss, or Regret, Table for the PDC Condominium Project ($ Millions)

Decision Analysis without Probabilities (Slide 9 of 10)

Table 15.5: Maximum Regret for Each PDC Decision Alternative

Decision Analysis without Probabilities (Slide 10 of 10)

Minimax Regret Approach (cont.):

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

Expected Value Approach

Risk Analysis

Sensitivity Analysis

Decision Analysis with Probabilities (Slide 1 of 8)

Expected Value Approach:

In decision-making situations where probability assessments for the states of nature are available, we can use the expected value approach to identify the best decision alternative.

The expected value (EV) of a decision alternative is the sum of weighted payoffs for the decision alternative.

Decision Analysis with Probabilities (Slide 2 of 8)

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

Decision Analysis with Probabilities (Slide 3 of 8)

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

Decision Analysis with Probabilities (Slide 4 of 8)

Expected Value Approach (cont.):

The weight for a payoff is the probability of the associated state of nature and therefore the probability that the payoff will occur.

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.

Decision Analysis with Probabilities (Slide 5 of 8)

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.

Decision Analysis with Probabilities (Slide 6 of 8)

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

Decision Analysis with Probabilities (Slide 7 of 8)

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.

Decision Analysis with Probabilities (Slide 8 of 8)

Sensitivity Analysis (cont.):

Illustration: Suppose that, in the PDC problem, the probability for a strong demand is 0.2 and the probability for a weak demand is 0.8.

With these probability assessments, the recommended decision alternative is to

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

Expected Value of Sample Information

Expected Value of Perfect Information

Decision Analysis with Sample Information (Slide 1 of 14)

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.

Decision Analysis with Sample Information (Slide 2 of 14)

Illustration: 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.

A decision strategy is a sequence of decisions and chance outcomes in which the decisions chosen depend on the yet-to-be-determined outcomes of chance events.

Decision Analysis with Sample Information (Slide 3 of 14)

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

Decision Analysis with Sample Information (Slide 4 of 14)

Figure 15.6: The PDC Decision Tree with Branch Probabilities

Decision Analysis with Sample Information (Slide 5 of 14)

Figure 15.7: PDC Decision Tree After Computing Expected Values at Chance Nodes 6 to 14

Decision Analysis with Sample Information (Slide 6 of 14)

Figure 15.8: PDC Decision Tree After Choosing Best Decisions at Nodes 3, 4, and 5

Decision Analysis with Sample Information (Slide 7 of 14)

Figure 15.9: PDC Decision Tree Reduced to Two Decision Branches

Decision Analysis with Sample Information (Slide 8 of 14)

If the market research is favorable, construct the large condominium complex.

If the market research is unfavorable, construct the medium condominium complex.

Decision Analysis with Sample Information (Slide 9 of 14)

Expected Value of Sample Information:

From Figure 15.9 we can conclude that the difference

is the expected value of sample information (EVSI).

Decision Analysis with Sample Information (Slide 10 of 14)

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.

Decision Analysis with Sample Information (Slide 11 of 14)

Expected Value of Perfect Information (cont.):

We can state PDC’s optimal decision strategy when the perfect information becomes available as follows:

Decision Analysis with Sample Information (Slide 12 of 14)

Table 15.6: Payoff Table for the PDC Condominium Project ($ Millions)

Decision Analysis with Sample Information (Slide 13 of 14)

Expected Value of Perfect Information (cont.):

The original probabilities for the states of nature:

From equation (15.2) the expected value of the decision strategy that

with perfect information (EVwPI)).

Earlier, we found the expected value approach is decision alternative

$14.2 million; this is referred to as the expected value without perfect information (EVwoPI).

Decision Analysis with Sample Information (Slide 14 of 14)

Expected Value of Perfect Information (cont.):

Illustration for PDF: Expected value of the perfect information (EVPI) is

In general, expected value for perfect information (EVPI) is computed as:

Computing Branch Probabilities with Bayes’ Theorem

Computing Branch Probabilities with Bayes’ Theorem (Slide 1 of 4)

Bayes’ theorem can be used to compute branch probabilities for decision trees.

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.

because they are conditional probabilities based on the outcome of the sample information.

Computing Branch Probabilities with Bayes’ Theorem (Slide 2 of 4)

Figure 15.10: The PDC Decision Tree

Computing Branch Probabilities with Bayes’ Theorem (Slide 3 of 4)

In performing the probability computations, we need to know PDC’s assessment of the probabilities of the two states of nature,

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.

Computing Branch Probabilities with Bayes’ Theorem (Slide 4 of 4)

In the PDC problem, we assume that the following assessments are available for these conditional probabilities:

Bayes’ Theorem restated is:

Utility Theory

Utility and Decision Analysis

Utility Functions

Exponential Utility Function

Utility Theory (Slide 1 of 18)

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.

Utility Theory (Slide 2 of 18)

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.

Utility Theory (Slide 3 of 18)

The three decision alternatives are:

The states of nature are:

Utility Theory (Slide 4 of 18)

Table 15.7: Payoff Table for Swofford, Inc.

Utility Theory (Slide 5 of 18)

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

Utility Theory (Slide 6 of 18)

Utility and Decision Analysis (cont.):

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

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:

Utility Theory (Slide 7 of 18)

Utility and Decision Analysis (cont.):

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:

Utility Theory (Slide 8 of 18)

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

Table 15.9: Utility Table for Swofford, Inc.

Utility Theory (Slide 9 of 18)

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.

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 (Slide 10 of 18)

Table 15.10: Revised Utilities for Swofford, Inc., Assuming a Risk Taker

Table 15.11: Payoff Table for Swofford, Inc.

Utility Theory (Slide 11 of 18)

Utility Functions (cont.):

The analysis recommends investment B, with the highest expected utility of 3.95.

Utility Theory (Slide 12 of 18)

Table 15.12: Utility Table of a Risk Taker for Swofford, Inc.

Utility Theory (Slide 13 of 18)

Utility Functions (cont.):

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).

Utility Theory (Slide 14 of 18)

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

Utility Theory (Slide 15 of 18)

Utility Functions (cont.):

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 (Slide 16 of 18)

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 (Slide 17 of 18)

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

Utility Theory (Slide 18 of 18)

Exponential Utility Function (cont.):

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% chance of winning $2,000 and a 50% chance of losing $1,000, but not with a gamble with a 50% chance of winning $3,000 and a 50% chance of losing $1,500, then we would use R = $2,000 in equation (15.7).

End of Chapter 15

a small complex with 30 condominiums.

a medium complex with 60 condominiums.

a large complex with 90 condominiums.

strong demand for the condominiums

weak demand for the condominiums.

We will use the notation to denote the p

ayoff associated with decision

alternative and state of nature .

Using Table 15.1, 20 indicates that a pa

yoff of $20 million occurs if the

decision is

to build a large complex () and the str

ong demand state of

nature () occurs.

Illustration: The topmost payoff of 8 in

dicates that an $8 million profit is

anticipated if PDC constructs a small co

ndominium complex ()

demand turns out to be strong ().

and

(

)

Using equation 15.1 and the payoffs in T

able 15.1, the regret associated

with each combination of decision altern

ative and state of nature

is computed.

(

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(

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(

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(

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EV0.280.877.2

EV0.2140.856.8

EV0.2200.893.2

=+=

=+-=-

(

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construct a small condominium complex wi

th an expected value of $7.

2 million

15.9314.201.73

and receive a payoff of $20 million.

If , select and receive a payoff of $7

million.

If , select

(

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(

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0.8 and 0.2

PsPs

(

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0.820

0.2

ses perf

717.4

ectinformation is i.e., expe

cted valu

(

$17.414.2$3.2 million.

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The notation in and is read as "given"

and

PsFPsF

(

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(

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are referred to as posterior probabiliti

and

PsFPsF

favorable market research report

unfavorable market research report

strong demand (state of nature 1)

weak demand (state of nature 2)

(

)

(

)

and .

PsPs

make investment A

make investment B

do not invest

real estate prices go up

real estate prices remain stable

real estate prices go down

u(best payoff) > u(worst payoff).

(

)

probability 1 of the worst payoff.

(

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(

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(

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(

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best payoff1worst payoff

UMpUpU

=+-

(

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(

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Using the state-of-nature probabilities

the expected utility for each decision a

lternative is:

0.3, 0.5, and

0.2,

PsPs

(

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(

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EU0.35.00.54.00.203.50

EU0.3100.51.50.21.03.95

EU0.32.50.52.50.22.52.50

=++=