Chapter 8Decision Analysis

Problem Formulation Decision Making without Probabilities Decision Making with Probabilities Risk Analysis and Sensitivity Analysis Decision Analysis with Sample Information Computing Branch Probabilities

Problem Formulation

A decision problem is characterized by decision alternatives, states of nature, and resulting payoffs.

The decision alternatives are the different possible strategies the decision maker can employ.

The states of nature refer to future events, not under the control of the decision maker, which may occur. States of nature should be defined so that they are mutually exclusive and collectively exhaustive.

Payoff Tables

The consequence resulting from a specific combination of a decision alternative and a state of nature is a payoff.

A table showing payoffs for all combinations of decision alternatives and states of nature is a payoff table.

Payoffs can be expressed in terms of profit, cost, time, distance or any other appropriate measure.

Decision Trees

A decision tree is a chronological representation of the decision problem.

Each decision tree has two types of nodes; round nodes correspond to the states of nature while square nodes correspond to the decision alternatives.

The branches leaving each round node represent the different states of nature while the branches leaving each square node represent the different decision alternatives.

At the end of each limb of a tree are the payoffs attained from the series of branches making up that limb.

Decision Tree Example

Decision Making without Probabilities

Three commonly used criteria for decision making when probability information regarding the likelihood of the states of nature is unavailable are: • the optimistic approach• the conservative approach• the minimax regret approach.

Optimistic Approach

The optimistic approach would be used by an optimistic decision maker.

The decision with the largest possible payoff is chosen.

If the payoff table was in terms of costs, the decision with the lowest cost would be chosen.

Conservative Approach

The conservative approach would be used by a conservative decision maker.

For each decision the minimum payoff is listed and then the decision corresponding to the maximum of these minimum payoffs is selected. (Hence, the minimum possible payoff is maximized.)

If the payoff was in terms of costs, the maximum costs would be determined for each decision and then the decision corresponding to the minimum of these maximum costs is selected. (Hence, the maximum possible cost is minimized.)

Minimax Regret Approach

The minimax regret approach requires the construction of a regret table or an opportunity loss table.

This is done by calculating for each state of nature the difference between each payoff and the largest payoff for that state of nature.

Then, using this regret table, the maximum regret for each possible decision is listed.

The decision chosen is the one corresponding to the minimum of the maximum regrets.

Example

Consider the following problem with three decision alternatives and three states of nature with the following payoff table representing profits:

States of Nature s1 s2 s3

d1 4 4 -2

Decisions d2 0 3 -1

d3 1 5 -3

Example: Optimistic Approach

An optimistic decision maker would use the optimistic (maximax) approach. We choose the decision that has the largest single value in the payoff table.

Maximum Decision Payoff

d1 4

d2 3

d3 5

Maximaxpayoff

Maximaxdecision

Example: Conservative Approach

A conservative decision maker would use the conservative (maximin) approach. List the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs.

Minimum Decision Payoff

d1 -2

d2 -1

d3 -3

Maximindecision

Maximinpayoff

For the minimax regret approach, first compute a regret table by subtracting each payoff in a column from the largest payoff in that column. In this example, in the first column subtract 4, 0, and 1 from 4; etc. The resulting regret table is:

s1 s2 s3

d1 0 1 1

d2 4 2 0

d3 3 0 2

Example: Minimax Regret Approach

For each decision list the maximum regret. Choose the decision with the minimum of these values.

Maximum Decision Regret d1 1

d2 4

d3 3

Example: Minimax Regret Approach

Minimaxdecision

Minimaxregret

Decision Making with Probabilities

Expected Value Approach• If probabilistic information regarding the

states of nature is available, one may use the expected value (EV) approach.

• Here the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring.

• The decision yielding the best expected return is chosen.

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

The expected value (EV) of decision alternative di is defined as:

where: N = the number of states of nature P(sj ) = the probability of state of

nature sj

Vij = the payoff corresponding to decision alternative di and state of nature sj

Expected Value of a Decision Alternative

EV( ) ( )d P s Vi j ijj

N

1

EV( ) ( )d P s Vi j ijj

N

1

Example: Burger Prince

Burger Prince Restaurant is considering opening a new restaurant on Main Street. It has three

different models, each with a different

seating capacity. Burger Prince

estimates that the average number of

customers per hour will be 80, 100, or

120. The payoff table for the three

models is on the next slide.

Payoff Table

Average Number of Customers Per Hour

s1 = 80 s2 = 100 s3 = 120

Model A $10,000 $15,000 $14,000

Model B $ 8,000 $18,000 $12,000

Model C $ 6,000 $16,000 $21,000

Expected Value Approach

Calculate the expected value for each decision. The decision tree on the next slide can assist in this calculation. Here d1, d2, d3

represent the decision alternatives of models A, B, C, and s1, s2, s3 represent the states of

nature of 80, 100, and 120.

Decision Tree

11

.2

.4

.4

.4

.2

.4

.4

.2

.4

d1

d2

d3

s1

s1

s1

s2

s3

s2

s2

s3

s3

Payoffs10,000

15,000

14,0008,000

18,000

12,000

6,000

16,000

21,000

22

33

44

Expected Value for Each Decision

Choose the model with largest EV, Model C.

33

d1

d2

d3

EMV = .4(10,000) + .2(15,000) + .4(14,000) = $12,600

EMV = .4(8,000) + .2(18,000) + .4(12,000) = $11,600

EMV = .4(6,000) + .2(16,000) + .4(21,000) = $14,000

Model A

Model B

Model C

22

11

44

Expected Value of Perfect Information

Frequently information is available which can improve the probability estimates for the states of nature.

The expected value of perfect information (EVPI) is the increase in the expected profit that would result if one knew with certainty which state of nature would occur.

The EVPI provides an upper bound on the expected value of any sample or survey information.

Expected Value of Perfect Information

EVPI Calculation• Step 1:

Determine the optimal return corresponding to each state of nature.

• Step 2: Compute the expected value of these

optimal returns.• Step 3:

Subtract the EV of the optimal decision from the amount determined in step (2).

Calculate the expected value for the optimum payoff for each state of nature and subtract the EV of the optimal decision.

EVPI= EVwPI - EVwoPI =.4(10,000) + .2(18,000) + .4(21,000) - 14,000 = $2,000

Expected Value of Perfect Information

Risk Analysis

Risk analysis helps the decision maker recognize the difference between:• the expected value of a decision alternative,

and• the payoff that might actually occur

The risk profile for a decision alternative shows the possible payoffs for the decision alternative along with their associated probabilities.

Risk Profile

Model C Decision Alternative

.10.10

.20.20

.30.30

.40.40

.50.50

5 10 15 20 255 10 15 20 25

Pro

bab

ility

Pro

bab

ility

Profit ($thousands)Profit ($thousands)

Sensitivity Analysis

Sensitivity analysis can be used to determine how changes to the following inputs affect the recommended decision alternative:• probabilities for the states of nature• values of the payoffs

If a small change in the value of one of the inputs causes a change in the recommended decision alternative, extra effort and care should be taken in estimating the input value.

Bayes’ Theorem and Posterior Probabilities

Knowledge of sample (survey) information can be used to revise the probability estimates for the states of nature.

Prior to obtaining this information, the probability estimates for the states of nature are called prior probabilities.

With knowledge of conditional probabilities for the outcomes or indicators of the sample or survey information, these prior probabilities can be revised by employing Bayes' Theorem.

The outcomes of this analysis are called posterior probabilities or branch probabilities for decision trees.

Computing Branch Probabilities

Branch (Posterior) Probabilities Calculation • Step 1:

For each state of nature, multiply the prior probability by its conditional probability for the indicator -- this gives the joint probabilities for the states and indicator.

Computing Branch Probabilities

Branch (Posterior) Probabilities Calculation • Step 2:

Sum these joint probabilities over all states -- this gives the marginal probability for the indicator.

• Step 3: For each state, divide its joint

probability by the marginal probability for the indicator -- this gives the posterior probability distribution.

Expected Value of Sample Information

The expected value of sample information (EVSI) is the additional expected profit possible through knowledge of the sample or survey information.

Expected Value of Sample Information

EVSI Calculation• Step 1:

Determine the optimal decision and its expected return for the possible outcomes of the sample using the posterior probabilities for the states of nature.

• Step 2: Compute the expected value of these optimal returns.

• Step 3: Subtract the EV of the optimal

decision obtained without using the sample information from the amount determined in step (2).

Efficiency of Sample Information

Efficiency of sample information is the ratio of EVSI to EVPI.

As the EVPI provides an upper bound for the EVSI, efficiency is always a number between 0 and 1.

Burger Prince must decide whether or not to purchase a marketing survey from Stanton Marketing for $1,000. The results of the survey are "favorable" or "unfavorable". The conditional probabilities are:

P(favorable | 80 customers per hour) = .2

P(favorable | 100 customers per hour) = .5

P(favorable | 120 customers per hour) = .9

Should Burger Prince have the survey performed by Stanton Marketing?

Sample Information

Favorable

State Prior Conditional Joint Posterior

80 .4 .2 .08 .148

100 .2 .5 .10 .185

120 .4 .9 .36 .667 Total .54 1.000

P(favorable) = .54

Posterior Probabilities

Unfavorable

State Prior Conditional Joint Posterior

80 .4 .8 .32 .696

100 .2 .5 .10 .217

120 .4 .1 .04 .087

Total .46 1.000

P(unfavorable) = .46

Posterior Probabilities

Decision Tree

Top Half

s1 (.148)

s1 (.148)

s1 (.148)s2 (.185)

s2 (.185)

s2 (.185)

s3 (.667)

s3 (.667)

s3 (.667)

$10,000

$15,000$14,000$8,000

$18,000

$12,000$6,000

$16,000

$21,000

I1(.54)

d1

d2

d3

22

44

55

66

11

Bottom Half

s1 (.696)

s1 (.696)

s1 (.696)

s2 (.217)

s2 (.217)

s2 (.217)

s3 (.087)

s3 (.087)

s3 (.087)

$10,000

$15,000

$18,000

$14,000$8,000

$12,000$6,000

$16,000

$21,000

I2(.46) d1

d2

d3

77

99

8833

11

Decision Tree

I2(.46)

d1

d2

d3

EMV = .696(10,000) + .217(15,000) +.087(14,000)= $11,433

EMV = .696(8,000) + .217(18,000) + .087(12,000) = $10,554

EMV = .696(6,000) + .217(16,000) +.087(21,000) = $9,475

I1(.54)

d1

d2

d3

EMV = .148(10,000) + .185(15,000) + .667(14,000) = $13,593

EMV = .148 (8,000) + .185(18,000) + .667(12,000) = $12,518

EMV = .148(6,000) + .185(16,000) +.667(21,000) = $17,855

44

55

66

77

88

99

22

33

11

$17,855

$11,433

Decision Tree

If the outcome of the survey is "favorable”,choose Model C. If it is “unfavorable”, choose Model A.

EVSI = .54($17,855) + .46($11,433) - $14,000 = $900.88

Since this is less than the cost of the survey, the survey should not be purchased.

Expected Value of Sample Information

Efficiency of Sample Information

The efficiency of the survey:

EVSI/EVPI = ($900.88)/($2000) = .4504