Decision Analysis Pertemuan 2

Matakuliah : K0442-Metode KuantitatifTahun : 2009

Material Outline

• Problem Formulation• Decision Making without Probabilities• Decision Making with Probabilities

• Expected Value of Perfect Information• Decision Analysis with Sample

Information• Developing a Decision Strategy• Expected Value of Sample Information

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.

Influence Diagram• An influence diagram is a graphical device

showing the relationships among the decisions, the chance events, and the consequences.

• Squares or rectangles depict decision nodes.• Circles or ovals depict chance nodes.• Diamonds depict consequence nodes.• Lines or arcs connecting the nodes show the

direction of influence.

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 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.

ExampleConsider 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 approach. All we really need to do is to choose the decision that has the largest single value in the payoff table. This largest value is 5, and hence the optimal decision is d3.

Maximum Decision Payoff

d1 4

d2 3

choose d3 d3 5 maximum

Example• Conservative Approach

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

Minimum Decision Payoff

d1 -2

choose d2 d2 -1 maximum

d3 -3

Example

• Minimax Regret ApproachFor 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; in the second column, subtract 4, 3, and 5 from 5; 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 (continued)

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

Decision Maximum Regret choose d1 d1 1 minimum

d2 4

d3 3

Decision Making with Probabilities

• Expected Value Approach– If probabilistic information regarding he 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.

Expected Value of a Decision Alternative

• 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

N

jijji VsPd

1

)()(EV

Example: Burger PrinceBurger Prince Restaurant is contemplating

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 as follows:

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

Example: Burger Prince• 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.

Example: Burger Prince• Decision Tree

1111

.2.2

.4.4

.4.4

..44

.2.2

..44

..44

..22

..44

dd11

dd22

dd33

ss11

ss11

ss11

ss22

ss33

ss22

ss22

ss33

ss33

Payoffs

10,00010,000

15,00015,000

14,00014,000

8,0008,000

18,00018,000

12,00012,000

6,0006,000

16,00016,000

21,00021,000

2222

3333

4444

Example: Burger Prince

Expected Value For Each DecisionExpected Value For Each Decision

Choose the model with largest EV, Model C.

3333

dd11

dd22

dd33

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

2222

1111

4444

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).

Example: Burger Prince• Expected Value of Perfect Information

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

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

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.

Example: Burger Prince• Risk Profile for the Model C Decision Alternative

.10.10

.20.20

.30.30

.40.40

.50.50

5 10 15 20 255 10 15 20 25

Pro

babili

tyPro

babili

ty

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 or 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.

– 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 or survey 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.

Example: Burger Prince• Sample Information

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) = .2P(favorable | 100 customers per hour) = .5 P(favorable | 120 customers per hour) = .9

Should Burger Prince have the survey performed by Stanton Marketing?

Example: Burger Prince• Influence Diagram

RestaurantRestaurantSizeSize Profit

Avg. Numberof Customers

Per Hour

MarketMarketSurveySurveyResultsResults

Legend:Legend:DecisionDecisionChanceChanceConsequenceConsequence

MarketMarketSurveySurvey

Example: Burger Prince• Posterior Probabilities

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

Example: Burger Prince• 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

Example: Burger Prince• 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

II11

(.54)(.54)

dd11

dd22

dd33

2222

4444

5555

6666

1111

Example: Burger Prince• Decision Tree (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

7777

9999

88883333

1111

Example: Burger Prince

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

4444

5555

6666

7777

8888

9999

2222

3333

1111

$17,855

$$11,433

Example: Burger Prince• Expected Value of Sample Information

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.

Example: Burger Prince• Efficiency of Sample Information

The efficiency of the survey:EVSI/EVPI = ($900.88)/($2000) = .4504