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Slides prepared by JOHN LOUCKS St. Edward’s University

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**Chapter 4 Decision Analysis**

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

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Problem Formulation The first step in the decision analysis process is problem formulation. We begin with a verbal statement of the problem. Then we identify: the decision alternatives the states of nature (uncertain future events) the payoff (consequences) associated with each specific combination of: decision alternative state of nature

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**Problem Formulation Example: Burger Prince Restaurant**

Burger Prince Restaurant is considering opening a new restaurant on Main Street. The company has three different building designs (A, B, and C), each with a different seating capacity. Burger Prince estimates that the average number of customers arriving per hour will be 40, 60, or 80.

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**Problem Formulation Decision Alternatives d1 = use building design A**

d2 = use building design B d3 = use building design C States of Nature s1 = an average of 40 customers arriving per hour s2 = an average of 60 customers arriving per hour s3 = an average of 80 customers arriving per hour

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**Problem Formulation Payoff Table**

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.

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**Problem Formulation Payoff Table (Payoffs are Profit Per Week)**

Average Number of Customers Per Hour s1 = s2 = s3 = 80 Design A Design B Design C $10, $15, $14,000 $ 8, $18, $12,000 $ 6, $16, $21,000

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**Problem Formulation 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.

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**Problem Formulation Influence Diagram Profit Average Number of**

Customers Per Hour States of Nature 40 customers per hour (s1) 60 customers per hour (s2) 80 customers per hour (s3) Decision Alternatives Design A (d1) Restaurant Design Profit Consequence Design B (d2) Profit Design C (d3)

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**Problem Formulation Decision Tree**

A decision tree is a chronological representation of the decision problem. A decision tree has two types of nodes: round nodes correspond to chance events square nodes correspond to decisions Branches leaving a round node represent the different states of nature; branches leaving a square node represent the different decision alternatives. At the end of a limb of the tree is the payoff attained from the series of branches making up the limb.

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**Problem Formulation Decision Tree 2 1 3 4 10,000 15,000 14,000 8,000**

40 customers per hour (s1) 10,000 Design A (d1) 60 customers per hour (s2) 2 15,000 80 customers per hour (s3) 14,000 40 customers per hour (s1) 8,000 Design B (d2) 60 customers per hour (s2) 1 3 18,000 80 customers per hour (s3) 12,000 40 customers per hour (s1) 6,000 Design C (d3) 60 customers per hour (s2) 4 16,000 80 customers per hour (s3) 21,000

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**Decision Making without Probabilities**

Criteria for Decision Making Three commonly used criteria for decision making when probability information regarding the likelihood of the states of nature is unavailable are: the optimistic (maximax) approach the conservative (maximin) approach the minimax regret approach.

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**Decision Making without Probabilities**

Optimistic (Maximax) Approach The optimistic approach would be used by an optimistic decision maker. The decision with the overall largest payoff is chosen. If the payoff table is in terms of costs, the decision with the overall lowest cost will be chosen (hence, a minimin approach).

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**Decision Making without Probabilities**

Optimistic (Maximax) Approach The decision that has the largest single value in the payoff table is chosen. States of Nature Decision (Customers Per Hour) Alternative s s s3 Design A d , , ,000 Design B d , , ,000 Design C d , , ,000 Maximaxdecision Maximax payoff

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**Decision Making without Probabilities**

Conservative (Maximin) Approach The conservative approach would be used by a conservative decision maker. For each decision the minimum payoff is listed. The decision corresponding to the maximum of these minimum payoffs is selected. If payoffs are in terms of costs, the maximum costs will be determined for each decision and then the decision corresponding to the minimum of these maximum costs will be selected. (Hence, a minimax approach)

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**Decision Making without Probabilities**

Conservative (Maximin) Approach List the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs. Maximindecision Maximin payoff Decision Minimum Alternative Payoff Design A d ,000 Design B d ,000 Design C d ,000

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**Decision Making without Probabilities**

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 corresponding to the minimum of the maximum regrets is chosen.

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**Decision Making without Probabilities**

Minimax Regret Approach First compute a regret table by subtracting each payoff in a column from the largest payoff in that column. The resulting regret table is: States of Nature Decision (Customers Per Hour) Alternative s s s3 Design A d , ,000 Design B d , ,000 Design C d , ,

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**Decision Making without Probabilities**

Minimax Regret Approach For each decision list the maximum regret. Choose the decision with the minimum of these values. Decision Maximum Alternative Regret Design A d ,000 Design B d ,000 Design C d ,000 Minimaxdecision Minimaxregret

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**Decision Making with Probabilities**

Assigning Probabilities Once we have defined the decision alternatives and states of nature for the chance events, we focus on determining probabilities for the states of nature. The classical, relative frequency, or subjective method of assigning probabilities may be used. Because only one of the N states of nature can occur, the probabilities must satisfy two conditions: P(sj) > 0 for all states of nature

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**Decision Making with Probabilities**

Expected Value Approach We use the expected value approach to identify the best or recommended decision alternative. The expected value of each decision alternative is calculated (explained on the next slide). The decision alternative yielding the best expected value is chosen.

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**Expected Value Approach**

The expected value of a decision alternative is the sum of the 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

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**Expected Value Approach**

Calculate the expected value (EV) for each decision. The decision tree on the next slide can assist in this calculation. Here d1, d2, d3 represent the decision alternatives of Designs A, B, and C. And s1, s2, s3 represent the states of nature of 40, 60, and 80 customers per hour. The decision alternative with the greatest EV is the optimal decision.

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**Expected Value Approach**

Decision Tree 40 customers (s1) P(s1) = .4 10,000 Design A (d1) 60 customers (s2) P(s2) = .2 2 15,000 80 customers (s3) P(s3) = .4 14,000 40 customers (s1) P(s1) = .4 8,000 Design B (d2) 60 customers (s2) P(s2) = .2 1 3 18,000 80 customers (s3) P(s3) = .4 12,000 40 customers (s1) P(s1) = .4 6,000 Design C (d3) 60 customers (s2) P(s2) = .2 4 16,000 80 customers (s3) P(s3) = .4 21,000

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**Expected Value Approach**

EV(d1) = .4(10,000) + .2(15,000) + .4(14,000) = $12,600 Design A d1 2 EV(d2) = .4(8,000) + .2(18,000) + .4(12,000) = $11,600 Design B d2 1 3 EV(d3) = .4(6,000) + .2(16,000) + .4(21,000) = $14,000 Design C d3 4 Choose the decision alternative with the largest EV: Design C

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**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.

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**Expected Value of Perfect Information**

Expected value of perfect information is defined as EVPI = |EVwPI – EVwoPI| where: EVPI = expected value of perfect information EVwPI = expected value with perfect information about the states of nature EVwoPI = expected value without perfect information about the states of nature

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**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).

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**Expected Value of Perfect Information**

EVPI Calculation 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

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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.

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**Risk Analysis Risk Profile for Decision Alternative d3 .50 .40**

Probability .30 .20 .10 Profit ($ thousands)

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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.

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**Sensitivity Analysis Resolving Using Different Values for the**

Probabilities of the States of Nature P(s1) P(s2) P(s3) EV(d1) EV(d2) EV(d3) Optimal d 12,900 12,987 12,700 12,250 11,800 13,500 13,800 14,100 12,400 12,654 12,200 11,500 10,800 14,000 14,800 15,600 14,250 14,319 13,500 12,250 11,000 14,750 15,000 15,250 d3 d1 and d3 d1 d2

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**Decision Analysis With Sample Information**

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.

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**Decision Analysis With Sample Information**

Example: Burger Prince Burger Prince must decide whether to purchase a marketing survey from Stanton Marketing for $1,000. The results of the survey are "favorable“ or "unfavorable". The branch probabilities corresponding to all the chance nodes are listed on the next slide.

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**Decision Analysis With Sample Information**

Branch Probabilities P(Favorable market survey) = .54 P(Unfavorable market survey) = .46 P(40 customers per hour | favorable) = .148 P(60 customers per hour | favorable) = .185 P(80 customers per hour | favorable) = .667 P(40 customers per hour | unfavorable) = .696 P(60 customers per hour | unfavorable) = .217 P(80 customers per hour | unfavorable) = .087

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**Decision Analysis With Sample Information**

Influence Diagram Decision Chance Consequence Market Survey Results Avg. Number of Customers Per Hour Market Survey Restaurant Design (seating capacity) Profit

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**Decision Analysis With Sample Information**

Decision Tree (top half) s1 P(s1|I1) = .148 10,000 d1 s2 P(s2|I1) = .185 4 15,000 s3 P(s3|I1) = .667 14,000 s1 P(s1|I1) = .148 8,000 d2 s2 P(s2|I1) = .185 2 5 18,000 P(I1) = .54 s3 P(s3|I1) = .667 12,000 I1 s1 P(s1|I1) = .148 6,000 d3 s2 P(s2|I1) = .185 6 16,000 1 s3 P(s3|I1) = .667 21,000

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**Decision Analysis With Sample Information**

Decision Tree (bottom half) s1 P(s1|I2) = .696 1 10,000 d1 s2 P(s2|I2) = .217 7 15,000 s3 P(s3|I2) = .087 I2 14,000 s1 P(s1|I2) = .696 8,000 P(I2) = .46 d2 s P(s2|I2) = .217 3 8 18,000 s P(s3|I2) = .087 12,000 s P(s1|I2) = .696 6,000 d3 s P(s2|I2) = .217 9 16,000 s P(s3|I2) = .087 21,000

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**Decision Analysis With Sample Information**

Decision Strategy A decision strategy is a sequence of decisions and chance outcomes. The sequence of decisions chosen depends on the yet to be determined outcomes of chance events. The optimal decision strategy is based on the Expected Value with Sample Information (EVwSI). The EVwSI is calculated by making a backward pass through the decision tree.

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**Decision Analysis With Sample Information**

Expected Value with Sample Information (EVwSI) Step 1: Determine the optimal decision strategy and its expected returns 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.

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**Decision Analysis With Sample Information**

EV = .148(10,000) (15,000) + .667(14,000) = $13,593 4 $17,855 d2 EV = .148 (8,000) (18,000) + .667(12,000) = $12,518 2 5 I1 (.54) d3 EV = .148(6,000) (16,000) +.667(21,000) = $17,855 6 1 d1 EV = .696(10,000) (15,000) +.087(14,000) = $11,433 7 I2 (.46) d2 EV = .696(8,000) (18,000) + .087(12,000) = $10,554 3 8 $11,433 d3 EV = .696(6,000) (16,000) +.087(21,000) = $9,475 9

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**Decision Analysis With Sample Information**

4 $13,593 $17,855 d2 2 5 $12,518 I1 (.54) d3 6 $17,855 EVwSI = .54(17,855) + .46(11,433) = $14,900.88 1 d1 I2 (.46) 7 $11,433 d2 3 8 $10,554 $11,433 d3 9 $ 9,475

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**Decision Analysis With Sample Information**

Expected Value with Sample Information (EVwSI) EVwSI = .54($17,855) + .46($11,433) = $14,900.88 Optimal Decision Strategy If the outcome of the survey is "favorable”, choose Design C. If the outcome of the survey is “unfavorable”, choose Design A.

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**Decision Analysis With Sample Information**

Risk Profile for Optimal Decision Strategy P(s|I) P(I) Payoff I1 = .54 P(s1|I1) = .148 P(s2|I1) = .185 P(s3|I1) = .667 .08 .10 .36 $ 6,000 $16,000 $21,000 Payoffs for d3 I2 =.46 P(s1|I2) = .696 P(s2|I2) = .217 P(s3|I2) = .087 .32 .10 .04 1.00 $10,000 $15,000 $14,000 Payoffs for d1

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**Decision Analysis With Sample Information**

Risk Profile for Optimal Decision Strategy .50 .40 .36 .32 Probability .30 .20 .10 .10 .10 .08 .04 Profit ($ thousands)

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**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. EVSI = |EVwSI – EVwoSI| where: EVSI = expected value of sample information EVwSI = expected value with sample information about the states of nature EVwoSI = expected value without sample information about the states of nature

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**Expected Value of Sample Information**

EVSI Calculation Subtract the EVwoSI (the value of the optimal decision obtained without using the sample information) from the EVwSI. EVSI = $14, $14,000 = $900.88 Conclusion Because the EVSI ($900.88) is less than the cost of the survey ($ ), the survey should not be purchased.

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**Efficiency of Sample Information**

The efficiency rating (E) of sample information is the ratio of EVSI to EVPI expressed as a percent. The efficiency rating (E) of the market survey for Burger Prince Restaurant is:

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**Computing Branch Probabilities**

Bayes’ Theorem can be used to compute branch probabilities for decision trees. For the computations we need to know: the initial (prior) probabilities for the states of nature, the conditional probabilities for the outcomes or indicators of the sample information, given each state of nature. A tabular approach is a convenient method for carrying out the computations.

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**Computing Branch Probabilities**

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.

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**Computing Branch Probabilities**

Example: Burger Prince Restaurant Recall that Burger Prince is considering purchasing a marketing survey from Stanton Marketing. The results of the survey are "favorable“ or "unfavorable". The conditional probabilities are: P(favorable |40 customers per hour) = .2 P(favorable |60 customers per hour) = .5 P(favorable |80 customers per hour) = .9 Compute the branch (posterior) probability distribution.

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**Computing Branch Probabilities**

Posterior Probability Distribution Favorable State Prior Conditional Joint Posterior 40 60 80 .4 .2 .2 .5 .9 .08 .10 .36 .148 .185 .667 .08/.54 Total 1.000 P(favorable) = .54

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**Computing Branch Probabilities**

Posterior Probability Distribution Unfavorable State Prior Conditional Joint Posterior 40 60 80 .4 .2 .8 .5 .1 .32 .10 .04 .696 .217 .087 .32/.46 Total 1.000 P(unfavorable) = .46

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End of Chapter 4

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Chapter 4 Decision Analysis.

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