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**Business 260: Managerial Decision Analysis**

Professor David Mease Lecture 4 Agenda: 1) Go over Midterm Exam 1 solutions 2) Assign Homework #2 (due Thursday 4/2) 3) Decision Analysis (QBA Book Chapter 4)

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**Homework #2 Homework #2 will be due Thursday 4/2**

We will have an exam that day after we review the solutions The homework is posted on the class web page: Questions 3 and 4 will be added as the due date gets closer The solutions will be posted so you can check your answers:

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**Quantitative Business Analysis Decision Analysis (Chapter 4)**

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**Chapter Topics Problem Formulation**

Decision Making without Probabilities Decision Making with Probabilities Decision Analysis with Sample Information

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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 the decision alternatives and the state of natures

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**Problem Formulation Example: 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**

States of Nature d1 = use building design A d2 = use building design B d3 = use building design C 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 Example (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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In class exercise #59: A food vendor can sell either ice cream or hot chocolate but not both. If it is warm, selling ice cream will make $250 and hot chocolate will make $40. If it is not warm, selling ice cream will make $90 and hot chocolate will make $200. There is a 40% chance it will be warm. Make a payoff table for this problem.

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In class exercise #60: The Islander Fishing Company purchases clams for $1.50 per pound from Peconic Bay fisherman and sells them to various New York restaurants for $2.50 per pound. Any clams not sold to the restaurants by the end of the week can be sold to a local soup company for $0.50 per pound. The probabilities of the various levels of demand are as follows: Make a payoff table for purchase levels 500, 1000 and 2000 pounds using profit as the payoff.

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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: 1) round nodes correspond to chance events 2) 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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In class exercise #61: Make a decision tree for the Burger Prince example.

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**Make a decision tree for the Burger Prince example. (ANSWER)**

In class exercise #61: Make a decision tree for the Burger Prince example. (ANSWER) 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**

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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In class exercise #62: What decision would the optimistic approach favor for the Burger Prince Restaurant example? 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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**Decision (Customers Per Hour)**

In class exercise #62: What decision would the optimistic approach favor for the Burger Prince Restaurant example? (ANSWER) 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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In class exercise #63: What decision would the optimistic approach favor for the ice cream/hot chocolate example?

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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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In class exercise #64: What decision would the conservative approach favor for the Burger Prince Restaurant example? 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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In class exercise #64: What decision would the conservative approach favor for the Burger Prince Restaurant example? (ANSWER) Maximindecision Maximin payoff Decision Minimum Alternative Payoff Design A d ,000 Design B d ,000 Design C d ,000

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In class exercise #65: What decision would the conservative approach favor for the ice cream/hot chocolate example?

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**Decision Making Without Probabilities**

Minimax Regret Approach -The minimax regret approach requires 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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In class exercise #66: What decision would the minimax regret approach favor for the Burger Prince Restaurant example? 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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**Decision (Customers Per Hour)**

In class exercise #66: What decision would the minimax regret approach favor for the Burger Prince Restaurant example? Regret table: (ANSWER) States of Nature Decision (Customers Per Hour) Alternative s s s3 Design A d , ,000 Design B d , ,000 Design C d , , Decision Maximum Alternative Regret Design A d ,000 Design B d ,000 Design C d ,000 Minimaxdecision Minimaxregret

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In class exercise #67: What decision would the minimax regret approach favor for the ice cream/hot chocolate example?

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**Decision Making With Probabilities**

Up until now we have ignored probabilities for the states of nature But usually you should have some reasonable estimate of these probabilities P(sj) > 0 for all states of nature

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**Decision Making With Probabilities**

We can use these probabilities to compute the expected value for each decision alternative Then the expected value approach to decision making is to choose the alternative solution that gives you the largest expected value

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In class exercise #68: Assuming probabilities of .4, .2 and .4 for 40, 60 and 80 customers respectively, what decision would the expected value approach favor for the Burger Prince Restaurant example? Show the simplified decision tree with the expected values. 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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**In class exercise #68: (WORK)**

Assuming probabilities of .4, .2 and .4 for 40, 60 and 80 customers respectively, what decision would the expected value approach favor for the Burger Prince Restaurant example? Show the simplified decision tree with the expected values. (WORK) 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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In class exercise #68: Assuming probabilities of .4, .2 and .4 for 40, 60 and 80 customers respectively, what decision would the expected value approach favor for the Burger Prince Restaurant example? Show the simplified decision tree with the expected values. (ANSWER) 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

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In class exercise #69: What decision would the expected value approach favor for ice cream/hot chocolate example?

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**Decision Making With Probabilities**

When choosing among alternative decisions with very similar expected values, often people will use the variance or standard deviation to help make their decisions. The alternative with the smaller variance or standard deviation is generally preferred. This is a form of risk analysis, but it differs from the type discussed in your text.

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In class exercise #70: Compute the standard deviation for only the ice cream alternative in the ice cream/hot chocolate example.

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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 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 EVPI = |EVwPI – EVwoPI|

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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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In class exercise #71: Compute the EVPI for the Burger Prince Restaurant example. What does the EVPI mean in this context? 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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In class exercise #71: Compute the EVPI for the Burger Prince Restaurant example. What does the EVPI mean in this context? (ANSWER) 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 EVPI= .4(10,000) + .2(18,000) + .4(21,000) - 14,000 = $2,000

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In class exercise #72: Compute the EVPI for the ice cream/hot chocolate example. What does the EVPI mean in this context?

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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. The updated probabilities are called posterior probabilities or branch probabilities for decision trees.

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**Decision Analysis With Sample Information**

You can choose decision alternatives in the decision tree based on the outcome of the sample information using the expected value approach. The expected value based on this is called the Expected Value with Sample Information (EVwSI). Finally, you can compute the expected value of sample information (EVSI) as the additional expected profit possible through knowledge of the sample information. EVSI = |EVwSI – EVwoSI|

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**Decision Analysis With Sample Information**

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 EVSI = |EVwSI – EVwoSI|

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In class exercise #73: A market research survey is available for the Burger Prince Restaurant example. It will report either favorable or not favorable. There is a 54% chance it will be favorable. Based on this, the posterior probabilities are given below. Using these numbers, compute the expected value of sample information and explain its meaning in this context. 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 Tree (top half): (WORK)**

In class exercise #73: Decision Tree (top half): (WORK) 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 Tree (bottom half): (WORK)**

In class exercise #73: Decision Tree (bottom half): (WORK) 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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**In class exercise #73: (ANSWER) d1 d2 d3 d1 d2 d3**

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 EVSI = $14, $14,000 = $900.88

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In class exercise #74: Show the complete decision tree for the Burger Prince Restaurant example (including a decision node for whether or not to obtain the survey information).

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