Quantitative Business Analysis Practice Exercises Chapter 3 - Solutions

QBA341

SUMMER I, 2014/15

SOLUTIONS TO PRACTICE EXERCISES Chapter 3: Decision Analysis Question 1 a) Maximax The decision is selected that will result in the maximum of the maximum payoffs. This is how this criterion derives its name--the maximum of the maxima. The maximax criterion is very optimistic. The decision maker assumes that the most favorable state of nature for each decision alternative will occur. Thus, for this example, the company would optimistically assume that good competitive conditions will prevail in the future, resulting in the following maximum payoffs and decisions:

Decision: Maintain status quo b) Maximin The maximin criterion is pessimistic. With the maximin criterion, the decision maker selects the decision that will reflect the maximum of the minimum payoffs. For each decision alternative, the decision maker assumes that the minimum payoff will occur; of these, the maximum is selected as follows:

Decision: Expand c) Hurwicz A compromise between the maximax and maximin criteria. The decision maker is neither totally optimistic (as the maximax criterion assumes) nor totally pessimistic (as the maximin criterion assumes). With the Hurwicz criterion, the decision payoffs are weighted by a coefficient of optimism, a measure of the decision maker's optimism. The coefficient of optimism, defined as , is between 0 and 1 (i.e., 0 <  < 1.0). If  = 1.0, then the decision maker is completely optimistic, and if a = 0, the decision maker is completely pessimistic. (Given this definition, 1 -  is the coefficient of pessimism.) For each decision alternative, the maximum payoff is multiplied by  and the minimum payoff is multiplied by 1 - . For our investment example, if  equals 0.3 (i.e., the company is slightly optimistic) and 1 -  = 0.7, the following decision will result:

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Decision: Expand d) Equal Likely The equal likelihood (or Laplace) criterion weights each state of nature equally, thus assuming that the states of nature are equally likely to occur. Since there are two states of nature in our example, we assign a weight of 0.50 to each one. Next, we multiply these weights by each payoff for each decision and select the alternative with the maximum of these weighted values. (You may also compute the row averages and select the maximum average)

Decision: Expand e) Minimax Regret The decision maker attempts to avoid regret by selecting the decision alternative that minimizes the maximum regret. A decision maker first selects the maximum payoff under each state of nature; then all other payoffs under the respective states of nature are subtracted from these amounts, as follows: Good Competitive Poor Competitive $1,300,000 – $800,000 = $500,000 $500,000 – $500,000 = 0 $1,300,000 – $1,300,000 = $0 $500,000 – (–$150,000) = $650,000 $1,300,000 – $320,000 = $980,000 $500,000 – $320,000 = $180,000 These values represent the regret for each decision that would be experienced by the decision maker if a decision were made that resulted in less than the maximum payoff. The maximum regret for each decision must be determined, and the decision corresponding to the minimum of these regret values is selected as follows:

Decision: Expand The decision to expand the plant was designated most often by four of the five decision criteria. The decision to sell was never indicated by any criterion. This is because the payoffs for expansion, under either set of future economic conditions, are always better than the payoffs for selling. Given any situation with these two 2|Page

QBA341

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alternatives, the decision to expand will always be made over the decision to sell. The sell decision alternative could have been eliminated from consideration under each of our criteria. The alternative of selling is said to be dominated by the alternative of expanding. In general, dominated decision alternatives can be removed from the payoff table and not considered when the various decisionmaking criteria are applied, which reduces the complexity of the decision analysis. Question 2 a) Payoff Table (profit measured in $ million) State of Nature Alternatives Low Demand High Demand Gradual introduction 1 4 Concentrated -5 10 introduction b) First you find the maximum profit for each action. For a gradual introduction to the market, the maximum profit is $4 million. For a concentrated introduction to the market, the maximum profit is $10 million. Because the maximum of the maximum profits is $10 million, you choose the action that involves a concentrated introduction to the market. State of Nature Maximum in a Low High Alternatives Row ($ million) Demand Demand Gradual introduction 1 4 4 Concentrated -5 10 10 (Maximax) introduction Decision: Concentrated Introduction c) First, you find the minimum profit for each action. For a gradual introduction to the market, the minimum profit is $1 million. For a concentrated introduction to the market, the minimum profit is million. Because the maximum of the minimum profits is $1 million, you choose the action that involves a gradual introduction to the market. Alternatives

State of Nature Low Demand High Demand 1 4 -5 10

Minimum in a Row ($ million) 1 (Maximin) -5

Gradual introduction Concentrated introduction Decision: Gradual Introduction d) The coefficient of realism (optimism) is 80%, so  = 0.8. The weighted average is computed as follows: =  (best in row) + (1  )(worst in row). Based on the following table, the best decision is choose the action that involves concentrated introduction to the market. Alternatives State of Nature Realism / Hurwicz Low Demand High Demand

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Gradual introduction 1 Concentrated -5 introduction Decision: Concentrated Introduction

SUMMER I, 2014/15

4 10

( = 0.8) 4(0.8)+1(0.2) = 3.4 10(0.8) + (-5)(0.2) = 7

Question 3 The expected monetary values for each decision alternative are computed as follows.

The decision according to this criterion is to maintain the status quo, since it has the highest expected value. Question 4 The problem is one of decision making under uncertainty. Before answering the specific questions, a decision table should be developed showing the alternatives, states of nature, and related consequences. State of Nature Alternative Favorable ($) Unfavorable ($) Strategy 1 10,000 8,000 Strategy 2 8,000 4,000 Strategy 3 0 0 a) Since Cal is a risk taker, he should use the maximax decision criteria. This approach selects the row that has the highest or maximum value. The $10,000 value, which is the maximum value from the table, is in row 1. Thus, Cal’s decision is to select strategy 1, which is an optimistic decision approach. b) Becky should use the maximin decision criteria because she wants to avoid risk. The minimum or worst outcome for each row, or strategy, is identified. These outcomes are –$8,000 for strategy 1, –$4,000 for strategy 2, and $0 for strategy 3. The maximum of these values is selected. Thus, Becky would select strategy 3, which reflects a pessimistic decision approach. c) If Cal and Becky are indifferent to risk, they could use the equally likely approach. This approach selects the alternative that maximizes the row averages. The row average for strategy 1 is ($1,000)(0.5)+ ($8,000)(0.5) = $1,000. The row average for strategy 2 is ($8,000)(0.5)+ ($4,000)(0.5) = $2,000, and the row average for strategy 3 is $0. Thus, using the equally likely approach, the decision is to select strategy 2, which maximizes the row averages. Question 5 4|Page

QBA341

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Since the decision-making environment is risk (probabilities are known), it is appropriate to use the EMV criterion. The problem can be solved by developing a payoff table that contains all alternatives, states of nature (market), and probability values. The EMV for each alternative is also computed, as in the following table: State of Nature Alternative EMV ($) Good Average Bad Small shop 75,000 25,000 15,500 40,000 Medium-sized shop 100,000 35,000 19,500 60,000 No shop 0 0 0 0 Probabilities 0.20 0.50 0.30 EMV (small shop) = (0.2)(75,000) + (0.5)(25,000) + (0.3)(40,000) = $15,500 EMV (medium shop) = (0.2)(100,000) + (0.5)(35,000) + (0.3)(60,000) = $19,500 EMV (no shop) = (0.2)(0) + (0.5)(0) + (0.3)(0) = $0 As can be seen, the best decision is to build the medium-sized shop. The EMV for this alternative is $19,500. a) EVwPI = (0.2)(100,000) + (0.5)(35,000) = (0.3)(0) = $37,500 EVPI = EVwPI – Maximum EMV = $37,500 - $19,500 = $18,000 b) Opportunity Lost Table Alternative Small shop Medium-sized shop No shop Probabilities

State of Nature Good Average Bad 25,000 0

10,000 0

40,000 60,000

100,000 0.20

35,000 0.50

0 0.30

Minimax Regret Maximum in row 40,000 60,000

22,000 18,000

0

37,500

EOL ($)

The best payoff in a good market is 100,000, so the opportunity losses in the first column indicate how much worse each payoff is than 100,000. The best payoff in an average market is 35,000, so the opportunity losses in the second column indicate how much worse each payoff is than 35,000. The best payoff in a bad market is 0, so the opportunity losses in the third column indicate how much worse each payoff is than 0. The minimax regret criterion considers the maximum regret for each decision, and the decision corresponding to the minimum of these is selected. The decision would be to build a small shop since the maximum regret for this is 40,000, while the maximum regret for each of the other two alternatives is higher as shown in the opportunity loss table.

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The decision based on the EOL criterion would be to build the medium-sized shop. Note that the minimum EOL ($18,000) is the same as the EVPI computed in part b. The calculations are: EOL (small shop) = (0.2)(25,000) + (0.5)(10,000) + (0.3)( 40,000) = $22,00 EOL (medium shop) = (0.2)(0) + (0.5)(0) + (0.3)(60,000) = $18,000 EOL (no shop) = (0.2)(100,000) + (0.5)(35,000) + (0.3)(0) = $37,500 Question 6 a) Stock

Demand

(Cases)

(Cases)

11

12

13

EMV ($)

11

385

385

385

385

12

329

420

420

379.05

13

273

364

455

341.25

Probabilities

0.45

0.35

0.20

Example (how to calculate profits at demand of 11 cases):Profits at stock 11 cases = $35*11 cases = $385 (at stock 11 cases) Profits at stock 12 cases = $35*11 cases = $385  $56 = $329 (unsold 1 case) Profits at stock 13 cases = $35*11 cases = $385  $56(2 cases) = $273 (unsold 2 cases) EMV(stock 11 cases) = (0.45)(385)+(0.35)(385)+(0.20)(385) = $385 EMV(stock 12 cases) = (0.45)(329)+(0.35)(420)+(0.20)(420) = $379.05 EMV(stock 13 cases) = (0.45)(273)+(0.35)(364)+(0.20)(455) = $341.25 b) Stock 11 cases. c) If no loss is involved in excess stock, the recommended course of action is to stock 13 cases and to replenish stock to this level each week. This follows from the following decision table. Stock

Demand

(Cases)

(Cases)

11

12

13

EMV ($)

11

385

385

385

385

12

385

420

420

404.25

13

385

420

455

411.25

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