MODULE 2 â&#x20AC;&#x201C; DECISION THEORY PROBLEMS 2.1

Damas Electronics is a manufacturer of computer components. At present, its facilities are working at full capacity and the company has to consider expanding the facilities. The company has decided that the only feasible alternatives are to continue with the present facilities, expand the facilities, or develop a remote facility to supplement the present facilities. A consultant has assisted the company in developing the accompanying table which represents the expected profit (in thousands \$) for each of three possible levels of sales. The consultant has estimated that the probabilities of low, normal, and high sales will be 0.4, 0.3, and 0.3, respectively. Alternatives State of Maintain present Expand present Add a remote Economy facilities facilities facilities Low sales 1,010 -500 -2,000 Normal sales 1,100 2,010 -500 High Sales 1,200 2,200 3,010 Construct an opportunity loss table and deduce the best action available to the company if it wishes to minimize its expected opportunity loss.

2.2

Swan Valley Cereals are considering the launch of a new range of cereals to replace the old range of products. The new range of cereals will have a five-year product lifetime. If the new range is introduced, it is thought that the demand will be high (with probability 0.425), average (with probability 0.25), or low (with probability 0.325). If demand is high, the company expects a contribution of \$800,000 a year for each of the five years. The annual contribution over the five-year period will be \$400,000 if demand is average. If demand is low, the annual contribution falls to \$120,000 for each of the five years. If a decision is made not to introduce the new range, the old range will be continued, with an annual contribution of \$320,000 for each of the five years. The fixed cost of advertising and promotion is \$1,000,000 for the new range and \$500,000 for the old range for the entire five-year period. (Note: The use of discounted cash flow is not required.) (a) Using profit as a measure of payoff, construct a payoff table for the problem. (b) Using the maximization of expected profit; determine the optimal action for Swan Valley Cereals. (c) What is the maximum amount that the company is willing to pay to learn of the demand of the new range of cereals?

2.3

Syed Rashid is considering opening a new bookstore and has started analyzing the situation. There are two possible locations to be considered. Location A is relatively small and location B is large. If he opens at location A and the demand is good, he will earn a profit of \$10,000. If the demand is low, he will lose \$10,000. If he opens at location B and the demand is good, he will earn a profit of \$80,000, but he will lose \$30,000 if the demand is low. He also has the option of not opening a bookstore at all. Irrespective of location, he believes that there is a 50% chance that demand will be good. Prior to selecting the location, he has the option of conducting a market research that will cost \$5,000. The probability of a good demand given a favourable study is 0.8. The probability of a good demand given unfavourable study is 0.1. There is a 60% chance that the study will be favourable.

(a) (b)

Construct a decision tree for the above problem. Advise Syed Rashid on the optimal strategy.

2.4

An investor has \$25,000 and cannot borrow any additional capital. An investment opportunity requiring \$25,000 is offered to the investor. This investment opportunity is estimated to have a return of either a profit of \$10,000 or a loss of \$10,000. The investment must be made tomorrow. The amount invested and the profit or loss will be returned in 25 days. The investor has assessed the probability of a profitable return as 0.6. Moreover, one month hence there is a second similar opportunity. It will require \$20,000 with potential returns of either \$10,000 profit or \$4,000 profit or a \$2,000 loss. The respective probabilities for the returns are estimated at 0.4, 0.4, and 0.2. Again the amount invested plus the returns will be redeemable in 25 days. (a) Draw a decision tree to represent the alternatives of action open to the investor. (b) Decide the best course of action for the investor. (c) If the investor can borrow \$5,000 what then is his best course of action? What is the maximum amount he would be willing to pay for the use of the loan?

2.5

A manager of Tongkat Ali Company (TAC) has to decide whether to offer a marketing agent to become the sole distributor of TACâ&#x20AC;&#x2122;s products. If he makes the offer, he can concentrate on his companyâ&#x20AC;&#x2122;s production and obtain a profit margin of \$15 per crate. The agent promises to order 100,000 crates per month. On the other hand, if he does not make the offer, he will come up with his own marketing plan. With \$0.5 million for an advertising campaign, he can obtain a profit margin of \$30 per crate. The sales will depend on the demand of the products. If the demand is good, his marketing group can sell as much as 100,000 crates per month, but if the demand were poor, the sales would only be 20,000 crates a month. The probability of having good demand is 0.5. He also considers the possibility of hiring a market research agent to help him in making decisions. From past experience, the probability that the demand is good given a favourable result of market research is 0.84, whereas the probability that the demand is good given unfavourable market research is 0.25. The probability of having a favourable result of market research is 0.55. (a) Construct a complete tree diagram for the above problem. (b) Calculate the expected value of sample information. (c) Determine the best decision for the company.

2.6

A man is considering three possible ways to invest the \$200,000 he has just inherited. (1) Some of his friends are considering financing a combined Laundromat, video game arcade, and pizzeria, where the young singles in the area can meet and play while doing their laundry. This venture is highly risked and could result in a major loss or substantial within a year. The man estimates he will make \$200,000 with probability 0.4. (2) He can invest in some new apartments that are being built in town within one year. This fairly conservative project will produce a profit of at least \$10,000 but it might yield a profit of \$15,000, \$20,000, \$25,000, or even \$30,000. The investor estimates the probability of these five returns at 0.20, 0.30, 0.25, 0.20, and 0.05, respectively. (3) He can invest in some government securities that have a current yield of 8.25%. (a) (b)

Construct a decision tree to help the investor to invest his money. Which investment will maximize his expected yearly profit?

(c) (d)

2.7

How high would the yield on the government bonds have to be before he would decide to invest in them? How much would he be willing to pay for perfect information about the success of the Laundromat?

The director of a national car dealer is going to introduce one of the two new models of national cars: Sedan and Coupe. The companyâ&#x20AC;&#x2122;s profit is determined by the market conditions and is presented in the table below: Market Conditions Model Favourable Stable Unfavourable Sedan \$100,000 \$80,000 -\$35,000 Coupe \$95,000 \$82,000 -\$30,000 From his records, the probability of a favourable market is estimated at 0.4 and that of unfavourable market 0.1. At the same time, he is considering hiring a market research firm to do a survey to determine future market conditions. The result of the survey will indicate either positive or negative market conditions. It is estimated that there is a 0.6 probability that the survey will be positive. If the survey is positive, the probabilities that the market will be favourable, stable, or unfavourable are 0.72, 0.26, and 0.02, respectively. On the other hand, if the survey is negative, the probabilities that the market will be favourable, stable, or unfavourable are 0.20, 0.66, and 0.14, respectively. Using decision tree analysis, determine: (a) The expected value of sample information. (b) The maximum amount the company should pay the market research firm. (c) The decision strategy the company should follow.