Decision theory Problems

INDUSTRIAL ENGINEERING DEPARTMENT Introduction to Operations Research III Decision Theory 1. The MBA Movie Studio is trying to decide to distribute its new movie “Claws”. The movie has the potential of being a great financial success (a “smash”), but the executives are not sure because the subject is controversial. And they have seen some films heralded as “smashes” become “flops” with disastrous financial consequences. The decision facing MBA is whether or not to put out the movie “Claws” on a limited first run basis. This means that the movie will show only in a few select theaters during the first six months. After six months it will be released generally. If the movie turns out to be a success, this is clearly the best approach because the studio makes considerable profit from these theaters. The other alternative is to release the film for wide distribution immediately. The profits for the two alternatives are given in the table below, classified in terms of whether the film is a “smash”, or “medium” success, or a “flop”. Profits from Film “Claws” Level of Success Probability Limited Initial Widespread Release Release (in millions) (in millions) Smash 0.3 22 12 Medium 0.4 8 8 Flop 0.3 -10 -2 There is considerable discussion in MBA about the potential of Claws. Management has finally agreed on the probabilities shown in the table. But which decision to make is still not clear. One possibility is to have a few sneak previews of the movie and get the audience’s opinions. The cost of such a process would be about Php 50,000. And several executives in the company feel it would be money wasted, since sneak preview audience tends to rate a movie as good or outstanding even when it later tuns out to be flop. To support this, the following table was produced, describing the company’s past experience with sneak preview audience reactions. Sneak Preview Audience Reaction Audience Smash Medium Flop Total Rating Outstanding 9 12 3 24 Good 1 6 5 12 Poor 0 2 2 4 Total 10 20 10 40 a. Draw the decision tree for this problem. b. Calculate the posterior probabilities for “smash”, “medium”, and “flop” given the various audience reactions. c. Assume that MBA is willing to base its decision on Expected Monetary Value. What decision should the MBA Movie Studio make about the movie “Claws”? 2. The Tarheel Manufacturing Company must decide whether to build a large plant or a small one to process a new product with an expected life of 10 years. Demand may be high during the first 2 years, but if many users find the product unsatisfactory, demand will be low for the remaining 8 years. High demand during the first 2 years may indicate high demand for the

next 8 years. If demand is high during the first 2 years and company does not expand within the first 2 years, competitive products will be introduced, thus lowering the benefits. If the company builds a large processing plant, it must keep it for 10 years. If its builds the small plant, the plant can be expanded in 2 years if demand is high, or the company can stay in the small plant while making smaller benefits on the small volume of sales. Estimates of demand are these: Probability Probability High demand (first 2 years) 0.5 High demand during first 2 0.6 followed by high demand years (next 8 years) High demand (first 2 years 0.1 followed by low demand (next 8 years) Low demand (first 2 years) 0.4 Low demand during first 2 0.4 followed by continuing low years demand (next 8 years) Low demand (first 2 years) 0 followed by high demand (next 8 years) Financial costs and profits are as follows: ♦ A large plant with high demand would yield Php 1 million annually in profits. ♦ A large plant with low demand would yield Php 200,000 annually because of production inefficiencies. ♦ A small plant, not expanded, with a low demand would yield annual profits of Php 250,000 for 10 years. ♦ A small plant during a 2-year period of high demand would yield Php 450,000 annually; if high demand continued and if the plant was not expanded, this would drop to Php 300,000 annually for the next 8 years as a result of competition. ♦ A small plant which was expanded after 2 years would yield Php 100,000 annually for 8 years if low demand occurred during that period. ♦ A large plant would cost Php 5 million to build and put into operation. ♦ A small plant would cost Php 1,500,000 to build and put into operation. ♦ Expanding a small plant after 2 years would cost Php 2,500,000. Under the conditions stated and with the information furnished, analyze the alternatives and choose the best decision. 3. A machine in a group of 50 machines is serviced when it breaks down. At the end of T period, preventive maintenance is performed by servicing all 50 machines. The cost of repairing a broken machine is Php 1000 while Php 100 is the preventive maintenance cost per machine. Determine the optimum T that minimizes the total cost per period.

Probability Distribution of Breakdown before T 0.03, t =1  P(t ) =  Pt −1 + 0.01, t = 2 10 0.125, t = 11,12  4. An automatic machine produces α (thousands of) units of a certain product per day. As α increases, the proportion of defective p goes up. The probability density function of p in terms of α is given by αp α −1 ,  0 ≤ p ≤1 f ( p) =  0,  otherwise Each defective item incurs a loss of Php 50. A good item produces a profit of Php 5. Determine the optimal value of α. 5. A doctor must diagnose the condition of one of her patients. She is certain that the condition is not life threatening, so there is not risk to the patient’s life. However, in diagnosing the condition the doctor would like to minimize the cost to the patient. There are three tests she could conduct on her patient. The doctor has narrowed the range of possibilities down to three disease conditions, and indicates her professional judgmental probability assessments of the three states as 40 percent, 25 percent, and 35 percent that the disease is type 1, type 2, and type 3, respectively. The cost (in Php) to the patient for diagnosis and treatment is given in the table of possibilities shown in the table below depending on the test employed. Answer the following by dealing directly with costs (the problem can also be solved by constructing a table of negative payoffs): a. Determine the Bayes decision (the test which minimizes the expected cost). b. Compute the EVPI. c. Suppose the doctor first administers a special blood test that could be used to refine her probability estimates. The cost of the blood test is Php 50 and it would indicate one of two results – O1 or O2. The likelihood probabilities are given in the following table: Disease Condition S=1 S=2 S=3 O1 0.80 0.05 0.40 P(O1/S), i = 1, 2, 3 O2 0.20 0.95 0.60 P(O2/S), i = 1, 2, 3 Should the doctor administer the blood test prior to selecting one of the three major tests? Cost of Diagnosis and Treatment Test 1 2 3 1 500 300 400 2 600 500 300 3 300 550 450 6. A large mill is faced with the problem of extending Php 100,000 credit to a new customer, a dress manufacturer. The mill classifies typical companies into the categories: poor risk, average risk, and good risk. Their experience indicates that 20 percent of similar companies are poor risks, 50 percent are average risks, and 30 percent are good risks .If credit is extended, the expected profit for poor risks is- Php 15,000, for average risks Php 10,000, and for good risks Php 20,000. If credit is not extended, the dress manufacturers will turn to

another mill. The mill is able to consult a credit rating organization for a fee of Php 2,000. Their experience with this credit –rating company is given by Credit company Actual credit rating % evaluation Poor Average Good Poor 50 40 20 Average 40 50 40 Good 10 10 40 a) What is the Bayes’ action, assuming the credit rating company is not used? b) How much money can be paid for “ perfect information”? c) What is the optimal expected loss if the credit rating company data is used? Does it pay to utilize these data? d) What is the Bayes’ action if the credit-rating company determines the dress manufacturer to be a poor risk? 7. The Breezy Breakfast Foods Company is considering marketing a new breakfast cereal. If the new cereal is successful, it will mean a Php 10 million profit (present value) over the life of the product. If unsuccessful, a Php 2 million, loss on investment will be incurred. Management currently feels there is a 50-50 chance that the product will be successful. Two market research firms have approached Breezy with proposals to obtain more information. Attitude Surveys collects data on consumer attitudes with respect to specific characteristics of a product, such as sweetness, caloric content, nutritive value, etc., and produces a forecast of “success” or “fail”. Of the studies this company has performed on similar products recently, their experience has been as follows: Attitude Surveys Experience Actual outcome Forecast Success Failure Success 20 5 Failure 5 20 A second company, Market Competition Inc. ; performs analysis in an entirely different, independent manner. This company performs extensive analysis on competitive products, and produce a recommendation of “success” or ”fail” based on the anticipated amount and quality of competitive products. Their experience with 50 studies has been as follows: Market Competition Experience Actual outcome Forecast Success Failure Success 22 3 Failure 0 25 Attitude Surveys charges Php 100,000 per survey, while Market competition charges Php 150,000. a. Consider only Attitude Surveys. Use a decision tree to decide whether or not Breezy should purchase this survey.

b. Consider only Market Competition, Inc. Use a decision tree to decide whether or not Breezy should purchase this survey. 8. Inventory Management. Blumberg’s Department Store will hold a one-month suit sale. The suits can be purchased in lots of 25 each, an the wholesale post per suit is a function of the number of suites ordered, as shown below: Number of suits ordered 25 50 75 100 Cost per suit (Php) 800 750 700 650 Each suit left over at the end of the month will be sold at the clearance sale for half the retail sales price of Php 1400. If a shopping arises during the month, nothing will be done to replenish inventory, The possible sales levels are 25, 50, 75, and 100, having probabilities of 0.20, 0.30, 0.40 and 0.10, respectively. a. Construct a payoff table for this decision problem. b. Construct a decision tree to represent the problem. c. Fold back the decision tree to determine the optical inventory ordering the decision. d. Marketing Analysis. Suppose Blumberg’s could purchase the services of a marketing consultant who would conduct a survey that could be summarized by one of two outcomes: O1 (a favorable market exists for the sale) or O 2 (an unfavorable market exists). The likelihoods, estimated by Blumberg’s from past experience with this consultant, are given in Table 2. What is the maximum amount Blumberg’s should pay for the consultant’s services? (Hint: conduct a preposterior analysis). Table 2 Sales (states of nature) P(0/S) S = 25 S = 50 S = 75 S = 100 O1 0.05 0.50 0.70 0.85 O2 0.95 0.50 0.30 0.15 9. The Profit & and Gambit Company has a major product that has been losing money recently because of declining sales. In fact, during the current quarter of the year sales will be 4 million units below the break-even point. Since the marginal revenue for each unit sold exceeds the marginal cost by Php 1. This amount to a loss of Php 4 million for shutting down. The other alternative is to undertake an intensive advertising campaign to increase sales and then abandon the product (at the cost of Php 4 million) only if the campaign is not sufficiently successful. Tentative plans for this advertising campaign have been developed and analyzed. It would extend over the next three quarters (subject to early cancellation), and the cost would be Php 6 million in each of the three quarters. It is estimated that the increase in sales would be approximately 3 million units in the first quarter, another 2 million units in the second quarter and another 1 million units in the third quarter. However, because of a number of unpredictable market variables, there is considerable uncertainty as to what impact the advertising actually would have, and careful analysis indicates that the estimate for each quarter could turn out to be off by as much as 2 million units in either direction. (To quantify this uncertainty, assume that the additional increase in sales in the three quarters are independent of random variables having a uniform distribution with a range from 1 to 5 million, from 0 to 4 million, and from –1 to 3 million, respectively). If the actual increases are too small, the advertising campaign can be discontinued and the product abandoned at the end of either of the next 2 quarters.

If the intensive advertising campaign were to be initiated and continued to its competition, it is estimated that the sales for some time thereafter would continue to be at about the same level as in the third (last) quarter of the campaign. Therefore, if the sales in that quarter still are below the break-even point, the product would be abandoned. Otherwise, it is estimated that the expected discounted profit thereafter would be Php 8 for each unit sold over the break-even point in the third quarter. Determine the optimal policy maximizing expected profit. 10. An oil company has some land that is purported to contain oil. The company classifies such land into four categories by the total number of barrels that are expected to be obtained from the well, i.e. a 500,000 – barrel well, a 200,000 barrel well, a 50,000- barrel well, or a dry well. The company is faced with deciding whether to drill for oil, to unconditionally lease the land, or to conditionally lease the land at a rate depending upon the oil strike. The cost of drilling a producing well is Php 100,000, and the cost of drilling a dry well is Php 75,000. For producing wells the profit per barrel of oil is Php 1.50 (after deducting all production costs). Under the unconditional lease agreement, the company receives Php 45,000 for the land, whereas under the conditional lease arrangement the company receives 50 cents for each barrel of oil extracted, provided the land yields a 200,000 – or 500,000- barrel strike; otherwise, it receives nothing. The possible profits for the oil company are shown below: a. What should the company’s decision be? Why? b. If the company has had some experience with well in similar geographical areas and has concluded that about 10% of the strikes are 500,000 barrel wells, 15% are 200,000 barrel wells, 25% are 50,000 barrel wells, and 50% are dry wells, what will be the decision maker’s decision? Why? (Please use different methods to justify you answer.) c. The decision-maker has the choice of not utilizing seismic soundings or utilizing seismic sounding which would mean a research study which would cost Php 12,000. Seismic soundings may reveal information on four possible seismic classification denoted as follows: 1. there is definitely a closed geological structure to the site ( a very favorable outcome if the presence of oil is desired); 2. there is probably a closed structure to the site; 3. there is a non-closed structure( a relatively unfavorable report); 4. there is no structure to the site an unfavorable condition). Based upon past examination of similar geological areas (100 such examinations), the data below are obtained: State of 500,000- barrel 200,000- barrel 50,000-barrel well Dry well seismic well well nature class f Prob. f Prob. f Prob. f Prob. 1 7 7/12 9 9/16 11 11/24 9 9/48 2 4 4/12 3 3/16 6 6/24 13 13/48 3 1 1/12 2 2/16 3 3/24 15 15/48 4 0 0/12 2 2/16 4 4/24 11 11/48 Use decision tree to come up with your decision analysis of the problem.

CASE I A manufacturer produces items having a probability p of being defective. These items are formed into lots of 150. Past experience indicates that p is either 0.05 or 0.25, and furthermore, in 80 percent of the lots produced p equals 0.05 (and in 20 percent of the lots p equals 0.25). These items are then used in an assembly, and ultimately, their quality is determined before the final assembly leaves the plant. The manufacturer can initially screen each item in a lot at a cost of P15 per item, replacing defective items found, or use the items directly without screening. If the latter action is chosen, the cost per lot can be calculated as: p = 0.05 p = 0.25 Screen 2,250 2,250 Do not screen 750 3,750 Because screening requires scheduling of inspectors and equipment, the decision to screen or not screen must be made 2 days before the potential screening takes place. However, one item can be taken from the lot and sent to a laboratory, and its quality (defective or non-defective) can be reported before the screen, no-screen decision must be made. The cost of this initial inspection is P125. a) What is the Bayes’ action without looking at the single item? b) How much money can be paid for “perfect information” c) What is the optimal expected cost if the quality of items is determined before the screen, no- screen decision is made? d) What is the Bayes’ action if the quality of one of the items is determined and found to be defective? CASE II In a manufacturing process, lot having 8,10, 12, or 14% defectives are produced according to the respective probabilities 0.4, 0.3, 0.25, and 0.05. Three customers have contracts to receive lots from the manufacturer. The contracts specify that percentages of defective in lots shipped to customers A, B, C should not exceed 8, 12, 14, respectively. If a lot has a higher percentage of defectives than stipulated, a penalty of Php 100 per percentage point is incurred. On the other hand, supplying better quality than required costs the manufacturer Php 50 per percentage point. A sample of rise n = 20 is inspected before each lot is shipped lot to customers. Suppose that 4 defectives are found in the sample. What would be optimal decision of the firm? CASE III A new type of airplane is to be purchased by the Air force, and the number of spare engines accompanying the order must be determined. The Air Force must order these spare engines in bathes of 5 and can only choose among 15, 20, or 25 spares. The supplier of these engines has two plants, and the Air Force must make its decision prior to knowing which plant will be used. From past experience it is known that the number of spare engines required when production takes place at plant A is approximated by a Poisson distribution with parameter θ = 21, whereas the number of spare engines when production takes place at plant B is approximated by Poisson distribution with parameter θ = 24. The cost of a spare engine purchased now is Php 400,000, whereas the cost and interest are to be neglected. Spares must always be supplied if they are demanded, and unused engines will be scrapped when the airplanes become obsolete.

The Air Force know from past experience that 2/3 of all types of airplane engines are produced in plant A and only 1/3 in plant B. Furthermore, it is known that a similar type of engine was produced for an earlier version for the current airplane under consideration. The order size for this earlier type was the same as for the current model. Furthermore, its non-obsolete life is identical with that planned for the present version. The engine for the current order will be produced in the same plant as the previous model, although the Air Force is not aware of which of the two plant this is. The reason for this lack of knowledge is due to the haste in which the spare engine decision must be made. The Air Force has access to the data which the spare engine decision must be made. The Air Force has access to the data on the number of spares actually required for the older version (which had a Poisson distribution), but it does not have time determine the production location. a. What is the Bayes’ action, assuming that the information on the old airplane model is unavailable? b. How much money can be paid for “perfect information”? c. Assuming that cost of data on the old airplane model is free and 30 spares were required, determine the Bayes’ action. CASE IV Carlo Steerio is a chain of retail outlets specializing in sound equipment. Because of its high volume, Charles stocks its inventory of stereo pickup cartridges by ordering lots of 100 units from various manufacturers. Carlo Hagano, the owner, decides randomly whether to accept or reject a particular lot from a supplier, KICO. Each cartridge in a rejected lot is thoroughly inspected by Carlo, and cartridges that are found to be defective are replaced by the supplier. An accepted lot is parceled without inspection to the retail stores for sale to customers, who are relied on to find the defective cartridge, which Charles replaces without charge from its inventory. KICO will not give Carlo credit for cartridges that have been used by retail customers, even if they were originally defective. One of Carlo’s employees suggested that the company sample incoming lots, using the information thereby obtained as a basis of deciding whether a lot should be accepted or rejected. Carlo is skeptical about the adventures of the sampling, since the test costs Php 300 per cartridge tested. Detailed records of KICO cartridge received by Carlo have been maintained, and the frequencies of lot proportions defective have been found. These frequencies serve as estimates of the following prior for values of p of .10, .20 and .30 Possible lot Proportion Defective p Prior Probability 0.10 0.40 0.20 0.30 0.30 0.30 Total 1.00 The cartridges are bought at P150 each by Carlo Steerio from KICO and sold for Php 20,000 per lot customers. Carlo calls on you to evaluate his employee’s suggestion but tells you that if sampling is beneficial, he is noting to sample up to 2 only.

What would your recommendation be? CASE V Artex Computers is going to purchase 10,000 units of a certain part that is to be assembled into the Artex products. The order is to be placed with the lowest bidder with the 10,000 units to be delivered at a rate of 1,000 per month over the next 10 months. The Frank Machine Shop (FMS) is considering bidding on this order. Two factors are puzzling Mr. Frank in his attempt to fix a bid price. The first factor deals with FMS’s chances of winning the bid. Mr. Frank finally decides to consider only two bids - either Php 12 per unit or Php 13 per unit. He estimates that the chances are two thirds of winning the former price and one third of winning the latter price. The second factor involved in the decision is the FMS unit manufacturing cost. Two production processes are available. The first, process A, is known to cost Php 10 per unit. The second, process B is one that FMS has not used before. The chief foreman says there is a one- fourth chance that the per unit cost will be P9; a one-half chance that the cost will be P10; and a one-fourth chance that the cost will be P11, if process B is used. The chief foreman has suggested that he conduct an experiment with the new process (B). He could produce 10 or 15 units; and from the experience gained, he believes he could estimate unit cost “ pretty well”. The cost of this test would be P500. Then asked to be more specific about how accurate his estimate of cost would be, the chief foreman provided the following table. Chance of Various Estimated Cost Foreman’s Actual per unit cost estimated cost Php 9 Php 10 Php 11 Php 9 0.8 0.1 0.1 Php 10 0.1 0.8 0.1 Php 11 0.1 0.1 0.8 1 1 1 Mr. Frank is not sure that this information is at all relevant to his problem. The controller has argued that the test suggested by the foreman may be valuable but should be performed after the bid is awarded. Otherwise, he argues, the firm may spend Php 500 and not win the contract. The foreman feels that the test should be done before the bid, since it may influence FMS’s bid price. Draw the complete Decision tree diagram. What specific sequence of actions should be FMS take? CASE VI Two batches of a product have had their labels torn off in error. One batch was produced on Machine A. The proportion of defectives of A is given by the following density function. 80,000 p f ( PA ) = , 0 < p < 0.015 9 = 0, otherwise

The other was produced by machine B whose density function given by f ( PA ) = 10,125 p 2 , 0 < p < 1 / 15 = 0, otherwise The output of Machine B is sol a “ as is” to a user who incurs very little cost when a defective unit is found. However, Machine A output is sold to a user who charges the manufacturer P5 for each defective unit, since it is installed in a complex piece of machinery. There are 100 units in each batch. The manufacturer must find out which batch is which, as he has an urgent order for a batch of Machine A output. A statistician has suggested has suggested taking a sample of 20 units from one batch. If there are one or more defectives in that batch, label it “Machine B output” while if there are zero defectives, label it “Machine A output”; and label the other batch in the opposite manner. Then ship the batch labeled “ Machine A output.” a. Compute the conditional probabilities of misleading the batches. b. Assuming that the two machines have been used with the same frequency (based on historical data), compute the unconditional expected cost of the statisticians’s suggestion. c. Suppose taking a sample of 20 costs P 2, but a complete batch of 100 units would cost only P 10. Based on your results from (b), should the manufacturer take 100% sample of one of the batches?

Decision theory Problems

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Decision theory Problems Document Transcript