University of Windsor Faculty of Business Administration Winter 2001 Mid Term Examination: units.

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1 Time: 1 hour 20 minutes University of Winsor Faculty of Business Aministration Winter 2001 Mi Term Examination: Instructors: Dr. Y. Aneja NAME: LAST (PLEASE PRINT) FIRST Stuent ID Number: Signature: 12 Q.1 The pro shop at a large sports complex must ecie how many cans of tennis balls to keep in inventory. Weekly eman is just about constant at 64 cans. The cost per can is $5 an the annual inventory holing cost is 32% of the value of the inventory. Orers are place by one of the junior tennis instructors who is pai $18 per hour. It takes a half hour to place an orer. The orer forms are faxe at a cost of $1.00. There is a one week lea time. There are 50 working weeks in a year. 4 a. Fin the optimal economic orer quantity. Annual Deman D = 64(50)=3200, Unit holing cost C h = 0.32(5)=$1.60, Setup Cost C 0 = $9+$1=$10. * DC ( )( ) Q = 2 0 = = C h 200 units. 4 b. What is the total annual inventory relate cost? Annual holing cost = (Q/2).C h = (200/2)(1.6) = $160 Annual orering cost = C 0.(D/Q) = (10)(3200 / 200) =$160 Total =$ c. What is the reorer point? r = eman uring lea time of 1 week = 64 units 2. What is the cycle time? Cycle time = Q * / eman = (200) / (64) = weeks.

2 Page 2 of 6 10 Q.2 A weekly sports magazine publishes a special eition for the Worl Series. The sales forecast is for the number of copies to be normally istribute with mean 800,000 copies an stanar eviation 60,000 copies. It costs 40 cents to print a copy, an the newsstan price is $2. 5 a. Suppose unsol copies will be scrappe. How many copies shoul be printe? C u = = $1.60, C 0 = 0.4. => (C u )/(C u + C 0 ) = Hence we want to fin Q * so that * P( D Q ) = 08.. Deman is normally istribute with mean of 800,000 an st.ev. of 60,000. From normal tables, P(Z < 0.84) = Hence Q * = (60000) = b. Suppose 40 cents is assigne as a goowill cost for any stockout. How many copies shoul now be printe? The only quantity that changes is C u. New C u = ol C u + goowill cost = Hence (C u )/(C u + C 0 ) = 2/(2+0.4) = Proceeing as before will provie a z-value of Hence new Q * = (60000) = units. 18 Q.3 Daily eman for packages of five vieo tapes at a warehouse store is foun to be ranom with an average of 40 units. The orering cost is $20. Each pack of tapes costs $10 an there is a 25% annual holing cost for inventory. Assume 250 working ays per year. The management policy is to continuously monitor the inventory level, an orer the same amount every time, if the inventory level goes below a specifie level. 3 a. What shoul be the EOQ base on the average eman? D = 40(250) = 10000, C 0 = $20, C h = 0.25(10) = $2.50. * DC ( )( ) Q = 2 0 = = 400 units. C h 6 b. Suppose the lea time is 2 ays an the aily eman is normally istribute with a mean of 40 boxes an a stanar eviation of 5 boxes. If the store wants the probability of stocking out to be no more than 10%, an eman each ay is inepenent of the ay before, what reorer point shoul be set? Let X 1 = eman on the first ay. Similarly X 2 = eman on the secon ay. Then, the eman uring lea time = X 1 + X 2. Hence,

3 Page 3 of 6 µ σ σ σ σ = E( ) = E( X ) + E( X ) = = 80. = + = ( 5) + ( 5) = 50. X X = 50 = P( Z > 128. ) = r = µ + zσ = = Hence, Hence, reorer point is: 3 c. Determine the expecte annual inventory relate costs. From (b), safety stock = = Annual holing cost = 200(2.5) = $500 Annual orering cost = $500. Holing cost for the safety stock = 9.05(2.50) = $ Total inventory relate costs = = $1, Suppose instea the lea time is one ay an the aily eman is uniformly istribute between 20 an 60 units. If the store wants the probability of stocking out to be no more than 10%, what reorer point shoul then be set? r = (40) = 56 units. 2 e. Determine the safety stock value in part (). Safety stock = = 16 units. 25 Q.4 An oil wilcatter must ecie either to rill ( 1 ), or not to rill ( 2 ) an unexplore oil-well that he owns. He is uncertain whether the well is Dry (S 1 ), Wet (S 2 ), or Soaking (S 3 ). His payoffs are given in the table below: States of Nature Decision 1 2 Dry (S 1 ) -$70,000 $10,000 Wet (S 2 ) $50,000 $10,000 Soaking (S 3 ) $200,000 $10,000

4 Page 4 of 6 5 a. If the ecision maker knows nothing about the probabilities of the ifferent states of nature, etermine the optimal ecision base on the minimax regret criterion. Regret Values Decision 1 2 Dry (S 1 ) $80,000 0 Wet (S 2 ) 0 $40,000 Soaking (S 3 ) 0 $190,000 Maximum regret $80,000 $190,000 Minimum of the maximum regrets = $80,000. Hence the optimal ecision is 1 - to rill. 5 b. Suppose, there is a 10% chance that the well is of the soaking type, an 40% chance that the well is ry. What is the optimal ecision if the wilcatter is risk neutral? EV( 1 ) = 0.4(-70000) + 0.5(50000) + 0.1(200000) = $17,000. EV( 2 ) =0.4(10000) + 0.5(10000) + 0.1(10000) = $10,000. Hence, the optimal ecision is to rill. 5 c. The wilcatter can ask an expert for seismic sounings (an experiment) that will help provie better estimates of chances of the three states of nature. What is the maximum limit on the amount that the wilcatter will be willing to pay for this experiment? Explain. EV w PI = 0.4(10000) + 0.5(50000) (200000) = $49,000. EV wo PI = $17,000 (from part (b). Hence, EVPI = $49,000 - $17,000 = $32, Suppose the wilcatter is not risk neutral. After careful analysis of his financial position, he has inicate the following inifference probabilities, in the basic reference lottery, for ifferent monetary values: Amount Inifference Probability (p) $50, $10, What woul be his optimal ecision base on the expecte utility approach?

5 Page 5 of 6 Utilities Decision 1 2 Dry (S 1 ) Wet (S 2 ) Soaking (S 3 ) EU( 1 ) = 0.4(0) + 0.5(0.5) + 0.1(1) = 0.35 EU( 2 ) = 0.4(0.25) + 0.5(0.25) + 0.1(0.25) = 0.25 Optimal ecision = 1. 5 e. Base on the p-values state in part () above, woul you that the wilcatter is acting like a risk neutral, risk avoier or a risk taker ecision maker? Explain. Given U($50,000) = 0.5. For 0.5-basic reference lottery, EV = 0.5(20000) + 0.5(-70000)=$65,000. Since this is more than $50,000, the wilcatter is acting as a risk-avoier. Given U($10,000) = For a 0.25-basic reference lottery, EV = 0.25(200000) (-70000)= - $2,500. In this case EV < $10,000. Hence the wilcatter is acting in this case as a risk-taker. Hence overall the wilcatter is neither a risk-taker, risk-avoier or risk-neutral. 35 Q.5 Consier the following payoff matrix showing profits (in thousans of ollars) for a DM : Decision Alternatives States of Nature s 1 s a. Initial assessment suggests that there is 60% chances that state s 1 woul occur. What is the optimal ecision base on the expecte value approach? EV( 1 ) = 0.6(150) + 0.4(50) = $110 thousan. EV( 2 ) = 0.6(400) + 0.4(-200) = $ 160 thousan. Hence the optimal ecision is 2.

6 Page 6 of 6 8 b. An agency, known for its preictive ability about the states of nature, will preict G or B base on an extensive stuy, with G representing as goo chances of state being s 1. Agency s track recor is provie through the following conitional probabilities: P(U s 1 ) = 0.8, P(B s 2 ) = 0.9. Using probability tree or otherwise, etermine posterior probabilities, base on the given prior an conitional probabilities. Using Bayes theorem (by probability tree, or otherwise), we get: P(S 1 G) = (0.48)/(0.52) = 0.92 => P(S 2 G) = P(S 1 B) = (0.12)/(0.48) = 0.25 => P(S 2 B) = Also, P(G) = 0.52 => P(B) = c. Develop a ecision tree an etermine the optimal strategy base on the information provie by the agency. Developing the ecision tree, we get: EV w SI = 0.52(352) (75) = $ Determine the expecte value of information provie by the agency. EV w SI = $ (from (c)), EV wo SI = $160 (from (a). Hence, EVSI = $ $160 = $ e. Consier using the expecte utility approach for the problem. Suppose the DM is inifferent between a payoff of 50 (thousan ollars) an a probability of 0.4 (using the basic reference lottery). To etermine his inifference probability for 150 (thousan ollars), he inicates that he is inifferent between getting 150 (thousan ollars) an playing the following lottery: on the toss of a fair coin, he gets 400 (thousan ollars) if the outcome is heas an gets 50 (thousan ollars) if it is tails. What is the inifference probability that shoul be assigne to 150 (thousan ollars)? Explain Given U(400) =1, U(-200) = 0, an P(50) = 0.4. The DM is inifferent between getting 150 (thousan ollars) an a lottery where DM gets 400 (utility 1) with a probability of 0.5, an getting 50 (utility 0.4) with 0.5 probability. Hence U(150) = 0.5(1) + 0.5(0.4) = 0.7