Final solved spring 07

Final solved spring 07 - Questions [46 points, 2 points...

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Unformatted text preview: Questions [46 points, 2 points each] 1. Solve the LP below: max 3X + 2Y s.t. 3X + Y ! 12 X + 3Y ! 12 X,Y"0 Optimal Solution: (X=3 , Y=3) Objective function value = 15 2. Considering the LP formulation below, which of the following is false? max 2X + Y s.t. -X + 2Y ! 4 X+ Y!3 Y"0 a) b) c) d) e) Optimal Solution: (X=2/3 , Y=7/3) Objective function value = 11/3 (-1,1) is a feasible solution. (2,1) is not an optimal solution. The objective function value of an optimal solution will be no less than 6. (1,2) is a feasible solution. None of the above. 3. Considering the LP formulation in (2) above, which of the following is true? a) b) c) d) If the objective function were x+2y then the optimal solution wouldn't be unique. If x ! 0 were added then the optimal solution would change. If y! 0 were removed then the feasible region would be unbounded. Changing any of the constraints would always result in either a different optimal solution or a different optimal objective value or both. e) None of the above. A printer manufacturer purchases 18,000 fusers for assembly annually. The carrying cost of these fusers is 10% of the unit cost of $45 per year. The fixed cost per purchase order is $4,750. 4. What is the EOQ for these fusers? EOQ = 2" D"S 2" D"S 2 "18000 " 4750 = = = 6164.41 # 6164 fusers H i" C 10% " 45 5. What is the corresponding monthly total inventory cost? ! TC = D "! + D "! + EOQ " H = 18000 " 45 + 18000 " 4750 + 6164 "10% " 45 = $837,739.86 C S EOQ 2 6164 2 6. How often should the manufacturer be placing orders for fusers? ! D 18000 ! = = 2.92 orders/year, or an order about every 125 days. Q 6164 7. What is the manufacturer's maximum inventory level for fusers? 6164 ! ! Page 1 out of 7 8. Suppose there is a 50% chance that, after investing $100 million in constructing a new factory, demand will be strong and the factory will produce an operating income stream with a present value of $150 million. Suppose also there is a 50% chance that demand will be weak and the present value of the operating income stream will be $20 million. The firm has the option to delay the decision for 1 year when demand will be known with certainty to be either strong or weak. However, delaying the factory construction for 1 year will increase costs by $30 million (because the timeframe for construction will be much compressed). Which of the following decisions maximizes expected revenue? i. Construct the factory now. ii. Construct the factory in 1 year. iii. Wait for 1 year and only construct the factory if demand is strong. iv. Never build the factory. v. There is not enough information given to answer the question. 9. Which of the following statements regarding capacity flexibility is false? i. Flexible capacity is better equipped to meet changing demand. ii. Flexible capacity may increase utilization rates. iii. Flexible capacity is generally less expensive than dedicated capacity. iv. Many car manufacturers have flexible assembly lines today. v. None of the above. 10. Which of the following statements is false? i. There are two main directions in which customer satisfaction with waits can be influenced: by changing customer expectation and by influencing customer perception. ii. When a customer has a negative experience in the early stages of service, even when the rest of his encounter is smooth, he tends to remain disgruntled. iii. Uncertain waits feel longer than known, finite waits. iv. Customers tend to tolerate longer waits for service perceived as more valuable. v. None of the above. For the following questions consider a call center with an hourly arrival rate of 18 calls and each caller spending 12 minutes on average with an agent. 11. What is the minimum number of agents the call center should employ? != " " 18 c/hr <1 ! M> = =3.6 ! M = 4 agents #$ 5 c/hr 12. Assuming exponential interarrival and service times, how many callers on average would then be on hold waiting to speak with an agent? ! ! ! From Exhibit TN7.11 on page 304 in the textbook we get Lq = 7.0893 callers 13. Assuming exponential interarrival and service times, what is the minimum number of agents the call center should employ in order for the average time callers spent on hold waiting to speak with an agent to be 3.5 minutes? 3.5 From Exhibit TN7.11 on page 304 in the textbook with Lq = ! " = 1.05 we get M = 5 agents. 60 ! Page 2 out of 7 14. Your new supervisor loves riddles. For instance, when you asked him what is the annual demand of an item he responded to you as follows: "We use the EOQ model and our monthly order quantity is 75 units for this item." Is this enough information to determine the annual demand for this item? If yes then compute it? If not then state what additional information you'd need. 900 units 15. An e-ticket check-in machine has an arrival rate of 5 customers per hour with each customer taking on average two minutes at the machine. Assume customers arrive according to a Poisson process and the time each customer spends at the machine follows an exponential distribution. What is the probability of at least one customer waiting-in-line to use the machine? $ # ' $ # ' 0 $ # ' $ # '1 Pr{at least two customers in the system} = 1-P0-P1 = 1 &1" ) * & ) &1" ) * & ) = 2.78% % ( %( % ( %( 16. How many minutes would you expect to take a customer to check-in using the e-ticket machine above? 1 Ws = = 2.4 min ! ! "# A coffee shop operates according to the following process flow diagram: ! Customers Order Taker: 60 customers per hour Customer Orders Coffee Maker: 40 customers per hour 17. Which of the following statements is true? i. 360 customer orders can be prepared each morning between 6 am and noon. ii. A coffee-loving customer makes the following offer: one free hour of work taking orders every morning in exchange for 3 shots of espresso. Accepting this offer would increase the number of orders prepared every morning. iii. A coffee-loving customer makes the following offer: one free hour of work making coffees every morning in exchange for 3 shots of espresso. Accepting this offer would increase the number of orders prepared every morning. iv. None of the above. 18. Suppose the coffee shop takes orders from 6 am to 3 pm every day. Assume the coffee makers must stay until they have finished making all customer coffees. What is the maximum number of coffee orders waiting to be made? 180 Page 3 out of 7 19. Regarding the seasonality method presented in the slides (and in class), which of the following statements is true? (1) The number of periods in each season can be different. (2) You do not need to forecast ( "smooth") the seasonality indices for the new season. (3) The forecasted deseasonalized value for the new season is not used in forecasting the new seasonal values. (4) None of the above. Actual Demand 42 53 34 45 26 67 3-MA 43 44 35 46 Deviation Exp. Sm. 42 42 45.85 41.7025 42.856625 36.956806 47.472 Deviation -11 11.85 -3.3 16.86 -30.04 January February March April May June July -2 18 -32 Use the demand data above for the following forecasting questions: 20. Using the 3-period Moving Average method, determine the forecast for July's demand. 46 21. What is the bias of the 3-period Moving Average method in (a) above? -5.333 22. Using exponential smoothing with #=0.35, determine the forecast for July's demand. 47.472 23. What is the MAD of the exponential smoothing method in (22) above? 14.61 Page 4 out of 7 Problem 1 [28 points, 4 points each] Farmer John travels to the Farmers Market every Sunday to sell his delicious organically grown apples. He sells each apple at $1.55 since he has estimated his cost per apple to be about $0.85. Any apples not sold at the end of the day Sunday he takes them back to his farm and are fed to his horses. a. If the demand for his apples each Sunday follows a normal distribution with =280 and !=20 then how many apples should he bring to the market each Sunday in order to maximize his profit? F(Q*) = Cu MP r - c $1.55 - $0.85 (r - c) + g = = = = = 45.16% Cu + Co $1.55 MP + ML r [c + h] + [(r - c) + g] Q* = + (z F(Q*) " !) = 280 + (-0.1216) " 20 = 277.568 ! 278 apples ! ! b. ! the demand for his apples each Sunday follows a uniform distribution between 175 and 425 then If ! ! how many apples should he bring to the market each Sunday in order to maximize his profit? Q* = 175 + (425-175) " 45.16% = 287.9 ! 288 apples c. If the demand for his apples each Sunday follows the discrete distribution below then how many apples should he bring to the market each Sunday in order to maximize his profit? 300 apples Demand 150 200 250 300 350 400 450 Probability 5% 10% 20% 30% 20% 10% 5% Cumulative Probability 5% 15% 35% 65% 85% 95% 100% Farmer John has received an offer from Whole Foods to sell his apples to them at $1.90 per apple. He is to deliver to them his delicious apples every morning and they have agreed to discard any unsold apples at the end of each day. d. If Whole Foods were to sell Farmer John's apples at $2.75 per apple, how many apples should they buy from him to maximize their expected profit, assuming the daily demand at Whole Foods for Farmer John's apples follows the distribution in part (c) above? 250 apples F(Q*) = Cu MP r - w $2.75 - $1.90 (r - w ) + g = = = = = 30.91% Cu + Co $2.75 MP + ML r [w + h " b] + [(r - w ) + g] Therefore, Whole Foods should buy 250 apples from Farmer John each day. ! ! ! e. ! What would then be the ! daily expected profit of Whole Foods? { If demand = 150 then profit = 150"(2.75-1.90) - (250-150) "1.90 = -$62.50 } " 5% + { If demand = 200 then profit = 200"(2.75-1.90) - (250-200) "1.90 = $75 } " 10% + { If demand ! 250 then profit = 250"(2.75-1.90) = $212.50 } " 85% = $185 f. What would then be Farmer John's daily profit? Farmer John's profit = 250 " (1.90-0.85) = $262.50 Page 5 out of 7 g. Whole Foods now consider purchasing Farmer John's apple farm. What would then be the total expected integrated channel profit? F(Q*) = Cu MP r - c $2.75 - $0.85 (r - c) + g = = = = = 69.10% Cu + Co $2.75 MP + ML r [c + h] + [(r - c) + g] Therefore, Whole Foods should get 350 apples from Farmer John's apple farm each day. The expected profit of the integrated channel is computed as follows: ! ! ! ! ! { If demand = 150 then profit = 150"(2.75-0.85) - (350-150) "0.85 = $115 } " 5% + { If demand = 200 then profit = 200"(2.75-0.85) - (350-200) "0.85 = $252.50 } " 10% + { If demand = 250 then profit = 250"(2.75-0.85) - (350-250) "0.85 = $390 } " 20% + { If demand = 300 then profit = 300"(2.75-0.85) - (350-300) "0.85 = $527.50 } " 30% + { If demand ! 350 then profit = 350"(2.75-0.85) = $665 } " 35% = $500 Page 6 out of 7 Problem 2 [28 points, 4 points each] Consider the following project data for the remaining questions: Task A B C D E F G H Immediate Optimistic Most Likely Pessimistic Predecessor(s) (in weeks) 2 2 5 A 2 3 4 B 2 3 7 C,F 6 7 8 3 4 5 E 3 3 6 E 3 4 5 C,F 7 8 9 0 A,2.5 0 A ,1.5 2.5 2.5 2.5 B,3 2.5 B , 2 5.5 0 E,4 1.5 E , 3 5.5 4 5.5 4 5.5 (in weeks) !2 0.25 0.1111 0.6944 0.1111 0.1111 0.25 0.1111 0.1111 9 C,3.5 Cost to crash (per week) 2.5 3 3.5 7 4 3.5 4 8 5.5 $32,750 $39,250 $37,650 $30,500 $38,750 $31,250 $35,250 $39,750 9 D,7 10 D , 2 17 9 H,8 9 H , 7 17 17 16 C ,2. 9 5 F,3.5 7.5 5.5 F ,2.5 9 4 8 G,4 13 G , 3 17 a. Which tasks are critical in the above project? A-B-C-H b. What is the expected project completion time? 17 weeks c. What is the late start time of task G? week 13 d. What is the slack of task F? 1.5 weeks e. What is the probability that this project will be completed within eighteen weeks? Pr{project will be completed within 18 weeks} = Pr{T!18} = Pr{z ! = Pr{z ! 0.9258} ! 82.27% f. What should be this project's bid date, so that the probability it will be late is no more than 10%? ! ! Bid Date = T + (z(1-10%) " !T) = 17 + 1.282 " 1.1667 = 18.38 weeks ! 19 weeks g. If you had to crash this project by two weeks, which tasks would you crash each by only one week (no task can be crashed by half a week) to minimize the crash cost and what would be the crash cost? ! A and H for a total of $72,500 Page 7 out of 7 18 " # 18 "17 } = Pr{z ! }= $# 1.1667 ...
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This note was uploaded on 10/15/2007 for the course BUAD 311 taught by Professor Vaitsos during the Spring '07 term at USC.

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