Operations Management I Fall 2003 Odette School of Business University of Windsor. Midterm Exam II Solution Thursday, November 20 Education Gym

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1 Last Name First Name ID Operations Management I 7-1 Fall 00 Odette School of usiness University of Windsor Midterm Exam II Solution Thursday, November 0 Education Gym Instructor: Mohammed Fazle aki Aids Permitted: Calculator, straightedge, and a one-sided formula sheet. Time available: 1 hour 0 min Instructions: This exam has 0 pages including this cover page and 8 pages of tables. Please be sure to put your name and student ID number on each odd numbered page. Show your work. State results up to four decimal places. It s not necessary to return tables and formula sheet. Grading: Question Marks: 1 /10 /1 /1 4 /10 5 /10 6 /10 Total: /65

2 Question 1: (10 points) Circle the most appropriate answer 1.1 With all units discount schedule, a. total cost of buying may be less if a larger quantity is ordered b. unit cost of buying may be less if a larger quantity is ordered c. both d. none 1. Which of the following is not a part of the ordering/setup cost? a. Salary paid to the purchasing clerk b. Order receiving costs c. Order processing fees d. None of the above 1. For the space constraint problem, an optimal solution can be obtained by reducing each EOQ value by a constant multiplication factor, a. if total space required by the EOQ solution is less than the total space available b. if total space required by the EOQ solution is more than the total space available c. if for each product space required per unit is a constant multiple of holding cost per unit d. b and c 1.4 Consider the single-period inventory problem. When to order? a. When the inventory on hand equals the reorder point, R b. When the inventory on hand equals safety stock, s c. When the inventory on hand equals R + s d. At the beginning of the period 1.5 In a ( Q, R) system, the same number of units is ordered each time an order is placed. Why? a. To get the same holding and ordering costs b. To achieve a zero stockout c. ecause demand characteristics, cost parameters and inventory on hand do not change between two successive orders d. ecause suppliers do not sell a different order size at a different time 1.6 If the order quantity is less than EOQ a. annual holding cost is more than annual ordering cost b. annual holding cost is less than annual ordering cost c. annual holding cost is equal to annual ordering cost d. annual cost of buying item decreases 1.7 Which of the following is not a part of the carrying cost? a. Transportation costs b. Obsolescence c. Cost of capital d. Storage and insurance on inventory

4 1.8 Consider the multi-item, multi-period, single-facility production-planning problem. If an optimal order quantity is computed separately for each product, the solution may not be acceptable because a. different holding costs may be charged to different products b. different product may have a different cycle time c. fund constraint may be violated d. space constraint may be violated 1.9 The standardized loss function is used to compute a. annual holding cost b. annual ordering cost c. annual stockout cost d. none of the above 1.10 In the ( Q, R) system it is assumed that the ending inventory of every period is a. salvaged b. scrapped c. repaired d. the beginning inventory of the following period. Question : (1 points) A manufacturer of greeting cards must determine the order size of a certain popular line of cards. Call it line A. The demand for these cards has been a fairly steady 60,000 per year. The manufacturer is currently using an order size of 40,000. The ordering cost is $600 each time an order is placed. Assume that for each card the cost is 48 cents. The accounting department of the firm has established an interest rate to represent the opportunity cost of alternative investment and storage costs at 5 percent of the value of each card. a. (4 points) What is the optimal value of the EOQ for line A of greeting cards? ( ) $0. 1 ( 600)( 60,000) h Ic per unit per year EOQ Kλ h ,000 units b. (5 points) Determine the additional annual cost resulting from using the wrong order size. Assume only one line of cards is carried. With Order size, hq Kλ + Q Q EOQ 60,000 units, the total annual cost , ,000 +,600 +,600 $7,00 ( points) 60,000 With order size, Q 40,000 units the total annual cost hq Kλ , , , ,400 $7,800 ( points) Q 40,000 Hence, the additional cost resulting from Q 40,000 units is 7,800-7,00 $600 (1 point) 4

5 Note: The annual cost of buying λ c 60, $17, 800 is the same for both cases. Adding the cost of buying, the annual costs are $180,000 for Q EOQ 60,000 units and $180,600 for Q 40,000 units. The difference between the two costs is $600. c. (4 points) Continue from part a. Suppose that another expensive line of cards, call it line, has an unit price of $.50 and an EOQ of 9,600 units. If the maximum investment in lines A and must be limited to $9,00, compute the order sizes of lines A and. Fund required by the EOQ order quantity of Item A 60,000(0.48) $8,800 Fund required by the EOQ order quantity of Item 9,600(.50) $4,000 Total fund required by the EOQ order quantities of Items A and 8,800+4,000 $5,800 (1 point) Fund available $9,00 fund available 9,00 Hence, m (1 point) fund required 5,800 Therefore, the optimal order quantity of Item A m EOQ A 0.744(60,000) 44, units and the optimal order quantity of Item m EOQ A 0.744(9,600) 7,17.7 units ( points) Question : (1 points) Suppose that item A has a unit cost of $.60, a production setup cost of $00, and a quarterly demand of 600 units. It is estimated that cost of capital is approximately 15 percent per year. Annual storage cost amounts to percent and breakage to percent of the value of the each item. Assume that item A has a production rate of,600 items per year. a. ( points) Compute EPQ of item A. I h Ic ( 0.0)(.60) $0. 7 per unit per year λ 4 600, units per year EPQ, Q ( )( ) 400 Kλ h' Kλ λ h 1 P ( 00)(,400 ) ( 00 )(,400 ), , ,000 units b. ( points) What is the percentage of downtime in each cycle if only item A is produced? Compute cycle time and downtime. Express downtime as a percentage of cycle time. Q,000 Cycle time, T 0. 8 year λ,400 Q,000 Uptime, T year, Downtime, T T year P,600 5

6 T Percentage of downtime in each cycle 0..% T 0.8 6

7 Now, suppose that Item has a production rate of,000 items per year, a unit cost of $10.00, a production setup cost of $100, and a monthly demand of 50 units. Use the same cost of capital, storage cost and breakage as in parts a and b. c. (4 points) What is the cycle time if both items A and are produced in a single facility? Assume negligible setup times for both items A and. T h' K j j λ j [ K + K ] [ ] h' λ A A A + h' λ + λ A λ ha 1 λa + h P 1 A P λ 00 +, ,400 + Ic,600 [ 100] [ 00] λ 1 λ P 0.4, ,000 ( 0.0)( 10) , years (4 points, including 1 for unit of time) d. ( points) Continue from part c. What is the percentage of idle time in each cycle? QA λat, ,500 Uptime of A, TA years P,600,600,600 Uptime of, T A Q P λ T,000 ( 50 1) 0.65, ,000 75,000 Idle time T T A T years in each cycle IdleTime 0.08 Hence, percentage of idle time % T 0.65 Question 4: (10 points) 0.15 years The buyer for Needles Markup, a famous high end department store, must decide on the quantity of a high-priced woman s handbag to procure in Italy for the following Christmas season. The unit cost of the handbag to the store is $70.00 and the handbag will sell for $85. Every handbag not sold by the end of the season is purchased by a discount firm for $5.00. a. ( points) What is the underage cost per unit? cu Selling price purchase price $15/unit b. ( points) What is the overage cost per unit? co Purchase price salvage value 70-5 $5/unit 7

8 c. ( points) If the demand is uniformly distributed between 40 and 00 units, find the optimal order quantity. cu 15 For the optimal order quantity Q, Probability(demand Q ), p 0. 0 c + c Hence, Q a + p( b a) ( 00 40) 88 units u o d. ( points) If the demand is normally distributed with a mean of 10 and a standard deviation of 40, find the optimal order quantity. cu 15 For the optimal order quantity Q, Probability(demand Q ), p 0. 0 c + c Find the standard normal z -value for which cumulative area on the left, p Using Table A-1 Table A-1 gives the area between z 0 and positive z -values. Since p 0.0 < 0.50, find z -value for which Table A-1 area is and consider z negative. Hence, z Using Table A-4 p find z -value for which 1 ( z) 0. 0 Since 0.0 < 0.50, From Table A-4, 1 ( z) 0. 0 F for z Hence, z (or, z 0. 5, z 0. 5) ( points including 1 point for ve value of z) u F and consider z negative. o Q µ + zσ 10 + ( 0.55 ) units (1 point) Question 5: (10 points) Comptek Computers wants to reduce a large stock of personal computers it is discontinuing. It has offered the University ookstore a quantity discount pricing schedule if the store will purchase the personal computers in volume, as follows: Quantity Price 1-14 $ The annual inventory holding cost is 5%, the ordering cost is $10, and annual demand for this particular model is estimated to be 40 units. Compute the optimal order size. 8

9 Consider the cheapest price level of c $1,50 per unit. h Ic 0.5 1,50 $ 80/unit/year Kλ ( 10)( 40) EOQ h units (1 point) Since the price level of c $1,50 is not available for an order quantity Q EOQ units, EOQ is infeasible and a candidate for optimal order quantity is Q 6, because 6 is the minimum order quantity for the price level of c $1,50. Consider the next price level, c $1,55 per unit. h Ic 0.5 1,55 $81. 5/unit/year Kλ ( 10)( 40) EOQ h units (1 point) Since the price level of c $1,55 is not available for an order quantity Q EOQ units, EOQ is infeasible and a candidate for optimal order quantity is Q 15, because 15 is the minimum order quantity for the price level of c $1,55. Consider the next price level, c 1 $1,600 per unit. h 1 Ic ,600 $ 400/unit/year EOQ1 Kλ h 1 ( )( 40) units (1 point) Since the price level of c 1 $1,600 is available for an order quantity Q EOQ 1 1 units, EOQ 1 is feasible and a candidate for optimal order quantity is Q1 1. Now, compute total cost for each candidate for optimal order quantity: j Candidate Q j Q 6 (1 point) 80 6 Holding cost h Q j j,400 Ordering cost Kλ Q j ( points) Cost of item λ c j (1 point) Total cost Holding cost + Ordering cost + Cost of item ,50 64, 800 $70, , Q 15 (1 point) , ,55 66, 000 $70, , Q 1 (1 point) 400 1, ,600 84, 000 $88,800,400 1 Conclusion: The total cost is minimum, $70,779.8 for Q 15. Therefore, an optimal order quantity is Q 15 units ar a price of $1,55 per unit. (1 point) 9

10 Question 6: (10 points) Mean and standard deviation of annual demand are 600 and 100 units respectively. Lead-time is Q,R policy is used with ( Q, R) ( 00, 00). months. Suppose that a ( ) a. ( points) Find the number of units of the safety stock. Mean lead-time demand, µ λτ units 1 Safety stock, s R µ units b. ( points) What is the probability of stockout during the lead time? Standard deviation of lead-time demand, σ σ y τ units 1 R µ z σ Probability of Stock out during lead time 1 F ( z) Using Table A-4, 1 F ( z) and using Table A-1, 1 F ( z) Hence, probability of Stock out during lead time 1 F ( z) c. ( points) What is the expected annual number of orders? λ 600 Annual number of orders orders per year Q 00 d. ( points) What is the expected annual number of units stockout? Number of units stockout per cycle, L( z ) n σ units 600 Annual number of units stockout n λ units Q 00 e. ( points) What is the average inventory? Q 00 Average inventory + s units 10

Operations Management I Winter 2005 Odette School of Business University of Windsor

Operations Management I Winter 2005 Odette School of Business University of Windsor Last Name First Name ID Operations Management I 7-1 Winter 005 Odette School of Business University of Windsor Midterm Exam II Solution Thursday, March 1 Last Name A-L: Room OB B0 Last Name M-Z: Room ED