Operations Management I Fall 2004 Odette School of Business University of Windsor

Transcription

1 Last Name First Name ID Operations Management I Fall 004 Odette School of Business University of Windsor Final Exam Solution Wednesday, December 15, 1:00 noon 3:00 p.m. Ambassador Auditorium: Areas C,D,E,F,G Instructor: Mohammed Fazle Baki Aids Permitted: Calculator, straightedge, and 3 one-sided formula sheets. Time available: 3 hours Instructions: This exam has 3 pages including this cover page, 1 blank page and 8 pages of Table It s not necessary to return tables and formula sheets Please be sure to put your name and student ID number on each odd numbered pages Show your results up to four decimal places Show your work Grading: Question Score Question Score 1 /15 /1 3 /1 4 /1 5 /1 6 /13 7 /10 8 /5 9 /9 Total /100

2 Question 1: (15 points) Circle the most appropriate answer 1.1 Forecasting is used in the context of a. make-to-order production system b. assemble-to-order production system c. make-to-stock production system d. all of the above 1. MRP is used in the context of a. make-to-order production system b. assemble-to-order production system c. make-to-stock production system d. all of the above 1.3 L4L performs best if a. holding cost is low b. ordering cost is low c. the costs change over time d. the costs do not change over time 1.4 Make sure you run out of inventory before you produce new components, is a condition necessary to minimize cost in the context of a. MRP with capacity constraints b. MRP with no capacity constraints c. both d. none 1.5 The essence of JIT is the willingness of the worker to a. produce as much as possible without halting the production at all b. specialize in a single function c. maximize the length of the time between maintenance d. none of the above 1.6 If the rate of learning, L is high, the learning is a. faster and the learning curve flatter b. faster and the learning curve steeper c. slower and the learning curve flatter d. slower and the learning curve steeper 1.7 The following method is used for forecasting a stationary series a. Exponential smoothing b. Holt s method c. Linear regression d. a and b 1.8 Which of the following forecast series is expected to be the smoothest?

3 a. A 4-period simple moving average b. A 8-period simple moving average c. An exponential smoothing method with α 0. 5 d. An exponential smoothing method with α Which of the following is not a part of the cost of hiring? a. Severance pay b. Cost of training c. Cost of advertising d. b and c 1.10 In a level strategy, cumulative production a. equals cumulative demand b. is more than cumulative demand c. is less than cumulative demand d. none of the above 1.11 Lead time is the amount of time between a. two successive order placement b. two successive updating of inventory on hand c. placing order and receiving delivery d. two successive capacity additions 1.1 The motivation for holding inventory includes a. continuous improvement b. economies of scale c. better quality d. less capital required 1.13 If three products A, B, C are produced in a rotation cycle policy, the following is a possible sequence a. A, B, C, B, C, A, C, A, B b. A, C, B, A, C, B, A, C, B c. A, B, C, A, B, C, B, B, B d. A, B, A, B, C, A, B, A, B 1.14 The best inventory policy satisfying the Type 1 service objective a. is easier to compute than the best inventory policy satisfying the Type service b. can accurately approximate a Type service c. should be used in place of the best inventory policy satisfying the Type service d. all of the above 1.15 Which of the following is an overage cost? a. The cost per unit of positive inventory remaining at the end of the period b. The cost per unit of the unsatisfied demand 3

4 c. The profit margin d. Depends whether backorder is allowed or not 4

5 Question : (1 points) A major oil company is considering the optimal timing for the construction of new refineries. From past experience, each doubling of the size of a refinery at a single location results in an increase in the construction costs of about 75 percent. Furthermore, a plant of size 6,000 barrels per day costs $5 million. Assume that the annual demand for the oil is increasing at a constant rate of 1.5 million barrels yearly and the discount rate for future costs is 0 percent. a a. (4 points) Find the values of k and a assuming a relationship of the form ( y) ky. is in units of barrels per day. f f ( y) ( y) ( y) a ( y) a k a ln a k ln ( y) 5 ( 1.75) ( ) ( points) a f 5 5,000,000 f ( y) ky k 0.03 a ( points) or, k, y f Assume that y b. (4 points) Determine the optimal timing of plant additions and the optimal size and cost of each plant addition. Figure 1-14 From Fig 1-14, locating a , rx rx 0.41 Optimal timing, x. 05 years r 0.0 ( points) Optimal size, ( ) xd million (1 point) Optimal cost, f barrels/ye ar a ( y) ky 0.03 $ million ,075,000 or,76.07 $3,880, (1 point) c. (4 points) Suppose that the largest single refinery that can be built with current technology is 7,500 barrels per day. Determine the optimal timing of plant additions and the optimal size and cost of each plant in this case. Optimal size, Optimal timing, * 3,075,000 x D min 7500, 365 x * * x D D barrels/day.7375 million 1.85 year (1 point) a Optimal cost, f ( y) ky 0.03 ( 7500 ) $ million Function a or,,76.07( 7500 ) $9, (1 point) u rx barrels/year ( point)

6 Question 3: (1 points) A popular brand of tennis shoe has had the following demand history by quarters over a two-year period. Quarter Demand Quarter Demand a. (4 points) Determine the seasonal factors for each quarter by the method of centered moving averages N 4 The demand is quarterly, there are 4 quarters in each year. Centered (B/D) Period Demand MA(4) MA Ratio A B C D E Final Seasonal Seasonal Period Factors Factors Total b. (4 points) Compute the deseasonalized demand series. Using the method of linear regression, determine the slope and intercept of the straight line that best fits the deseasonalized series. 6

7 x Deseasonalized Demand y Demand xy x Sum Average n n n n xi yi xi yi i 1 i 1 i 1 8( ) 36( ) Slope n n 8(04 ) (36) n xi xi i 1 i 1 Intercept y slope( x) ( )(4.5) c. (4 points) Predict the demand of all quarters of 005. Plot the original demand of and predicted demand of 005. Deseasonalized demand, First quarter of 004: 9 y x x, y ( 9) To get the predicted demand, reseasonalize, y Second quarter of 004: 10 x, y ( 10) To get the predicted demand, reseasonalize, y Third quarter of 004: 11 x, y ( 11) To get the predicted demand, reseasonalize, y Fourth quarter of 004: 1 x, y ( 1) To get the predicted demand, reseasonalize, y Demand Period 7

8 Question 4: (1 points) A start-up firm has kept careful records of the time required to manufacture its product, a shutoff valve used in gasoline pipelines. Cumulative Number of Units Produced, u Number of Hours Required for the Next Unit, Y ( u) a. ( points) Compute the logarithms of the numbers in each column. Use natural logs. ln y u ln ( u) ( ( ) b. (4 points) Graph the ln(hours) against the ln(cumulative units) and eyeball a straightline fit of the data. Using your approximate fit, estimate slope and intercept. ln(u) vs ln(y(u)) From graph: Intercept 3.00 Slope Accept intercept.5 to 3.5 Accpet slope -0.3 to -0.5 Precise answers (not required): Intercept 3.0 Slope -0.4 ln(y(u)) 3.50 Intercept Slope ln(u) Graph: points, slope 1 point, intercept 1 point 8

9 (Continued ) c. (4 points) Using the results of part (b), estimate the time required to produce the first unit and the appropriate percentage learning curve that fits these data. Time required to produce the first unit, a e intercept e hours b -slope 0.40 Rate of learning, L e slope(ln()) e -0.40(0.6931) % d. ( points) Consider the learning curve derived in part (c). How much time will be required to produce the,000 th unit, assuming the learning curve remains accurate. The time required for the,000 th b 0.40 unit is Y ( u) au (000) hours Question 5: (1 points) Green s Buttons of Rolla, Missouri, supplies all the New Jersey Fabric stores with three different styles of buttons for men s dress shirts. The plastic injection molding machine can produce only one button style at a time and requires substantial time to reconfigure the machine for different button styles. As Green s has contracted to supply fixed quantities of buttons for the next four years, its demand can be treated as fixed and known. The relevant data for this problem are: Button Type Annual Sales Production Rate (units/day) Setup Time (hours) Variable Cost A 30, $0.30 B 40, $0.0 C 15, $0.40 Assume 8 hours in a day and 50 working days per year. Green s accounting department established a 17 percent annual interest rate for the cost of capital and a 3 percent interest rate to account for storage space. Setup costs are $5 per hour required to reconfigure the equipment for a new style. Suppose that the firm decides to use a rotation cycle policy for production of the buttons. a. (5 points) What is the optimal rotation cycle time? Cycle time 9

10 T * h' λ 1 ( K + K + K ) ( K + K K ) h' λ 3 + h' 3 ( ) 1 + λ1 λ h1 1 λ1 + h 1 λ P1 P ( ) years days days λ 3 ( ) ,000 40, , , , ,000 15,000 15, , , ,17 + 1,00 30,000 40,000 15, , , , , , ,000 3 λ3 + h3 1 λ P Again, cycle time * s1 + s + s3 T λ1 λ λ 1 P P P days days ,000 40,000 15, hours 30,000 15, , ,000 15, ,0000 * Hence, cycle time, T max( ,5.858 ) days years years b. ( points) How large should the lots be? * Q λ T 30, ,0 units 1 1 * T 40, Q λ 18,960 units * Q λ T 15, ,110 units 3 3 c. (5 points) Compute the annual cost of holding and setups for each button type at the optimal solution. Compute the total annual cost of holding and setups at the optimal solution. Button Type A: Q 14,0 Annual holding cost 1 h' $ K1λ ,000 Annual setup cost $ Q 14,0 1 Annual total cost $ Button Type B: 10

11 Q 18,960 Annual holding cost h' $ K λ 75 40,000 Annual setup cost $ Q 18,960 Annual total cost $ Button Type C: Q 7,110 Annual holding cost 3 h' $ K 3λ ,000 Annual setup cost $ Q 7,110 3 Annual total cost $ Hence, total annual cost $1,

12 Question 6: (13 points) Bobbi s Restaurant in Boise, Idaho is a popular place for weekend brunch. The restaurant serves real maple syrup with French toast and pancakes. Bobbi buys the maple syrup from a company in Maine that requires three weeks for delivery. The syrup costs Bobbi $4 a bottle and may be purchased in any quantity. Fixed costs of ordering amount to about $75 for bookkeeping expenses and holding costs are based on a 0 percent annual rate. Bobbi estimates that the loss of customer goodwill for not being able to serve the syrup when requested amounts to $5. Based on past experience, the weekly demand for the syrup is normal with mean 1 and variance 16 bottles. For the purpose of your calculations, you may assume that there are 5 weeks in a year and that all excess demand is backordered. a. (5 points) How large an order should Bobbi be placing with her supplier for the maple syrup and when should the orders be placed? Find Q and R. λ λ w units, µ λ τ w units, σ h Ic $0.80 per unit per year, K $75, p $5 Iteration 1 Kλ (75)(64) Step 1: Q EOQ units h 0.80 Qh (0.80) Step : 1 F ( z) pλ 5(64) z.11 (Table A-4) R µ + zσ Step 3: L ( z) (Table A-4) n σ L( z) 6.98 (0.0063) σ w τ units λ (64) Step 4: Q [ np + K] [ ] h 0.80 (near , the process may stop after finding R ) Qh (0.80) Step 5: 1 F ( z) pλ 5(64) z.105 (Table A-4) R µ + zσ An optimal policy is Q345, R51 (rounded to the nearest integer) b. (1 point) What level of Type 1 service is being provided by the policy you found in part a? Type 1 service, α F( z ) 1 [ 1 F( z) ] % c. (1 point) What level of Type service is being provided by the policy you found in part a? n Type service, β % Q

13 d. ( points) What policy should Bobbi use if the stock-out cost is replaced with a Type 1 service objective of 95 percent? Step 1: Set Q EOQ units Step : F( z) 0.95, so z (From Table A-4) Step 3: R µ + zσ units An optimal policy is Q34, R47 e. ( points) What policy should Bobbi use if the stock-out cost is replaced with a Type service objective of 95 percent? (You may assume an EOQ lot size.) Step 1: Set Q EOQ units Step : n Q( 1 β ) (1 0.95) n L ( z).4686 σ 6.98 From Table A-4, L ( z). 463 for z. 46 L ( z). 47 for z. 47 So, z. 465 R µ + zσ An optimal policy is Q34, R19 ( ) 90 f. ( points) Suppose that Bobbi s supplier requires a minimum order size of 500 bottles. Find the reorder level that Bobbi should use if she wishes to satisfy 99 percent of her customer demands for the syrup. Step 1: Set Q 500 units (as given) Step : n Q( 1 β ) 500(1 0.99) 5 n 5 L ( z) σ 6.98 From Table A-4, L ( z) for z L ( z) for z So, z R µ + zσ An optimal policy is Q500, R3 ( )

14 Question 7: (10 points) Each unit of A is composed of two units of B and one unit of C. Each unit of B is composed of three units of C and four units of D. Items A, B and C have on-hand inventories of 40, 60 and 80 units respectively. Item B has a scheduled receipt of 50 units in period 1, and D has a scheduled receipt of 160 units in Period 1. Lot-for-lot (L4L) is used for Item A. Item B requires a minimum lot size of 50 units. Each of the Items C and D is required to be purchased in multiples of 100. Lead times are one period for each of the Items A, B and C, and two periods for Item D. The gross requirements for A are 0 in Period 3, 50 in Period 6, and 90 in Period 9. Find the planned order releases for all items to meet the requirements over the next 10 periods. a. ( points) Construct a product structure tree. A B() C(3) D(4) C(1) b. ( points) Consider Item A. Find the planned order releases and on-hand units in period 10 Period Item Gross Requirements A Scheduled receipts On hand from LT prior period 1 Net requirements Q L4L Time-phased Net Requirements Planned order releases Planned order delivery (Continued ) 14

15 c. ( points) Consider Item B. Find the planned order releases and on-hand units in period 10. Period Item Gross Requirements B Scheduled receipts 50 On hand from LT 1 prior period Net Requirements 130 Time-phased Net 130 Q > Requirements 50 Planned order 130 releases Planned order 130 delivery d. ( points) Consider Item C. Find the planned order releases and on-hand units in period 10. Period Item Gross Requirements C Scheduled receipts On hand from LT 1 prior period Net requirements Q Time-phased Net Requirements Planned order releases Planned order delivery e. ( points) Consider Item D. Find the planned order releases and on-hand units in period 10. Period Item Gross Requirements 50 D Scheduled receipts 160 On hand from LT prior period Net requirements 360 Time-phased Net 360 Q Requirements 100 Planned order 400 releases 15

16 Planned order delivery

17 Question 8: (5 points) A single inventory item is ordered from an outside supplier. The anticipated demand for this item over the next 7 months is 14, 1, 13, 15, 10, 9, 8. Current inventory of this item is 5, and the ending inventory should be 6. Assume a holding cost of $ per unit per month and a setup cost of $70. Assume a zero lead time. Determine the order policy for this item over the next 7 months. Use the Silver-Meal heuristic. Net requirements: r , r 1, r3 13, r4 15, r5 10, r6 9, r Months Q I1 I I3 I4 I5 I6 I7 Holding Cost Ordering Cost Per Period Cost 1 to to to stop 3 to to to to stop 6 to to a. (3 points) Use the table above to show your computation and summarize your order policy here: Month Lot size to order b. ( points) Use the table below to show the ending inventory that results from your order policy at the end of each of the 7 months: Month Gross Requirements Beginning Inventory Net Requirements Time-phased Net Requirements Planned order Release Planned Deliveries

18 Ending Inventory

19 Question 9: (9 points) Consider Question 8 again. The question is re-written below: A single inventory item is ordered from an outside supplier. The anticipated demand for this item over the next 7 months is 14, 1, 13, 15, 10, 9, 8. Current inventory of this item is 5, and the ending inventory should be 6. Assume a holding cost of $ per unit per month and a setup cost of $70. Assume a zero lead time. Determine the order policy for this item over the next 7 months. ( a ) (3 points) Suppose that the maximum order size is 1 per month. Does there exist a feasible solution? If there does not exist a feasible solution, what is first month when there will be a shortage? Month Production Requirement Capacity Cumulative Production Requirement 19 Cumulative capacity < < < > 48 stop No, there is no feasible solution. Month 4 is the first month when there will be a shortage. Consider Question 8 again. The question is re-written below: A single inventory item is ordered from an outside supplier. The anticipated demand for this item over the next 7 months is 14, 1, 13, 15, 10, 9, 8. Current inventory of this item is 5, and the ending inventory should be 6. Assume a holding cost of $ per unit per month and a setup cost of $70. Assume a zero lead time. Determine the order policy for this item over the next 7 months. (b) (3 points) Suppose that the maximum order size is 13 per month. Use lot-shifting technique to obtain a feasible solution (without using holding and setup cost). Show your final solution in the table given below. Month Production Requirement Actual Production Production Capacity A feasible solution obtained by lot-shifting technique: Month

20 Actual Production

21 Consider Question 8 again. The question is re-written below: A single inventory item is ordered from an outside supplier. The anticipated demand for this item over the next 7 months is 14, 1, 13, 15, 10, 9, 8. Current inventory of this item is 5, and the ending inventory should be 6. Assume a holding cost of $ per unit per month and a setup cost of $70. Assume a zero lead time. Determine the order policy for this item over the next 7 months. (c) (3 points) Improve the solution obtained in Part (b). Assuming a maximum order size of 14 units per month and using the back-shifting technique, find another solution that has less total holding and setup cost than the solution obtained in Part (b). Show your final solution in the table given below. Back-shift 13 units of Month 7? Check if it s better to backshift 13 units of Month 7 as follows: o 4 units 1 month back to Month 6 o 4 units months back to Month 5 o 1 unit 3 months back to Month 4 o 1 unit 4 months back to Month 3 o 1 unit 5 months back to Month and o units 6 months back to Month 1 Additional holding cost [4(1)+4()+1(3)+1(4)+1(5)+(6)] 7 > 70 savings in ordering cost. No, do not back-shift Back-shift 10 units of Month 6? Check if it s better to backshift 10 units of Month 6 as follows: o 4 units 1 month back to Month 5 o 1 unit months back to Month 4 o 1 unit 3 months back to Month 3 o 1 unit 4 months back to Month and o 3 units 5 months back to Month 1 Additional holding cost [4(1)+1()+1(3)+1(4)+3(5)] 56 < 70 savings in ordering cost. Yes, back-shift Final answer to part ( c ). An improved solution: Month Actual Production