Ch 6 Select HW Solutions

Ch 6 Select HW Solutions - 6.1 6.2 Chapter 6 Answers to...

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Unformatted text preview: 6.1 6.2 Chapter 6 Answers to End-of-Chapter Problems . Using equation 6.4 the variability coefficient is _1:2(212 +292 +...+32=) VC_ _1 =12x24,115 (21+ 29 + + 3:2)2 (433)2 1 0'24 Since VC > 0.2, the appropriate approach is the Silver-Meal heuristic and the associated strategy is to replenish 74 units at the start of January, 155 units at the start of April, 115 units at the start of July, and 139 units at the start of October. The TRC is $205.39. The EOQ, if it were used, would be approximately 128 units. The associated strategy (covering an integer number of periods each time) would be 160 units in J anuary, 114 units in May, 148 in August and 61 in November, with a resulting TRC of $257.98. (The large discrepancy is partly due to the horizon cutting short the last replenishment). In contrast, the 7‘qu = constant (i.e. fixed time supply) would give the same result as the SM heuristic on this example. Here, w = ($4.75)(0.24 $f$iyr.) 1“ = $0.0951mo. 12 mo. Cumulative. Cumulative T DU) A (111)me Row Sum Sum Sum +7" 1 18 $35.00 0 $35.00 $35.00 $35.00 2 31 1(31).095 2.95 37.95 18.97 3 23 2(23).095 4.37 42.32 14.114: 4 95 3(95).095 27.08 69.40 17.35 - for January, order 72 units (a 3 month supply). 1 95 $35.00 0 $35.00 $35.00 $35.00 2 29 1(29).095 2.76 37.76 18.88 3 37 2(37).095 7.03 44.79 14.93 4 50 3(50).095 14.25 59.04 14.76: 5 39 4(39).095 14.82 73.86 14.77 - for April, order 211 units (a 4 month supply). 1 39 $35.00 0 $35.00 $35.00 $35.00 2 30 1(30).095 2.85 37.85 18.93 119 3 as 2(88).095 16.72 54.57 13.19 4 22 3(22).09_5 6.27 60.84 15.21 5 36 4(36).095 13.68 74.52 14.90 - for August, order 215 units (a 5 month supply). 6.3 a. W = 0.1 Slunitfpen'od T DU) A (T-l) D(T) w- Row Cumulative Cumulative Sum Sum Q Sum 1 200 $50.00 $0 $50.00 $50.00 200 $0.25funit 2 300 l(300)0.l 30.00 80.00 500 0.164: 3 500 2(500)0.1 100.00 180.00 1,000 0.18 The first replenishment is Q=500 units, a 2 period supply. b. The Silver-Meal result, which is also to use T = 2 in the example above, is independent of the size of D(1). However, making D(1) large enough, namer greater than 500 here, would eventually switch the “least unit cost” results to T = 1. 6.4 a. The fixed EOQ as an integer number of months supply: 5 = 85 unitsfmo. r = 0.24 $f$lyr. = 0.02 $I$fmo. 2.45 W' EOQ = = 292 units Clearly the first order will cover either three months (220) units, or four months (340 units). \ Since the latter is closer to the EOQ, (2(1) = 340. Similarly, 9(5) = 310 units, {2(8) = 270 units and Q(11) is constrained to be 100 units. In Summary, Starting Ending Month Q Invento Demand Invento 1 340 340 50 ' 290 2 290 70 220 3 220 100 120 4 120 120 0 5 310 310 1 10 200 120 6 200 100 100 7 100 100 0 8 270 270 80 190 9 190 120 70 10 70 70 0 11 100 100 60 40 12 40 40 0 gm Note that the total amount of inventory carried between periods is simply the sum of all the ending inventories. The carrying charge per unit per month is vr = $0.04lunitfmonth. Thus, total costs = 4A + 1230($0.04) = $80.00 +. 49.20 = $129.20 b. EOQ as a fixed time supply: 292 units = —-— = 3.43 . T500 85 unitsfmo. m The best time supply will be either 3 or 4 months. Using the EOQ model, TRC(T) = é+ BTW = 39+1.70T T 2 T TRC(3) = $11.77fmo. while TRC(4) = $11.80/mo, so the TEOQ= 3 months. This means orders for months 1,4,7 and 10 of 220 units, 330 units, 300 units and 170 units respectively. In summary, Starting Ending Month Q lnventog Demand Inventog 1 220 220 50 170 2 170 70 100 3 100 100 O 4 330 330 120 210 5 210 I 10 100 6 100 100 0 7 300 300 100 200 8 200 80 120 9 120 120 0 121 10 170 170 70 100 11 100 60 40 12 4o 40 0 - LE1! Total costs = 4A + 1,040($0.04) = $80 + 41.60 = $121.60 I c. The least unit cost method: TC(T) = A + z vr(j —1)D(j) 1:1 T 13(7) A (T-1)D( v T—1)D(7) Tom Q TC(T)!Q 1 50 $20.00 0 $0.00 $20.00 50 $0.401unit 2 1'0 70 2.80 22.80 120 0.19 3 100 200 8.00 30.80 220 0.14 4 120 "360 14.00 45.20 340 0.1331: 5 110 440 17.60 62.80 450 0.139 - for month 1, order 340 (a 4 month supply). I 110 $20.00 0 $0.00 $20.00 110 $0.182funit 2 100 100 4.00 24.00 210 0.114 3 100 200 8.00 32.00 310 0.103: 4 80 240 9.60 41.60 390 0.107 - for month 5, order 310 (a 3 month supply). 1 80 $20.00 0 $0.00 $20.00 80 $0.2SOIunit 2 120 120 4.80 24.80 200 0.124 3 70 140 5.60 30.40 270 0.113: 4 60 180 7.20 37.60 330 0.114 — for month 8, order 270 (a 3 month supply) T D T) A (T-1)D w(1"-1)D(:r) TC(T) ram! 1 60 $20.00 0 - $0.00 $20.00 60 $0.3331'unit 2 40 40 $1.60 21.60 100 0.216 In summary, Month Q Starting Demand Ending Inventor: Inventory 1 340 340 50 290 2 ' 290 70 220 3 220 100 120 122 \DOOQONLh-D- 10 11 12 Total cost 310 270 100 120 120 310 110 200 100 100 - 100 270 80 I90 120 70 70 100 60 40 40 0 200 100 0 1 90 70 ' 0 40 0 fl = 4A + 1230($0.04) = $80.00 + 49.20 = $129.20 (1. The Silver-Meal Heuristic: T DU) A (T-1)D('I) w Row Cumulative Cumulative Sum Sum Sum +1“ 1 50 $20.00 0 $20.00 $20.00 $20.00 2 70 100104 2.80 22.80 11.40 3 100 2(100).04 8.00 30.80 10.27:: 4 120 3(120).04 14.40 45.20 11.30 - for January, order 220 units (a 3 month supply). T D(T) A (T—l) D(T)Vr Row Cumulative Cumulative Sum Sum Sum +1" 1 120 $20.00 0 $20.00 $20.00 $20.00 2 110 1(110).04 4.40 24.40 12.20 - 3 100 2(100).04 8.00 32.40 10.80: 4 100 3(100).04 12.00 44.40 11.10 - for April, order 330 units (a 3 month supply) 1 100 $20.00 0 $20.00 $20.00 $20.00 2 80 1(80).04 3.20 23.30 11.60 3 120 2(120).04 9.60 32.80 10.93 4 70 3(70).04 8.40 41.20 10.30 5 60 4(60).04 9.60 50.80 10.16 6 40 5(40).04 8.00 58.80 9.80 - for July, order 470 units (a 6 month supply). In summary, Month Q Starting Demand Ending Inventogz Inventoxl 1 220 220 50 170 2 170 70 100 123 3 100 100 0 4 330 330 120 210 5 210 110 100 6 100 100 0 7 470 470 100 " 370 8 370 80 290 9 290 120‘ 170 10 170 70 100 11 100 60 40 12 40 40 0 1229 Total cost = 3A + 1550($0.04) = $60.00 + 62.00 = $122.00 Note that in this example, where. the demand pattern is not hi ghly variable, the fixed time supply does $0.40 better than the Silver-Meal. However, the fixed time supply can do very poorly with a pattern of high variability. e. One replenishment to cover all requirements: Month Q Starting Demand Ending Inventory Inventor! 1 1020 1020 50 970 2 970 70 900 3 900 100 800 4 800 120 680 5 680 1 10 . 570 6 570 100 420 7 470 100 320 8 370 80 290 9 290 120 170 10 1'70 70 100 1 1 100 60 40 12 40 40 0 5.16.0 Total cost = 1A + 5360(5004) = $20.00 + $114.40 = $234.40 f. A replenishment each month. Month Q Starting Demand Ending - Inventor! Inventor"! 1 50 50 50 0 2 70 70 70 0 3 100 100 100 0 124 6.5 4 120 120 120 0 5 110 - 110 1 10 0 6 100 100 100 0 7 100 100 100 0 8 80 80 30 0 9 120 . 120 120 0 10 70 70 70 O 1 1 60 60 60 0 12 4O 40 40 O Q Total Cost = 12A + 0($0.04) = $240 a. The fixed EOQ as an integer number of months supply: 5 = 9s unitslwk. 2A5 W‘ EOQ = = 169 units The first order will cover either 2 weeks (130 units) or 3 weeks (310 units). Since the former is closer to the EOQ, Q(1)=130. Similarly, Q(3)=180. Q(4)=260, Q(8)=160, Q(10)=100, Q(11)=180, and Q(12)=130. In summary, Week Q Starting Demand Ending Invento ' Invento 1 130 130 50 80 2 80 80 O 3 180 180 180 0 4 80 80 80 0 5 O 0 0 6 O 0 0 '7 180 180 180 0 8 160 160 150 10 9 10 10 0 10 100 100 100 0 11 180 180 180 0 12 130 130 130 0 m 125 Carrying Costs = 90(5020) = $18.00 Total Costs = 8 A + $18 = $258.00 b. EOQ as a fixed time supply: Taco = M— = 1.8 weeks 95 unitsl'wk. The best time supply will be either 1 or 2 weeks. Using the EOQ model, TRC(T) =§+ DEW =%3+9.5T Thus TRCU) = $39.50 while TRC(2) = $34.00 Therefore It"Em = 2 weeks In summary, Week Q Starting Demand Ending Inventogg Inventogg 1 130 130 50 80 2 80 80 0 3 260 260 180 80 4 80 80 0 5 0 O 0 6 O 0 0 7 330 330 180 150 8 150 150 O 9 110 110 10 100 10 100 100 0 11 310 310 180 130 12 130 130 0 ' £0 Total Cost = 5A + 540(S0.20) = $258.00 c. The least unit cost method: TC(T) = A+ivr(j—1)D(j) ;=: - 126 T 13(2) A (T—l 13(7) vriT-i)D(T) TC TC 8 ‘z 1 50 $30.00 0 $0.00 $30.00 50 $0.60 2 80 80 16.00 46.00 130 0.35.: r 3 180 360 72.00 118.00 310 0.38 k - for week 1, order 130 (a 2 week supply). K T D A (T-l D v T—l D TC Q TcayQ 1 180 $30.00 0 $0.00 $30.00 180 $0.17: 2 80 80 16.00 46.00 260 0.18 - for week 3, order 180 (a 1 week supply). 1 80 $30.00 0 $0.00 $30.00 80 $0.38<= 4 180 _ 540 - 108.00 138.00 260 0.53 - for week 4, order 80 (a 1 week supply). 1 180 $30.00 0 $0.00 $30.00 180 $0.17: 2 150 150 30.00 60.00 330 0.18 - for week 7, order 180 (a 1 week supply) 1 150 $30.00 0 $0.00 $30.00 150 $0.20 2 10 10 2.00 32.00 160 0.20: 3 100 200 40.00 72.00 260 0.28 - for week 8, order 160 (a 2 week supply) 1 100 $30.00 0 $0.00 $30.00 100 $0.30 2 180 180 36.00 66.00 280 0.244: 3 130 260 52.00 118.00 410 0.29 - for week 10, order 280 (a 2 week supply). - for week 12, order 130 (a 1 week supply). In sununary, Week Q Starting Demand Ending Inventogy Inventory 1 1360 130 50 80 2 80 80 0 3 180 180 180 0 4 80 80 80 0 127 5 0 6 0 7 180 180 8 160 160 9 10 10 280 280 11 180 12 130 130 Total Cost = 7A + 270($0.20) = $264.00 (:1. The Silver-Meal Heuristic: T .00) A (m) 0(1) W 50 $30.00 0 80 1(80).2 180 2(130).2 - for week 1, order 130 units. 1 180 $30.00 0 4 80 1(80).2 5 180 4(180).2 - for week 3, order 260 units. 1 180 $30.00 0 2 150 1(150).2 3 10 2(10).2 4 100 3(100).2 - for week 7, order 340 units. 100 $30.00 0 180 1(180).2 - for week 10, order 180 units. 180 $30.00 0 130 1(130).2 — for week 11, order 310 units. In summary, 128 0 0 180 150 10 100 180 130 Row Sum $30.00 16.00 72.00 $30.00 16.00 144.00 $30.00 30.00 4.00 60.00 $30.00 36.00 $30.00 26.00 0 0 0 10 0 180 0 0 £0. Cumulative Cumulative Sum Sum +T $30.00 $30.00 46.00 23.00: I 18.00 39.33 $30.00 $30.00 46.00 11.50<= 190.00 38.00 $30.00 $30.00 60.00 30.00 64.00 21.33: 124.00 31.00 $30.00 $30.00: 66.00 33.00 $30.00 $30.00 56.00 28.00: Week IQ Starting Demand Ending Inventor! Inventory 1 130 130 50 80 2 80 80 0 3 260 260 180 80 4 80 80 - 0 5 0 0 0 6 0 0 0 7 340 340 180 160 3 160 150 10 9 . 10 10 0 10 100 100 100 O 11 310 310 180 130 12 . 130 130 0 Q Total Cost = 5A + 460($0.20) = $150 + $92 = $242. e. One replenishment to cover all requirements: Starting Ending Week Q Inventog Demand Inventor-X 1 l 140 1 140 50 1090 2 1090 80 1010 3 1010 180 830 4 830 80 750 5 750 0 750 6 750 0 750 7 750 180 570 8 570 150 420 9 420 10 410 10 410 100 310 11 310 180 130 12 130 0 - 0 10211 Total Cost = A + 7.020(5020) = $1434.00 f. A replenishment each week: Week Q Starting Demand Ending Inventory Inventory 1 50 50 50 0 2 80 80 80 0 3 1 80 l 80 1 80 0 129 10 11 12 Total Cost = 10A + 0($0.20) = $300.00 80 4 5 6 7 . 8 9 0 0 180 150 10 100 180 130 30 0 0 180 150 10 100 180 130 80 0 180 150 10 100 180 130 IOOOOOOOOOO 6.10 a. The fixed EOQ: 5 = 67 unitsfperiod 2.45 V?" EOQ = = 129 units The first production quantity will be either 60 or 170 units, to cover the first two or three months. Because 170 is closer to 129 than 60 is, use QC 1) = 170. For other production quantities, the demands in whole periods are totaled until a quantity as close as possible to the EOQ is obtained. In sunnnary, Starting Ending Period Q Invento Demand Invento 1 170 O 20 150 2 O 150 _ 40 110 3 0 110 110 0 4 120 0 120 0 S 140 0 60 80 6 O 80 30 50 7 0 50 20 . 30 8 0 30 30 0 9 80 0 80 0 10 120 O 120 0 11 130 0 130 0 12 40 O 40 0 fl Total Cost = ?A + (420)($0.2) = $259 b. The Wagner-Whitin table follows: Replenishments are made in months 1, 3, 4, 8, 10, and 11 for the intervening months, that is, with quantities 60, 110, 230, 110, 120 and 170. The total inventory is 340 units._ The total cost is 6(25) + 340(0.2) = $218. 0. The Silver-Meal heuristic: Starting Ending Period g2 Inventog Demand Inventogx 1 60 O 20 40 2 0 4O 40 0 3 340 0 110 230 4 0 230 120 110 5 0 1 10 60 50 6 0 50 30 20 7 0 20 20 0 8 1 10 0 30 80 9 0 80 80 0 10 120 0 120 0 l 1 170 0 130 40 12 O 40 40 0 Q 136 Total cost is 5(25) + 570(02) = $239. d. Least unit cost: - Starting Ending Period ' Q Inventogg Demand Inventor}: 1 170 0 20 150 2 0 150 40 1 10 3 0 1 10 1 10 0 4 180 0 120 _ 60 5 0 60 60 0 6 80 0 30 50 7 0 50 2O 30 8 0 30 30 0 9 200 I 0 80 120 10 0 120 120 0 1 1 1 30 O 1 30 0 12 40 0 40 O Q Total cost is 6(25) + 520(0.2) = $254. e. Part Period Balancing: Starting Ending Period Q Inventory Demand Inventor! I 60 0 20 40 2 0 40 40 0 3 230 O 110 120 4 O 120 120 0 5 140 0 6O 80 6 0 80 30 50 7 0 50 20 30 8 0 30 30 0 9 200 0 80 120 10 0 120 120 0 1 1 170 0 130 _ 4O 12 0 40 40 0 Q Total cost is 5(25) + 480(0.2) = $221. f. Periodic Order Quantity: 137 The EOQ is 129 units, and the average demand is 67. So the EOQ would last, on average around 2 periods. Therefore, use a period of 2, or order every other period. Starting Ending Period Q Inventory Demand Inventor}: 1 60 0 20 4O 2 0 40 40 0 3 230 0 110 120 4 O 120 120 O 5 90 0 60 30 6 0 30 30 0 7 50 0 20 3O 8 0 30 30 0 9 200 0 80 120 10 0 ' 120 120 O 11 170 O 130 40 1 2 0 4O 4O 0 3 80 M The total cost is 6(25) + 380(0.2) = $226. 6.17 a. The fixed EOQ: E = 262.5 unitsfperiod 2A5 vr EOQ = = 20 units Because the EOQ is so small, we order every period and the EOQ strategy is identical to a Iot-for—lot strategy. Ten replenishments are made (with no production during periods of zero demand) and carrying costs are zero. The total cost is 100550) = $500. b. The Wagner-Whitin table follows: Demand 350 200 0 _ 150 500 600 1 2 3 4 5 6 7 3 9 10 1 1 12 50 260 260 733 2.333 5.933 3.313 11.390 13.070 13.070 14.545 16,955 100 100 415 1.990 4.510 6.873 9.078 10.543 10.548 11.965 14,065 150 303 1.353 3.243 5.138 5,975 3.235 8.235 9,495 11.335 150 675 1.935 3.353 4.823 5.373 5.373 6.975 3.655 200 330 1.775 2.373 3,713 3.713 4.663 6.133 250 723 1.453 2,083 2.033 2,375 4.135 300 663 1 .033 1.038 1,713 2.763 350 560 550 1.033 1.373 400 400 715 1 .345 ' 450 603 1 .023 450 660 500 The result is identical to the EOQ. c. The Silver-Meal heuristic must be modified using the adjustment in footnote 9 of Chapter 6. When this is done, the result is the same as in parts a and 1). Note that neglecting this adjustment yields extremely high costs because batches are produced in periods of zero demand and held for one period. d. Least unit cost again yields the same result. e. Part Period Balancing also is the same. f. Periodic Order Quantity: The EOQ is 20 units, and the average demand is 262.5. So the EOQ would last even less than one period. Assuming that we can only produce once per period, the periodic order quantity would be the same as the EOQ and the other heuristics. 149 ...
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This note was uploaded on 11/03/2008 for the course IE 251 taught by Professor Wilson during the Spring '08 term at Lehigh University .

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Ch 6 Select HW Solutions - 6.1 6.2 Chapter 6 Answers to...

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