ISyE 3104: Introduction to Supply Chain Modeling: Manufacturing and Warehousing Instructor : Spyros Reveliotis Summer 2006

Solutions for Homework #6 ISYE 3104 Summer 2006 Homework 6 Solution Section 3.3

7. A rolling production schedule implies that the schedule may be revised at the start of a new planning period. However, this does not mean that there is no value in knowing future demands. For instance, for a company adopting a constant workforce / production capacity policy, if high demands are anticipated in some future periods, production may have to be ramped up well in advance to build the anticipatory inventories that will enable the satisfaction of those demands. In fact, even in the case that capacity adjustments like employee hiring or subcontracting are allowed, knowing the upcoming surges in advance will allow the company to take early on the necessary actions that will enable the hiring of the extra employees or the provision of the subcontracted quantities (e.g., by advertising the available positions, in the case of hiring, or by finding the subcontractors and signing the necessary contracts, in the case of subcontracting).

Section 3.4

14. k = $60,000/250 = $240 per worker per day. After making the necessary adjustments to the demand vector in order to account for the initial and the desired ending inventory, we can computed the required workforce and the cost of the resulting plan through the following table: (A) Cum # *Cum # Units/ Units/ (B) Cum Net *Monthly Monthly Worker Worker Net Demand Demand [B/A] Product Product Working (in (in Forecast Forecast Min # (in (in *End Invent. Month Days $10,000) $10,000) (in $10,000) (in $10,000) Workers $10,000) $10,000) (in $10,000) 1 22 0.528 0.528 328 328 622 410.256 410.256 82.256 2 16 0.384 0.912 380 708 777 298.368 708.624 0.624 3 21 0.504 1.416 220 928 656 391.608 1100.232 172.232 4 19 0.456 1.872 100 1028 550 354.312 1454.544 426.544 5 23 0.552 2.424 490 1518 627 428.904 1883.448 365.448 6 20 0.480 2.904 625 2143 738 372.96 2256.408 113.408 7 24 0.576 3.480 375 2518 724 447.552 2703.96 185.96 8 12 0.288 3.768 310 2828 751 223.776 2927.736 99.736 9 19 0.456 4.224 175 3003 711 354.312 3282.048 279.048 10 22 0.528 4.752 145 3148 663 410.256 3692.304 544.304 11 20 0.480 5.232 120 3268 625 372.96 4065.264 797.264 12 16 0.384 5.616 175 3443 614 298.368 4363.632 920.632 Max = 777 Total 3987.456

* Note: These figures assume the minimum constant workforce of 777 workers each month.

a) Minimum constant work force = 777 workers. b) CH = $200, CF = $400, Initial # workers = 675, Workers added = 102 Beg. Inv = $120,000. End inv = $100,000. Total ending inventory = $39,874,560 + 100,000 = $39,974,560. Inventory costs per month = 0.25/12 = 0.0208 per $ per month.

Hence total hiring + inventory costs for the constant work force plan are (200)(102) + (0.0208)(39,874,560+100,000) = $853,190

2 ISYE 3104 Summer 2006 Homework 6 Solution

15. Net Cum # Units/ Demand Monthly Monthly Cum Net *End Worker Forecast Product Product Demand Invent. Working (in (in Min # Workers Workers (in (in Forecast (in Month Days $10,000) $10,000) Workers Hired Fired $10,000) $10,000) (in $10,000) $10,000) 1 22 0.528 328 622 0 53 328.416 328.416 328 0.416 2 16 0.384 380 990 368 0 380.160 708.576 708 0.576 3 21 0.504 220 437 0 553 220.248 928.824 928 0.824 4 19 0.456 100 220 0 217 100.320 1029.144 1028 1.144 5 23 0.552 490 888 668 0 490.176 1519.320 1518 1.320 6 20 0.480 625 1303 415 0 625.440 2144.760 2143 1.760 7 24 0.576 375 652 0 651 375.552 2520.312 2518 2.312 8 12 0.288 310 1077 425 0 310.176 2830.488 2828 2.488 9 19 0.456 175 384 0 693 175.104 3005.592 3003 2.592 10 22 0.528 145 275 0 109 145.200 3150.792 3148 2.792 11 20 0.480 120 250 0 25 120.000 3270.792 3268 2.792 12 16 0.384 175 456 206 0 175.104 3445.896 3443 2.896 Totals 2082 2301 Total 21.912

CH = $200, CF = $400 Total cost of hiring, firing and inventory = (200)(2082) + (400)(2301) + (0.0208) (219,120+100,000) = $1,343,448

Chapter 7

41. Gross requirement schedule for each of the components: Week 5 6 7 8 9 10 11 12 13 14 15 16 17 MPS 1200 1200 800 1000 1000 300 2200 1400 1800 600 Gross reqts. processors 1200 1200 800 1000 1000 300 2200 1400 1800 600 solar cells 4800 4800 3200 4000 4000 1200 8800 5600 7200 2400 display 1200 1200 800 1000 1000 300 2200 1400 1800 600 buttons 48000 48000 32000 40000 40000 12000 88000 56000 72000 24000 42. K = 12.00 h = (0.24)(0.02)/48 = 0.0001

Gross requirements are given by last line of solution to problem 41. For convenience, we multiply h by 10,000 and divide each demand by 10,000. Hence, h = 1 and r = (4.8, 4.8, 3.2, 4.0, 4.0, 1.2, 8.8, 5.6, 7.2, 2.4)

Start in period 1: C(1) = 12 C(2) = (12 + 4.8)/2 = 8.4 C(3) = [(2)(8.4) + (2)(3.2)]/3 = 7.733

3 ISYE 3104 Summer 2006 Homework 6 Solution C(4) = [(3)(7.733) + (3)(4)]/4 = 8.8 Stop (since the average cost per period increased with respect to its previous value). y1 = r1 + r2 + r3 = 12.8

Start in period 4: C(1) = 12 C(2) = (12 + 4)/2 = 8 C(3) = [(2)(8) + (2)(1.2)]/3 = 6.133 C(4) = [(3)(6.133) + (3)(8.8)]/4 = 11.2 Stop. y4 = r4 + r5 + r6 = 9.2

Start in period 7: C(1) = 12 C(2) = (12 + 5.6)/2 = 8.8 C(3) = [(2)(8.8) + (2)(7.2)]/3 = 10.67 Stop. y7 = r7 + r8 = 14.4

Finally, we will obtain y9 = r9 + r10 = 9.6

Hence, order policy recommended by Silver-Meal is (128,000; 0; 0 ; 92,000; 0; 0; 144,000; 0; 96,000; 0)

The cost of this solution is: (4)(12) + (8 + 3.2 + 5.2 + 1.2 + 5.6 + 2.4) = $73.60

48. According to the product structure, we find the item level for each component: Level 0: EP1, EP2 Level 1: A, B, C, D, E Level 2: F Level 3: G, H

The following table shows the material requirement for components that use F, G and H. Since in this case, the initial inventories and the scheduled receipts are equal to zero, and the applied lot-sizing policy is lot-for-lot, for all items, we skip all the intermediate steps and provide only the grossed requirements for each item and the planned order releases.

4 ISYE 3104 Summer 2006 Homework 6 Solution

Week Lead Low- Lot Time level Item 12 13 14 15 16 17 18 19 20 21 22 23 24 size (wks) Code ID - 0 EP1 MPS 120 112 76 22 56 90 210

- 0 EP2 MPS 62 68 90 77 26 30 54

Lot- Gross for EP1 120 112 76 22 56 90 210 for- 1 1 A Req. lot POR 120 112 76 22 56 90 210

Lot- Gross for EP1 240 224 152 44 112 180 420 for- 2 1 B Req. lot POR 240 224 152 44 112 180 420

Lot- Gross for EP2 62 68 90 77 26 30 54 for- 1 1 D Req. lot POR 62 68 90 77 26 30 54

for A 360 336 228 66 168 270 630 Lot- Gross for- 2 2 F Req. for D 124 136 180 154 52 60 108 lot Total 484 472 408 220 220 330 738 POR 484 472 408 220 220 330 738

Gross for B 480 448 304 88 224 360 840 Lot- Req. for F 1452 1416 1224 660 660 990 2214 1 for B for- 3 G for B 480 448 304 88 224 360 840 3 for F lot POR for F 1452 1416 1224 660 660 990 2214 Total 1452 1416 1224 1140 1108 1294 2302 224 360 840

for B 240 224 152 44 112 180 420 Lot- Gross for- 2 3 H Req. for F 968 944 816 440 440 660 1476 lot Total 968 1184 1040 592 484 772 1656 420 POR 968 1184 1040 592 484 772 1656 420

5 ISYE 3104 Summer 2006 Homework 6 Solution Extra Credit

Section 3.4

16. The following graph plots the cumulative net demand for each month.

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Cum Net Demand Forecast (in $10,000)

We need to determine a profile for the cumulative production that  is piece-wise linear with at most four distinct segments (since the production level can change no more than three times over the planning horizon);  it remains above the cumulative net demand profile at every month (since we want to meet the net forecasted demand every month); and  it also stays as close as possible to the cumulative net demand profile (since we want to minimize the excess inventories).

One such profile is plotted below:

6 ISYE 3104 Summer 2006 Homework 6 Solution

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Cum Net Demand Forecast Cum Production Under New Plan The changes in (inproduction $10,000) rate occur between periods 2 and 3, 4 and 5, and 7 and 8. These breakpoints are chosen according to the change of the slope of the first plot.

The four production rates are chosen by trial and error to meet the demands for each month without accumulating much excess inventory to subsequent periods.

The following table shows the proposed solution:

Net # Units/ Demand Actual Cum Cum Net Worker Forecast Prod. Product. Demand Working (in Desired (in Min # Workers Workers (in (in Forecast *End Invent. Month Days $10,000) Prod. $10,000) Workers Hired Fired $10,000) $10,000) (in $10,000) (in $10,000) 1 22 0.528 354.00 328 671 0 4 354.288 354.288 328 26.288 2 16 0.384 354.00 380 922 251 0 354.048 708.336 708 0.336 3 21 0.504 220.00 220 437 0 485 220.248 928.584 928 0.584 4 19 0.456 220.00 100 483 46 0 220.248 1148.832 1028 120.832 5 23 0.552 502.00 490 910 427 0 502.320 1651.152 1518 133.152 6 20 0.480 502.00 625 1046 136 0 502.080 2153.232 2143 10.232 7 24 0.576 502.00 375 872 0 174 502.272 2655.504 2518 137.504 8 12 0.288 175.00 310 608 0 264 175.104 2830.608 2828 2.608 9 19 0.456 175.00 175 384 0 224 175.104 3005.712 3003 2.712 10 22 0.528 175.00 145 332 0 52 175.296 3181.008 3148 33.008 11 20 0.480 175.00 120 365 33 0 175.200 3356.208 3268 88.208 12 16 0.384 175.00 175 456 91 0 175.104 3531.312 3443 88.312 Totals 984 1203 Total 643.776

Hence, the hiring, firing, and inventory cost of this plan is:

7 ISYE 3104 Summer 2006 Homework 6 Solution (984)(200)+(1203)(400)+(0.208)(6,437,760+100,000) = $814,203.33

Remark: Notice that the above plan still involves substantial hiring and firing, due to the variation of the working days in every month. If we want to support the proposed production plan with constant workforce at every phase of the plan where production rate is constant, and without experiencing any stockouts at any month, we must fix the number of workers at every phase of the plan to the minimum level that will enable the achievement of the monthly production rate in every month of that phase. This number will essentially be determined by the month in the phase with the smaller number of days. This arrangement implies that during the months with a higher number of working days, the employed workers will be under-utilized.

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