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:
Month
1
2
3
4
5
6
7
8
9
10
11
12
Working
Days
22
16
21
19
23
20
24
12
19
22
20
16
# Units/
Worker
(in
$10,000)
0.528
0.384
0.504
0.456
0.552
0.480
0.576
0.288
0.456
0.528
0.480
0.384
(A) Cum #
Units/
(B) Cum Net
*Monthly
Worker Net Demand
Demand
[B/A]
Product
(in
Forecast
Forecast
Min #
(in
$10,000) (in $10,000) (in $10,000) Workers
$10,000)
0.528
328
328
622
410.256
0.912
380
708
777
298.368
1.416
220
928
656
391.608
1.872
100
1028
550
354.312
2.424
490
1518
627
428.904
2.904
625
2143
738
372.96
3.480
375
2518
724
447.552
3.768
310
2828
751
223.776
4.224
175
3003
711
354.312
4.752
145
3148
663
410.256
5.232
120
3268
625
372.96
5.616
175
3443
614
298.368
Max = 777
*Cum
Monthly
Product
(in
*End Invent.
$10,000) (in $10,000)
410.256
82.256
708.624
0.624
1100.232
172.232
1454.544
426.544
1883.448
365.448
2256.408
113.408
2703.96
185.96
2927.736
99.736
3282.048
279.048
3692.304
544.304
4065.264
797.264
4363.632
920.632
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
2
ISYE 3104 Summer 2006
Homework 6 Solution
(200)(102) + (0.0208)(39,874,560+100,000) = $853,190
15.
Month
1
2
3
4
5
6
7
8
9
10
11
12
Working
Days
22
16
21
19
23
20
24
12
19
22
20
16
Net
Cum
# Units/ Demand
Monthly Monthly
Worker Forecast
Product Product
(in
(in
Min # Workers Workers
(in
(in
$10,000) $10,000) Workers Hired
Fired $10,000) $10,000)
0.528
328
622
0
53 328.416 328.416
0.384
380
990
368
0 380.160 708.576
0.504
220
437
0
553 220.248 928.824
0.456
100
220
0
217 100.320 1029.144
0.552
490
888
668
0 490.176 1519.320
0.480
625
1303
415
0 625.440 2144.760
0.576
375
652
0
651 375.552 2520.312
0.288
310
1077
425
0 310.176 2830.488
0.456
175
384
0
693 175.104 3005.592
0.528
145
275
0
109 145.200 3150.792
0.480
120
250
0
25 120.000 3270.792
0.384
175
456
206
0 175.104 3445.896
Totals
2082
2301
Cum Net
*End
Demand
Invent.
Forecast
(in
(in $10,000) $10,000)
328
0.416
708
0.576
928
0.824
1028
1.144
1518
1.320
2143
1.760
2518
2.312
2828
2.488
3003
2.592
3148
2.792
3268
2.792
3443
2.896
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
MPS
1200 1200 800 1000 1000
Gross reqts.
processors 1200 1200 800 1000 1000 300 2200 1400
solar cells
4800 4800 3200 4000 4000 1200
display
1200 1200 800 1000 1000 300
buttons
48000 48000 32000 40000 40000 12000 88000
13
14
15
16 17
300 2200 1400 1800 600
1800 600
8800 5600 7200 2400
2200 1400 1800 600
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
3
ISYE 3104 Summer 2006
Homework 6 Solution
C(3) = [(2)(8.4) + (2)(3.2)]/3 = 7.733
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 LowLot Time level Item
size (wks) Code ID
0
EP1 MPS
-
0
Lotforlot
1
1
Lotforlot
2
Lotforlot
1
Lotforlot
2
1
1
2
12
Lotforlot
2
3
3
14
15
16
17
EP2 MPS
62
Gross
for EP1
A Req.
POR
POR
22 23 24
22 56 90 210
77 26 30 54
120 112 76
22 56 90 210
240 224 152
62
21
76 22
56 90 210
44 112 180 420
44 112 180 420
62
68 90
77 26 30 54
68
90 77
26 30 54
for A
360 336 228 66 168 270 630
for D
Total
124 136 180 154 52 60 108
484 472 408 220 220 330 738
484 472 408 220 220 330 738
Gross for B
Req. for F
G
for B
POR for F
Total
Gross
H Req.
20
240 224 152
Gross
for EP2
D Req.
POR
F
19
68 90
120 112
Gross
for EP1
B Req.
POR
Gross
Req.
18
120 112 76
POR
Lot1 for B
for3 for F
lot
13
for B
for F
Total
1452
480
1452 1416 1224 660
1452 1416 1224 1140
480
1416
448
660
1108
448
1224
304
990
1294
304 88 224 360 840
660 660 990 2214
88 224 360 840
2214
2302 224 360 840
240 224 152
44 112 180 420
968 944 816 440 440 660 1476
968 1184 1040 592 484 772 1656 420
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.
4000
3500
3000
2500
2000
1500
1000
500
11
10
9
8
7
6
5
4
3
2
1
M
on
th
0
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
4000
3500
3000
2500
2000
1500
1000
500
Cum Net Demand Forecast
11
10
9
8
7
6
5
4
3
2
1
M
on
th
0
Cum Production Under New Plan
The changes in (in
production
rate occur between periods 2 and 3, 4 and 5, and 7 and 8. These
$10,000)
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:
Month
1
2
3
4
5
6
7
8
9
10
11
12
Working
Days
22
16
21
19
23
20
24
12
19
22
20
16
Net
# Units/
Demand
Actual
Cum
Cum Net
Worker
Forecast
Prod.
Product.
Demand
(in
Desired
(in
Min # Workers Workers
(in
(in
Forecast
$10,000) Prod. $10,000) Workers Hired
Fired $10,000) $10,000) (in $10,000)
0.528 354.00
328
671
0
4 354.288 354.288
328
0.384 354.00
380
922
251
0 354.048 708.336
708
0.504 220.00
220
437
0
485 220.248 928.584
928
0.456 220.00
100
483
46
0 220.248 1148.832
1028
0.552 502.00
490
910
427
0 502.320 1651.152
1518
0.480 502.00
625
1046
136
0 502.080 2153.232
2143
0.576 502.00
375
872
0
174 502.272 2655.504
2518
0.288 175.00
310
608
0
264 175.104 2830.608
2828
0.456 175.00
175
384
0
224 175.104 3005.712
3003
0.528 175.00
145
332
0
52 175.296 3181.008
3148
0.480 175.00
120
365
33
0 175.200 3356.208
3268
0.384 175.00
175
456
91
0 175.104 3531.312
3443
Totals
984
1203
Total
*End Invent.
(in $10,000)
26.288
0.336
0.584
120.832
133.152
10.232
137.504
2.608
2.712
33.008
88.208
88.312
643.776
7
ISYE 3104 Summer 2006
Homework 6 Solution
Hence, the hiring, firing, and inventory cost of this plan is:
(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.
8