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Parameters

dit 1 2 3 4 5 6

1 22000 30000 42000 26000 31000 210002 12000 13000 15000 12000 14000 150003 20000 30000 38000 24000 29000 190004 13000 18000 24000 15000 18000 120005 13000 15000 17000 14000 16000 18000

Ui Lri L

oi L

si hi total component cost

1 1.00 $8.00 $12.50 $14.00 $0.56 158620002 2.19 $17.50 $27.00 $28.00 $1.19

3 0.94 $7.50 $11.00 $13.00 $0.554 1.00 $8.00 $12.50 $14.50 $0.585 2.38 $19.00 $29.00 $30.50 $1.18

pit 1 2 3 4 5 6

$0.50 $0.50 $5.00 $0.50 $0.50 $0.50

Aggregation 1 (aggregate products 1, 3, and 4 into regular watches and aggregate products 2 and 5 into regular watches)

dit 1 2 3 4 5 6 Ratio in re Ratio in slim watch

1 55000 78000 104000 65000 78000 52000 39.8%2 25000 28000 32000 26000 30000 33000 46.6%

37.0%23.1%

53.4%

Ui Lri L

oi L

si hi

1 0.98 $7.81 $11.94 $13.75 $0.562 2.29 $18.30 $28.07 $29.34 $1.18

pit 1 2 3 4 5 6

$0.50 $0.50 $5.00 $0.50 $0.50 $0.50

Aggregation 2 (aggregate all products into a single product, say watch)

dit 1 2 3 4 5 6 Ratio in watch

1 80000 106000 136000 91000 108000 85000 28.4%13.4%

26.4%16.5%15.3%

Ui Lri L

oi L

si hi

1 $1.35 $10.83 $16.57 $18.22 $0.74

pit 1 2 3 4 5 6

$0.50 $0.50 $5.00 $0.50 $0.50 $0.50

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The LP formulation without aggregation

Parameters

dit 1 2 3 4 5 6

1 22000 30000 42000 26000 31000 210002 12000 13000 15000 12000 14000 150003 20000 30000 38000 24000 29000 190004 13000 18000 24000 15000 18000 120005 13000 15000 17000 14000 16000 18000

Ui Lri L

oi L

si hi

1 1.00 $8.00 $12.50 $14.00 $0.562 2.19 $17.50 $27.00 $28.00 $1.193 0.94 $7.50 $11.00 $13.00 $0.554 1.00 $8.00 $12.50 $14.50 $0.585 2.38 $19.00 $29.00 $30.50 $1.18

pit 1 2 3 4 5 6

$0.50 $0.50 $5.00 $0.50 $0.50 $0.50

Decision variables

Xrit 1 2 3 4 5 6 subtotal

1 22000 30000 42000 26000 31000 21000 1720002 12000 13000 7171 12000 14000 15000 731713 0 0 4000 6533 0 0 105334 13000 18000 24000 15000 18000 12000 1000005 20526 14132 10342 15421 12789 18605 91816

Xoit 1 2 3 4 5 6 subtotal

1 0 0 0 0 0 0 02 0 0 0 0 0 0 03 20000 32000 32000 17467 29000 19000 149467

4 0 0 0 0 0 0 05 0 0 0 0 1184 0 1184

Xsit 1 2 3 4 5 6 subtotal

1 0 0 0 0 0 0 02 0 0 7829 0 0 0 78293 0 0 0 0 0 0 04 0 0 0 0 0 0 05 0 0 0 0 0 0 0

I+

it 1 2 3 4 5 6 subtotal

1 0 0 0 0 0 0 02 0 0 0 0 0 0 03 0 2000 0 0 0 0 2000

4 0 0 0 0 0 0 05 7526 6658 0 1421 0 0 15605

I-it 1 2 3 4 5 6 subtotal

1 0 0 0 0 0 0 02 0 0 0 0 0 0 03 0 0 0 0 0 0 04 0 0 0 0 0 0 05 0 0 0 0 605 0 605

Constraints

regular time used limited by110000 110000 110000 110000 110000 110000 110000

overtime used limited by

18750 30000 30000 16375 30000 17813 30000

Inventory balance equals0 0 0 0 0 0 0

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0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

I+

i6, I-i6 = 0

Nonnegativity

Objective funstion

23059492

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The LP formulation of aggregation into regular and slim watches

Parameters (aggregated) Parameters (original)

dit 1 2 3 4 5 6 dit 1 2 3 4

1 55000 78000 104000 65000 78000 52000 1 22000 30000 42000 260002 25000 28000 32000 26000 30000 33000 2 12000 13000 15000 12000

3 20000 30000 38000 240004 13000 18000 24000 150005 13000 15000 17000 14000

Ui Lri L

oi L

si hi Ui L

ri L

oi

1 0.98 $7.81 $11.94 $13.75 $0.56 1 1.00 $8.00 $12.502 2.29 $18.30 $28.07 $29.34 $1.18 2 2.19 $17.50 $27.00

3 0.94 $7.50 $11.004 1.00 $8.00 $12.505 2.38 $19.00 $29.00

pit 1 2 3 4 5 6 pit 1 2 3 4

$0.50 $0.50 $5.00 $0.50 $0.50 $0.50 $0.50 $0.50 $5.00 $0.50

Decision variables Disaggregation

Xrit 1 2 3 4 5 6 subtotal X

rit 1 2 3 4

1 53802 47289 73289 51717 47289 30383 303769 1 21421 18828 29180 205912 25109 27891 16789 26000 27891 35109 158789 2 11689 12984 7815 12103

3 19927 17514 27144 191544 12454 10947 16965 119715 13421 14907 8973 13897

Xoit 1 2 3 4 5 6 subtotal X

oit 1 2 3 4

1 1198 30711 30711 13283 30711 21617 128231 1 477 12227 12227 52892 0 0 0 0 0 0 0 2 0 0 0 0

3 444 11374 11374 49204 277 7109 7109 30755 0 0 0 0

Xsit 1 2 3 4 5 6 subtotal X

sit 1 2 3 4

1 0 0 0 0 0 0 0 1 0 0 0 02 0 0 15211 0 0 0 15211 2 0 0 7081 0

3 0 0 0 04 0 0 0 05 0 0 8130 0

I+

it 1 2 3 4 5 6 subtotal I+

it 1 2 3 4

1 0 0 0 0 0 0 0 1 0 954 361 2412 109 0 0 0 0 0 109 2 0 0 0 0

3 370 0 0 0

4 0 0 0 05 421 328 431 328

I-it 1 2 3 4 5 6 subtotal I-it 1 2 3 4

1 0 0 0 0 0 0 0 1 102 0 0 02 0 0 0 0 2109 0 2109 2 311 328 431 328

3 0 741 222 1484 269 213 139 935 0 0 0 0

Constraints

regular time used limited by110000 110000 110000 110000 110000 110000 110000

overtime used limited by

1170 30000 30000 12976 30000 21117 30000

Inventory balance equals0 0 0 0 0 0 00 0 0 0 0 0

I+

i6, I-i6 = 0

Nonnegativity

Objective funstion Actual total cost

23121083 23129346

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5 6 Ratio in reg Ratio in slim

31000 21000 39.8%14000 15000 46.6%

29000 19000 37.0%18000 12000 23.1%

16000 18000 53.4%

Lsi hi

$14.00 $0.56

$28.00 $1.19$13.00 $0.55

$14.50 $0.58$30.50 $1.18

5 6

$0.50 $0.50

5 6 subtotal

18828 12097 120945

12984 16344 73919

17514 11253 11250710947 7033 70317

14907 18765 84870

5 6 subtotal

12227 8607 51055

0 0 011374 8006 47493

7109 5004 296830 0 0

5 6 subtotal

0 0 00 0 7081

0 0 00 0 0

0 0 8130

5 6 subtotal

296 0 1852

0 0 0

0 0 3700 0 0

0 0 1507

5 6 subtotal

0 0 102

1344 0 2741259 0 1370

37 0 750765 0 765

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The LP formulation of aggregation into regular and slim watches

Parameters (aggregated) Parameters (original)

dit 1 2 3 4 5 6 dit 1 2 3 4

1 80000 106000 136000 91000 108000 85000 1 22000 30000 42000 260002 12000 13000 15000 120003 20000 30000 38000 240004 13000 18000 24000 150005 13000 15000 17000 14000

Ui Lri L

oi L

si hi Ui L

ri L

oi

1 1.35 $10.83 $16.57 $18.22 $0.74 1 1.00 $8.00 $12.502 2.19 $17.50 $27.003 0.94 $7.50 $11.004 1.00 $8.00 $12.505 2.38 $19.00 $29.00

pit 1 2 3 4 5 6 pit 1 2 3 4

$0.50 $0.50 $5.00 $0.50 $0.50 $0.50 $0.50 $0.50 $5.00 $0.50

Decision variables Disaggregation

Xrit 1 2 3 4 5 6 subtotal X

rit 1 2 3 4

1 81286 81286 81286 81286 81286 81286 487719 1 23071 23071 23071 230712 10865 10865 10865 108653 21462 21462 21462 214624 13414 13414 13414 134145 12475 12475 12475 12475

Xoit 1 2 3 4 5 6 subtotal X

oit 1 2 3 4

1 22169 22169 22169 9714 22169 8258 106648 1 6292 6292 6292 27572 2963 2963 2963 12983 5853 5853 5853 25654 3658 3658 3658 16035 3402 3402 3402 1491

Xsit 1 2 3 4 5 6 subtotal X

sit 1 2 3 4

1 0 0 11633 0 0 0 11633 1 0 0 3302 02 0 0 1555 03 0 0 3072 04 0 0 1920 05 0 0 1785 0

I+it 1 2 3 4 5 6 subtotal I+

it 1 2 3 4

1 23456 20911 0 0 0 0 44367 1 7364 6727 0 02 1828 2656 3040 32033 7315 4630 0 0

4 4072 3144 0 05 2877 3754 4416 4381

I-it 1 2 3 4 5 6 subtotal I

-it 1 2 3 4

1 0 0 0 0 4544 0 4544 1 0 0 2607 27792 0 0 0 03 0 0 2983 29574 0 0 1865 18485 0 0 0 0

Constraints

regular time used limited by110000 110000 110000 110000 110000 110000 110000

overtime used limited by

30000 30000 30000 13145 30000 11175 30000

Inventory balance equals0 0 0 0 0 0 0

I+

i6, I-i6 = 0

Nonnegativity

Objective funstion Actual total cost

23156694 23226823

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5 6 Ratio in watch

31000 21000 28.4%14000 15000 13.4%29000 19000 26.4%

18000 12000 16.5%16000 18000 15.3%

Lsi hi

$14.00 $0.56$28.00 $1.19

$13.00 $0.55$14.50 $0.58

$30.50 $1.18

5 6

$0.50 $0.50

5 6 subtotal

23071 23071 13842810865 10865 65190

21462 21462 128771

13414 13414 8048212475 12475 74848

5 6 subtotal

6292 2344 302702963 1104 14255

5853 2180 28158

3658 1363 175993402 1267 16367

5 6 subtotal

0 0 3302

0 0 15550 0 3072

0 0 19200 0 1785

5 6 subtotal

0 0 140913031 0 13758

0 0 119450 0 7216

4258 0 19686

5 6 subtotal

4415 0 98010 0 0

4642 0 105832776 0 6489

0 0 0