The wafer probing scheduling problem (WPSP) is a practical version of the parallel-machine scheduling problem, which has many real-world applications including the integrated circuit (IC) manufacturing industry and other industries containing the manufacturing process with parallel machines. In this paper, we consider WPSP with the objective to minimize the maximum completion time and
formulate the WPSP with minimum makespan as an integer-programming problem.
To solve the WPSP with minimum makespan effectively, we proposed improving heuristics and the hybrid GA for our cases. The computational results show that improving heuristics and hybrid GA are efficient tools for solving our testing
problem by
improving heuristics can make scheduling solutions outperform scheduling ones of improving heuristics. From now on, the collection of initial population satisfying the WPSP with minimum makespan is our studying point because it not only expands the variety of genetic composition but affects the average and best solutions of GA.
s of WPSP with minimum makespan. And GA with initial population
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Appendix
able A1. Processing times of jobs with product types and product families under low total processing time level.
T
Table A2. Processing times of jobs with product types and product families under high total processing time level.
Product
Table A3. Tightness of due dates in 16 testing problems.
14 320 372 372 0 108000 54.10% 59.41% 61.81%
15 49 148 791 16066 34459 54126 36000 72000 108000 8.23% 15.45% 50.85%
16 53 160 851 3416 13404 66379 36000 72000 108000 9.64% 18.84% 62.25%
Problem No.
Expected Setup Time Total Processing Time Available Capacity Tightness of Due Dates Due dates of jobs Due dates of jobs Due dates of jobs Due dates of jobs
1440 2880 4320 1440 2880 4320 1440 2880 4320 1440 2880 4320
1 169 198 198 16066 34459 54126 36000 72000 108000 45.10% 48.13% 50.30%
2 192 225 225 19157 42402 66379 36000 72000 108000 53.75% 59.20% 61.67%
3 28 85 452 2913 10973 54126 36000 72000 108000 8.17% 15.36% 50.54%
4 32 96 514 3416 13404 66379 36000 72000 108000 9.58% 18.75% 61.94%
5 309 361 361 16066 34459 54126 36000 72000 108000 45.49% 48.36% 50.45%
6 332 388 388 19157 42402 66379 36000 72000 108000 54.14% 59.43% 61.82%
7 51 155 825 2913 10973 54126 36000 72000 108000 8.23% 15.46% 50.88%
8 55 166 887 3416 13404 66379 36000 72000 108000 9.64% 18.85% 62.28%
9 156 183 183 16066 34459 54126 36000 72000 108000 45.06% 48.11% 50.29%
10 180 209 209 19157 42402 66379 36000 72000 108000 53.71% 59.18% 61.66%
11 26 78 418 2913 10973 54126 36000 72000 108000 8.16% 15.35% 50.50%
12 30 89 479 3416 13404 66379 36000 72000 108000 9.57% 18.74% 61.91%
13 296 346 346 16066 34459 54126 36000 72000 108000 45.45% 48.34% 50.44%
19157 42402 66379 36000 7200
Table A4. All jobs with product types and due dates while tightness of due dates is stable and total processing time level is low.
Job IDProduct
Type Due Date Job ID Product
Type Due Date Job ID Product
Type Due Date Job ID Product
Type Due Date Job ID Product Type Due Date
Table A5. All jobs with product types and due dates while tightness of due dates is increasing and total processing time level is low.
Job IDProduct
Type Due Date Job IDProduct
Type Due Date Job IDProduct
Type Due Date Job IDProduct
Type Due Date Job IDProduct Type Due Date
Table A6. All jobs with product types and due dates while tightness of due dates is stable and total processing time level is high.
Job IDProduct
Type Due Date Job ID Product
Type Due Date Job IDProduct
Type Due Date Job IDProduct
Type Due Date Job IDProduct Type Due Date
Table A7. All jobs with product types and due dates while tightness of due dates is increasing and total processing time level is high.
20
16 16 2880 36 28 4320 56 8 1440 76 29 4320 96 14 20
17 17 2880 37 18 2880 57 23 2880 77 16 4320 97 10 1440
Job ID Product
Type Due Date Job IDProduct
Type Due Date Job ID Product
Type Due Date Job IDProduct
Type Due Date Job ID Product Type Due Date
Table A8. Setup time with product types while product family ratio is 2 and temperature changing is not considered.
5
Table A9. Setup time with product types while product family ratio is 2 and temperature changing is considered.
Type U 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Table A10. Setup time with product types while product family ratio is 6 and temperature changing is not considered.
225 225 185 225 225 185 225 225 225 185 225 225 225 185 225 185 1 0 0 125 85 85 125 125 85 5 125 5 125 125 125 5 125 125 5 125 125 85 125 125 125 85 125 125 125 5 125 85
Table A11. Setup time with product types while product family ratio is 6 and temperature changing is considered.
165
29 5
30 0 145 185 145 5 125 185 5 125 65 125 125 185 145 125 0
Type U 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 U 0 245 285 245 185 225 285 185 245 285 245 225 225 285 245 225 285 245 225 285 185 285 285 225 245 225 225 285 245 225 185 1 0 0 265 225 165 205 265 165 145 265 145 205 205 265 145 205 265 145 205 265 165 265 265 205 225 205 205 265 145 205 165 2 0 225 0 225 165 85 145 165 225 145 225 205 205 145 225 85 145 225 85 265 165 145 145 85 225 85 85 265 225 85 165 3 0 225 265 0 165 205 265 85 225 265 225 205 205 265 225 205 265 225 205 265 85 265 265 205 225 205 205 265 225 205 165 4 0 145 185 145 0 125 185 85 145 185 145 125 125 185 145 125 185 145 125 185 85 185 185 125 65 125 125 185 145 125 5 5 0 145 65 145 85 0 65 85 145 65 145 125 125 65 145 5 65 145 5 185 85 65 65 5 145 5 5 185 145 5 85 6 0 225 145 225 165 85 0 165 225 145 225 205 205 145 225 85 145 225 85 265 165 145 145 85 225 85 85 265 225 85 165 7 0 145 185 65 85 125 185 0 145 185 145 125 125 185 145 125 185 145 125 185 5 185 185 125 145 125 125 185 145 125 85 8 0 145 265 225 165 205 265 165 0 265 145 205 205 265 145 205 265 145 205 265 165 265 265 205 225 205 205 265 145 205 165 9 0 225 145 225 165 85 145 165 225 0 225 205 205 145 225 85 145 225 85 265 165 145 145 85 225 85 85 265 225 85 165 10 0 145 265 225 165 205 265 165 145 265 0 205 205 265 145 205 265 145 205 265 165 265 265 205 225 205 205 265 145 205 165 11 0 145 185 145 85 125 185 85 145 185 145 0 5 185 145 125 185 145 125 65 85 185 185 125 145 125 125 65 145 125 85 12 0 145 185 145 85 125 185 85 145 185 145 5 0 185 145 125 185 145 125 65 85 185 185 125 145 125 125 65 145 125 85 13 0 225 145 225 165 85 145 165 225 145 225 205 205 0 225 85 145 225 85 265 165 145 145 85 225 85 85 265 225 85 165 14 0 145 265 225 165 205 265 165 145 265 145 205 205 265 0 205 265 145 205 265 165 265 265 205 225 205 205 265 145 205 165 15 0 145 65 145 85 5 65 85 145 65 145 125 125 65 145 0 65 145 5 185 85 65 65 5 145 5 5 185 145 5 85 16 0 225 145 225 165 85 145 165 225 145 225 205 205 145 225 85 0 225 85 265 165 145 145 85 225 85 85 265 225 85 165 17 0 145 265 225 165 205 265 165 145 265 145 205 205 265 145 205 265 0 205 265 165 265 265 205 225 205 205 265 145 205 165 18 0 145 65 145 85 5 65 85 145 65 145 125 125 65 145 5 65 145 0 185 85 65 65 5 145 5 5 185 145 5 85 19 0 225 265 225 165 205 265 165 225 265 225 85 85 265 225 205 265 225 205 0 165 265 265 205 225 205 205 145 225 205 165 20 0 145 185 65 85 125 185 5 145 185 145 125 125 185 145 125 185 145 125 185 0 185 185 125 145 125 125 185 145 125 85 21 0 225 145 225 165 85 145 165 225 145 225 205 205 145 225 85 145 225 85 265 165 0 145 85 225 85 85 265 225 85 165 22 0 225 145 225 165 85 145 165 225 145 225 205 205 145 225 85 145 225 85 265 165 145 0 85 225 85 85 265 225 85 165 23 0 145 65 145 85 5 65 85 145 65 145 125 125 65 145 5 65 145 5 185 85 65 65 0 145 5 5 185 145 5 85 24 0 225 265 225 85 205 265 165 225 265 225 205 205 265 225 205 265 225 205 265 165 265 265 205 0 205 205 265 225 205 85 25 0 145 65 145 85 5 65 85 145 65 145 125 125 65 145 5 65 145 5 185 85 65 65 5 145 0 5 185 145 5 85 26 0 145 65 145 85 5 65 85 145 65 145 125 125 65 145 5 65 145 5 185 85 65 65 5 145 5 0 185 145 5 85 27 0 225 265 225 165 205 265 165 225 265 225 85 85 265 225 205 265 225 205 145 165 265 265 205 225 205 205 0 225 205 165 28 0 145 265 225 165 205 265 165 145 265 145 205 205 265 145 205 265 145 205 265 165 265 265 205 225 205 205 265 0 205
0 145 65 145 85 5 65 85 145 65 145 125 125 65 145 5 65 145 5 185 85 65 65 5 145 5 5 185 145 0 8 85 145 185 145 125 125 185 145 125 185 145 125 185 85 185 18
2950 3000 3050 3100 3150 3200 3250 3300 3350 3400 3450
0 200 400 600 800 1000 1200 1400 1600
generation
makespan
pm0gen1500scan30 pm0.25gen1500scan30 pm0.5gen1500scan30 pm0.75gen1500scan30 pm1gen1500scan30
Figure 7. The trend of average solutions population of hybrid GA repeated four times in problem No. 7.
in
2970 3020 3070 3120 3170 3220
0 200 400 600 800 1000 1200 1400 1600
generation
makespan
pm0gen1500scan30 pm0.25gen1500scan30 pm0.5gen1500scan30 pm0.75gen1500scan30 pm1gen1500scan30
Figure 8. The trend of best solution in population of hybrid GA repeated four times in problem No. 7.
3600 3650 3700 3750 3800 3850 3900 3950 4000
0 200 400 600 800 1000 1200 1400 1600
makespan
generation
pm0gen1500scan30 pm0.25gen1500scan30 pm0.5gen1500scan30 pm0.75gen1500scan30 pm1gen1500scan30
Figure 9. The trend of average solutions in population of hybrid GA repeated four times in problem No. 8.
3620 3640 3660 3680 3700 3720 3740 3760
0 200 400 600 800 1000 1200 1400 1600
generation
makespan
pm0gen1500scan30 pm0.25gen1500scan30 pm0.5gen1500scan30 pm0.75gen1500scan30 pm1gen1500scan30
Figure 10. The trend of average so ation of hybrid GA repeated four times in problem No. 8.
lutions in popul
3155 3160 3165 3170 3175 3180 3185 3190 3195 3200
0 200 400 600 800 1000 1200 1400 1600
makespan
pm0.5gen1500scan30 pm0.75gen1500scan30 pm1gen1500scan30
generation
he .
Figure 11. The further improvement of hybrid GA with initialization by improving uristics in problem No. 8
3180 3185 3190 3195 3200 3205 3210 3215 3220 3225
0 200 400 600 800 1000 1200 1400 1600
generation
makespan
pm0.5gen1500scan30 pm0.75gen1500scan30 pm1gen1500scan30
Figure 12. The further improvement of hybrid GA with initialization by improving heuristics in problem No. 16.