No-Wait Job Shop Scheduling Using a Population-Based Iterated Greedy Algorithm
Abstract
:1. Introduction
2. No-Wait Job Shop Scheduling Problem (NWJSP) with Makespan Minimization
2.1. Problem Statement
2.2. Problem Formulation
2.3. Timetabling Methods
3. Population-Based Iterated Greedy Algorithm
3.1. Iterated Greedy Procedure
3.1.1. Destruction and Construction
Algorithm 1. The destruction and construction (DC) operator. |
1: choose d unrepeated jobs s1, …, sd randomly, delete them from π, and a sequence with n – d jobs is obtained. 2: for i from 1 to d 3: insert si into the n – d + i positions of πF, evaluate the obtained n – d + i sequences, and replace πF with the best one. 4: endfor. |
3.1.2. Local Search
Algorithm 2. Insertion-based local search (IBLS). |
1: = a permutation generated randomly 2: i = 0, h = 1 3: while (i < n) 4: let s = 5: find 6: if ( is better than π) 7: 8: i = 1 9: else 10: i = i + 1 11: endif 12: h = (h + 1) % n 13: endwhile |
3.2. Initialization
3.3. Competitive Co-Evolutionary Scheme
Algorithm 3. Competitive Strategy. |
1: randomly select three solutions from all and find the worst one. 2: if (rand < pb) 3: πC := πG 4: md(k*) := mdbest 5: else if (mdbest) 6: πC := πL 7: md(k*) := true 8: else 9: πC := πI 10: md(k*) := false 11: endif 12: endif 13: perform DC on with parameter D and obtain a perturbation solution 14: := |
3.4. Procedure of the Population-Based Iterated Greedy (PBIG) Algorithm
Algorithm 4. Procedure of the PBIG. |
1: set parameters d, p, D, pb. 2: initialize πk (k = 1, …, p), πL, πI, πG, md, mdbest, Temp. 3: while (not termination) 4: for (each πk) //perform each IG procedure 5: perform the DC operator on πk and then the IBLS, and obtain a new solution . If is better than πk, then let πk := and update πL, πI, πG, mdbest if possible. 6: endfor 7: perform competitive strategy. 8: endwhile |
4. Computational Results and Comparisons
4.1. Calibration of the PBIG Algorithm
4.2. Comparisons with Other Metaheuristics
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Notation. | Description | Notation | Description |
---|---|---|---|
m | number of machines | ti | start time of Ji |
n | number of jobs | job permutation | |
M = {M1, M2, …, Mm} | set of m machines | t[i] | start time of |
J = {J1, J2, …, Jn} | set of n jobs | operation of on | |
oi,u | u-th operation of Ji | processing time of | |
Mi,u | machine on which oi,u is processed | start time difference set | |
pi,u | processing time of oi,u | makespan of | |
Pi,u = | cumulated processing time of Ji when oi,u is finished | start time table of | |
= {(u,v)| Mi,u = Mj,v} | pairs of operations processed on the same machine | cumulated processing time of when is finished | |
Li | total processing times of Ji |
Source | Sum of Squares | Df | Mean Square | F-Ratio | p-Value |
---|---|---|---|---|---|
MAIN EFFECTS | |||||
A: p | 54.9931 | 3 | 18.3310 | 16.8500 | 0.0000 |
B: d | 343.396 | 3 | 114.465 | 105.220 | 0.0000 |
C: pb | 5.45246 | 3 | 1.81749 | 1.67000 | 0.1737 |
D: D | 142.747 | 3 | 47.5825 | 43.7400 | 0.0000 |
E: instance | 7962.73 | 6 | 1327.12 | 1219.90 | 0.0000 |
REDIDUAL | 9797.72 | 8941 | 1.09582 | ||
TOTAL (CORRECTED) | 18307.0 | 8959 |
Instance. | n, m | BKS | MCLM | HABC | PBIG | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
RPDB | ARPD | TA(s) | RPDB | ARPD | TA(s) | RPDB | ARPD | TA(s) | |||
Ft06 | 6, 6 | 73 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.15 | 0.00 | 0.00 | 0.65 |
La01 | 10, 5 | 971 | 0.00 | 0.00 | 4.55 | 0.41 | 0.41 | 0.78 | 0.00 | 0.02 | 1.50 |
La02 | 10, 5 | 937 | 0.00 | 0.00 | 7.60 | 2.56 | 2.56 | 1.41 | 0.00 | 0.00 | 1.50 |
La03 | 10, 5 | 820 | 0.00 | 0.00 | 3.10 | 0.00 | 0.00 | 0.70 | 0.00 | 0.00 | 1.50 |
La04 | 10, 5 | 887 | 0.00 | 0.00 | 6.25 | 0.00 | 0.00 | 0.87 | 0.00 | 0.00 | 1.50 |
La05 | 10, 5 | 777 | 0.51 | 0.90 | 3.90 | 0.51 | 0.51 | 1.14 | 0.00 | 0.00 | 1.50 |
Ft10 | 10, 10 | 1607 | 0.00 | 0.00 | 7.85 | 0.00 | 0.00 | 9.86 | 0.00 | 0.00 | 3.00 |
Orb01 | 10, 10 | 1615 | 0.00 | 0.00 | 6.65 | 0.00 | 0.00 | 8.00 | 0.00 | 0.00 | 3.00 |
Orb02 | 10, 10 | 1485 | 2.16 | 2.16 | 6.70 | 0.00 | 0.00 | 6.86 | 0.00 | 0.00 | 3.00 |
Orb03 | 10, 10 | 1599 | 0.00 | 0.00 | 13.75 | 0.00 | 0.00 | 10.07 | 0.00 | 0.00 | 3.00 |
Orb04 | 10, 10 | 1653 | 0.00 | 0.12 | 7.85 | 0.00 | 0.00 | 5.28 | 0.00 | 0.00 | 3.00 |
Orb05 | 10, 10 | 1365 | 0.00 | 0.00 | 8.50 | 0.37 | 0.37 | 4.77 | 0.15 | 0.15 | 3.00 |
Orb06 | 10, 10 | 1555 | 0.00 | 0.00 | 3.55 | 0.00 | 0.00 | 5.76 | 0.00 | 0.00 | 3.00 |
Orb07 | 10, 10 | 689 | 0.00 | 0.00 | 7.25 | NA | NA | NA | 0.00 | 0.00 | 3.00 |
Orb08 | 10, 10 | 1319 | 0.00 | 0.00 | 6.40 | 0.00 | 0.00 | 8.99 | 0.00 | 0.00 | 3.00 |
Orb09 | 10, 10 | 1445 | 0.00 | 0.00 | 4.25 | 0.00 | 0.28 | 4.10 | 0.00 | 0.00 | 3.00 |
Orb10 | 10, 10 | 1557 | 0.00 | 0.00 | 11.85 | 0.00 | 0.00 | 5.51 | 0.00 | 0.00 | 3.00 |
La16 | 10, 10 | 1575 | 1.84 | 1.84 | 5.65 | 0.00 | 0.00 | 5.43 | 0.00 | 0.00 | 3.00 |
La17 | 10, 10 | 1371 | 0.00 | 0.12 | 11.85 | 0.00 | 0.00 | 4.62 | 0.00 | 0.00 | 3.00 |
La18 | 10, 10 | 1417 | 2.82 | 2.82 | 6.15 | 0.00 | 5.22 | 7.13 | 0.00 | 0.00 | 3.00 |
La19 | 10, 10 | 1482 | 0.00 | 0.61 | 5.00 | 0.00 | 0.43 | 3.44 | 0.00 | 0.00 | 3.00 |
La20 | 10, 10 | 1526 | 1.31 | 1.31 | 5.40 | 0.00 | 0.00 | 3.00 | 0.00 | 0.00 | 3.00 |
Average | 0.39 | 0.45 | 6.55 | 0.23 | 0.47 | 4.63 | 0.01 | 0.01 | 2.55 |
Instance | n, m | BKS | MCLM | HABC | PBIG | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Best | RPDB | ARPD | TA(s) | Best | RPDB | ARPD | TA(s) | Best | RPDB | ARPD | TA(s) | |||
La06 | 15, 5 | 1248 | 1248 | 0.00 | 0.75 | 90.00 | 1248 | 0.00 | 0.00 | 39.99 | 1248 | 0.00 | 0.00 | 67.50 |
La07 | 15, 5 | 1172 | 1178 | 0.51 | 2.46 | 83.00 | 1172 | 0.00 | 0.92 | 90.19 | 1172 | 0.00 | 0.44 | 67.50 |
La08 | 15, 5 | 1244 | 1244 | 0.00 | 0.47 | 80.00 | 1244 | 0.00 | 0.15 | 85.63 | 1244 | 0.00 | 0.48 | 67.51 |
La09 | 15, 5 | 1358 | 1365 | 0.52 | 0.73 | 106.00 | 1362 | 0.29 | 0.67 | 34.43 | 1362 | 0.29 | 0.61 | 67.50 |
La10 | 15, 5 | 1287 | 1287 | 0.00 | 0.04 | 69.00 | 1294 | 0.54 | 0.70 | 65.42 | 1294 | 0.54 | 0.84 | 67.50 |
La11 | 20, 5 | 1671 | 1635 | −2.15 | −0.58 | 439.00 | 1627 | −2.63 | −1.97 | 259.37 | 1621 | −2.99 | −1.08 | 120.02 |
La12 | 20, 5 | 1452 | 1429 | −1.58 | 0.57 | 593.00 | 1434 | −1.24 | 0.00 | 168.97 | 1425 | −1.86 | 0.17 | 120.02 |
La13 | 20, 5 | 1624 | 1605 | −1.17 | −0.15 | 303.00 | 1580 | −2.71 | −1.47 | 222.40 | 1582 | −2.59 | −0.09 | 120.02 |
La14 | 20, 5 | 1691 | 1648 | −2.54 | −1.16 | 314.00 | 1640 | −3.02 | −2.24 | 156.58 | 1640 | −3.02 | −1.96 | 120.02 |
La15 | 20, 5 | 1694 | 1685 | −0.53 | 1.09 | 424.00 | 1679 | −0.89 | −0.09 | 240.56 | 1677 | −1.00 | 0.35 | 120.02 |
La21 | 15, 10 | 2048 | 2048 | 0.00 | 0.11 | 78.00 | 2043 | −0.24 | 0.27 | 71.42 | 2043 | −0.24 | −0.04 | 135.01 |
La22 | 15, 10 | 1887 | 1902 | 0.80 | 0.99 | 142.00 | 1852 | −1.85 | −1.12 | 91.66 | 1852 | −1.85 | −1.19 | 135.01 |
La23 | 15, 10 | 2032 | 2022 | −0.49 | 1.47 | 50.00 | 2032 | 0.00 | 0.71 | 120.56 | 2032 | 0.00 | 0.14 | 135.01 |
La24 | 15, 10 | 2015 | 2015 | 0.00 | 0.77 | 98.00 | 1994 | −1.04 | −0.02 | 97.64 | 1994 | −1.04 | −0.30 | 135.01 |
La25 | 15, 10 | 1917 | 1930 | 0.68 | 2.07 | 71.00 | 1906 | −0.57 | −0.57 | 92.53 | 1906 | −0.57 | −0.57 | 135.01 |
La26 | 20, 10 | 2553 | 2532 | −0.82 | 1.91 | 349.00 | 2506 | −1.84 | 0.30 | 223.46 | 2506 | −1.84 | 1.66 | 240.02 |
La27 | 20, 10 | 2747 | 2715 | −1.17 | 0.28 | 388.00 | 2674 | −2.66 | −2.62 | 154.83 | 2673 | −2.69 | −2.19 | 240.02 |
La28 | 20, 10 | 2624 | 2560 | −2.44 | 1.77 | 313.00 | 2560 | −2.44 | 0.62 | 197.33 | 2581 | −1.64 | 0.77 | 240.03 |
La29 | 20, 10 | 2489 | 2367 | −4.90 | −2.48 | 445.00 | 2389 | −4.02 | −2.70 | 531.94 | 2405 | −3.37 | −2.18 | 240.03 |
La30 | 20, 10 | 2665 | 2544 | −4.54 | −1.51 | 376.00 | 2452 | −7.99 | −3.00 | 248.25 | 2452 | −7.99 | −3.50 | 240.03 |
La31 | 30, 10 | 3745 | 3575 | −4.54 | −1.40 | 3099.00 | 3592 | −4.09 | −2.60 | 1716.18 | 3479 | −7.10 | −2.35 | 540.16 |
La32 | 30, 10 | 4028 | 3835 | −4.79 | 0.56 | 3314.00 | 3913 | −2.86 | −0.35 | 1590.40 | 3877 | −3.75 | −0.68 | 540.19 |
La33 | 30, 10 | 3749 | 3574 | −4.67 | −1.52 | 3003.00 | 3529 | −5.87 | −3.37 | 1544.34 | 3560 | −5.04 | −2.51 | 540.18 |
La34 | 30, 10 | 3824 | 3684 | −3.66 | −0.88 | 3375.00 | 3610 | −5.60 | −2.76 | 1405.05 | 3615 | −5.47 | −2.55 | 540.20 |
La35 | 30, 10 | 3760 | 3698 | −1.65 | 1.27 | 3083.00 | 3593 | −4.44 | −0.41 | 1797.55 | 3687 | −1.94 | 0.43 | 540.19 |
La36 | 15, 15 | 2685 | 2736 | 1.90 | 4.98 | 189.00 | 2685 | 0.00 | 0.29 | 76.42 | 2685 | 0.00 | 0.00 | 202.51 |
La37 | 15, 15 | 2962 | 2962 | 0.00 | 0.11 | 93.00 | 2938 | −0.81 | 0.98 | 184.41 | 2831 | −4.42 | 0.06 | 202.51 |
La38 | 15, 15 | 2617 | 2525 | −3.52 | −1.73 | 91.00 | 2525 | −3.52 | −1.08 | 311.66 | 2525 | −3.52 | −2.80 | 202.51 |
La39 | 15, 15 | 2697 | 2729 | 1.19 | 2.43 | 116.00 | 2703 | 0.22 | 0.88 | 146.87 | 2687 | −0.37 | 0.34 | 202.51 |
La40 | 15, 15 | 2594 | 2580 | −0.54 | −0.54 | 61.00 | 2594 | 0.00 | 0.00 | 190.51 | 2594 | 0.00 | 0.00 | 202.51 |
Swv01 | 20, 10 | 2328 | 2333 | 0.22 | 0.37 | 516.00 | 2318 | −0.43 | −0.02 | 874.70 | 2318 | −0.43 | −0.17 | 240.02 |
Swv02 | 20, 10 | 2418 | 2418 | 0.00 | 0.38 | 488.00 | 2417 | −0.04 | −0.02 | 961.12 | 2417 | −0.04 | −0.02 | 240.02 |
Swv03 | 20, 10 | 2415 | 2381 | −1.41 | −0.41 | 517.00 | 2381 | −1.41 | −0.92 | 1018.02 | 2381 | −1.41 | −0.89 | 240.02 |
Swv04 | 20, 10 | 2506 | 2462 | −1.76 | −0.30 | 426.00 | 2506 | 0.00 | 0.29 | 1728.64 | 2462 | −1.76 | −0.10 | 240.02 |
Swv05 | 20, 10 | 2333 | 2333 | 0.00 | 0.00 | 285.00 | 2333 | 0.00 | 0.00 | 535.69 | 2333 | 0.00 | 0.00 | 240.02 |
Swv06 | 20, 15 | 3291 | 3291 | 0.00 | 1.69 | 747.00 | 3291 | 0.00 | 0.40 | 885.67 | 3291 | 0.00 | 0.05 | 360.02 |
Swv07 | 20, 15 | 3271 | 3219 | −1.59 | −0.95 | 584.00 | 3188 | −2.54 | −2.54 | 829.20 | 3188 | −2.54 | −2.43 | 360.01 |
Swv08 | 20, 15 | 3530 | 3423 | −3.03 | −2.21 | 413.00 | 3423 | −3.03 | −1.78 | 1150.09 | 3423 | −3.03 | −2.56 | 360.02 |
Swv09 | 20, 15 | 3307 | 3270 | −1.12 | −0.28 | 448.00 | 3246 | −1.84 | −0.86 | 1531.08 | 3246 | −1.84 | −1.12 | 360.02 |
Swv10 | 20, 15 | 3488 | 3451 | −1.06 | 0.12 | 520.00 | 3462 | −0.75 | −0.22 | 1425.36 | 3462 | −0.75 | −0.64 | 360.02 |
Average | −1.25 | 0.28 | 654.48 | −1.73 | −0.64 | 577.40 | −1.88 | −0.64 | 238.16 |
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Xu, M.; Zhang, S.; Deng, G. No-Wait Job Shop Scheduling Using a Population-Based Iterated Greedy Algorithm. Algorithms 2021, 14, 145. https://doi.org/10.3390/a14050145
Xu M, Zhang S, Deng G. No-Wait Job Shop Scheduling Using a Population-Based Iterated Greedy Algorithm. Algorithms. 2021; 14(5):145. https://doi.org/10.3390/a14050145
Chicago/Turabian StyleXu, Mingming, Shuning Zhang, and Guanlong Deng. 2021. "No-Wait Job Shop Scheduling Using a Population-Based Iterated Greedy Algorithm" Algorithms 14, no. 5: 145. https://doi.org/10.3390/a14050145