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October  2017, 13(4): 2015-2031. doi: 10.3934/jimo.2017029

An efficient Tabu Search neighborhood based on reconstruction strategy to solve the blocking job shop scheduling problem

Adel Dabah 1,2, , Ahcene Bendjoudi 1, and  Abdelhakim AitZai 2,

1. 

CERIST Research Center 3 rue des freres Aissou, 16030 Ben-Aknoun, Algiers, Algeria
2. 

University of Sciences and Technology Houari Boumedienne (USTHB), Bab Ezzouar 16111, Algiers, Algeria

Received  December 2015 Revised  September 2016 Published  April 2017

The blocking job shop scheduling problem (BJSS) is a version of the classical job shop scheduling with no intermediate buffer between machines. The BJSS is known to be NP-hard in the strong sense. A known way to solve such a problem is to use the Tabu Search algorithm (TS) which is a higher level heuristic procedure for solving optimization problems, designed to guide other methods to escape the trap of local optimality. However, the use of the classical TS neighborhood on BJSS problem produces infeasible solutions in most cases (98% of cases). This leads to waste valuable time in exploring infeasible solutions. To overcome this drawback, we propose a new tabu search neighborhood based on reconstruction strategy. This neighborhood consists to remove arcs causing the infeasibility and rebuild the neighbor solutions by using heuristics. Experiments on the reference benchmark instances show that the TS algorithm using the proposed neighborhood improves most of the known results in the literature and gives new upper bounds for more than 52 benchmarks in both BJSS cases (BJSS with Swap and BJSS no-Swap). Moreover, the proposed approach reaches much faster the optimal solution for most of the optimally solved benchmarks.

Keywords: Scheduling Problem, blocking Job shop, swap situation, tabu search neighborhood.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Adel Dabah, Ahcene Bendjoudi, Abdelhakim AitZai. An efficient Tabu Search neighborhood based on reconstruction strategy to solve the blocking job shop scheduling problem. Journal of Industrial and Management Optimization, 2017, 13 (4) : 2015-2031. doi: 10.3934/jimo.2017029
References:
[1]

J. Adams, E. Balas and D. Zawack, The shifting bottleneck procedure for job shop scheduling, Management Science, 34 (1988), 391-401.  doi: 10.1287/mnsc.34.3.391.

[2]

A. AitZai, B. Benmedjdoub and M. Boudhar, A branch and bound and parallel genetic algorithm for the job shop scheduling problem with blocking, International Journal of Operational Research, 14 (2012), 343-365.  doi: 10.1504/IJOR.2012.047094.

[3]

D. Applegate and W. Cook, A computational study of the job-shop scheduling problem, ORSA Journal on Computing, 3 (1991), 149-156. 

[4]

A. Dabah, A. Bendjoudi and A. AitZai, Efficient Multi Start Parallel Tabu Search for the Blocking Job Shop Scheduling Problem, forthcoming.

[5]

F. Glover, Future paths for integer programming and links to artificial intelligence, Computers Operations Research, 13 (1986), 533-549.  doi: 10.1016/0305-0548(86)90048-1.

[6]

F. Glover, Tabu search-part Ⅰ, ORSA Journal on Computing, 1 (1989), 190-206. 

[7]

F. Glover, Tabu search-part Ⅱ, ORSA Journal on Computing, 2 (1990), 4-32. 

[8]

H. Groflin and A. Klinkert, A new neighborhood and tabu search for the blocking job shop, Discrete Applied Mathematics, 157 (2009), 3643-3655.  doi: 10.1016/j.dam.2009.02.020.

[9]

H. Gröflin, D. N. Pham and R. Bürgy, The flexible blocking job shop with transfer and set-up times, Journal of Combinatorial Optimization, 22 (2011), 121-144.  doi: 10.1007/s10878-009-9278-x.

[10]

N. G. Hall and C. Sriskandarajah, A survey of machine scheduling problems with blocking and no-wait in process, Operations Research, 44 (1996), 510-525.  doi: 10.1287/opre.44.3.510.

[11]

V. Laarhoven, J. M. Peter, E. H. L. Aarts and J. K. Lenstra, Job shop scheduling by simulated annealing, Operations Research, 40 (1992), 113-125.  doi: 10.1287/opre.40.1.113.

[12]

S. Lawrence, Resource constrained project scheduling: An experimental investigation of heuristic scheduling techniques (supplement), Graduate School of Industrial Administration, 1984.

[13]

A. Mascis and D. Pacciarelli, Job-shop scheduling with blocking and no-wait constraints, European Journal of Operational Research, 143 (2002), 498-517.  doi: 10.1016/S0377-2217(01)00338-1.

[14]

Y. Mati, N. Rezg and X. Xie, A taboo search approach for deadlock-free scheduling of automated manufacturing systems, Journal of Intelligent Manufacturing, 12 (2001), 535-552. 

[15]

Y. Mati and X. Xie, Multiresource shop scheduling with resource flexibility and blocking, Automation Science and Engineering, IEEE Transactions on, 8 (2011), 175-189. 

[16]

C. Meloni, D. Pacciarelli and M. Pranzo, A rollout metaheuristic for job shop scheduling problems, Annals of Operations Research, 131 (2004), 215-235.  doi: 10.1023/B:ANOR.0000039520.24932.4b.

[17]

J. F. Muth and G. L. Thompson, eds. Industrial Scheduling, Prentice-Hall, 1963.

[18]

A. Oddi, R. Rasconi, A. Cesta and S. F. Smith, Iterative Improvement Algorithms for the Blocking Job Shop, In ICAPS. 2012.

[19]

D.-N. Pham and A. Klinkert, Surgical case scheduling as a generalized job shop scheduling problem, European Journal of Operational Research, 185 (2008), 1011-1025.  doi: 10.1016/j.ejor.2006.03.059.

[20]

M. Pranzo and D. Pacciarelli, An iterated greedy metaheuristic for the blocking job shop scheduling problem, Journal of Heuristics, (2013), 1-25. 

[21]

B. Roy and B. Sussmann, Les problemes d'ordonnancement avec contraintes disjonctives, Note ds, 9(1964).

show all references

References:
[1]

J. Adams, E. Balas and D. Zawack, The shifting bottleneck procedure for job shop scheduling, Management Science, 34 (1988), 391-401.  doi: 10.1287/mnsc.34.3.391.

[2]

A. AitZai, B. Benmedjdoub and M. Boudhar, A branch and bound and parallel genetic algorithm for the job shop scheduling problem with blocking, International Journal of Operational Research, 14 (2012), 343-365.  doi: 10.1504/IJOR.2012.047094.

[3]

D. Applegate and W. Cook, A computational study of the job-shop scheduling problem, ORSA Journal on Computing, 3 (1991), 149-156. 

[4]

A. Dabah, A. Bendjoudi and A. AitZai, Efficient Multi Start Parallel Tabu Search for the Blocking Job Shop Scheduling Problem, forthcoming.

[5]

F. Glover, Future paths for integer programming and links to artificial intelligence, Computers Operations Research, 13 (1986), 533-549.  doi: 10.1016/0305-0548(86)90048-1.

[6]

F. Glover, Tabu search-part Ⅰ, ORSA Journal on Computing, 1 (1989), 190-206. 

[7]

F. Glover, Tabu search-part Ⅱ, ORSA Journal on Computing, 2 (1990), 4-32. 

[8]

H. Groflin and A. Klinkert, A new neighborhood and tabu search for the blocking job shop, Discrete Applied Mathematics, 157 (2009), 3643-3655.  doi: 10.1016/j.dam.2009.02.020.

[9]

H. Gröflin, D. N. Pham and R. Bürgy, The flexible blocking job shop with transfer and set-up times, Journal of Combinatorial Optimization, 22 (2011), 121-144.  doi: 10.1007/s10878-009-9278-x.

[10]

N. G. Hall and C. Sriskandarajah, A survey of machine scheduling problems with blocking and no-wait in process, Operations Research, 44 (1996), 510-525.  doi: 10.1287/opre.44.3.510.

[11]

V. Laarhoven, J. M. Peter, E. H. L. Aarts and J. K. Lenstra, Job shop scheduling by simulated annealing, Operations Research, 40 (1992), 113-125.  doi: 10.1287/opre.40.1.113.

[12]

S. Lawrence, Resource constrained project scheduling: An experimental investigation of heuristic scheduling techniques (supplement), Graduate School of Industrial Administration, 1984.

[13]

A. Mascis and D. Pacciarelli, Job-shop scheduling with blocking and no-wait constraints, European Journal of Operational Research, 143 (2002), 498-517.  doi: 10.1016/S0377-2217(01)00338-1.

[14]

Y. Mati, N. Rezg and X. Xie, A taboo search approach for deadlock-free scheduling of automated manufacturing systems, Journal of Intelligent Manufacturing, 12 (2001), 535-552. 

[15]

Y. Mati and X. Xie, Multiresource shop scheduling with resource flexibility and blocking, Automation Science and Engineering, IEEE Transactions on, 8 (2011), 175-189. 

[16]

C. Meloni, D. Pacciarelli and M. Pranzo, A rollout metaheuristic for job shop scheduling problems, Annals of Operations Research, 131 (2004), 215-235.  doi: 10.1023/B:ANOR.0000039520.24932.4b.

[17]

J. F. Muth and G. L. Thompson, eds. Industrial Scheduling, Prentice-Hall, 1963.

[18]

A. Oddi, R. Rasconi, A. Cesta and S. F. Smith, Iterative Improvement Algorithms for the Blocking Job Shop, In ICAPS. 2012.

[19]

D.-N. Pham and A. Klinkert, Surgical case scheduling as a generalized job shop scheduling problem, European Journal of Operational Research, 185 (2008), 1011-1025.  doi: 10.1016/j.ejor.2006.03.059.

[20]

M. Pranzo and D. Pacciarelli, An iterated greedy metaheuristic for the blocking job shop scheduling problem, Journal of Heuristics, (2013), 1-25. 

[21]

B. Roy and B. Sussmann, Les problemes d'ordonnancement avec contraintes disjonctives, Note ds, 9(1964).

Figure 1.  Alternative pairs between blocking and ideal operations
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Figure 2.  Swap situation between three operations (jobs)
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Figure 3.  Alternative graph for BJSS instance of table 1
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Figure 4.  Schedule for BJSS in Table 1 whit Cmax=26
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Figure 5.  Gantt chart of the schedule in figure 4
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Figure 6.  The proposed neighborhood steps
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Figure 7.  Variation of the relative errors from the optimal solution over times for La17 instance
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Figure 8.  The behaviour of both TS methods over times for the La17 instance
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Table 1.  BJSS instance with two jobs and three machines
jobsequence processing times
$J1$ $M1, M2, M3 $ 5, 3, 8
$J2$ $M2, M1, M3 $ 8, 2, 7
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jobsequence processing times
$J1$ $M1, M2, M3 $ 5, 3, 8
$J2$ $M2, M1, M3 $ 8, 2, 7
Table 2.  The description of the symbols used in our TS
SymbolDescription
sa feasible schedule (solution) for the BJSS problem.
$s*$ The best solution found by TS.
$Cost(s)$ The Cost (makespan) of the solution s.
$sc$ Best solution in Cand(s).
$TL$ Tabu List.
$Cand(s)$ a set of unforbidden neighbor solutions of the schedule s.
$N(s)$ a set of all neighbor solutions of schedule s.
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SymbolDescription
sa feasible schedule (solution) for the BJSS problem.
$s*$ The best solution found by TS.
$Cost(s)$ The Cost (makespan) of the solution s.
$sc$ Best solution in Cand(s).
$TL$ Tabu List.
$Cand(s)$ a set of unforbidden neighbor solutions of the schedule s.
$N(s)$ a set of all neighbor solutions of schedule s.
Table 3.  Comparison of the obtained Makespan results with the optimal solutions of the (10 $\times$ 10) Instances
Instance BWS BNS
$C_{optimal}$ $C_{mean}$ $C_{best}$ $C_{optimal}$ $C_{mean}$ $C_{best}$
Abz5 1468 1487 1468 1641 1641 1641
Abz6 1145 1190 1160 1249 1254 1249
Mt10 1068 1095 1071 1158 1168 1158
Orb1 1175 1182 1175 1256 1265 1259
Orb2 1041 1062 1041 1144 1154 1146
Orb3 1160 1162 1160 1311 1311 1311
Orb4 1146 1187 1146 1246 1246 1246
Orb5 995 1008 995 1203 1203 1203
Orb6 1199 1199 1199 1266 1266 1266
Orb7 483 490 483 527 535 527
Orb8 995 1003 995 1139 1139 1139
Orb9 1039 1076 1045 1130 1131 1130
Orb10 1146 1146 1146 1367 1367 1367
La16 1060 1090 1060 1148 1151 1148
La17 929 943 930 968 979 968
La18 1025 1038 1025 1077 1087 1077
La19 1043 1067 1053 1102 1104 1102
La20 1060 1085 1060 1118 1125 1118
MRE 1.7% 0.18% 0.36% 0, 02%
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Instance BWS BNS
$C_{optimal}$ $C_{mean}$ $C_{best}$ $C_{optimal}$ $C_{mean}$ $C_{best}$
Abz5 1468 1487 1468 1641 1641 1641
Abz6 1145 1190 1160 1249 1254 1249
Mt10 1068 1095 1071 1158 1168 1158
Orb1 1175 1182 1175 1256 1265 1259
Orb2 1041 1062 1041 1144 1154 1146
Orb3 1160 1162 1160 1311 1311 1311
Orb4 1146 1187 1146 1246 1246 1246
Orb5 995 1008 995 1203 1203 1203
Orb6 1199 1199 1199 1266 1266 1266
Orb7 483 490 483 527 535 527
Orb8 995 1003 995 1139 1139 1139
Orb9 1039 1076 1045 1130 1131 1130
Orb10 1146 1146 1146 1367 1367 1367
La16 1060 1090 1060 1148 1151 1148
La17 929 943 930 968 979 968
La18 1025 1038 1025 1077 1087 1077
La19 1043 1067 1053 1102 1104 1102
La20 1060 1085 1060 1118 1125 1118
MRE 1.7% 0.18% 0.36% 0, 02%
Table 4.  Our $TS_{IN}$ makespan results Vs. the best known solutions for both BJSS cases (BWS, BNS)
Instance $Size$ BWS BNS
$Best$ $TS_{gr} $ $C_{mean}$ $C_{best}$ $Best$ $C_{mean}$ $C_{best}$
La01 10$\times$5 [18,20] 793 820 780 793 [20,15] 881 881 881
La02 10$\times$5 [8,18,20] 793 793 793 793 [20,15] 900 901 900
La03 10$\times$5 [18,20] 715 740 715 715 [20,15] 808 810 808
La04 10$\times$5 [18,20] 743 764 743 743 [20,15] 859 864 859
La05 10$\times$5 [18,20] 664 666 670 664 [20,15] 732 732 732
La06 15$\times$5 [18] 1064 1180 1120 1076 [20] 1225 1207 1194
La07 15$\times$5 [20] 1020 1084 1052 1029 [20] 1133 1138 1130
La08 15$\times$5 [18] 1062 1125 1089 1060 [15] 1216 1203 1173
La09 15$\times$5 [20] 1162 1223 1209 1188 [20] 1311 1324 1314
La10 15$\times$5 [18] 1110 1203 1142 1110 [20] 1237 1252 1232
La11 20$\times$5 [18] 1466 1584 1504 1475 [20] 1641 1621 1577
La12 20$\times$5 [20] 1271 1391 1326 1276 [20] 1465 1448 1401
La13 20$\times$5 [18] 1465 1541 1474 1456 [20] 1627 1586 1547
La14 20$\times$5 [18] 1506 1620 1520 1472 [20] 1686 1648 1586
La15 20$\times$5 [18,20] 1517 1630 1526 1490 [20] 1680 1651 1620
La16 10$\times$10 [20] 1060 1142 1090 1060 [20,15] 1148 1151 1148
La17 10$\times$10 [20] 929 977 943 930 [20,15] 968 979 968
La18 10$\times$10 [20] 1025 1078 1038 1025 [20,15] 1077 1087 1077
La19 10$\times$10 [20] 1043 1093 1067 1053 [20,15] 1102 1104 1102
La20 10$\times$10 [20] 1060 1154 1085 1060 [20,15] 1118 1125 1118
La21 15$\times$10 [20] 1490 1554 1499 1467 [20] 1627 1540 1501
La22 15$\times$10 [20] 1339 1458 1369 1347 [20] 1426 1421 1368
La23 15$\times$10 [20] 1445 1570 1477 1442 [20] 1574 1564 1537
La24 15$\times$10 [20] 1434 1546 1433 1398 [20] 1502 1510 1447
La25 15$\times$10 [20] 1392 1499 1420 1373 [20] 1533 1497 1453
La26 20$\times$10 [20] 1989 2125 1950 1929 [20] 2146 2023 1968
La27 20$\times$10 [20] 2017 2175 2007 1960 [20] 2191 2113 2047
La28 20$\times$10 [18] 2027 2071 1959 1880 [20] 2245 2090 2046
La29 20$\times$10 [20] 1846 1990 1846 1803 [20] 2030 1942 1857
La30 20$\times$10 [20] 2049 2097 1982 1965 [20] 2242 2090 2033
La31 30$\times$10 [18] 2921 3137 2790 2715 [20] 3219 2978 2942
La32 30$\times$10 [18] 3237 3316 3019 2987 [20] 3567 3163 3114
La33 30$\times$10 [18] 2844 3061 2755 2672 [20] 3201 2873 2845
La34 30$\times$10 [18] 2848 3146 2800 2729 [20] 3202 2959 2862
La35 30$\times$10 [18] 2923 3171 2828 2776 [15] 3373 2961 2871
La36 15$\times$15 [20] 1755 1919 1741 1713 [20] 1835 1796 1767
La37 15$\times$15 [20] 1870 2029 1840 1802 [20] 1931 1917 1871
La38 15$\times$15 [18] 1708 1828 1652 1630 [20] 1813 1770 1747
La39 15$\times$15 [20] 1731 1882 1719 1697 [20] 1811 1791 1758
La40 15$\times$15 [20] 1743 1925 1730 1692 [20] 1815 1802 1780
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Instance $Size$ BWS BNS
$Best$ $TS_{gr} $ $C_{mean}$ $C_{best}$ $Best$ $C_{mean}$ $C_{best}$
La01 10$\times$5 [18,20] 793 820 780 793 [20,15] 881 881 881
La02 10$\times$5 [8,18,20] 793 793 793 793 [20,15] 900 901 900
La03 10$\times$5 [18,20] 715 740 715 715 [20,15] 808 810 808
La04 10$\times$5 [18,20] 743 764 743 743 [20,15] 859 864 859
La05 10$\times$5 [18,20] 664 666 670 664 [20,15] 732 732 732
La06 15$\times$5 [18] 1064 1180 1120 1076 [20] 1225 1207 1194
La07 15$\times$5 [20] 1020 1084 1052 1029 [20] 1133 1138 1130
La08 15$\times$5 [18] 1062 1125 1089 1060 [15] 1216 1203 1173
La09 15$\times$5 [20] 1162 1223 1209 1188 [20] 1311 1324 1314
La10 15$\times$5 [18] 1110 1203 1142 1110 [20] 1237 1252 1232
La11 20$\times$5 [18] 1466 1584 1504 1475 [20] 1641 1621 1577
La12 20$\times$5 [20] 1271 1391 1326 1276 [20] 1465 1448 1401
La13 20$\times$5 [18] 1465 1541 1474 1456 [20] 1627 1586 1547
La14 20$\times$5 [18] 1506 1620 1520 1472 [20] 1686 1648 1586
La15 20$\times$5 [18,20] 1517 1630 1526 1490 [20] 1680 1651 1620
La16 10$\times$10 [20] 1060 1142 1090 1060 [20,15] 1148 1151 1148
La17 10$\times$10 [20] 929 977 943 930 [20,15] 968 979 968
La18 10$\times$10 [20] 1025 1078 1038 1025 [20,15] 1077 1087 1077
La19 10$\times$10 [20] 1043 1093 1067 1053 [20,15] 1102 1104 1102
La20 10$\times$10 [20] 1060 1154 1085 1060 [20,15] 1118 1125 1118
La21 15$\times$10 [20] 1490 1554 1499 1467 [20] 1627 1540 1501
La22 15$\times$10 [20] 1339 1458 1369 1347 [20] 1426 1421 1368
La23 15$\times$10 [20] 1445 1570 1477 1442 [20] 1574 1564 1537
La24 15$\times$10 [20] 1434 1546 1433 1398 [20] 1502 1510 1447
La25 15$\times$10 [20] 1392 1499 1420 1373 [20] 1533 1497 1453
La26 20$\times$10 [20] 1989 2125 1950 1929 [20] 2146 2023 1968
La27 20$\times$10 [20] 2017 2175 2007 1960 [20] 2191 2113 2047
La28 20$\times$10 [18] 2027 2071 1959 1880 [20] 2245 2090 2046
La29 20$\times$10 [20] 1846 1990 1846 1803 [20] 2030 1942 1857
La30 20$\times$10 [20] 2049 2097 1982 1965 [20] 2242 2090 2033
La31 30$\times$10 [18] 2921 3137 2790 2715 [20] 3219 2978 2942
La32 30$\times$10 [18] 3237 3316 3019 2987 [20] 3567 3163 3114
La33 30$\times$10 [18] 2844 3061 2755 2672 [20] 3201 2873 2845
La34 30$\times$10 [18] 2848 3146 2800 2729 [20] 3202 2959 2862
La35 30$\times$10 [18] 2923 3171 2828 2776 [15] 3373 2961 2871
La36 15$\times$15 [20] 1755 1919 1741 1713 [20] 1835 1796 1767
La37 15$\times$15 [20] 1870 2029 1840 1802 [20] 1931 1917 1871
La38 15$\times$15 [18] 1708 1828 1652 1630 [20] 1813 1770 1747
La39 15$\times$15 [20] 1731 1882 1719 1697 [20] 1811 1791 1758
La40 15$\times$15 [20] 1743 1925 1730 1692 [20] 1815 1802 1780
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