Abstract
This paper presents the problem of batching and scheduling jobs belonging to incompatible job families on unrelated-parallel machines. More specifically, we investigate cost-efficient approaches for solving batching and scheduling problems concerning the desired lower bounds on batch sizes (\({LB}_{b}\)), which indirectly has a considerable impact on the production cost. Batch scheduling is a more realistic extension of the traditional group scheduling approach, in which the jobs belonging to a job family can be processed as multiple batches. The objective is to minimize the total weighted job completion time and tardiness subject to a machine- and sequence-dependent setup time, dynamic machine availability and job release times, customer segments and job priority, and different machine capability and eligibility criteria for processing. Solving this type of batch scheduling problem is a big challenge due to the high computational complexity incurred by both the sequencing assignment and batching composition. A machine learning random forest classification algorithm is used for the \({LB}_{b}\) determination. Then, an efficient mixed-integer linear programming model (MILP) is developed based on the flow conservation constraints of jobs on machines to reduce the computational complexity. By mapping the MILP model onto a network formulation, an equivalent integer set partitioning type formulation is developed for a branch-and-price optimization algorithm. Computational experiments carried out over different sets of instances, indicate the efficiency and effectiveness of the optimization algorithm, compared to the linear relaxation and relaxed MILP models. Regarding the only available benchmark in the literature, the optimization algorithm yields optimal solutions with affordable computational time.
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Lower and upper bounds on batch sizes determine the minimum and maximum number of jobs assigned to batches, respectively, while the capacity on batch sizes restricts the total size of “jobs’ attributes”, like weight or volume.
\({C}_{\rm max}\): makespan; \({E}_{max}\): maximum earliness; \({T}_{max}\): maximum tardiness; \({L}_{max}\): maximum lateness.
\({C}_{j}\): job completion time; \({E}_{j}\): job earliness; \({T}_{j}\): job tardiness; \({U}_{j}\): number of tardy jobs; \({w}_{j}\): job weight.
\({TP}_{h}\): batch processing time; \({TP}_{l}\): batch performing time; \(\# batch\): limitation on the number of developed batches; \(OC\): outsourcing cost.
POA: Pareto Optimality Algorithm; VNS-GSA: Variable Neighborhood Search—Gravitational Search algorithm; SV-VNS: Society and Civilization—Variable Neighborhood Search algorithm; B&B: Branch-and-Bound algorithm; VNS-NKEA: Variable Neighborhood Search—Neighborhood Knowledge-Based Evolutionary algorithm; BRKGA-DE: Biased Random-Key Genetic—Differential Evolution algorithm; TS: Tabu Search; ABS-TS: Artificial Bee Colony—Tabu Search algorithm; SFLA-VNS: Shuffle Frog Leap—Variable Neighborhood Search algorithm; B&P: Branch-and-Price algorithm; PSO: Particle Swarm Optimization; BA-VNS: Bat Algorithm—Variable Neighborhood Search algorithm; TS/PR: Tabu Search/Path-Relinking; EDA-DE: Estimation Of Distribution Algorithm—Differential Evolution algorithm; JIT: Just In Time.
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Acknowledgements
The authors would like to thank the anonymous reviewers and the editors for their insightful comments and suggestions. Dr. Madjid Tavana is grateful for the partial support he received from the Czech Science Foundation (GAˇCR19-13946S) for this research.
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Appendices
Appendix A
Illustrations for group scheduling and batch scheduling were scaled down to smaller sizes in s. 1.
-
One time unit represents 2.5-time units.
# of groups | \(g=8\) | |
|---|---|---|
# of jobs of all groups | \(n=25\) | |
# of machines | \(m=5\) | |
\(\alpha =60\%\) | ||
\(\beta =40\%\) | ||
# of jobs in group \(i\) | Availability time of machine \(h\) (\({a}^{h}\)) | |
\({n}_{1}=3\) | \({a}^{1}=0\) | |
\({n}_{2}=4\) | \({a}^{2}=0\) | |
\({n}_{3}=3\) | \({a}^{3}=0\) | |
\({n}_{4}=3\) | \({a}^{4}=2\) | |
\({n}_{5}=3\) | \({a}^{5}=0\) | |
\({n}_{6}=3\) | ||
\({n}_{7}=3\) | ||
\({n}_{8}=3\) |
Jobs of group \({i}\) | Job \({j}\) | Weight of Job \({j}\) (\({{\varvec{w}}}_{{j}}\)) | Due date of Job \({j}\) ( \({{\varvec{d}}}_{{j}}\) ) | Release time of Job \({j}\) (\({{r}}_{{j}}\)) | Processing time of job \({j}\) on machine \({\varvec{h}}\) (\({{\varvec{t}}}_{{j}}^{{\varvec{h}}}\)) | ||||
|---|---|---|---|---|---|---|---|---|---|
\(h=1\) | \(h=2\) | \(h=3\) | \(h=4\) | \(h=5\) | |||||
\({i}=1\) | \(j=1\) | 4 | 4 | 2 | 1 | 4 | 2 | 2 | \(\infty \) |
\(j=2\) | 3 | 4 | 3 | 2 | 1 | 3 | 3 | \(\infty \) | |
\(j=3\) | 2 | 5 | 3 | 2 | \(\infty \) | 2 | 5 | \(\infty \) | |
\({i}=2\) | \(j=4\) | 3 | 11 | 8 | 2 | 3 | 3 | 2 | 4 |
\(j=5\) | 3 | 13 | 10 | 2 | 5 | 2 | \(\infty \) | 4 | |
\(j=6\) | 2 | 15 | 15 | 1 | 4 | 1 | 3 | \(\infty \) | |
\(j=7\) | 2 | 15 | 11 | 3 | \(\infty \) | 4 | 4 | \(\infty \) | |
\({i}=3\) | \(j=8\) | 3 | 7 | 5 | \(\infty \) | 1 | 5 | 3 | \(\infty \) |
\(j=9\) | 4 | 4 | 2 | \(\infty \) | 3 | 5 | 4 | 1 | |
\(j=10\) | 2 | 7 | 5 | 2 | 1 | 3 | \(\infty \) | \(\infty \) | |
\({i}=4\) | \(j=11\) | 3 | 12 | 8 | \(\infty \) | 1 | 4 | 1 | 2 |
\(j=12\) | 2 | 13 | 12 | \(\infty \) | 1 | \(\infty \) | 3 | \(\infty \) | |
\(j=13\) | 2 | 15 | 10 | 2 | 2 | 5 | 3 | \(\infty \) | |
\({i}=5\) | \(j=14\) | 3 | 16 | 5 | \(\infty \) | 3 | 3 | \(\infty \) | 4 |
\(j=15\) | 3 | 12 | 8 | 4 | 2 | 3 | 3 | 5 | |
\(j=16\) | 2 | 19 | 11 | \(\infty \) | \(\infty \) | 2 | 2 | \(\infty \) | |
\({i}=6\) | \(j=17\) | 3 | 8 | 6 | 2 | 4 | 3 | 2 | 2 |
\(j=18\) | 2 | 10 | 8 | \(\infty \) | \(\infty \) | 2 | 1 | \(\infty \) | |
\(j=19\) | 2 | 10 | 8 | 3 | \(\infty \) | 1 | 2 | \(\infty \) | |
\({i}=7\) | \(j=20\) | 4 | 10 | 3 | \(\infty \) | 3 | 2 | 1 | 5 |
\(j=21\) | 4 | 12 | 4 | \(\infty \) | 3 | 1 | 1 | \(\infty \) | |
\(j=22\) | 1 | 18 | 13 | 4 | 4 | 5 | 4 | \(\infty \) | |
\({i}=8\) | \(j=23\) | 2 | 15 | 1 | 2 | \(\infty \) | \(\infty \) | 2 | 1 |
\(j=24\) | 3 | 8 | 3 | \(\infty \) | 4 | \(\infty \) | 3 | 4 | |
\(j=25\) | 3 | 10 | 5 | \(\infty \) | 4 | 4 | 4 | 5 | |
M|C h: Machine h
M|C 1 | Group \({i}^{{\prime}}\) | |||||||
|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
Group \(i\) | ||||||||
0 | 1 | 1 | 3 | 9 | 1 | 1 | 3 | 7 |
1 | 0 | 4 | 3 | 4 | 9 | 9 | 4 | 2 |
2 | 6 | 0 | 10 | 4 | 4 | 2 | 2 | 8 |
3 | 6 | 6 | 0 | 2 | 3 | 6 | 9 | 1 |
4 | 9 | 8 | 9 | 0 | 6 | 9 | 4 | 2 |
5 | 10 | 7 | 7 | 1 | 0 | 1 | 10 | 3 |
6 | 7 | 1 | 9 | 7 | 9 | 0 | 4 | 3 |
7 | 7 | 2 | 1 | 5 | 5 | 7 | 0 | 3 |
8 | 2 | 1 | 7 | 5 | 3 | 10 | 10 | 0 |
M|C 2 | Group \({i}^{{\prime}}\) | |||||||
|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
Group \(i\) | ||||||||
0 | 2 | 1 | 4 | 1 | 5 | 9 | 2 | 1 |
1 | 0 | 7 | 4 | 5 | 4 | 2 | 3 | 9 |
2 | 9 | 0 | 1 | 4 | 4 | 3 | 7 | 4 |
3 | 10 | 2 | 0 | 2 | 5 | 3 | 10 | 9 |
4 | 5 | 3 | 6 | 0 | 5 | 5 | 2 | 2 |
5 | 2 | 9 | 6 | 8 | 0 | 2 | 1 | 1 |
6 | 8 | 3 | 4 | 10 | 1 | 0 | 6 | 9 |
7 | 9 | 4 | 7 | 8 | 10 | 7 | 0 | 8 |
8 | 2 | 10 | 10 | 3 | 5 | 9 | 2 | 0 |
M|C 3 | Group \({i}^{{\prime}}\) | |||||||
|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
Group \(i\) | ||||||||
0 | 8 | 10 | 2 | 9 | 2 | 5 | 2 | 2 |
1 | 0 | 2 | 7 | 4 | 10 | 10 | 6 | 7 |
2 | 1 | 0 | 7 | 3 | 3 | 3 | 6 | 3 |
3 | 10 | 8 | 0 | 9 | 10 | 4 | 2 | 7 |
4 | 5 | 10 | 1 | 0 | 4 | 3 | 8 | 4 |
5 | 5 | 9 | 6 | 6 | 0 | 7 | 10 | 4 |
6 | 2 | 7 | 9 | 5 | 1 | 0 | 7 | 10 |
7 | 7 | 7 | 8 | 2 | 3 | 7 | 0 | 3 |
8 | 10 | 8 | 8 | 2 | 10 | 3 | 6 | 0 |
M|C 4 | Group \({i}^{{\prime}}\) | |||||||
|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
Group \(i\) | ||||||||
0 | 6 | 1 | 9 | 3 | 8 | 8 | 2 | 8 |
1 | 0 | 5 | 7 | 3 | 8 | 8 | 7 | 7 |
2 | 1 | 0 | 7 | 6 | 5 | 2 | 3 | 10 |
3 | 1 | 7 | 0 | 10 | 6 | 9 | 5 | 8 |
4 | 10 | 7 | 7 | 0 | 1 | 9 | 1 | 10 |
5 | 9 | 9 | 5 | 1 | 0 | 7 | 8 | 6 |
6 | 2 | 3 | 4 | 9 | 6 | 0 | 3 | 9 |
7 | 6 | 7 | 9 | 6 | 2 | 2 | 0 | 6 |
8 | 4 | 9 | 9 | 2 | 5 | 4 | 3 | 0 |
M|C 5 | Group \({i}^{{\prime}}\) | |||||||
|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
Group \(i\) | ||||||||
0 | \(\infty \) | 2 | 1 | 7 | 8 | 2 | 3 | 1 |
1 | \(\infty \) | \(\infty \) | \(\infty \) | \(\infty \) | \(\infty \) | \(\infty \) | \(\infty \) | \(\infty \) |
2 | 7 | 0 | 1 | 1 | 5 | 8 | 9 | 9 |
3 | 10 | 6 | 0 | 10 | 10 | 2 | 1 | 1 |
4 | 5 | 5 | 4 | 0 | 9 | 4 | 7 | 5 |
5 | 4 | 2 | 6 | 3 | 0 | 10 | 9 | 5 |
6 | 4 | 8 | 10 | 6 | 2 | 0 | 2 | 10 |
7 | 5 | 2 | 10 | 2 | 1 | 2 | 0 | 2 |
8 | 8 | 8 | 4 | 4 | 7 | 7 | 8 | 0 |
\({LB}_{i}^{h}\) | Machine \(h\) | ||||
|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | |
Group \(i\) | |||||
1 | 2 | 1 | 3 | 3 | 2 |
2 | 4 | 3 | 4 | 4 | 3 |
3 | 3 | 2 | 3 | 2 | 1 |
4 | 3 | 3 | 3 | 2 | 3 |
5 | 2 | 3 | 3 | 3 | 2 |
6 | 2 | 3 | 2 | 3 | 1 |
7 | 3 | 3 | 2 | 1 | 3 |
8 | 2 | 3 | 3 | 3 | 3 |
Appendix B
Table B-8: CPLEX Comparison for loosely-loaded problems at the three levels of problem complexity.
\(g\) | \(m\) | Low complexity | Medium complexity | High complexity | ||||||
|---|---|---|---|---|---|---|---|---|---|---|
\(IP\) | \(LP\) | \(RP\) | \(IP\) | \(LP\) | \(RP\) | \(IP\) | \(LP\) | \(RP\) | ||
5 | 3 | 1525.59 | 1450.83 | 1515.19 | 1447.38 | 1339.91 | 1437.61 | 2248.29 | 1999.55 | 2231.14 |
4 | 1365.21 | 1323.12 | 1354.38 | 1093.95 | 1055.18 | 1078.72 | 2400.75 | 2306.31 | 2370.33 | |
5 | 1431.54 | 1378.58 | 1419.48 | 1776.06 | 1700.53 | 1761.67 | 2582.91 | 2466.48 | 2554.67 | |
6 | 1552.32 | 1518.97 | 1531.00 | 925.65 | 895.97 | 908.47 | 1986.93 | 1925.77 | 1931.60 | |
6 | 4 | 1139.49 | 1100.24 | 1126.34 | 2016.63 | 1933.93 | 1992.13 | 2935.35 | 2775.26 | 2893.98 |
5 | 1584.00 | 1517.78 | 1575.82 | 1584.99 | 1513.74 | 1575.36 | 877.14 | 840.07 | 872.02 | |
6 | 718.74 | 701.27 | 707.20 | 2084.94 | 2021.61 | 2030.05 | 1440.45 | 1398.01 | 1402.91 | |
7 | 4 | 856.35 | 822.63 | 843.24 | 1187.01 | 1140.74 | 1153.36 | 2033.46 | 1957.36 | 1956.21 |
5 | 1106.82 | 1063.05 | 1103.67 | 1813.68 | 1736.55 | 1805.97 | 2585.88 | 2455.02 | 2573.23 | |
6 | 1712.70 | 1687.33 | 1711.08 | 1216.71 | 1189.40 | 1215.27 | 3148.20 | 3062.00 | 3142.66 | |
8 | 5 | 1507.77 | 1461.82 | 1478.46 | 1480.05 | 1434.49 | 1444.08 | 3264.03 | 3134.43 | 3156.95 |
6 | 847.44 | 802.67 | 835.84 | 1446.39 | 1362.43 | 1421.01 | 2845.26 | 2599.16 | 2791.83 | |
9 | 5 | 693.00 | 641.65 | 667.16 | 1476.09 | 1373.86 | 1416.90 | 2267.10 | 2072.41 | 2151.65 |
6 | 1224.63 | 1156.93 | 1184.79 | 1955.25 | 1836.47 | 1892.89 | 2271.06 | 2066.87 | 2177.62 | |
10 | 6 | 693.99 | 661.34 | 690.62 | 1342.44 | 1265.42 | 1333.38 | 1411.74 | 1338.84 | 1398.21 |
11 | 6 | 1715.67 | 1640.86 | 1679.84 | 1514.70 | 1433.16 | 1472.52 | 2923.47 | 2762.95 | 2820.90 |
Table B-9: CPLEX Comparison for moderately-loaded problems at the three levels of problem complexity.
\(g\) | \(m\) | Low complexity | Medium complexity | High complexity | ||||||
|---|---|---|---|---|---|---|---|---|---|---|
\(IP\) | \(LP\) | \(RP\) | \(IP\) | \(LP\) | \(RP\) | \(IP\) | \(LP\) | \(RP\) | ||
6 | 3 | 1782.99 | 1723.31 | 1699.58 | 2661.12 | 2504.84 | 2547.95 | 1603.80 | 1495.49 | 1501.49 |
7 | 3 | 1925.55 | 1814.14 | 1797.81 | 1147.41 | 1063.38 | 1076.93 | 2710.62 | 2522.03 | 2469.19 |
8 | 3 | 1534.50 | 1435.06 | 1502.14 | 2522.52 | 2359.06 | 2447.60 | 3075.93 | 2866.64 | 2993.71 |
4 | 1250.37 | 1179.78 | 1207.30 | 1582.02 | 1473.10 | 1514.74 | 3363.03 | 3052.76 | 3215.72 | |
9 | 4 | 1735.47 | 1655.29 | 1717.09 | 2380.95 | 2169.41 | 2353.83 | 3165.03 | 2805.09 | 3112.39 |
10 | 4 | 1131.57 | 1084.51 | 1087.43 | 2293.83 | 2113.83 | 2160.28 | 2323.53 | 2148.50 | 2128.73 |
5 | 1646.37 | 1598.76 | 1594.36 | 1144.44 | 1079.55 | 1098.09 | 3601.62 | 3340.23 | 3423.66 | |
11 | 4 | 1752.30 | 1660.57 | 1644.71 | 1870.11 | 1765.97 | 1760.76 | 2546.28 | 2404.48 | 2375.06 |
5 | 1526.58 | 1421.06 | 1482.30 | 1258.29 | 1177.01 | 1219.05 | 3257.10 | 2998.31 | 3107.78 | |
12 | 5 | 1287.99 | 1171.87 | 1275.60 | 1184.04 | 1083.34 | 1166.78 | 1461.24 | 1292.22 | 1437.16 |
6 | 1761.21 | 1660.21 | 1732.86 | 1966.14 | 1775.03 | 1906.42 | 2734.38 | 2410.12 | 2628.89 | |
13 | 5 | 1738.44 | 1655.65 | 1697.39 | 1872.09 | 1739.03 | 1808.02 | 2736.36 | 2555.48 | 2600.57 |
6 | 1804.77 | 1651.51 | 1706.97 | 2311.65 | 2063.16 | 2179.80 | 3414.51 | 2930.01 | 3206.12 | |
14 | 5 | 1642.41 | 1545.61 | 1540.71 | 2561.13 | 2321.52 | 2357.82 | 3134.34 | 2709.15 | 2858.16 |
6 | 1504.80 | 1463.56 | 1491.11 | 1347.39 | 1283.73 | 1331.47 | 3353.13 | 3169.34 | 3299.26 | |
Table B-10: CPLEX Comparison for tightly-loaded problems at the three levels of problem complexity.
\(g\) | \(m\) | Low complexity | Medium complexity | High complexity | ||||||
|---|---|---|---|---|---|---|---|---|---|---|
\(IP\) | \(LP\) | \(RP\) | \(IP\) | \(LP\) | \(RP\) | \(IP\) | \(LP\) | \(RP\) | ||
9 | 3 | 1818.63 | 1730.46 | 1745.93 | 2398.77 | 2260.33 | 2296.76 | 1970.10 | 1806.37 | 1871.24 |
10 | 3 | 1630.53 | 1577.15 | 1510.17 | 2672.01 | 2581.83 | 2428.52 | 3345.21 | 3233.44 | 2961.11 |
11 | 3 | 1500.84 | 1392.86 | 1397.59 | 1932.48 | 1796.17 | 1810.51 | 3677.85 | 3397.67 | 3329.66 |
12 | 3 | 1493.91 | 1440.91 | 1403.88 | 1930.50 | 1836.68 | 1805.40 | 3584.79 | 3382.69 | 3336.23 |
4 | 1651.32 | 1513.68 | 1530.94 | 1927.53 | 1640.64 | 1751.88 | 3597.66 | 2901.55 | 3148.51 | |
13 | 3 | 1867.14 | 1822.90 | 1845.27 | 1584.00 | 1514.50 | 1563.15 | 1823.58 | 1704.36 | 1788.53 |
4 | 1500.84 | 1341.32 | 1475.06 | 1954.26 | 1735.61 | 1896.38 | 2138.40 | 1856.09 | 2079.50 | |
14 | 3 | 1936.44 | 1709.16 | 1780.80 | 1531.53 | 1271.01 | 1336.14 | 2948.22 | 2396.57 | 2579.61 |
4 | 1595.88 | 1415.13 | 1572.81 | 2024.55 | 1687.35 | 1967.15 | 3522.42 | 2642.39 | 3412.57 | |
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Shahvari, O., Logendran, R. & Tavana, M. An efficient model-based branch-and-price algorithm for unrelated-parallel machine batching and scheduling problems. J Sched 25, 589–621 (2022). https://doi.org/10.1007/s10951-022-00729-7
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DOI: https://doi.org/10.1007/s10951-022-00729-7