在生產管理中，在有限的資源下，盡可能滿足顧客的需求是最主要的目的。因此，本研究以最小化延遲成本為目標，探討了多機台環境下之困難生產排程問題。在本研究中，所有工作的開始生產時間皆被限制不得早於其可以開始生產的時間。在理想情況下，工作的完工時間需早於其承諾交付予顧客之時間，倘若不可實現，將因延遲交貨而必須支付額外的賠償費用。此外，由於每個工作的生產條件皆不同，並非每台機器均能生產某些特定的工作。因此，此研究亦須將機台限制的特性納入考量。
本研究採用三階段方法以求解此類問題。在第一階段，利用派工法則產生起始解。第二階段，測試四種平行模擬退火法，該方法利用多執行緒的平行運算去加速求解問題的效率。第三階段，測試與比較五種不同的混整數規劃模型，利用混整數規劃模型對第二階段的最終解進行優化。由於第一階段的求解速度非常快，但第二階段與第三階段都需要大量的計算時間，針對第二階段與第三階段，本研究完整利用了中央處理器中的八個執行緒以求得更好的解。本研究也會探討如何分配第二階段與第三階段的時間才是最好的組合。實驗結果顯示，混合平行模擬退火法以及混整數規劃所找到的解優於單獨使用混整數規劃或是平行模擬退火法所求得的解。所以三階段平行計算方法在此困難排程問題中的效果最為傑出。

The purpose of production management is to fulfill customer demands as much and as on-time as possible under limited resources. This study addresses a NP-hard scheduling problem with the objective of minimizing total tardiness on multiple-machine environments. In this study, each job is restricted to be started later than its ready date, and it is desired to be completed before its due date. If a delay cannot be avoided, a penalty for tardiness occurs. Besides, machine eligibility is taken into account; that is, only the machines in the eligible machine set of a job can be used to process the job.
This study adopts parallel-computation three-stage approach to solve the scheduling problem. In Stage 1, an initial solution is generated by applying dispatching rule. For Stage 2, four parallel simulated annealing (SA) search on multi-thread CPU computing platform are tested. Five mixed integer programming (MIP) models are tested and compared for Stage 3, which further optimize the solution from Stage 2. However, both SA search and solving MIP take extensive amount of computation time, Stages 2 and 3 are set to execute using 8 threads, which fully utilize the parallel computation capability of the CPU used. Under a given fixed amount of computation time, how to distribute the computation time between Stages 2 and 3 is investigated. According to experiment results, the hybrid of SA search and MIP model is better than only SA search or only MIP model. That is, the three-stage approach is an effective method in solving the NP-hard problem.

摘要 I
Abstract II
LIST OF FIGURES V
LIST OF TABLES VII
1. Introduction 1
1.1. Problem Statement and Research Approach 1
1.1.1. Problem Statement 1
1.1.2. Research Approach 3
2. Literature Review 6
2.1. Scheduling Problems on Parallel Machine 6
2.1.1. Time Window 7
2.1.2. Machine Eligibility 7
2.1.3 Modelling Techniques 8
2.2. Parallel Computing for Scheduling Problems 9
2.3. The Application of Simulated Annealing Method 10
2.3.1 Simulated Annealing Method 10
2.3.2 Parallel Simulated Annealing Method 10
3. Solution Methods 12
3.1. Stage 1 – Initial Solution by Dispatching Rule 12
3.2. Stage 2 – Four Parallel SA Search Algorithms 14
3.2.1. Neighborhood Solution 15
3.2.2. Simulated Annealing Algorithm 19
3.2.3. Parallel Simulated Annealing (PSA) 23
3.3. Stage 3 – Five MIP Models 28
3.3.1. Notations 29
3.3.2. Network MIP (NMIP) 30
3.3.3. Immediate Precedence MIP (IPMIP) 32
3.3.4. Linear Ordering MIP (LOMIP) 34
3.3.5. Modified Linear Ordering MIP (MLOMIP) 35
3.3.6. Assignment and Positional Date MIP (APMIP) 37
4. Computation Experiment 40
4.1. Experimental Parameters 40
4.2. Problem Generation Procedure 41
4.3. Parameter Setting 43
4.4. Experimental Results 44
4.4.1. Comparison of Four PSAs with Initial Solution 45
4.4.2. Comparison of Five MIP Models without Initial Solution 47
4.4.3. Comparison of Five MIP Models with Initial Solution 50
4.4.4. The Performance of the Three Stage Approach 52
4.5. Factorial Analysis of Three Stage Approach 56
5. Conclusions and Future Research 62
Appendix A: Time-Based Partition Parallel Search Algorithm (TBP) 64
Reference 78

Aarts, E. H., Bont, F. M., Habers, E. H., and Laarhoven, P. J., (1986),” Parallel implementations of the statistical cooling algorithm”, Integration 4, pp. 209-238.
Afzalirad, M., and Rezaeian, J., (2016), “Resource-constrained unrelated parallel machine scheduling problem with sequence dependent setup times, precedence constraints and machine eligibility restrictions”, Computers & Industrial Engineering, Vol. 98 pp. 40-52.
AitZai, A., and Boudhar, M., (2013), “Parallel branch-and-bound and parallel PSO algorithms for job shop scheduling problem with blocking”, International Journal of Operational Research, Vol. 16, Issue. 1, pp. 14-37.
AitZai, A., Boudhar, M., and Dabah, A., (2013), Parallel CPU and GPU Computations to Solve the Job Shop Scheduling Problem with Blocking, IEEE High Performance Extreme Computing Conference, Waltham, USA.
Akhshabi, M., Haddadnia, J., and Akhshabi, M., (2012), “Solving flow shop scheduling problem using a parallel genetic algorithm”, Procedia Technology, Vol. 1, pp. 351-355.
Allahverdi, A., (2015), “The third comprehensive survey on scheduling problems with setup times/costs”, European Journal of Operational Research, Vol. 246, No. 2, pp. 345-378.
Anghinolfi, D., and Paolucci, M., (2007), “Parallel machine total tardiness scheduling with a new hybrid. metaheuristic approach”, Computers & Operations Research, Vol. 34, No. 11, pp. 3471–3490.
Bevilacqua, A., (2002), “A methodological approach to parallel simulated annealing on an SMP system”, Journal of Parallel and Distributed Computing, Vol. 62, No. 10, pp. 1548-1570.
Bozejko, W., and Wodecki, M., (2002), Solving the flow shop problem by parallel tabu search. Parallel Computing in Electrical Engineering, Proceedings. International Conference, IEEE, pp. 189-194.
Bożejko, W., and Wodecki, M., (2003), Parallel genetic algorithm for the flow shop scheduling problem. International Conference on Parallel Processing and Applied Mathematics, Heidelberg, GERMANY, pp. 566-571.
Bożejko, W., and Wodecki, M., (2008). “Parallel scatter search algorithm for the flow shop sequencing problem”, Parallel Processing and Applied Mathematics, pp. 180-188.
Bozejko, W., and Wodecki, M., (2011), “The methodology of parallel memetic algorithms designing”, ICAART-2011 3rd International Conference on Agents and Artificial Intelligence, Rome, ITALY, pp. 643-648.
Bozorgirad, M. A. and Logendran, R., (2011), “Sequence-dependent group scheduling problem on unrelated-parallel machines”, Expert Systems with Applications, Vol. 39, pp. 9021-9030.
Cakici, E., and Mason, S. J., (2007), “Parallel machine scheduling subject to auxiliary resource constraints”, Production Planning & Control, Vol. 18, No. 3, pp. 217-225.
Centeno, G., and Armacost, R.L., (2004), “Minimizing makespan on parallel machines with release time and machine eligibility restrictions” International Journal of Production Research, Vol. 42, pp. 1243-1256.
Centeno, G., Armacost, R. L., (1997), “Parallel machine scheduling with release time and machine eligibility restrictions”, Computers & Industrial Engineering, Vol. 33 No. 1-2, pp. 273-276.
Chen, C-L., (2008), “An iterated local search for unrelated parallel machines problem with unequal ready times”, Proceedings of the IEEE International Conference on Automation and Logistics, Qingdao, China.
Chen, J., (2005), “Unrelated parallel machine scheduling with secondary resource Constraints”, International Journal of Advanced Manufacturing Technology, Vol. 26, No. 3, pp. 285-292.
Chen, J., (2006), “Minimization of maximum tardiness on unrelated parallel machines with process restrictions and setups”, International Journal of Advanced Manufacturing Technology, Vol. 29, No. 5, pp. 557-563.
Cheng, T. C. E., and Sin, C. C. S., (1990) “A state-of-the-art review of parallel-machine scheduling research”, European Journal of Operational Research, Vol. 47, pp. 271-292.
Defersha, F. M., and Chen, M., (2010), “A parallel genetic algorithm for a flexible job-shop scheduling problem with sequence dependent setups”, The international Journal of Advanced Manufacturing Technology, Vol. 49, Issue. 1, pp. 263-279.
Drobouchevitch, I. G., and Sidney, J. B., (2012), “Minimization of earliness, tardiness
and due date penalties on uniform parallel machines with identical jobs”, Computers and Industrial Engineering, Vol. 39, No. 9, pp. 1919-1926.
Gao, J. (2005), A Parallel Hybrid Genetic Algorithm for Solving a Kind of Non-Identical Parallel Machine Scheduling Problems, 2005 IEEE High-Performance Computing Conference, Beijing, China, pp. 469-472.
Gao, J., He, G., and Wang, Y., (2009), “A new parallel genetic algorithm for solving multiobjective scheduling problems subjected to special process constraint”, The International Journal of Advanced Manufacturing Technology, Vol. 43, Issue. 1, pp. 151-160.
Gharehgozli, A. H., Tavakkoli-Moghaddam, R., Zaerpour, N., (2009), “A fuzzy-mixed-integer goal programming model for a parallel-machine scheduling problem with sequence-dependent setup times and release dates”, Robotics and Computer-Integrated Manufacturing, Vol. 25, No. 4-5, pp. 853-859.
Gokhale, R., and Mathirajan, M., (2012), “Scheduling identical parallel machines with machine eligibility restrictions to minimize total weighted flowtime in automobile gear manufacturing”, International Journal of Advanced Manufacturing Technology, Vol. 60, pp. 1099-1110.
Graham, R.L., Lawler, E.L., Lenstra, J.K., and Rinnooy Kan, A.H.G., (1979) "Optimization and approximation in deterministic sequencing and scheduling: a survey", Annals of discrete mathematics Vol. 2, pp. 287-326.
Hu, T. C. (1961), “Parallel sequencing and assembly line problems”, Operations Research, Vol. 9, No. 6, pp. 841-948.
Hung, Y.F., Bao, J.S., and Cheng, Y.E. (2016), “Minimizing earliness and tardiness costs in scheduling jobs with time windows”, Computer and Industrial Engineering, Accepted.
Hung, Y. F., Chen, W. C., and Chen, J. C., (2012), “Search algorithms in the selection of warehouses and transshipment arrangement for high-end low-volume products”, Journal of Advanced Engineering, Vol. 7, No. 2, pp. 51-60.
Janiak, A., Janiak, W., Kovalyov, M. Y., Kozan, E., and Pesch, E., (2013), “Parallel machine scheduling and common due window assignment with job independent earliness and tardiness costs”, Information Sciences, Vol. 224, No. 1, pp. 109-117.
Kleinberg, J., and Tardos, E., (2005), Algorithm Design, 1th edition, Cornell, New York,
U.S.A..
Keha, A.B., Khowala, K., and Fowler, J.W., (2009), “Mixed integer programming formulations for single machine scheduling problems”, Computers & Industrial Engineering, Vol. 56, pp. 357-367.
Khowala, K., Keha, A. B., and Fowler, J.W., (2005), “A comparison of different formulations for the non-preemptive single machine total weighted tardiness scheduling problem” The Second Multidisciplinary International Conference on Scheduling: Theory & Application (MISTA).
Kim, D. W., Kim, K. H., Jang, W., and Chen, F. F., (2002), “Unrelated parallel machine scheduling with setup times using simulated annealing”, Robotics and Computer Integrated Manufacturing, Vol. 18, pp. 223–231.
Kirkpatrick, S., Gelatt, Jr., C.D. and Vecchi, M.P., (1983), “Optimization by simulated annealing”, Science, Vol. 200, No. 4598, pp. 671–680.
Knopman, J., and Aude, J. S., (1997), Parallel simulated annealing: An adaptive approach, IEEE 11th International Parallel Processing Symposium, Geneva, SWITZERLAND.
Lawler, E. L., (1977). “A pseudo polynomial time algorithm for sequencing jobs to minimize total tardiness”. Annals of Discrete Mathematics, Vol. 1, pp. 331-342.
Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A.H.G., and Shmoys, D.B. (1993) “Sequencing and scheduling: Algorithms and complexity”, Handbook in Operations Research and Management Science 4: Logistics of Production and Inventory, Vol. 4, pp.445-521.
Lee, C. Y., and Kim, S. J., (1995), ”Parallel genetic algorithms for the earliness-tardiness job scheduling problem with general penalty weights”, Computers & Industrial Engineering, Vol. 28, No. 2, pp. 231-243.
Lee, K., Leung, J.Y.T., and Pinedo, M., (2011), “Scheduling jobs with equal processing times subject to machine eligibility constraints”, Journal of Scheduling, Vol. 14, No. 1, pp. 27–38.
Lee, K., Leung, J.Y.T., and Pinedo, M. L., (2009), “Online scheduling on two uniform machines subject to eligibility constraints”, Theoretical Computer Science, Vol. 410, No. 38, pp. 3975-3981.
Lee, W.C., Wu, C.C., and Chen, P., (2006), “A simulated annealing approach to makespan minimization on identical parallel machines”, International Journal of Advanced Manufacturing Technology, Vol. 31, pp. 328-334.
Li, C. L., (2006), “Scheduling unit-length jobs with machine eligibility restrictions”, European Journal of Operational Research, Vol.174, No.2, pp.1325-1328.
Li, H.L., Huang, Y.H., and Fang, S.C., (2013), “A logarithmic method for reducing binary variables and inequality constraints in solving task assignment problems”, INFORMS Journal on Computing, Vol. 25, Issue. 4, pp. 643-653.
Lopes, M. J. P., and Carvalho, J. M. V., (2007), “A branch-and-price algorithm for scheduling parallel machines with sequence dependent setup times”, European Journal of Operational Research, Vol.176, No. 3, pp. 1508-1527.
Lundy, M., and Mess, A., (1986), “Convergence of an annealing algorithm,” Mathematical Programming, Vol. 34, pp. 111-124.
Ma, Y., Chu, C. and Zuo, C., (2010), “A survey of scheduling with deterministic
machine availability constraints”, Computers and Industrial Engineering, Vol. 58, No. 2, pp. 199-211.
Malek, M., Guruswamy, M., Pandya, M., and Owens, H., (1989), “Serial and parallel simulated annealing and tabu search algorithms for the traveling salesman problem”, Annals of Operations Research, Vol. 21, No. 1, pp. 59-84.
McNaughton, R. (1959) “Scheduling with deadlines and loss functions”, Management Science, Vol. 6, pp. 1-12.
Mehravaran, Y., and Logendran, R., (2011), “Non-permutation flowshop scheduling in a supply chain with sequence-dependent setup times”, International Journal of Production Economics, Vol. 135, No. 2, pp. 953-963.
Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A. and Teller, E., (1953), “Equation of state calculations by fast computing machines”, Journal of Chemical Physics, Vol. 21, No. 6, pp. 1087–1092.
Mokotoff, E., (1999), “Scheduling to Minimize the Makespan on Identical Parallel Machines: An LP-Based Algorithm”, Investigacion Operativa, Vol. 8 No. 1-3, pp. 97-107.
Park, M.W., and Kim, Y.D., (1997), “Search heuristics for a parallel machine scheduling problem with ready times and due dates”, Computers & Operations Research, Vol. 33, Issue. 3-4, pp. 793-796.
Perregaard, M., and Clausen, J., (1998), “Parallel branch-and-bound methods for the job-shop scheduling problem”, Annals of Operations Research, Vol. 83, pp. 137-160.
Pfund, M., Fowlwr, J. W., and Gupta, J. N. D., (2004), “A Survey of Algorithms for Single Machine and Multi-Objective Unrelated Parallel-Machine Deterministic Scheduling Problems”, Journal of the Chinese Institute of Industrial Engineers, Vol. 21 No. 3, pp. 230-241.
Pinedo, M.L., (2002), Scheduling: Theory, Algorithm and System, 2nded., Prentice Hall.
Rashidi, E., Jahandar, M., and Zandieh, M., (2010), “An improved hybrid multi-objective parallel genetic algorithm for hybrid flow shop scheduling with unrelated parallel machines”, The International Journal of Advanced Manufacturing Technology, Vol. 49, Issue. 9, pp. 1129-1139.
Rosa, B. F., Souza, M. J. F., Souza, S. R., Filho, M. F. F., Ales, Z., Michelon, P. Y. P., (2017), “Algorithms for job scheduling problems with distinct time windows and general earliness/tardiness penalties”, Computers & Operations Research, Vol. 81, pp. 203-215.
Safaei, N., Banjevic, D., and Jardine, A. K., (2012), “Multi-threaded simulated annealing for a bi-objective maintenance scheduling problem”, International Journal of Production Research, Vol. 50, No. 1, pp. 63-80.
Sheen, G. J., and Liao, L.W., (2007), “Scheduling machine-dependent jobs to minimize lateness on machines with identical speed under availability constraints” Computers & Operations Research, Vol. 34, pp. 2266-2278.
Tavakkoli-Moghaddam, R., Taheri, F., Bazzazi, M., Izadi, M., and Sassani, F., (2009), “Design of a genetic algorithm for bi-objective unrelated parallel machines scheduling with sequence-dependent setup times and precedence constraints”, Computers & Operations Research, Vol. 36, pp. 3224-3230.
Thiruvady, D., Ernst, A. T., and Singh, G., (2016), “Parallel ant colony optimization for resource constrained job scheduling”, Annals of Operations Research, Vol. 2, No. 242, pp. 355-372.
Unlu, Y., and Mason, S.J., (2010), “Evaluation of mixed integer programming formulations for non-preemptive parallel machine scheduling problems” Computers & Industrial Engineering, Vol. 58, pp. 785-800.
Vielma, J. P., and Nemhauser, G. L., (2011), “Modeling disjunctive constraints with a logarithmic number of binary variables and constraints” Math. Programming, Vol. 128, No. 2, pp. 49-72.
Wan, G., and Yen, B. P. C., (2002), “Tabu search for single machine scheduling with distinct due windows and weighted earliness/tardiness penalties”, European Journal of Operational Research, Vol. 142, No. 2, pp. 271-281.
Wang, C., Mu, D., Zhao, F., and Sutherland, J. W., (2015), “A parallel simulated annealing method for the vehicle routing problem with simultaneous pickup–delivery and time windows”, Computers & Industrial Engineering, Vol. 83, pp. 111-122.
Xi, Y. and Jang, J., (2012), “Scheduling jobs on identical parallel machines with unequal future ready time and sequence dependent setup An experimental study”, International Journal of Production Economics, Vol. 137, No. 1, pp. 1-10.