Abstract
This work deals with a very generic class of scheduling problems with identical/uniform/unrelated parallel machine environment. It considers well-known attributes such as release dates, deadlines, or sequence-dependent setup times and accepts any objective function defined over job completion times. Non-regular objectives are also supported. We introduce a branch-cut-and-price algorithm for such problems that makes use of non-robust cuts, i.e., cuts which change the structure of the pricing problem. This is the first time that such cuts are employed for machine scheduling problems. The algorithm also embeds other important techniques such as strong branching, reduced cost fixing and dual stabilization. Computational experiments over literature benchmarks showed that the proposed algorithm is indeed effective and could solve many instances to optimality for the first time.
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References
Achterberg, T. (2007). Constraint integer programming. PhD thesis, Technische Universitat Berlin.
Avella, P., Boccia, M., & D’Auria, B. (2005). Near-optimal solutions of large-scale single-machine scheduling problems. INFORMS Journal on Computing, 17(2), 183–191.
Baldacci, R., Mingozzi, A., & Roberti, R. (2011). New route relaxation and pricing strategies for the vehicle routing problem. Operations Research, 59(5), 1269–1283.
Bigras, L. P., Gamache, M., & Savard, G. (2008). The time-dependent traveling salesman problem and single machine scheduling problems with sequence dependent setup times. Discrete Optimization, 5(4), 685–699.
Bitar, A., Dauzère-Pérès, S., Yugma, C., & Roussel, R. (2016). A memetic algorithm to solve an unrelated parallel machine scheduling problem with auxiliary resources in semiconductor manufacturing. Journal of Scheduling, 19(4), 367–376.
Bülbül, K., & Şen, H. (2017). An exact extended formulation for the unrelated parallel machine total weighted completion time problem. Journal of Scheduling, 20(4), 373–389.
Chen, Z. L., & Powell, W. B. (1999). Solving parallel machine scheduling problems by column generation. INFORMS Journal on Computing, 11(1), 78–94.
Contardo, C., & Martinelli, R. (2014). A new exact algorithm for the multi-depot vehicle routing problem under capacity and route length constraints. Discrete Optimization, 12, 129–146.
Gauvin, C., Desaulniers, G., & Gendreau, M. (2014). A branch-cut-and-price algorithm for the vehicle routing problem with stochastic demands. Computers & Operations Research, 50, 141–153.
Graham, R., Lawler, E., Lenstra, J., & Kan, A. (1979). Optimization and approximation in deterministic sequencing and scheduling: A survey. In P. L. Hammer, E. L. Johnson, & B. H. Korte (Eds.), Discrete optimization II: Proceedings of the advanced research institute on discrete optimization and systems applications of the systems science panel of NATO and of the discrete optimization symposium co-sponsored by IBM Canada and SIAM Banff, Aha. and Vancouver, Annals of discrete mathematics, Vol 5. Elsevier, pp. 287–326.
Ioachim, I., Gélinas, S., Soumis, F., & Desrosiers, J. (1998). A dynamic programming algorithm for the shortest path problem with time windows and linear node costs. Networks, 31(3), 193–204.
Jepsen, M., Petersen, B., Spoorendonk, S., & Pisinger, D. (2008). Subset-row inequalities applied to the vehicle-routing problem with time windows. Operations Research, 56(2), 497–511.
Jepsen, M., Spoorendonk, S., & Ropke, S. (2013). A branch-and-cut algorithm for the symmetric two-echelon capacitated vehicle routing problem. Transportation Science, 47(1), 23–37.
Jouglet, A., & Savourey, D. (2011). Dominance rules for the parallel machine total weighted tardiness scheduling problem with release dates. Computers & Operations Research, 38(9), 1259–1266.
Kowalczyk, D., & Leus, R. (2018). A branch-and-price algorithm for parallel machine scheduling using ZDDs and generic branching. INFORMS Journal on Computing, 30(4), 768–782.
Kramer, A. (2015). Um método heurístico para a resolução de uma classe de problemas de sequenciamento da produção envolvendo penalidades por antecipação e atraso. Master’s thesis, Programa de Pós-Graduação em Engenharia de Produção, Universidade Federal da Paraíba, João Pessoa, Brazil (in Portuguese).
Kramer, A., & Subramanian, A. (2019). A unified heuristic and an annotated bibliography for a large class of earliness-tardiness scheduling problems. Journal of Scheduling, 22(1), 21–57.
Lenstra, J., Kan, A. R., & Brucker, P. (1977). Complexity of machine scheduling problems. In P. L. Hammer, E. L. Johnson, & G. Nemhauser (Eds.), Studies in integer programming. Annals of discrete mathematics (Vol. 1, pp. 343–362). Amsterdam: Elsevier.
Liaw, C. F., Lin, Y. K., Cheng, C. Y., & Chen, M. (2003). Scheduling unrelated parallel machines to minimize total weighted tardiness. Computers & Operations Research, 30(12), 1777–1789.
Nessah, R., Yalaoui, F., & Chu, C. (2008). A branch-and-bound algorithm to minimize total weighted completion time on identical parallel machines with job release dates. Computers & Operations Research, 35(4), 1176–1190.
Oliveira, D., & Pessoa, A. (2019). An improved branch-cut-and-price algorithm for parallel machine scheduling problems. INFORMS Journal on Computing Articles in Advance. https://doi.org/10.1287/ijoc.2018.0854.
Pan, Y., & Shi, L. (2008). New hybrid optimization algorithms for machine scheduling problems. IEEE Transactions on Automation Science and Engineering, 5(2), 337–348.
Pecin, D., Contardo, C., Desaulniers, G., & Uchoa, E. (2017a). New enhancements for the exact solution of the vehicle routing problem with time windows. INFORMS Journal on Computing, 29(3), 489–502.
Pecin, D., Pessoa, A., Poggi, M., & Uchoa, E. (2017b). Improved branch-cut-and-price for capacitated vehicle routing. Mathematical Programming Computation, 9(1), 61–100.
Pereira Lopes, M. J., & Valério de Carvalho, J. (2007). A branch-and-price algorithm for scheduling parallel machines with sequence dependent setup times. European Journal of Operational Research, 176(3), 1508–1527.
Pessoa, A., Uchoa, E., Poggi, M., & Rodrigues, R. (2010). Exact algorithm over an arc-time-indexed formulation for parallel machine scheduling problems. Mathematical Programming Computation, 2(3–4), 259–290.
Pessoa, A., Sadykov, R., Uchoa, E., & Vanderbeck, F. (2018). Automation and combination of linear-programming based stabilization techniques in column generation. INFORMS Journal on Computing, 30(2), 339–360.
Poggi, M., & Uchoa, E. (2003). Integer program reformulation for robust branch-and-cut-and-price. In L. Wolsey (Ed.), Annals of mathematical programming in Rio, Búzios, Brazil (pp. 56–61). Rio de Janeiro, Brazil: COPPE Sistemas.
Righini, G., & Salani, M. (2006). Symmetry helps: Bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints. Discrete Optimization, 3(3), 255–273.
Røpke, S. (2012). Branching decisions in branch-and-cut-and-price algorithms for vehicle routing problems. Presentation In Column Generation 2012.
Schaller, J. E. (2014). Minimizing total tardiness for scheduling identical parallel machines with family setups. Computers & Industrial Engineering, 72, 274–281.
Şen, H., & Bülbül, K. (2015). A strong preemptive relaxation for weighted tardiness and earliness/tardiness problems on unrelated parallel machines. INFORMS Journal on Computing, 27(1), 135–150.
Shim, S. O., & Kim, Y. D. (2007a). Minimizing total tardiness in an unrelated parallel-machine scheduling problem. Journal of the Operational Research Society, 58(3), 346–354.
Shim, S. O., & Kim, Y. D. (2007b). Scheduling on parallel identical machines to minimize total tardiness. European Journal of Operational Research, 177(1), 135–146.
Sourd, F. (2005). Earliness-tardiness scheduling with setup considerations. Computers & Operations Research, 32(7), 1849–1865.
Sourd, F., & Kedad-Sidhoum, S. (2003). The one-machine problem with earliness and tardiness penalties. Journal of Scheduling, 6(6), 533–549.
Sourd, F., & Kedad-Sidhoum, S. (2008). A faster branch-and-bound algorithm for the earliness-tardiness scheduling problem. Journal of Scheduling, 11(1), 49–58.
Tanaka, S., & Araki, M. (2008). A branch-and-bound algorithm with Lagrangian relaxation to minimize total tardiness on identical parallel machines. International Journal of Production Economics, 113(1), 446–458.
Tanaka, S., & Araki, M. (2013). An exact algorithm for the single-machine total weighted tardiness problem with sequence-dependent setup times. Computers & Operations Research, 40(1), 344–352.
Tanaka, S., & Fujikuma, S. (2008). An efficient exact algorithm for general single-machine scheduling with machine idle time. In: IEEE international conference on automation science and engineering, 2008. CASE 2008, pp. 371–376.
Tanaka, S., & Fujikuma, S. (2012). A dynamic-programming-based exact algorithm for general single-machine scheduling with machine idle time. Journal of Scheduling, 15(3), 347–361.
Tanaka, S., Fujikuma, S., & Araki, M. (2009). An exact algorithm for single-machine scheduling without machine idle time. Journal of Scheduling, 12(6), 575–593.
van den Akker, J., Hurkens, C., & Savelsbergh, M. (2000). Time-indexed formulations for machine scheduling problems: Column generation. INFORMS Journal on Computing, 12(2), 111–124.
Vanderbeck, F., Sadykov, R., & Tahiri, I. (2017). Bapcod—A generic branch-and-price code. Technical report. https://realopt.bordeaux.inria.fr/?page_id=2.
Yalaoui, F., & Chu, C. (2006). New exact method to solve the \({P}_m|r_j|\sum {C}_j\) schedule problem. International Journal of Production Economics, 100(1), 168–179.
Acknowledgements
We would like to thank Dr. Artur Pessoa and Dr. Rafael Martinelli for the valuable comments and suggestions. This research was partially funded by Programa Institucional de Internacionalização CAPES PrInt UFF No. 88881. AS received Grants CNPq 305223/2015-1 and 428549/2016-0. EU received grants CNPq 313601/2018-6 and Faperj E-26/202.887/2017. Experiments presented in this paper were carried out using the PlaFRIM (Federative Platform for Research in Computer Science and Mathematics), created under the Inria PlaFRIM development action with support from Bordeaux INP, LABRI and IMB and other entities: Conseil Régional d’Aquitaine, Université de Bordeaux, CNRS and ANR in accordance with the “Programme d’Investissements d’Avenir” (see www.plafrim.fr/en/home).
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Bulhões, T., Sadykov, R., Subramanian, A. et al. On the exact solution of a large class of parallel machine scheduling problems. J Sched 23, 411–429 (2020). https://doi.org/10.1007/s10951-020-00640-z
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DOI: https://doi.org/10.1007/s10951-020-00640-z