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Journal of Scheduling

, Volume 20, Issue 4, pp 373–389 | Cite as

An exact extended formulation for the unrelated parallel machine total weighted completion time problem

  • Kerem BülbülEmail author
  • Halil Şen
Article

Abstract

The plethora of research on \(\mathcal {NP}\)-hard parallel machine scheduling problems is focused on heuristics due to the theoretically and practically challenging nature of these problems. Only a handful of exact approaches are available in the literature, and most of these suffer from scalability issues. Moreover, the majority of the papers on the subject are restricted to the identical parallel machine scheduling environment. In this context, the main contribution of this work is to recognize and prove that a particular preemptive relaxation for the problem of minimizing the total weighted completion time (TWCT) on a set of unrelated parallel machines naturally admits a non-preemptive optimal solution and gives rise to an exact mixed integer linear programming formulation of the problem. Furthermore, we exploit the structural properties of TWCT and attain a very fast and scalable exact Benders decomposition-based algorithm for solving this formulation. Computationally, our approach holds great promise and may even be embedded into iterative algorithms for more complex shop scheduling problems as instances with up to 1000 jobs and 8 machines are solved to optimality within a few seconds.

Keywords

Unrelated parallel machines Weighted completion time Benders decomposition Cut strengthening Exact method Preemptive relaxation Transportation problem 

Supplementary material

References

  1. Azizoglu, M., & Kirca, O. (1999a). On the minimization of total weighted flow time with identical and uniform parallel machines. European Journal of Operational Research, 113(1), 91–100.CrossRefGoogle Scholar
  2. Azizoglu, M., & Kirca, O. (1999b). Scheduling jobs on unrelated parallel machines to minimize regular total cost functions. IIE Transactions, 31(2), 153–159.Google Scholar
  3. Barnes, J. W., & Brennan, J. (1977). An improved algorithm for scheduling jobs on identical machines. AIIE Transactions, 9(1), 25–31.CrossRefGoogle Scholar
  4. Belouadah, H., & Potts, C. N. (1994). Scheduling identical parallel machines to minimize total weighted completion time. Discrete Applied Mathematics, 48(3), 201–218.CrossRefGoogle Scholar
  5. Benders, J. F. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4(1), 238–252.CrossRefGoogle Scholar
  6. Biskup, D., Herrmann, J., & Gupta, J. N. (2008). Scheduling identical parallel machines to minimize total tardiness. International Journal of Production Economics, 115(1), 134–142.CrossRefGoogle Scholar
  7. Blazewicz, J., Ecker, K. H., Pesch, E., Schmidt, G., & Weglarz, J. (2007). Handbook on scheduling: from theory to applications. New York: Springer.Google Scholar
  8. Bruno, J., Coffman, E. G, Jr., & Sethi, R. (1974). Scheduling independent tasks to reduce mean finishing time. Communications of the ACM, 17(7), 382–387.CrossRefGoogle Scholar
  9. Bülbül, K., Kaminsky, P., & Yano, C. (2007). Preemption in single machine earliness/tardiness scheduling. Journal of Scheduling, 10(4–5), 271–292.CrossRefGoogle Scholar
  10. Burkard, R., Dell’Amico, M., & Martello, S. (2009). Assignment problems. Philadelphia: Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
  11. Chekuri, C., & Khanna, S. (2004). Approximation algorithms for minimizing average weighted completion time. In J. Y. Leung (Ed.), Handbook of scheduling: algorithms, models, and performance analysis. Boca Raton: CRC Press.Google Scholar
  12. Chen, Z.-L., & Powell, W. B. (1999). Solving parallel machine scheduling problems by column generation. INFORMS Journal on Computing, 11(1), 78–94.CrossRefGoogle Scholar
  13. Cheng, T., & Sin, C. (1990). A state-of-the-art review of parallel-machine scheduling research. European Journal of Operational Research, 47(3), 271–292.CrossRefGoogle Scholar
  14. Detienne, B., Dauzère-Pérès, S., & Yugma, C. (2011). Scheduling jobs on parallel machines to minimize a regular step total cost function. Journal of Scheduling, 14, 523–538.CrossRefGoogle Scholar
  15. Dyer, M., & Wolsey, L. (1990). Formulating the single machine sequencing problem with release dates as a mixed integer program. Discrete Applied Mathematics, 26(2–3), 255–270.CrossRefGoogle Scholar
  16. Elmaghraby, S. E., & Park, S. H. (1974). Scheduling jobs on a number of identical machines. AIIE Transactions, 6(1), 1–13.CrossRefGoogle Scholar
  17. Fischetti, M., Salvagnin, D., & Zanette, A. (2010). A note on the selection of Benders’ cuts. Mathematical Programming, 124(1–2), 175–182.CrossRefGoogle Scholar
  18. Goemans, M. X., Queyranne, M., Schulz, A. S., Skutella, M., & Wang, Y. (2002). Single machine scheduling with release dates. SIAM Journal on Discrete Mathematics, 15(2), 165–192.CrossRefGoogle Scholar
  19. Graham, R., Lawler, E., Lenstra, J., Rinnooy Kan, A., & Hammer, P. L. (1979). Optimization and approximation in deterministic sequencing and scheduling: a survey. In P. L. Hammer & B. Korte (Eds.), Discrete optimization II (Vol. 5, pp. 287–326)., Annals of discrete mathematics New York: Elsevier.Google Scholar
  20. IBM ILOG CPLEX (2012). IBM ILOG CPLEX Optimization Studio 12.5 Information Center. Retrieved 08 April 2014 from http://pic.dhe.ibm.com/infocenter/cosinfoc/v12r5/index.jsp.
  21. Kedad-Sidhoum, S., Solis, Y. R., & Sourd, F. (2008). Lower bounds for the earliness-tardiness scheduling problem on parallel machines with distinct due dates. European Journal of Operational Research, 189(3), 1305–1316.CrossRefGoogle Scholar
  22. Lawler, E. L., & Moore, J. M. (1969). A functional equation and its application to resource allocation and sequencing problems. Management Science, 16(1), 77–84.CrossRefGoogle Scholar
  23. Lee, C.-Y., & Uzsoy, R. (1992). A new dynamic programming algorithm for the parallel machines total weighted completion time problem. Operations Research Letters, 11(2), 73–75.CrossRefGoogle Scholar
  24. Li, K., & Yang, S.-L. (2009). Non-identical parallel-machine scheduling research with minimizing total weighted completion times: Models, relaxations and algorithms. Applied Mathematical Modelling, 33(4), 2145–2158.CrossRefGoogle Scholar
  25. Lin, Y., Pfund, M., & Fowler, J. (2011). Heuristics for minimizing regular performance measures in unrelated parallel machine scheduling problems. Computers and Operations Research, 38(6), 901–916.CrossRefGoogle Scholar
  26. Magnanti, T. L., & Wong, R. T. (1981). Accelerating Benders decomposition: Algorithmic enhancement and model selection criteria. Operations Research, 29(3), 464–484.CrossRefGoogle Scholar
  27. Mokotoff, E. (2001). Parallel machine scheduling problems: A survey. Asia Pacific Journal of Operational Research, 18(2), 193–242.Google Scholar
  28. 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.CrossRefGoogle Scholar
  29. Pan, Y., & Shi, L. (2007). On the equivalence of the max-min transportation lower bound and the time-indexed lower bound for single-machine scheduling problems. Mathematical Programming, 110(3), 543–559.Google Scholar
  30. Pinedo, M. (2008). Scheduling: theory, algorithms, and systems (3rd ed.). New York: Springer.Google Scholar
  31. Plateau, M.-C., & Rios-Solis, Y. A. (2010). Optimal solutions for unrelated parallel machines scheduling problems using convex quadratic reformulations. European Journal of Operational Research, 201(3), 729–736.CrossRefGoogle Scholar
  32. Posner, M. E. (1985). Minimizing weighted completion times with deadlines. Operations Research, 33(3), 562–574.CrossRefGoogle Scholar
  33. Rodriguez, F., Blum, C., García-Martínez, C., & Lozano, M. (2012). GRASP with path-relinking for the non-identical parallel machine scheduling problem with minimising total weighted completion times. Annals of Operations Research, 201(1), 383–401.CrossRefGoogle Scholar
  34. Rodriguez, F. J., Lozano, M., Blum, C., & García-Martínez, C. (2013). An iterated greedy algorithm for the large-scale unrelated parallel machines scheduling problem. Computers & Operations Research, 40(7), 1829–1841.CrossRefGoogle Scholar
  35. Rubin, P. (2011). Benders decomposition then and now. Retrieved 24 April 2013 from http://orinanobworld.blogspot.com/2011/10/benders-decomposition-then-and-now.html.
  36. Sarin, S. C., Ahn, S., & Bishop, A. B. (1988). An improved branching scheme for the branch and bound procedure of scheduling n jobs on m parallel machines to minimize total weighted flowtime. International Journal of Production Research, 26(7), 1183–1191.CrossRefGoogle Scholar
  37. Şen, H. & Bülbül, K. (2012). A simple, fast, and effective heuristic for the single-machine total weighted tardiness problem. In E. Demeulemeester & W. Herroelen (Eds.) Proceedings of the 13th International Conference on Project and Scheduling (PMS 2012), pp. 282–286, Leuven: Belgium.Google Scholar
  38. 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.CrossRefGoogle Scholar
  39. Shim, S.-O., & Kim, Y.-D. (2007). Minimizing total tardiness in an unrelated parallel-machine scheduling problem. Journal of the Operational Research Society, 58(3), 346–354.Google Scholar
  40. Skutella, M. (2001). Convex quadratic and semidefinite programming relaxations in scheduling. Journal of the ACM (JACM), 48(2), 206–242.CrossRefGoogle Scholar
  41. Smith, W. E. (1956). Various optimizers for single-stage production. Naval Research Logistics Quarterly, 3(1–2), 59–66.CrossRefGoogle Scholar
  42. Sourd, F., & Kedad-Sidhoum, S. (2003). The one-machine problem with earliness and tardiness penalties. Journal of Scheduling, 6(6), 533–549.CrossRefGoogle Scholar
  43. Unlu, Y., & Mason, S. J. (2010). Evaluation of mixed integer programming formulations for non-preemptive parallel machine scheduling problems. Computers & Industrial Engineering, 58(4), 785–800.CrossRefGoogle Scholar
  44. van den Akker, J. M., Hoogeveen, J. A., & van de Velde, S. L. (1999). Parallel machine scheduling by column generation. Operations Research, 47(6), 862–872.CrossRefGoogle Scholar
  45. Vredeveld, T., & Hurkens, C. (2002). Experimental comparison of approximation algorithms for scheduling unrelated parallel machines. INFORMS Journal on Computing, 14(2), 175–189.CrossRefGoogle Scholar
  46. Yalaoui, F., & Chu, C. (2006). New exact method to solve the \({P}m/r_j/\sum _j {C}_j\) schedule problem. International Journal of Production Economics, 100(1), 168–179.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Industrial EngineeringSabancı UniversityİstanbulTurkey

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