Annals of Operations Research

, Volume 201, Issue 1, pp 383–401 | Cite as

GRASP with path-relinking for the non-identical parallel machine scheduling problem with minimising total weighted completion times

  • F. J. RodriguezEmail author
  • C. Blum
  • C. García-Martínez
  • M. Lozano


In this work, we tackle the problem of scheduling a set of jobs on a set of non-identical parallel machines with the goal of minimising the total weighted completion times. GRASP is a multi-start method that consists of two phases: a solution construction phase, which randomly constructs a greedy solution, and an improvement phase, which uses that solution as an initial starting point. In the last few years, the GRASP methodology has arisen as a prospective metaheuristic approach to find high-quality solutions for several difficult problems in reasonable computational times. With the aim of providing additional results and insights along this line of research, this paper proposes a new GRASP model that combines the basic scheme with two significant elements that have been shown to be very successful in order to improve GRASP performance. These elements are path-relinking and evolutionary path-relinking. The benefits of our proposal in comparison to existing metaheuristics proposed in the literature are experimentally shown.


Non-identical parallel machine scheduling problem with minimising total weighted completion times Metaheuristics GRASP Path-relinking 



This work was supported by grant TIN2011-24124 of the Spanish government and by grant P08-TIC-4173 of the Andalusian regional goverment.


  1. Aiex, R., Binato, S., & Resende, M. (2003). Parallel GRASP with path-relinking for job shop scheduling. Parallel Computing, 29(4), 393–430. CrossRefGoogle Scholar
  2. Aiex, R., Resende, M., Pardalos, P., & Toraldo, G. (2005). GRASP with path relinking for three-index assignment. INFORMS Journal on Computing, 17(2), 224–247. CrossRefGoogle Scholar
  3. Allahverdi, A., Gupta, J., & Aldowaisan, T. (1999). A review of scheduling research involving setup considerations. Omega, 27(2), 219–239. CrossRefGoogle Scholar
  4. Anghinolfi, D., & Paolucci, M. (2007). Parallel machine total tardiness scheduling with a new hybrid metaheuristic approach. Computers & Operations Research, 34(11), 3471–3490. CrossRefGoogle Scholar
  5. Arnaout, J. P., Rabadi, G., & Musa, R. (2010). A two-stage ant colony optimization algorithm to minimize the makespan on unrelated parallel machines with sequence-dependent setup times. Journal of Intelligent Manufacturing, 21, 693–701. CrossRefGoogle Scholar
  6. 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
  7. 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
  8. Baker, K., & Merten, A. (1973). Scheduling with parallel machines and linear delay costs. Naval Research Logistics Quarterly, 20, 793–804. CrossRefGoogle Scholar
  9. Belouadah, H., & Potts, C. (1994). Scheduling identical parallel machines to minimize total weighted completion time. Discrete Applied Mathematics, 48(3), 201–218. CrossRefGoogle Scholar
  10. Blazewicz, J., Ecker, K., Pesch, E., Schmidt, G., & Weglarz, J. (2007). International handbooks on information systems. Handbook on scheduling: models and methods for advanced planning. Secaucus: Springer. Google Scholar
  11. Brucker, P., & Hurink, J. (2000). Solving a chemical batch scheduling problem by local search. Annals of Operations Research, 96(1), 17–38. CrossRefGoogle Scholar
  12. Bruno, J., Coffman, E., & Sethi, R. (1974). Scheduling independent tasks to reduce mean finishing time. Communications of the ACM, 17(7), 382–387. CrossRefGoogle Scholar
  13. Buxey, G. (1989). Production scheduling: practice and theory. European Journal of Operational Research, 39, 17–31. CrossRefGoogle Scholar
  14. Chen, C. L., & Chen, C. L. (2009). Hybrid metaheuristics for unrelated parallel machine scheduling with sequence-dependent setup times. The International Journal of Advanced Manufacturing Technology, 43(1), 161–169. CrossRefGoogle Scholar
  15. Chen, Z. L., & Powell, W. (1999). Solving parallel machine scheduling problems by column generation. INFORMS Journal on Computing, 11(1), 78–94. CrossRefGoogle Scholar
  16. Cheng, R., Gen, M., & Tozawa, T. (1995). Minmax earliness/tardiness scheduling in identical parallel machine system using genetic algorithms. Computers & Industrial Engineering, 29(1–4), 513–517. CrossRefGoogle Scholar
  17. 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
  18. Croce, F. D., Tadei, R., & Asioli, P. (1999). Scheduling a round robin tennis tournament under courts and players availability constraints. Annals of Operations Research, 92, 349–361. CrossRefGoogle Scholar
  19. Dodin, B., & Chan, K. H. (1991). Application of production scheduling methods to external and internal audit scheduling. European Journal of Operational Research, 52(3), 267–279. CrossRefGoogle Scholar
  20. Elmaghraby, S., & Park, S. (1974). Scheduling jobs on a number of identical machines. AIIE Transactions, 6(1), 1–13. CrossRefGoogle Scholar
  21. Fanjul-Peyro, L., & Ruiz, R. (2010). Iterated greedy local search methods for unrelated parallel machine scheduling. European Journal of Operational Research, 207(1), 55–69. CrossRefGoogle Scholar
  22. Fanjul-Peyro, L., & Ruiz, R. (2011). Size-reduction heuristics for the unrelated parallel machines scheduling problem. Computers & Operations Research, 38(1), 301–309. CrossRefGoogle Scholar
  23. Fanjul-Peyro, L., & Ruiz, R. (2012). Scheduling unrelated parallel machines with optional machines and jobs selection. Computers & Operations Research, 39(7), 1745–1753. CrossRefGoogle Scholar
  24. Feng, G., & Lau, H. (2008). Efficient algorithms for machine scheduling problems with earliness and tardiness penalties. Annals of Operations Research, 159, 83–95. CrossRefGoogle Scholar
  25. Feo, T., & Resende, M. (1989). A probabilistic heuristic for a computationally difficult set covering problem. Operations Research Letters, 8(2), 67–71. CrossRefGoogle Scholar
  26. Feo, T., & Resende, M. (1995). Greedy randomized adaptive search procedures. Journal of Global Optimization, 6(2), 109–133. CrossRefGoogle Scholar
  27. Garcia, S., Molina, D., Lozano, M., & Herrera, F. (2008). A study on the use of non-parametric tests for analyzing the evolutionary algorithms’ behaviour: A case study on the CEC’2005 special session on real parameter optimization. Journal of Heuristics, 15, 617–644. CrossRefGoogle Scholar
  28. Glass, C. A., Potts, C. N., & Shade, P. (1994). Unrelated parallel machine scheduling using local search. Mathematical and Computer Modelling, 20(2), 41–52. CrossRefGoogle Scholar
  29. Glover, F. (1996). Tabu search and adaptive memory programing—advances, applications and challenges. In Interfaces in computer science and operations research (pp. 1–75). Norwell: Kluwer Academic. Google Scholar
  30. Hall, L. A., Shmoys, D. B., & Wein, J. (1996). Scheduling to minimize average completion time: off-line and on-line algorithms. In Proceedings of the seventh annual ACM-SIAM symposium on discrete algorithms, SODA’96 (pp. 142–151). Philadelphia: Society for Industrial and Applied Mathematics. Google Scholar
  31. Hall, L., Schulz, A., Shmoys, D., & Wein, J. (1997). Scheduling to minimize average completion time: off-line and on-line approximation algorithms. Mathematics of Operations Research, 22(3), 513–544. CrossRefGoogle Scholar
  32. Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6, 65–70. Google Scholar
  33. Iman, R., & Davenport, J. (1980). Approximations of the critical region of the Friedman statistic. Communications in Statistics. Theory and Methods, 9(6), 571–595. CrossRefGoogle Scholar
  34. Jarrah, A. I. Z., Bard, J. F., & de Silva, A. H. (1992). A heuristic for machine scheduling at general mail facilities. European Journal of Operational Research, 63(2), 192–206. CrossRefGoogle Scholar
  35. Kendall, G., Tan, K., Burke, E., & Smith, S. (2010). Preface for the special volume on computational intelligence in scheduling. Annals of Operations Research, 180, 1–2. CrossRefGoogle Scholar
  36. Laguna, M., & Marti, R. (1999). GRASP and path relinking for 2-layer straight line crossing minimization. INFORMS Journal on Computing, 11(1), 44–52. CrossRefGoogle Scholar
  37. Lenstra, J., Rinnooy-Kan, A., & Brucker, P. (1977). Complexity of machine scheduling problems. In B. K. P. L. Hammer, E. L. Johnson & G. Nemhauser (Eds.), Studies in integer programming, annals of discrete mathematics (Vol. 1, pp. 343–362). Amsterdam: Elsevier. Google Scholar
  38. 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
  39. Lin, S. W., Lu, C. C., & Ying, K. C. (2011a). Minimization of total tardiness on unrelated parallel machines with sequence- and machine-dependent setup times under due date constraints. The International Journal of Advanced Manufacturing Technology, 53, 353–361. CrossRefGoogle Scholar
  40. Lin, Y., Pfund, M., & Fowler, J. (2011b). Heuristics for minimizing regular performance measures in unrelated parallel machine scheduling problems. Computers & Operations Research, 38(6), 901–916. CrossRefGoogle Scholar
  41. McNaughton, R. (1959). Scheduling with deadlines and loss functions. Management Science, 6(1), 1–12. CrossRefGoogle Scholar
  42. Mokotoff, E. (2001). Parallel machine scheduling problems: a survey. Asia-Pacific Journal of Operational Research, 18(2), 193–242. Google Scholar
  43. Pendharkar, P., & Rodger, J. (2000). Nonlinear programming and genetic search application for production scheduling in coal mines. Annals of Operations Research, 95(1), 251–267. CrossRefGoogle Scholar
  44. Phillips, C., Stein, C., & Wein, J. (1997). Parallel machine scheduling problems: a survey. SIAM Journal on Discrete Mathematics, 10(4), 573–598. CrossRefGoogle Scholar
  45. Resende, M., Marti, R., Gallego, M., & Duarte, A. (2010). GRASP and path relinking for the max-min diversity problem. Computers & Operations Research, 37(3), 498–508. CrossRefGoogle Scholar
  46. Resende, M., & Ribeiro, C. (2003). Greedy randomized adaptive search procedures. In F. Glover & G. Kochenberger (Eds.), Handbook of metaheuristics (pp. 219–249). Norwell: Kluwer Academic. Google Scholar
  47. Resende, M., & Ribero, C. (2003). A GRASP with path-relinking for private virtual circuit routing. Networks, 41, 104–114. CrossRefGoogle Scholar
  48. Resende, M., & Werneck, R. (2004). A hybrid heuristic for the p-median problem. Journal of Heuristics, 10(1), 59–88. CrossRefGoogle Scholar
  49. Ribeiro, C., Uchoa, E., & Werneck, R. (2002). A hybrid GRASP with perturbations for the steiner problem in graphs. INFORMS Journal on Computing, 14(3), 228–246. CrossRefGoogle Scholar
  50. Rochat, Y. (1998). A genetic approach for solving a scheduling problem in a robotized analytical system. Journal of Heuristics, 4, 245–261. CrossRefGoogle Scholar
  51. Rosenbloom, E., & Goertzen, N. (1987). Cyclic nurse scheduling. European Journal of Operational Research, 31, 19–23. CrossRefGoogle Scholar
  52. Sarin, S., Ahn, S., & Bishop, A. (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
  53. Schulz, A., & Skutella, M. (1997). Random-based scheduling: new approximations and LP lower bounds. In Proceedings of the first international symposium on randomization and approximation techniques in computer science (Random’97) (pp. 119–133). Berlin: Springer. CrossRefGoogle Scholar
  54. Schulz, A. S., & Skutella, M. (2002). Scheduling unrelated machines by randomized rounding. SIAM Journal on Discrete Mathematics, 15, 450–469. CrossRefGoogle Scholar
  55. Skutella, M. (2001). Convex quadratic and semidefinite programming relaxations in scheduling. Journal of the ACM, 48, 206–242. CrossRefGoogle Scholar
  56. 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
  57. Waligora, G. (2009). Tabu search for discrete-continuous scheduling problems with heuristic continuous resource allocation. European Journal of Operational Research, 193(3), 849–856. CrossRefGoogle Scholar
  58. Weng, M., Lu, J., & Ren, H. (2001). Unrelated parallel machine scheduling with setup consideration and a total weighted completion time objective. International Journal of Production Economics, 70(3), 215–226. CrossRefGoogle Scholar
  59. Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics, 1, 80–83. CrossRefGoogle Scholar
  60. Zaidi, M., Jarboui, B., Loukil, T., & Kacem, I. (2010). Hybrid meta-heuristics for uniform parallel machine to minimize total weighted completion time. In Proc. of 8th international conference of modeling and simulation (MOSIM’10). Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • F. J. Rodriguez
    • 1
    Email author
  • C. Blum
    • 2
  • C. García-Martínez
    • 3
  • M. Lozano
    • 1
  1. 1.Department of Computer Science and Artificial IntelligenceUniversity of GranadaGranadaSpain
  2. 2.ALBCOM Research GroupTechnical University of CataloniaBarcelonaSpain
  3. 3.Department of Computing and Numerical AnalysisUniversity of CórdobaCórdobaSpain

Personalised recommendations