Journal of Heuristics

, Volume 17, Issue 6, pp 637–658 | Cite as

Matching based very large-scale neighborhoods for parallel machine scheduling

Open Access


In this paper we study very large-scale neighborhoods for the minimum total weighted completion time problem on parallel machines, which is known to be strongly \(\mathcal{NP}\)-hard. We develop two different ideas leading to very large-scale neighborhoods in which the best improving neighbor can be determined by calculating a weighted matching. The first neighborhood is introduced in a general fashion using combined operations of a basic neighborhood. Several examples for basic neighborhoods are given. The second approach is based on a partitioning of the job sets on the machines and a reassignment of them. In a computational study we evaluate the possibilities and the limitations of the presented very large-scale neighborhoods.


Scheduling Parallel machines Total weighted completion time Very large-scale neighborhoods Local search 


  1. Agarwal, R., Ergun, Ö., Orlin, J.B., Potts, C.N.: Solving parallel machine scheduling problems with very-large scale neighborhood search. Working paper (2007)., J. Sched., to appear
  2. Ahuja, R.K., Özlem, E., Orlin, J.B., Punnen, A.P.: A survey of very large-scale neighborhood search techniques. Discrete Appl. Math. 123, 75–102 (2002) MathSciNetMATHCrossRefGoogle Scholar
  3. van den Akker, J.M., Hoogeveen, J.A., van de Velde, S.L.: Parallel machine scheduling by column generation. Oper. Res. 47, 862–872 (1999) MathSciNetMATHCrossRefGoogle Scholar
  4. Baker, K.R., Merten, A.G.: Scheduling with parallel processors and linear delay costs. Nav. Res. Logist. Q. 20, 793–804 (1973) MATHCrossRefGoogle Scholar
  5. Barnes, J.W., Laguna, M.: Solving the multiple-machine weighted flow time problem using tabu search. IIE Trans. 25(2), 121–128 (1993) CrossRefGoogle Scholar
  6. Belouadah, H., Potts, C.N.: Scheduling identical parallel machines to minimize total weighted completion time. Discrete Appl. Math. 48(3), 201–218 (1994) MathSciNetMATHCrossRefGoogle Scholar
  7. Brucker, P.: Scheduling Algorithms, 4th edn. Springer, Berlin (2004) MATHGoogle Scholar
  8. Brueggemann, T., Hurink, J.L.: Two exponential neighborhoods for single machine scheduling. OR Spectrum 29, 513–533 (2007) MathSciNetMATHCrossRefGoogle Scholar
  9. Chen, Z.L., Powell, W.B.: Solving parallel machine scheduling problems by column generation. INFORMS J. Comput. 11, 78–94 (1999) MathSciNetMATHCrossRefGoogle Scholar
  10. Congram, R.K., Potts, C.P., van de Velde, S.L.: An iterated dynasearch algorithm for the single machine total weighted tardiness problem. INFORMS J. Comput. 14, 52–67 (2002) MathSciNetCrossRefGoogle Scholar
  11. Deineko, V., Woeginger, G.J.: A study of exponential neighborhoods for the traveling salesman problem and the quadratic assignment problem. INFORMS J. Comput. 14, 52–67 (2000) Google Scholar
  12. Eastman, W.L., Even, S., Isaacs, I.M.: Bounds for the optimal scheduling of n jobs on m processors. Manag. Sci. 11, 268–279 (1964) MathSciNetCrossRefGoogle Scholar
  13. Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices. J. Res. Natl. Bureau Stand. B 69, 125–130 (1965) MathSciNetMATHGoogle Scholar
  14. Elmaghraby, S.E., Park, S.H.: Scheduling jobs on a number of identical machines. Trans. Am. Inst. Ind. Eng. 6, 1–12 (1974) MathSciNetGoogle Scholar
  15. Ergun, Ö., Orlin, J.B., Steele-Feldman, A.: Creating very large-scale neighborhoods out of smaller ones by compounding moves. J. Heuristics 12, 115–140 (2006) MATHCrossRefGoogle Scholar
  16. Gabow, H.N.: Implementation of algorithms for maximum matching on nonbipartite graphs. Ph.D. Thesis, Department of Computer Science, Stanford University, Stanford, California (1973) Google Scholar
  17. Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-Completeness. Freeman, New York (1979) MATHGoogle Scholar
  18. Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discrete Math. 5, 287–326 (1979) MathSciNetMATHCrossRefGoogle Scholar
  19. Hoede, C.: Private communication (2006) Google Scholar
  20. Hurink, J.: An exponential neighborhood for a one machine batching problem. OR Spektrum 21, 461–476 (1999) MathSciNetMATHCrossRefGoogle Scholar
  21. Kawaguchi, T., Kyan, S.: Worst case bound of an LRF schedule for the mean weighted flow-time problem. SIAM J. Comput. 15(4), 1119–1129 (1986) MathSciNetMATHCrossRefGoogle Scholar
  22. Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York (1976) MATHGoogle Scholar
  23. Lenstra, J.K., Rinnooy Kan, A.H.G., Brucker, P.: Complexity of machine scheduling problems. Ann. Discrete Math. 1, 343–362 (1977) MathSciNetCrossRefGoogle Scholar
  24. Potts, C.N., van de Velde, S.L.: Dynasearch-iterative local improvement by dynamic programming: Part 1. The traveling salesman problem. Technical Report, University of Twente, Enschede, The Netherlands (1995) Google Scholar
  25. Sahni, S.K.: Algorithms for scheduling independent tasks. J. Assoc. Comput. Mach. 23, 116–127 (1976) MathSciNetMATHGoogle Scholar
  26. Skutella, M., Woeginger, G.J.: A PTAS for minimizing the total weighted completion time on identical parallel machines. Math. Oper. Res. 25, 63–75 (2000) MathSciNetMATHCrossRefGoogle Scholar
  27. Smith, W.E.: Various optimizers for single-stage production. Nav. Res. Logist. Q. 3, 59–66 (1956). Math. Oper. Res. 25, 63–75 CrossRefGoogle Scholar

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© The Author(s) 2010

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of TwenteEnschedeThe Netherlands

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