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Exact algorithm over an arc-time-indexed formulation for parallel machine scheduling problems

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Abstract

This paper presents an exact algorithm for the identical parallel machine scheduling problem over a formulation where each variable is indexed by a pair of jobs and a completion time. We show that such a formulation can be handled, in spite of its huge number of variables, through a branch cut and price algorithm enhanced by a number of practical techniques, including a dynamic programming procedure to fix variables by Lagrangean bounds and dual stabilization. The resulting method permits the solution of many instances of the P||∑w j T j problem with up to 100 jobs, and having 2 or 4 machines. This is the first time that medium-sized instances of the P||∑w j T j have been solved to optimality.

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References

  1. Avella P., Boccia M., D’Auria B.: Near-optimal solutions of large scale single-machine scheduling problems. ORSA J. Comput. 17, 183–191 (2005)

    MathSciNet  Google Scholar 

  2. Ben Amor H., Frangioni A., Desrosiers J.: On the choice of explicit stabilizing terms in column generation. Discrete Appl. Math. 157, 1167–1184 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bigras L., Gamache M., Savard G.: Time-indexed formulations and the total weighted tardiness problem. INFORMS J. Comput. 1, 133–142 (2008)

    Article  MathSciNet  Google Scholar 

  4. Dash, S., Fukasawa, R., Gunluk, O.: On the generalized master knapsack polyhedron. In: Proceedings of the 12th IPCO Conference, Lecture Notes in Computer Science, vol. 4513, pp. 197–209 (2007)

  5. Dyer M., Wolsey L.: Formulating the single machine sequencing problem with release dates as a mixed integer program. Discrete Appl. Math. 26, 255–270 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Fukasawa R., Longo H., Lysgaard J., Poggi de Aragão M., Reis M., Uchoa E., Werneck R.F.: Robust branch-and-cut-and-price for the capacitated vehicle routing problem. Math. Program. 106, 491–511 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Irnich S., Desaulniers G., Desrosiers J., Hadjar A.: Path-reduced costs for eliminating arcs in routing and scheduling. INFORMS J. Comput. 22, 297–313 (2010)

    Article  MathSciNet  Google Scholar 

  8. Lawler E.: A pseudopolynomial algorithm for sequencing jobs to minimize total tardiness. Ann. Discrete Math. 1, 331–342 (1977)

    Article  MathSciNet  Google Scholar 

  9. Lemaréchal C.: Lagrangean relaxation. In: Juenger, M., Naddef, D. (eds) Computational Combinatorial Optimization, pp. 115–160. Springer, Berlin (2001)

    Google Scholar 

  10. du Merle O., Villeneuve D., Desrosiers J., Hansen P.: Stabilized column generation. Discrete Math. 194, 229–237 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Oukil A., Ben Amor H., Desrosiers J., El Gueddari H.: Stabilized column generation for highly degenerate multiple-depot vehicle scheduling problems. Comput. Oper. Res. 34, 817–834 (2007)

    Article  MATH  Google Scholar 

  12. Pan Y., Shi L.: On the equivalence of the max- min transportation lower bound and the time-indexed lower bound for single machine scheduling problems. Math. Program. 110, 543–559 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Pessoa A., Poggi de Aragão M., Uchoa E.: Robust branch-cut-and-price algorithms for vehicle routing problems. In: Golden, B., Raghavan, S., Wasil, E. (eds) The Vehicle Routing Problem: Latest Advances and New Challenges, pp. 297–326. Springer, Berlin (2008)

    Chapter  Google Scholar 

  14. Pessoa A., Uchoa E., Poggi de Aragão M.: A robust branch-cut-and-price algorithm for the heterogeneous fleet vehicle routing problem. Networks 54, 167–177 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Pessoa, A., Uchoa, E., Poggi de Aragão, M., Rodrigues, R.: Algorithms over arc-time indexed formulations for single and parallel machine scheduling problems. Tech. Rep. RPEP Vol.8 no.8, Universidade Federal Fluminense, Engenharia de Produção, Niterói, Brazil (2008)

  16. Pinedo M.: Scheduling: Theory, Algorithms, and Systems. Prentice-Hall, USA (2002)

    MATH  Google Scholar 

  17. Poggi de Aragão M., Uchoa E.: Integer program reformulation for robust branch-and-cut-and-price. In: Wolsey, L. (eds) Annals of Mathematical Programming in Rio, pp. 56–61. Búzios, Brazil (2003)

    Google Scholar 

  18. Potts C., Wassenhove L.: A branch-and-bound algorithm for the total weighted tardiness problem. Oper. Res. 32, 363–377 (1985)

    Article  Google Scholar 

  19. Queyranne, M., Schulz, A.: Polyhedral approaches to machine scheduling. Tech. Rep. 408, University of Berlin (1997)

  20. Rodrigues, R., Pessoa, A., Uchoa, E., Poggi de Aragão, M.: Heuristics for multi-machine weighted tardiness problems. Tech. Rep. RPEP Vol.8 no.11, Universidade Federal Fluminense, Engenharia de Produção, Niterói, Brazil (2008)

  21. Sadykov, R.: Integer programming-based decomposition approaches for solving machine scheduling problems. Ph.D. thesis, Universite Catholique de Louvain (2006)

  22. Souayah N., Kacem I., Haouari M., Chu C.: Scheduling on parallel identical machines to minimise the total weighted tardiness. Int. J. Adv. Oper. Manage. 1(1), 30–69 (2009)

    Article  Google Scholar 

  23. Sourd F.: New exact algorithms for one-machine earliness-tardiness scheduling. INFORMS J. Comput. 21, 167–175 (2009)

    Article  MathSciNet  Google Scholar 

  24. Sousa J., Wolsey L.: A time indexed formulation of non-preemptive single machine scheduling problems. Math. Program. 54, 353–367 (1990)

    Article  MathSciNet  Google Scholar 

  25. Tanaka S., Araki M.: A branch and bound algorithm with lagrangian relaxation to minimize total tardiness on identical parallel machines. Int. J. Prod. Econ. 113, 446–458 (2008)

    Article  Google Scholar 

  26. Tanaka S., Fujikuma S., Araki M.: An exact algorithm for single-machine scheduling without machine idle time. J. Sched. 12, 575–593 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Uchoa E., Fukasawa R., Lysgaard J., Pessoa A., Poggi de Aragão M., Andrade D.: Robust branch-cut-and-price for the capacitated minimum spanning tree problem over a large extended formulation. Math. Program. 122, 443–472 (2008)

    Google Scholar 

  28. Van der Akker J., Hurkens C., Savelsbergh M.: Time-indexed formulations for machine scheduling problems: column generation. INFORMS J. Comput. 12(2), 111–124 (2000)

    Article  MathSciNet  Google Scholar 

  29. Van der Akker J., Van Hoesel C., Savelsbergh M.: A polyhedral approach to single-machine scheduling problems. Math. Program. 85, 541–572 (1999)

    Article  MathSciNet  Google Scholar 

  30. Wentges P.: Weighted dantzig-wolfe decomposition for linear mixed-integer programming. Int. Trans. Oper. Res. 4, 151–162 (1997)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Engenharia de Produção, Universidade Federal Fluminense, Niterói, Brazil

    Artur Pessoa & Eduardo Uchoa

  2. Departamento de Informática, PUC Rio de Janeiro, Rio de Janeiro, Brazil

    Marcus Poggi de Aragão

  3. Departamento de Ciência da Computação, Universidade Federal do Amazonas, Manaus, Brazil

    Rosiane Rodrigues

  4. COPPE-Engenharia de Sistemas e Computação, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Rosiane Rodrigues

Authors
  1. Artur Pessoa
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  2. Eduardo Uchoa
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  3. Marcus Poggi de Aragão
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  4. Rosiane Rodrigues
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Correspondence to Eduardo Uchoa.

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Pessoa, A., Uchoa, E., de Aragão, M.P. et al. Exact algorithm over an arc-time-indexed formulation for parallel machine scheduling problems. Math. Prog. Comp. 2, 259–290 (2010). https://doi.org/10.1007/s12532-010-0019-z

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  • DOI: https://doi.org/10.1007/s12532-010-0019-z

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