Annals of Operations Research

, Volume 43, Issue 5, pp 285–293 | Cite as

Penalty computations for the set partitioning problem

  • Brigitte Jaumard
  • Marcelo Prais
  • Celso Carneiro Ribeiro
Section V Set Covering And Set Partitioning


The computation of penalties associated with the continuous relaxation of integer programming problems can be useful to derive conditional and relational tests which allow to fix some variables at their optimal value or to generate new constraints (cuts). We study in this paper the computation and the use of penalties as a tool to improve the efficiency of algorithms for solving set partitioning problems. This leads to a preprocessing scheme which can be embedded within any exact or approximate algorithm. The strength of these penalties is illustrated through computational results on some real-world set partitioning problems.


Set partitioning penalties integer programming 


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  1. [1]
    R.D. Armstrong and P. Sinha, Improved penalty calculations for a mixed integer branch-and-bound algorithm, Math. Progr. 6(1974)212–223.Google Scholar
  2. [2]
    E. Balas and P.R. Landweer, Traffic assignment in communication satellites, Oper. Res. Lett. 2(1983) 141–147.Google Scholar
  3. [3]
    E. Beale and R. Small, Mixed integer programming by a branch-and-bound technique,Proc. 3rd IFIP Congress 2 (1965) pp. 450–451.Google Scholar
  4. [4]
    E. Brandt, Optimisation des ressources pour des transmissions par satellite utilisant l'AMRT, Thèse de Doctorat de 3ème Cycle, Université Paris-Sud, Paris (1982).Google Scholar
  5. [5]
    A.V. Cabot and S.S. Erenguc, Some branch-and-bound procedures for fixed-cost transportation problems, Naval Res. Log. Quart. 31(1984)145–154.Google Scholar
  6. [6]
    A.V. Cabot and S.S. Erenguc, Improved penalties for fixed cost linear programs using Lagrangian relaxation, Manag. Sci. 32(1986)856–869.Google Scholar
  7. [7]
    T.J. Chan, J.C. Bean and C.A. Yano, A multiplier-adjustment-based branch-and-bound algorithm for the set partitioning problem, Technical Report 87-OR-08, Department of Operations Research and Engineering Management, Southern Methodist University, Dallas (1987).Google Scholar
  8. [8]
    N.J. Driebeek, An algorithm for the solution of mixed integer programming problems, Manag. Sci. 12(1966)576–587.Google Scholar
  9. [9]
    P. Hansen, Programmes mathématiques en variables 0–1, Thèse d'Agrégation, Université de Bruxelles (1974).Google Scholar
  10. [10]
    N. Mannur, M.H. Karwan and S. Zionts, A comparison of variable setting methods in 0–1 linear integer programming,TIMS/ORSA Joint National Meeting, Washington DC (1988).Google Scholar
  11. [11]
    R.E. Marsten, The design of the XMP linear programming library, ACM Trans. Math. Software 7(1981)481–497.Google Scholar
  12. [12]
    U.S. Palekar, M.H. Karwan and S. Zionts, A branch-and-bound method for the fixed charge transportation problem, Manag. Sci. 36(1990)1092–1105.Google Scholar
  13. [13]
    C.C. Ribeiro, M. Minoux and M.C. Penna, An optimal column-generation-with-ranking algorithm for very large scale set partitioning problems in traffic assignment, Eur. J. Oper. Res. 41(1989)232–239.Google Scholar
  14. [14]
    J.R. Schaffer and D.E. O'Leary, Use of penalties in a branch-and-bound procedure for the fixed charge transportation problem, Eur. J. Oper. Res. 43(1989)305–312.Google Scholar
  15. [15]
    L. Schrage,LINDO - User's Manual, 4th ed. (The Scientific Press, Redwood City, 1989).Google Scholar
  16. [16]
    J.A. Tomlin, Branch-and-bound methods for integer and non-convex programming, in:Integer and Nonlinear Programming, ed. J. Abadie (North-Holland, Amsterdam, 1970) pp. 419–436.Google Scholar
  17. [17]
    J.A. Tomlin, An improved branch-and-bound method for integer programming, Oper. Res. 19(1971)1070–1074.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1993

Authors and Affiliations

  • Brigitte Jaumard
    • 1
  • Marcelo Prais
    • 2
  • Celso Carneiro Ribeiro
    • 1
  1. 1.GERAD and École Polytechnique de MontréalMontréalCanada
  2. 2.ELETROBRAS, Rua Visconde de InhaúmaBrazil

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