Abstract
Dynamic constraint aggregation is an iterative method that was recently introduced to speed up the linear relaxation solution process of set partitioning type problems. This speed up is mostly due to the use, at each iteration, of an aggregated problem defined by aggregating disjoint subsets of constraints from the set partitioning model. This aggregation is updated when needed to ensure the exactness of the overall approach. In this paper, we propose a new version of this method, called the multi-phase dynamic constraint aggregation method, which essentially adds to the original method a partial pricing strategy that involves multiple phases. This strategy helps keeping the size of the aggregated problem as small as possible, yielding a faster average computation time per iteration and fewer iterations. We also establish theoretical results that provide some insights explaining the success of the proposed method. Tests on the linear relaxation of simultaneous bus and driver scheduling problems involving up to 2,000 set partitioning constraints show that the partial pricing strategy speeds up the original method by an average factor of 4.5.
Similar content being viewed by others
References
Balinski M.L., Gomory R.E.: A mutual primal-dual simplex method. In: Graves, R.L., Wolfe, P. (eds) Recent Advances in Mathematical Programming., pp. 17–28. McGraw-Hill, New York (1963)
Balinski M.L., Quandt R.E.: On an integer program for a delivery problem. Oper. Res. 12, 300–304 (1964)
Bland R.G.: New finite pivoting rule for simplex method. Math. Oper. Res. 2, 103–107 (1977)
Charnes A.: Optimality and degeneracy in linear programming. Econometrica 20, 160–170 (1952)
Cordeau J.-F., Desaulniers G., Lingaya N., Soumis F., Desrosiers J.: Simultaneous locomotive and car assignment at VIA Rail Canada. Transp. Res. B 35, 767–787 (2001)
Dantzig G.B., Orden G.B., Wolfe P.: The generalized simplex method for minimizing a linear form under linear inequality restraints. Pac. J. Math. 5, 183–195 (1955)
Desrochers M., Desrosiers J., Soumis F.: A new optimization algorithm for the vehicle routing problem with time Windows. Oper. Res. 40, 342–354 (1992)
Desrochers M., Soumis F.: A generalized permanent labeling algorithm for the shortest path problem with time Windows. INFOR 26, 191–212 (1988)
Desrochers M., Soumis F.: A column generation approach to the urban transit crew scheduling problem. Transp. Sci. 23, 1–13 (1989)
Elhallaoui I., Villeneuve D., Soumis F., Desaulniers G.: Dynamic aggregation of set partitioning constraints in column generation. Oper. Res. 53, 632–645 (2005)
Fletcher R.: A new degeneracy method and steepest-edge-based conditioning for LP. SIAM J. Optim. 8, 1038–1059 (1998)
Gal, T., (ed.): Degeneracy in optimization problems. Ann. Oper. Res. 46/47, 1–7 (1993)
Gamache M., Soumis F., Marquis G., Desrosiers J.: A column generation approach for large scale aircrew rostering problems. Oper. Res. 47, 247–263 (1999)
Geoffrion A.M.: Lagrangean relaxation for integer programming. Math. Program. Study 2, 82–114 (1974)
Glover F.: Surrogate constraints. Oper. Res. 16, 741–749 (1968)
Goldfarb D., Reid J.K.: A practicable steepest-edge simplex algorithm. Math. Program. 12, 361–371 (1977)
Haase K., Desaulniers G., Desrosiers J.: Simultaneous vehicle and crew scheduling in urban mass transit systems. Transp. Sci. 35, 286–303 (2001)
Harris P.M.: Pivot selection methods of the devex LP code. Math. Program. 5, 1–28 (1973)
Hoffman K.L., Padberg M.: Solving ariline crew scheduling problems by branch-and-cut. Manage. Sci. 39, 657–682 (1993)
Irnich S., Desaulniers G.: Shortest path problems with resource constraints. In: Desaulniers, G., Desrosiers, J., Solomon, M.M. (eds) Column Generation., pp. 33–65. Springer, New York (2005)
Mendelssohn R.: An iterative aggregation procedure for Markov decision process. Oper. Res. 30, 62–73 (1982)
Pan P.-Q.: A basis deficiency-allowing variation of the simplex method for linear programming. Comput. Math. Appl. 36(3), 33–53 (1998)
Rogers D.F., Plante R.D., Wong R.T., Evans J.R.: Aggregation and disaggregation techniques and methodology in optimization. Oper. Res. 39, 553–582 (1991)
Ryan D.M., et Osborne M.: On the solution of highly degenerate linear programmes. Math. Program. 41, 385–392 (1988)
Shetty C.M., Taylor R.W.: Solving large-scale linear programs by aggregation. Comput. Oper. Res. 14, 385–393 (1987)
Terlaky T., Sushong Z.: Pivot rules for linear programming: a survey on recent theoretical developments. Ann. Oper. Res. 46, 203–233 (1993)
Villeneuve D.: Logiciel de Génération de Colonnes, Ph.D. Dissertation. Université de Montréal, Canada (1999)
Wolfe P.: A technique for resolving degeneracy in LP. SIAM J. 2, 205–211 (1963)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Elhallaoui, I., Metrane, A., Soumis, F. et al. Multi-phase dynamic constraint aggregation for set partitioning type problems. Math. Program. 123, 345–370 (2010). https://doi.org/10.1007/s10107-008-0254-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-008-0254-5