We present an exact method for integer linear programming problems that combines branch and bound with column generation at each node of the search tree. For the case of models involving binary column vectors only, we propose the use of so-called geometrical cuts to be added to the subproblem in order to eliminate previously generated columns. This scheme could be applied to general integer problems without specific structure. We report computational results on a successful application of this approach to a telecommunications network planning problem.
@article{RO_2003__37_2_67_0,
author = {Maculan, Nelson and Passini, Marcos de Mendon\c{c}a and Brito, Jos\'e Andr\'e de Moura and Loiseau, Irene},
title = {Column-generation in integer linear programming},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {67--83},
publisher = {EDP-Sciences},
volume = {37},
number = {2},
year = {2003},
doi = {10.1051/ro:2003014},
mrnumber = {2010413},
zbl = {1036.90076},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ro:2003014/}
}
TY - JOUR AU - Maculan, Nelson AU - Passini, Marcos de Mendonça AU - Brito, José André de Moura AU - Loiseau, Irene TI - Column-generation in integer linear programming JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2003 SP - 67 EP - 83 VL - 37 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ro:2003014/ DO - 10.1051/ro:2003014 LA - en ID - RO_2003__37_2_67_0 ER -
%0 Journal Article %A Maculan, Nelson %A Passini, Marcos de Mendonça %A Brito, José André de Moura %A Loiseau, Irene %T Column-generation in integer linear programming %J RAIRO - Operations Research - Recherche Opérationnelle %D 2003 %P 67-83 %V 37 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ro:2003014/ %R 10.1051/ro:2003014 %G en %F RO_2003__37_2_67_0
Maculan, Nelson; Passini, Marcos de Mendonça; Brito, José André de Moura; Loiseau, Irene. Column-generation in integer linear programming. RAIRO - Operations Research - Recherche Opérationnelle, Tome 37 (2003) no. 2, pp. 67-83. doi : 10.1051/ro:2003014. http://www.numdam.org/articles/10.1051/ro:2003014/
[1] , and, Column generation and the airline crew pairing problem. DOC. Math. J. DMV (1998) 677-686. | MR | Zbl
[2] , and, Using branch-and-price and cut to solve origin destination integer multicommodity flow problems. Oper. Res. 48 (2000) 318-326.
[3] ,,, and, Branch-and-price: Column generation for solving huge integer programs. Oper. Res. 46 (1998) 316-329. | MR | Zbl
[4] , and, Scheduling duties by adptive column generation. ZIB-Report 01-02 (2001).
[5] , and, A combinatorial column generation algorithm for the maximun stable set problem. Oper. Res. Lett. (1997). | MR | Zbl
[6] , Um modelo de otimização para dimensionamento de uma rede de telecomunicações, Tese de mestrado, COPPE. Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil (1999).
[7] and, On the effectiveness of set covering formulations for the vehicle routing problem with time windows. Oper. Res. 45 (1997) 295-301. | Zbl
[8] and, An optimal solution procedure for the multiple tour maximum collec tion problem using column generation. Comp. Oper. Res. 26 (1999) 427-441. | MR | Zbl
[9] and, A column generation based descomposition algorithm for a parallel machine scheduling problem. EJOR 116 (1999) 220-232. | Zbl
[10] , Linear programming. W.H. Freeman and Company, New York, San Francisco (1983). | MR | Zbl
[11] and, Approximation algorithms for integer covering problems via greedy column generation. RAIRO: Oper. Res. 28 (1994) 283-302. | EuDML | Numdam | MR | Zbl
[12] , A tree search algorithm for mixed integer programming problems. Comput. J. 8 (1965) 250-255. | MR | Zbl
[13] , Upper bounds, secondary constraints and block triangulary in linear programming. Econometrica 23 (1995) 174-183. | MR | Zbl
[14] , Linear programming and extensions. Princeton University Press, New Jersey, USA (1963). | MR | Zbl
[15] and, Decomposition principle for linear programming. Oper. Res. 8 (1960) 101-111. | Zbl
[16] , Exact solution of cutting stock problems using column generation and branch-and-bound. Int. Trans. Oper. Res. 5 (1998) 35-43. | Zbl
[17] , Contributions à la résolution du problème de recouvrement : méthode de troncature, Docteur-ingenieur dissertation. Université Paris VI, Paris, France (1974).
[18] , and, Accelerating strategies in Column Generation methods for vehicle routing and crew scheduling. Cahiers du Gerad G-99-36 (1999).
[19] ,, and, Daily Aircraft routing and Scheduling. Cahiers du Gerad G-94-21 (1994).
[20] , and, Accelerating strategies in Column Generation Methods for Vehicle Routing and crew scheduling problems. Cahiers du Gerad G-99-36 (1994).
[21] , and, A new optimization algorithm for the vehicle routing problem with time windows. Oper. Res. 40 (1992) 342-353. | MR | Zbl
[22] ,, and, Time constrained routing and scheduling, edited by M.O. Ball, T.L. Magnanti, C.L. Monma and G.L. Nemhauser, Network Routing. INFORMS - North Holland, Handbooks Oper. Res. Management Sci. 8 (1995) 35-139. | MR | Zbl
[23] and, A column-generation approach to the urban transit crew scheduling problem. Transportation Sci. 23 (1989) 1-13. | Zbl
[24] , and, Routing with time-windows by column generation. Networks 14 (1984) 545-565. | Zbl
[25] , and, Circuit partitioning via set partitioning and column generation. Oper. Res. 44 (1996) 65-76. | Zbl
[26] ,, and, A column generation approach for large-scale aircrew rostering problems. Oper. Res. 48 (1992) 247-263. | Zbl
[27] and, A linear programming approach to the cutting stock problem. Oper. Res. 9 (1961) 849-859. | MR | Zbl
[28] and, A linear programming approach to the cutting stock problem - Part II. Oper. Res. 11 (1963) 863-888. | Zbl
[29] , and, Un algorithme primal de programmation linéaire généralisée pour les programmes mixtes. C. R. Acad. Sci. Paris Sér. I Math. 313 (1991) 557-560. | MR | Zbl
[30] , and, A Column Generation Aproach to Cell Formation Problems in Cellular Manufacturing. Cahiers du Gerad G-99-20 (1999).
[31] , and, How to combine a column and row generation method with a column or row elimination procedures - Application to a channel Assiggnment problem. Cahiers du Gerad G-99-18 (1999).
[32] , and, Column/Row Generation and Elimination Methods. Cahiers du Gerad G-99-34 (1999).
[33] , and, Min-cut clustering. Math. Programming 62 (1993) 133-152. | MR | Zbl
[34] and, Crew Scheduling for Netherlands Railways “Destination Customer” (2001). | Zbl
[35] , Optimization Theory for Large Systems. Macmillan, New York, USA (1970). | MR | Zbl
[36] , Vehicle scheduling in public transit and Lagrangian pricing. Management Sci. 44 (1998) 1637-1649. | Zbl
[37] , The cutting stock problem and integer rounding. Math. Programming 13 (1985) 82-92. | MR | Zbl
[38] , and, Programação linear e inteira. Notes - COPPE/Universidade Federal do Rio de Janeiro (1999).
[39] , and, Column generation in linear programming with bounding variable constraints and its applications in integer programming. Pesquisa Operacional 12 (1992) 45-57.
[40] ,, and, Column generation method for network design, Transportation and Network Analysis Current Trends, edited by M. Gendreau and P. Marcotte. Kluwer Academic Publishers (2002) 165-179. | MR | Zbl
[41] and, A column generation approach for graph coloring. INFORMS J. Comput. 8 (1996) 344-353. | Zbl
[42] , Optimal traffic assignment in a SS/TDMA frame: A new approach by set covering and column generation. RAIRO: Oper. Res. 20 (1986) 273-286. | EuDML | Numdam | Zbl
[43] , A class of combinatorial problems with polynomially solvable large scale set covering/partioning relaxations. RAIRO: Oper. Res. 21 (1987) 105-136. | EuDML | Numdam | MR | Zbl
[44] and, A polyhedral approach to edge coloring. Oper. Res. Lett. 10 (1991) 315-322. | MR | Zbl
[45] and, Integer and Combinatorial Optimization. John Wiley & Sons Inc. (1988). | MR | Zbl
[46] , Um modelo de otimização combinatória para o dimensionamento de uma rede urbana de telecomunicações, M.Sc. Thesis. COPPE, Universidade Federal do Rio de Janeiro (1996).
[47] and, An integer programming approach to scheduling, in Computer Scheduling of Public Transport Urban Passenger Vehicle and Crew Scheduling. North Holland (1981). | MR
[48] , and, An optimal column-generation-with-ranking algorithm for very large scale partitioning problems in traffic assignment. Eur. J. Oper. Res. 41 (1989) 232-239. | MR | Zbl
[49] and, A column generation approach to the multiple-depot vehicle scheduling problem. Oper. Res. 42 (1994) 41-52. | Zbl
[50] , A branch-and-price algorithm for the generalized assignment problem. Oper. Res. 46 (1997) 831-841. | MR | Zbl
[51] , Geração de colunas para o problema de empacotamento de árvores de Steiner, M.Sc. dissertation. Universidade Federal do Rio de Janeiro (2003).
[52] , and, Optimal placemente of add/drop multiplexers: Heuristic and exact algorithms. Oper. Res. 46 (1998) 719-728. | Zbl
[53] , A heuristic generation method for the heterogeneous fleet VRP. RAIRO: Oper. Res. 33 (1999) 1-14. | Numdam | MR | Zbl
[54] , Decomposition and Column Generation for Integer Programming, Thèse de doctorat en Sciences Appliquées. Université Catholique de Louvain, Louvain, Belgique (1994).
[55] , Branch-and-price algorithms for the one-dimensional cutting stock problem. Comput. Optim. Appl. 9 (1998) 211-228. | MR | Zbl
[56] ,, and, Solving binary cutting stock problems by column generation and branch-and-bound. Comput. Optim. Appl. 3 (1994) 111-130. | MR | Zbl
[57] ,, and, Airline crew scheduling: A new formulation and decomposition algorithm. Oper. Res. 45 (1997) 188-200. | Zbl
[58] , Computational study of a column generation algorithm for bin packing and cut stocking problems. Math. Programming 86 (1999) 565-594. | MR | Zbl
[59] , On Dantzig-Wolfe decomposition in integer programming and ways to perform branching in a branch-and-price algorithm. Oper. Res. 48 (2000) 111-128. | Zbl
[60] , and, Parallel machine scheduling by column generation. Oper. Res. 47 (1999) 862-872. | MR | Zbl
[61] and, An exact algorithm for IP column generation. Oper. Res. Lett. 19 (1996) 151-159. | MR | Zbl
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