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A bi-objective column generation algorithm for the multi-commodity minimum cost flow problem

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  • Moradi, Siamak
  • Raith, Andrea
  • Ehrgott, Matthias

Abstract

We present a column generation algorithm for solving the bi-objective multi-commodity minimum cost flow problem. This method is based on the bi-objective simplex method and Dantzig–Wolfe decomposition. The method is initialised by optimising the problem with respect to the first objective, a single objective multi-commodity flow problem, which is solved using Dantzig–Wolfe decomposition. Then, similar to the bi-objective simplex method, our algorithm iteratively moves from one non-dominated extreme point to the next one by finding entering variables with the maximum ratio of improvement of the second objective over deterioration of the first objective. Our method reformulates the problem into a bi-objective master problem over a set of capacity constraints and several single objective linear fractional sub-problems each over a set of network flow conservation constraints. The master problem iteratively updates cost coefficients for the fractional sub-problems. Based on these cost coefficients an optimal solution of each sub-problem is obtained. The solution with the best ratio objective value out of all sub-problems represents the entering variable for the master basis. The algorithm terminates when there is no entering variable which can improve the second objective by deteriorating the first objective. This implies that all non-dominated extreme points of the original problem are obtained. We report on the performance of the algorithm on several directed bi-objective network instances with different characteristics and different numbers of commodities.

Suggested Citation

  • Moradi, Siamak & Raith, Andrea & Ehrgott, Matthias, 2015. "A bi-objective column generation algorithm for the multi-commodity minimum cost flow problem," European Journal of Operational Research, Elsevier, vol. 244(2), pages 369-378.
    • Handle: RePEc:eee:ejores:v:244:y:2015:i:2:p:369-378
    DOI: 10.1016/j.ejor.2015.01.021
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    1. Current, John & Marsh, Michael, 1993. "Multiobjective transportation network design and routing problems: Taxonomy and annotation," European Journal of Operational Research, Elsevier, vol. 65(1), pages 4-19, February.
    2. D. Klingman & A. Napier & J. Stutz, 1974. "NETGEN: A Program for Generating Large Scale Capacitated Assignment, Transportation, and Minimum Cost Flow Network Problems," Management Science, INFORMS, vol. 20(5), pages 814-821, January.
    3. Current, John & Min, HoKey, 1986. "Multiobjective design of transportation networks: Taxonomy and annotation," European Journal of Operational Research, Elsevier, vol. 26(2), pages 187-201, August.
    4. Sang M. Lee & Marc J. Schniederjans, 1983. "A Multicriteria Assignment Problem: A Goal Programming Approach," Interfaces, INFORMS, vol. 13(4), pages 75-81, August.
    5. Y. P. Aneja & K. P. K. Nair, 1979. "Bicriteria Transportation Problem," Management Science, INFORMS, vol. 25(1), pages 73-78, January.
    6. M. Ehrgott & J. Puerto & A. M. Rodríguez-Chía, 2007. "Primal-Dual Simplex Method for Multiobjective Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 483-497, September.
    7. Hamacher, Horst W. & Pedersen, Christian Roed & Ruzika, Stefan, 2007. "Multiple objective minimum cost flow problems: A review," European Journal of Operational Research, Elsevier, vol. 176(3), pages 1404-1422, February.
    8. J. A. Tomlin, 1966. "Minimum-Cost Multicommodity Network Flows," Operations Research, INFORMS, vol. 14(1), pages 45-51, February.
    9. Eusébio, Augusto & Figueira, José Rui, 2009. "On the computation of all supported efficient solutions in multi-objective integer network flow problems," European Journal of Operational Research, Elsevier, vol. 199(1), pages 68-76, November.
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    4. Oğuz, Murat & Bektaş, Tolga & Bennell, Julia A., 2018. "Multicommodity flows and Benders decomposition for restricted continuous location problems," European Journal of Operational Research, Elsevier, vol. 266(3), pages 851-863.
    5. Gülesin Sena Daş & Fatma Gzara, 2024. "Column generation based solution for bi-objective gate assignment problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 123-151, August.
    6. Rajabighamchi, Farzaneh & van Hoesel, Stan & Defryn, Christof, 2023. "Bus service for cargo," Research Memorandum 005, Maastricht University, Graduate School of Business and Economics (GSBE).
    7. Hela Masri & Saoussen Krichen, 2018. "Exact and approximate approaches for the Pareto front generation of the single path multicommodity flow problem," Annals of Operations Research, Springer, vol. 267(1), pages 353-377, August.
    8. Khodakaram Salimifard & Sara Bigharaz, 2022. "The multicommodity network flow problem: state of the art classification, applications, and solution methods," Operational Research, Springer, vol. 22(1), pages 1-47, March.
    9. François Lamothe & Emmanuel Rachelson & Alain Haït & Cedric Baudoin & Jean-Baptiste Dupé, 2021. "Randomized rounding algorithms for large scale unsplittable flow problems," Journal of Heuristics, Springer, vol. 27(6), pages 1081-1110, December.
    10. Kuan-Min Lin & Matthias Ehrgott & Andrea Raith, 2017. "Integrating column generation in a method to compute a discrete representation of the non-dominated set of multi-objective linear programmes," 4OR, Springer, vol. 15(4), pages 331-357, December.
    11. Archetti, Claudia & Peirano, Lorenzo & Speranza, M. Grazia, 2022. "Optimization in multimodal freight transportation problems: A Survey," European Journal of Operational Research, Elsevier, vol. 299(1), pages 1-20.

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