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Exact hybrid algorithms for solving a bi-objective vehicle routing problem. (English) Zbl 1245.90010
Summary: The paper investigates a capacitated vehicle routing problem with two objectives: (1) minimization of total travel cost and (2) minimization of the length of the longest route. We present algorithmic variants for the exact determination of the Pareto-optimal solutions of this bi-objective problem. Our approach is based on the adaptive ε-constraint method. For solving the resulting single-objective subproblems, we apply a branch-and-cut technique, using (among others) a novel implementation of Held-Karp-type bounds. Incumbent solutions are generated by means of a single-objective genetic algorithm and, alternatively, by the multi-objective NSGA-II algorithm. Experimental results for a benchmark of 54 test instances from the TSPLIB are reported.
90B06Transportation, logistics
90C27Combinatorial optimization
90C29Multi-objective programming; goal programming
90B10Network models, deterministic (optimization)
90C57Polyhedral combinatorics, branch-and-bound, branch-and-cut
[1]Achuthan NR, Caccetta L, Hill SP (1996) A new subtour elimination constraint for the vehicle routing problem. Eur J Oper Res 91: 573–586 · Zbl 0924.90057 · doi:10.1016/0377-2217(94)00332-7
[2]Ascheuer N, Fischetti M, Grötschl M (2000) A polyhedral study of the asymmetric traveling salesman problem with time windows. Networks 36(2): 69–79 · doi:10.1002/1097-0037(200009)36:2<69::AID-NET1>3.0.CO;2-Q
[3]Ascheuer N, Fischetti M, Grötschl M (2001) Solving the asymmetric travelling salesman problem with time windows by branch-and-cut. Math Program 90: 475–506 · doi:10.1007/PL00011432
[4]Augerat P, Belenguer JM, Benavant E, Corberán A, Naddef D (1999) Seperating capacity inequalities in the CVRP using tabu search. Eur J Oper Res 106: 546–557 · Zbl 0991.90028 · doi:10.1016/S0377-2217(97)00290-7
[5]Baldacci R, Hadjiconstantinou E, Mingozzi A (2004) An exact algorithm for the capacitated vehicle routing problem based on a two-commodity network flow formulation. Oper Res 52(5): 723–738 · Zbl 1165.90353 · doi:10.1287/opre.1040.0111
[6]Bektas T (2006) The multiple traveling salesman problem: an overview of formulations and solution procedures. Omega 34(3): 209–219 · doi:10.1016/j.omega.2004.10.004
[7]Bérubé J-F, Gendreau M, Potvin J-Y (2009) An exact ϵ-constraint method for bi-objective combinatorial problems: application to the traveling salesman problem with profits. Eur J Oper Res 194: 39–50 · Zbl 1179.90274 · doi:10.1016/j.ejor.2007.12.014
[8]Chu F, Labadi N, Prins C (2006) A scatter search for the periodic capacitated arc routing problem. Eur J Oper Res 169(2): 586–605 · Zbl 1079.90028 · doi:10.1016/j.ejor.2004.08.017
[9]Deb K, Goel T (2001) Controlled elitist non-dominated sorting genetic algorithms for better convergence. In: First international conference on evolutionary multi-criterion optimization (EMO-2001)
[10]Deb K, Jain S (2002) Running performance metrics for evolutionary multi-objective optimization. KanGAL Report, 2002004
[11]Deb K, Pratap A, Meyarivan T (2000a) Constrained test problems for multi-objective evolutionary optimization. In: First international conference on evolutionary multi-criterion optimization, Springer, pp 284–298
[12]Deb K, Pratap A, Agarwal S, Meyarivan T (2000) A fast elitist multi-objective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6: 182–197 · doi:10.1109/4235.996017
[13]Fukasawa R, Longo H, Lysegaard J, Poggide Aragao M, Reis M, Uchoa E, Werneck R (2006) Robust branch-and-cut-and-price for the capacitated vehicle routing problem. Math Program 106(3): 491–511 · Zbl 1094.90050 · doi:10.1007/s10107-005-0644-x
[14]Haimes Y, Lasdon L, Wismer D (1971) On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Trans Syst Man Cybern 1: 296–297 · Zbl 0224.93016 · doi:10.1109/TSMC.1971.4308298
[15]Held M, Karp RM (1970) The traveling salesman problem and minimum spanning trees. Oper Res 18: 1135–1162
[16]Jozefowiez N, Semet F, Talbi EG (2002) Parallel and hybrid models for multi-objective optimization: application to the vehicle routing problem. In: Guervos J (eds) Parallel problem solving from nature–PPSN VII, LNCS. Springer, Granada
[17]Jozefowiez N, Semet F, Talbi EG (2006) Enhancements of NSGA-II and its application to the vehicle routing problem with route balancing. In: Talbi E (eds) Proceedings of the 7th international conference artificial evolution-EA 2005, number 3871 in LNCS, Springer, pp 131–142
[18]Jozefowiez N, Semet F, Talbi EG (2008) Multi-objective vehicle routing problems. Eur J Oper Res 189(2): 293–309 · Zbl 1148.90338 · doi:10.1016/j.ejor.2007.05.055
[19]Laporte G, Desrochers M, Nobert Y (1984) Two exact algorithms for the distance constrained vehicle routing problem. Networks 14: 161–172 · Zbl 0538.90093 · doi:10.1002/net.3230140113
[20]Laporte G, Mercure H, Nobert Y (1986) An exact algorithm for the asymmetrical capacitated vehicle routing problem. Networks 16
[21]Laumanns M, Thiele L, Zitzler E (2006) An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method. Eur J Oper Res 169(3)
[22]Lysgaard J (2003) CVRPSEP: A package of seperations routines for the capacitated vehicle routing problem. http://www.hha.dk/lys
[23]Lysgaard J, Letchford AN, Eglese RW (2003) A new branch-and-cut algorithm for the capacitated vehicle routing problem. Math Program 100: 2004
[24]Pasia JM, Doerner KF, Hartl RF, Reimann M (2007a) Solving a bi-objetive vehicle routing problem by pareto-ant colony optimization, vol 4638 of LNCS, pp 187–191. SLS 2007
[25]Pasia JM, Doerner KF, Hartl RF, Reimann M (2007b) A population-based local search for solving a bi-objective vehicle routing problem, vol 4446 of LNCS, pp 166–175. EvoCOP
[26]Prins C (2004) A simple and effective evolutionary algorithm for the vehicle routing problem. Comput Oper Res 31: 1985–2002 · Zbl 1100.90504 · doi:10.1016/S0305-0548(03)00158-8
[27]Ralphs TK, Saltzman M, Wiecek M (2004) An improved algorithm for biobjective integer programming and its application to network routing problems. Ann Oper Res
[28]Toth P, Vigo D (eds) (2001) The vehicle routing problem, vol 9 of SIAM monographs on discrete mathematics and applications, SIAM
[29]Tricoire F, Doerner KF, Hartl RF, Iori M (2009) Heuristic and exact algorithms for the multi-pile vehicle routing problem. Submitted to OR Spectrum
[30]Valenzuela CL, Jones AJ (1997) Estimating the Held-Karp lower bound for the geometric TSP. Eur J Oper Res 102(1): 157–175 · Zbl 0948.90034 · doi:10.1016/S0377-2217(96)00214-7
[31]Volgenant T, Jonker R (1982) A branch and bound algorithm for the symmetric traveling salesman problem based on 1-tree relaxation. Eur J Oper Res 9: 83–89 · Zbl 0471.90088 · doi:10.1016/0377-2217(82)90015-7
[32]Zitzler E, Thiele L (1998) Multiobjective optimization using evolutionary algorithms-a comparative case study. Lect Notes Comput Sci 292–304
[33]Zitzler E, Thiele L, Laumanns M, Fonseca CM, da Fonseca VG (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 7(2): 117–132 · doi:10.1109/TEVC.2003.810758
[34]Zitzler E, Brockhoff D, Thiele L (2007) The hypervolume indicator revisited: on the design of pareto-compliant indicators via weighted integration. Lect Notes Comput Sci 4403: 862 · doi:10.1007/978-3-540-70928-2_64