##
**A hybrid multiobjective evolutionary algorithm for solving vehicle routing problem with time windows.**
*(English)*
Zbl 1111.90022

Summary: Vehicle routing problem with time windows (VRPTW) involves the routing of a set of vehicles with limited capacity from a central depot to a set of geographically dispersed customers with known demands and predefined time windows. The problem is solved by optimizing routes for the vehicles so as to meet all given constraints as well as to minimize the objectives of traveling distance and number of vehicles. This paper proposes a hybrid multiobjective evolutionary algorithm (HMOEA) that incorporates various heuristics for local exploitation in the evolutionary search and the concept of Pareto’s optimality for solving multiobjective optimization in VRPTW. The proposed HMOEA is featured with specialized genetic operators and variable-length chromosome representation to accommodate the sequence-oriented optimization in VRPTW. Unlike existing VRPTW approaches that often aggregate multiple criteria and constraints into a compromise function, the proposed HMOEA optimizes all routing constraints and objectives simultaneously, which improves the routing solutions in many aspects, such as lower routing cost, wider scattering area and better convergence trace. The HMOEA is applied to solve the benchmark Solomon’s 56 VRPTW 100-customer instances, which yields 20 routing solutions better than or competitive as compared to the best solutions published in literature.

### MSC:

90B20 | Traffic problems in operations research |

90C29 | Multi-objective and goal programming |

90C59 | Approximation methods and heuristics in mathematical programming |

### Software:

VRP
PDF
BibTeX
XML
Cite

\textit{K. C. Tan} et al., Comput. Optim. Appl. 34, No. 1, 115--151 (2006; Zbl 1111.90022)

Full Text:
DOI

### References:

[1] | H.F. Dias Alexandre and A. de Vasconcelos Jõao, ”Multiobjective genetic algorithms applied to solve optimization problems,” IEEE Transactions on Magnetic, vol. 38, no. 2, pp. 1133–1136, 2002., |

[2] | T.Bäck, Evolutionary Algorithms in Theory and Practice, Oxford University Press: New York, 1996., |

[3] | T.P. Bagchi, Multiobjective Scheduling by Genetic Algorithms, Kluwer Academic Publishers:Boston 1999., · Zbl 1155.90300 |

[4] | J.F. Bard, G. Kontoravdis, and G. Yu, ”A branch-and-cut procedure for the vehicle routing problem with time windows,” Transportation Science, vol. 36, no. 2, pp. 250–269, 2002., · Zbl 1134.90547 |

[5] | J.E. Beasley and N. Christofides, ”Vehicle routing with a sparse feasibility graph,” European Journal of Operational Research, vol. 98, no. 3, pp. 499–511, 1997., · Zbl 0930.90008 |

[6] | P. Ben, R.C. Rankin, A. Cumming, and T.C. Fogarty, ”Timetabling the classes of an entire university with an evolutionary algorithm,” Parallel Problem Solving From Nature V, Lecture Notes in Computer Science No. 1498, A. E. Eiben, T. Back, M. Schoenauer and H. Schwefel, Springer-Verlag:Amsterdam, 1998., |

[7] | R. Bent and P. VanHentenryck, ”A two-stage hybrid local search for the vehicle routing problem with time windows,” Computer Science Department, Brown University, RI, Technical Report CS-01–06, Sept. 2001., |

[8] | J. Berger, M. Barkaoui, and O. Bräysy, ”A parallel hybrid genetic algorithm for the vehicle routing problem with time windows,” Defense Research Establishment Valcartier, Canada, Working Paper, 2001., · Zbl 1100.90503 |

[9] | D. Bertsimas and D. Simchi-Levi, ”A new generation of vehicle routing research: robust algorithms, addressing uncertainty,” Operations Research, vol. 44, no. 2, pp. 286–304, 1993., · Zbl 0855.90053 |

[10] | J.L. Blanton Jr. and R.L. Wainwright, ”Multiple vehicles routing with time and capacity constraints using genetic algorithms,” Fifth International Conference on Genetic Algorithms, 1993, pp. 452–459., |

[11] | O. Bräysy, ”A reactive variable neighborhood search algorithm for the vehicle routing problem with time windows,” INFORMS Journal on Computing, vol. 15, no. 4, 2003., · Zbl 1238.90136 |

[12] | O. Bräysy and M. Gendreau, ”Genetic algorithms for the vehicle routing problem with time windows,” SINTEF Applied Mathematics, Department of Optimisation, Oslo, Norway, Internal Report STF42 A01021, 2001., |

[13] | O. Bräysy and M. Gendreau, ”Tabu search heuristics for the vehicle routing problem with time windows,” SINTEF Applied Mathematics, Department of Optimisation, Oslo, Norway, Internal Report STF42 A01022, 2001., · Zbl 1038.90005 |

[14] | A.V. Breedam, ”Comparing descent heuristic and metaheuristic for the vehicle routing problem,” Computer & Operations Research, vol. 28, no. 4, pp. 289–315, 2001., · Zbl 0976.90018 |

[15] | E.K. Burke and J.P. Newall, ”A multi-stage evolutionary algorithm for the timetable problem,” IEEE Transactions on Evolutionary Computation, vol. 3, no. 1, pp. 63–74, 1999., · Zbl 05451925 |

[16] | Y. Caseau and F. Laburthe, ”Heuristics for large constrained vehicle routing problems,” Journal of Heuristics, vol. 5, no. 3, pp. 281–303, 1999., · Zbl 1064.90508 |

[17] | J.M. Chambers, W.S. Cleveland, B. Kleiner, and P.A. Turkey, Graphical Methods for Data Analysis, Wadsworth:Belmont, CA, 1983., · Zbl 0532.65094 |

[18] | W. Chavalitwongse, D. Kim, and P.M. Pardalos, ”GRASP with a new local search scheme for vehicle routing problems with time windows,” Journal of Combinatorial Optimization, vol. 7, no. 2, pp. 179–207, 2003., · Zbl 1035.90005 |

[19] | C.L. Chen, R.V. Neppalli, and N. Aljabel, ”Genetic algorithms applied to the continuous flowshop problem,” Computers and Industrial Engineering, vol. 30, no. 4, pp. 919–929, 1996., |

[20] | W.C. Chiang and R.A. Russel, ”Simulated annealing metaheuristic for the vehicle routing problem with time windows,” Annals of Operations Research, vol. 63, pp. 3–27, 1996., · Zbl 0849.90054 |

[21] | W.C. Chiang and R.A. Ruseel, ”A reactive tabu search metaheuristics for the vehicle routing problem with time windows,” INFORMS Journal on Computing, vol. 9, pp. 417–430, 1997., · Zbl 0901.90088 |

[22] | N. Christofides, A. Mingozzi and P. Toth, ”Exact algorithms for the vehicle routing problem based on spanning tree and shortest path relaxations,” Math. Programming, vol. 20, no. 3, pp. 255–282, 1981., · Zbl 0461.90067 |

[23] | C.A. Coello Coello, ”A comprehensive survey of evolutionary-based multiobjective optimization techniques,” Knowledge and Information Systems, vol. 1, no. 3, pp. 269–308, 1999., |

[24] | C.A. Coello Coello, D.A. VanVeldhuizen, and G.B. Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic/Plenum Publishers, 2002., · Zbl 1130.90002 |

[25] | W. Cook and J.L. Rich, ”A parallel cutting plan algorithm for the vehicle routing problem with time windows,” Computational and Applied Mathematics Department, Rice University, Houston, TX, Technical Report, 1999., |

[26] | J.F. Cordeau, M. Gendreau, G. Laporte, J.Y. Potvin, and F. Semet, ”A guide to vehicle routing heuristics,” Journal of the Operational Research Society, vol. 53, no. 5, pp. 512–522, 2002., · Zbl 1099.90506 |

[27] | J.-F. Cordeau, G. Laporte, and A. Mercier, ”A unified tabu search heuristic for vehicle routing problems with time windows,” Journal of the Operational Research Society, vol. 52, no. 8, pp. 928–936, 2001., · Zbl 1181.90034 |

[28] | R. Cordone and R. Wolfler-Calvo, ”A heuristic for the vehicle routing problem with time windows,” Journal of Heuristics, vol. 7, no. 2, pp. 107–129, 2001., · Zbl 0994.90036 |

[29] | D. Cvetkovic and I.C. Parmee, ”Preferences and their application in evolutionary multiobjective optimization,” IEEE Transactions on Evolutionary Computation, vol. 6, no. 1, pp. 42–57, 2002., · Zbl 05452056 |

[30] | Z.J. Czech and P. Czarnas, ”A parallel simulated annealing for the vehicle routing problem with time windows,” Proc. 10th Euromicro Workshop on Parallel, Distributed and Network-based Processing, Canary Islands, Spain, pp. 376–383, 2002., |

[31] | B. De Backer, V. Furnon, P. Kilby, P. Prosser, and P. Shaw, ”Solving vehicle routing problems using constraint programming and metaheuristics,” Journal of Heuristics, vol. 6, no. 4, pp. 501–523, 2002., · Zbl 0972.68631 |

[32] | K. Deb, Multi-objective Optimization using Evolutionary Algorithms, John Wiley & Sons:London, 2001., · Zbl 0970.90091 |

[33] | M. Desrochers, J. Desrosiers, and M. Solomon, ”A new optimization algorithm for the vehicle routing problem with time windows,” Operational Research, vol. 40, no. 2, pp. 342–354, 1992., · Zbl 0749.90025 |

[34] | J. Desrosier, Y. Dumas, M. Solomon, and F. Soumis, ”Time constraint routing and scheduling,” Handbooks in Operations Research and Management Science 8: Network Routing, M. Ball, no. (ed.), Elsevier Science Publishers:Amsterdam, 1995, pp. 35–139., · Zbl 0861.90052 |

[35] | U. Dorndorf and E. Pesch, ”Evaluation based learning in a jobshop scheduling environment,” Computers and Operations Research, vol. 22, pp. 25–40, 1995., · Zbl 0815.90089 |

[36] | W. Dullaert, G.K. Janssens, K. Sörensen, and B. Vernimmen, ”New heuristics for the fleet size and mix vehicle routing problem with time windows,” Journal of the Operational Research Society, vol. 53, no. 11, pp. 1232–1238, 2002., · Zbl 1139.90336 |

[37] | M.L. Fisher, ”Vehicle routing,” in Handbooks in Operations Research and Management Science 8: Network Routing, M. Ball, (ed), Elsevier Science Publishers:Amsterdam, 1995, pp. 1–33., · Zbl 0870.90058 |

[38] | C.M. Fonseca, Multiobjective Genetic Algorithms with Application to Control Engineering Problems, Dept. Automatic Control and Systems Eng., University of Sheffield, Sheffield, UK, Ph.D. Thesis, 1995., |

[39] | C.M. Fonseca and P.J. Fleming, ”Genetic algorithm for multiobjective optimization, formulation, discussion and generalization,” in Genetic Algorithms: Proceeding of the Fifth International Conference. Morgan Kaufmann:San Mateo, CA, pp. 416–423, 1993., |

[40] | H. Gehring and J. Homberger, ”A parallel two phase metaheuristic for routing problems with time windows,” Asia-Pacific Journal of Operation Research, vol. 18, no. 1, pp. 35–47, 2001., |

[41] | H. Gehring and J. Homberger, ”Parallelization of a two-phase metaheuristic for routing problems with time windows,” Journal of Heuristics, vol. 3, no. 8, pp. 251–276, 2002., · Zbl 1012.68793 |

[42] | M. Gendreau, G. Laporte, and J.Y. Potvin, ”Metaheuristics for the vehicle routing problem,” University of Montreal, Canada, Les Cahiers du GERAD G-98-52, 1999., · Zbl 1076.90545 |

[43] | A. Gezdur and M. Türkay, ”MILP solution to the vehicle routing problem with time windows and discrete vehicle capacities,” XXIII National Operations Research and Industrial Engineering Congress, Istanbul, Turkey, July 2002., |

[44] | D.E. Goldberg and R. Lingle, ”Alleles, loci and the traveling salesman problem,” First International Conference on Genetic Algorithms. Lawrence Erlbaum:Hillsdale, NJ, 1985, pp. 154–159., · Zbl 0674.90095 |

[45] | B.L. Golden and A.A. Assad, Vehicle Routing: Methods and Studies, North-Holland:Amsterdam 1988., |

[46] | J.J. Grefenstette, R. Gopal, B. Rosmaita, and D. VanGucht, ”Genetic algorithms for the traveling salesman problem,” First International. Conf. Genetic Algorithms and Their Applications, pp. 160–168, 1985., |

[47] | W.K. Ho, H.J. Chin, and A. Lim, ”A hybrid search algorithm for the vehicle routing problem with time windows,” International Journal on Artificial Intelligence Tools, vol. 10, pp. 431–449, 2001., · Zbl 05421385 |

[48] | J. Homberger and H. Gehring, ”Two evolutionary metaheuristic for the vehicle routing problem with time windows,” INFOR, vol. 37, no. 1, pp. 297–318, 1999., |

[49] | G. Ioannou, M. Kritikos, and G. Prastacos, ”A greedy look ahead heuristic for the vehicle routing problem with time windows,” Journal of the Operational Research Society, vol. 52, no. 5, pp. 523–537, 2001., · Zbl 1114.90317 |

[50] | H. Ishibashi, H. Aguirre, K. Tanaka, and T. Sugimura, ”Multi-objective optimization with improved genetic algorithm,” IEEE International Conference on Systems, Man, and Cybernetics (SMC2000), Nashville, pp. 3852–3857, 2000., |

[51] | A. Jaszkiewicz, ”Multiple objective metaheuristic algorithms for combinatorial optimization,” Poznań University of Technology, Poznan, Habilitation thesis 360, 2001., · Zbl 1018.90049 |

[52] | A. Jaszkiewicz, ”Genetic local search for multiple objective combinatorial optimization,” Institute of Computing Science, Poznań University of Technology, Research Report RA-014/98, 1998., · Zbl 1002.90051 |

[53] | N. Jozefowiez, F. Semet, and E. Talbi, ”Parallel and hybrid models for multi-objective optimization: application to the vehicle routing problem,” Parallel Problem Solving from Nature, Lecture Notes in Computer Science, Springer-Verlag:New York, 2002, pp. 271–282., |

[54] | S. Jung and B.R. Moon, ”A hybrid genetic algorithm for the vehicle routing problem with time windows,” The Genetic And Evolutionary Computation Conference, Morgan Kaufmann Publishers:San Francisco, 2002, pp 1309–1316., |

[55] | B. Kallehauge, J. Larsen and O.B.G. Madsen, ”Lagrangean duality applied on vehicle routing with time windows,” IMM, Technical University of Denmark, Technical Report IMM-TR-2001–9, 2001., · Zbl 1126.90038 |

[56] | P.J. Kilby, P. Prosser, and P. Shaw, ”Guided local search for the vehicle routing problem with time windows,” in Meta Heuristics: Advances and Trends in Local Search Paradigms for Optimisation. Kluwer Academic Publishers, 1999, pp. 473–486., · Zbl 0985.90019 |

[57] | P.J. Kilby, P. Prosser, and P. Shaw, ”A comparison of traditional and constraint-based heuristic methods on vehicle routing problems with side constraints,” Journal of Constraints, vol. 5, no. 4, pp. 389–414, 2000., · Zbl 0985.90038 |

[58] | J.D. Knowles and D.W. Corne, ”Approximating the nondominated front using Pareto archived evolutionary strategy,” Evolutionary Computation, vol. 8, no. 2, pp. 149–172, 2000., · Zbl 05412708 |

[59] | N. Kohl, J. Desrosiers, O.B.G. Madsen, M.M. Solomon, and F. Soumis, ”2 path cuts for the vehicle routing problem with time windows,” Transportation Science, vol. 33, no. 1, pp. 101–116, 1999., · Zbl 1004.90015 |

[60] | G. Laporte, ”The vehicle routing problem: An overview of exact and approximate algorithms,” Europe Journal of Operational Research, vol. 59, no. 3, pp. 345–358, 1992., · Zbl 0761.90034 |

[61] | G. Laporte, M. Gendreau, J.Y. Potvin, and F. Semet, ”Classical and modern heuristics for the vehicle routing problem,” International Transaction in Operational Research, vol. 7, pp. 285–300, 2000., |

[62] | H.C. Lau, Y.F. Lim and Q.Z. Liu, ”Diversification of Search Neighborhood via Constraint-Based Local Search and Its Applications to VRPTW,” 3rd International Workshop on Integration of AI and OR Techniques (CP-AI-OR), Kent, United Kingdom, pp. 1–15, 2001., |

[63] | H.C. Lau, M. Sim and K.M. Teo, ”Vehicle routing problem with time windows and a limited number of vehicles,” European Journal of Operational Research, vol. 148, no. 3, pp. 559–569, 2003., · Zbl 1035.90014 |

[64] | L.H. Lee, K.C. Tan, K. Ou, and Y.H. Chew, ”Vehicle Capacity Planning System (VCPS): A case study on vehicle routing problem with time windows,” IEEE Transactions on Systems, Man and Cybernetics: Part A (Systems and Humans), vol. 33, no. 2, pp. 169–178, 2003., |

[65] | H. Li and A. Lim, ”Local search with annealing-like restarts to solve the vehicle routing problem with time windows,” ACM Symposium on Applied Computing (SAC 2002), pp. 560–565, 2002., |

[66] | S. Lin, ”Computer Solutions for Traveling Salesman Problem,” Bell System Technical Journal, vol. 44, pp. 2245–2269, 1965., · Zbl 0136.14705 |

[67] | S.J. Louis, X. Yin, and Z.Y. Yuan, ”Multiple vehicle routing with time windows using genetic algorithms,” Proceedings of the Congress on Evolutionary Computation, pp. 1804–1808, 1999., |

[68] | Z. Michalewicz, D.B. Fogel, and A. Michaelewica, How to Solve It: Modern Heuristics, Springer-Verlag, 1999., · Zbl 1058.68105 |

[69] | T. Murata and H. Ishibuchi, ”Performance evaluation of genetic algorithms for flow shop scheduling problems,” Computers and Industrial Engineering, vol. 30, no. 4, pp. 1061–1071, 1996., |

[70] | I.M. Oliver, D.J. Smith, and J.R.C. Holland, ”A study of permutation crossover operators on the traveling salesman problem,” in Proceedings of The Second ICGA, Lawrence Erlbaum Associates:New Jersey, 1987 pp. 224–230., |

[71] | I.H. Osman and N. Christofides, ”Simulated annealing and descent algorithms for capacitated clustering problem,” Imperial College, University of London, Research Report, 1989., |

[72] | L.F. Paquete and C.M. Fonseca, ”A study of examination timetabling with multiobjective evolutionary algorithms,” Metaheuristics International Conference, Porto:Portugal, 2001., |

[73] | M. Pinaki and M.R. Elizabeth, Genetic Algorithms for VLSI Design, Layout & Test Automation, Prentice hall:New Jersey, 1999., |

[74] | J.Y. Potvin, T. Kervahut, B. Garcia, and J.M. Rousseau, ”A Tabu search for the vehicle routing problem with time window,” Centre de Recherche sur les Transports, University de Montreal, Canada, Technical Report CRT-855, 1993., |

[75] | J.Y. Potvin, T. Kervahut, B. Garcia, and Rousseau J.M. Garcia, ”The vehicle routing problem with time windows–part I: Tabu search,” INFORMS Journal on Computing, vol. 8, no. 2, pp. 158–164, 1996., · Zbl 0866.90057 |

[76] | J.Y. Potvin and S. Bengio, ”The vehicle routing problem with time windows–part II: genetic search,” INFORMS Journal on Computing, vol. 8, no. 2, pp. 165–172, 1996., · Zbl 0866.90058 |

[77] | P.M.R. Prinetoo and M.S. Reorda, ”Hybrid genetic algorithms for the traveling salesman problem,” Fifth International Conference on Genetic Algorithms, pp. 559–566, 1993., |

[78] | C. Rego, ”Node ejection chains for the vehicle routing problem: sequential and parallel algorithms,” Parallel Computing, vol. 27, no. 3, pp. 201–222, 2001., · Zbl 0969.68176 |

[79] | Y. Rochat and E.D. Tailard, ”Probabilistic diversification and intensification in local search for vehicle routing problem,” Journal of Heuristic, vol. 1, no. 1, pp. 147–167, 1995., · Zbl 0857.90032 |

[80] | L.M. Rousseau, M. Gendreau, and G. Pesant, ”Using constraint-based operators to solve the vehicle routing with time windows,” Journal of Heuristics, vol. 8, no. 1, pp. 43–58, 2002., · Zbl 1073.90056 |

[81] | M.W.P. Savelsbergh, ”Local Search for routing problems with time windows,” Annals of Operations Research, vol. 4, pp. 285–305, 1985., |

[82] | J. Schulze and T. Fahle, ”A parallel algorithm for the vehicle routing problem with time window constraints,” Annals of Operations Research, vol. 86, pp. 585–607, 1999., · Zbl 0922.90059 |

[83] | P. Shaw, ”Using constraint programming and local search methods to solve vehicle routing problems,” Principles and Practice of Constraint Programming–CP98, Lecture Notes in Computer Science, M. Maher and J.-F. Puget (eds), Springer-Verlag:New York, 1998, pp. 417–431., |

[84] | M.M. Solomon, ”Algorithms for vehicle routing and scheduling problem with time window constraints,” Operations Research, vol. 35, no. 2, pp. 254–265, 1987., · Zbl 0625.90047 |

[85] | E. Taillard, P. Badeau, M. Gendreau, F. Guertin, and J.Y. Potvin, ”A Tabu search heuristic for the vehicle routing problem with soft time windows,” Transportation Science, vol. 31, no. 2, pp. 170–186, 1997., · Zbl 0886.90070 |

[86] | K.C. Tan, E.F. Khor, J. Cai, C.M. Heng, and T.H. Lee, ”Automating the drug scheduling of cancer chemotherapy via evolutionary computation,” Artificial Intelligence in Medicine, vol. 25, pp. 169–185, 2002., · Zbl 05391142 |

[87] | K.C. Tan, T.H. Lee, and E.F. Khor, ”Evolutionary algorithm with dynamic population size and local exploration for multiobjective optimization,” IEEE Transactions on Evolutionary Computation, vol. 5, no. 6, pp. 565–588, 2001., · Zbl 05452163 |

[88] | K.C. Tan, L.H. Lee, and K. Ou, ”Hybrid genetic algorithms in solving vehicle routing problems with time window constraints,” Asia-Pacific Journal of Operational Research, vol. 18, no. 1, pp. 121–130, 2001., · Zbl 1042.90654 |

[89] | K.C. Tan, L.H. Lee, and K. Ou, ”Artificial intelligence techniques in solving vehicle routing problems with time window constraints,” Engineering Applications of Artificial Intelligence, vol. 14, pp. 825–837, 2001., |

[90] | K.C. Tan, T.H. Lee, K. Ou, and L.H. Lee, ”A messy genetic algorithm for the vehicle routing problem with time window constraints,” IEEE Congress on Evolutionary Computation, pp. 679–686, 2001., |

[91] | K.C. Tan, L.H. Lee, Q.L. Zhu, and K. Ou, ”Heuristic methods for vehicle routing problem with time windows,” Artificial Intelligence in Engineering, vol. 15, no. 3, pp. 281–295, 2001., · Zbl 05388398 |

[92] | J. Tavares, F.B. Pereira, P. Machado, and E. Costa, ”On the influence of GVR in vehicle routing,” ACM Symposium on Applied Computing (SAC 2003), Florida, USA, March, 2003., · Zbl 1018.68784 |

[93] | S.R. Thangiah, ”An adaptive clustering method using a geometric shape for vehicle routing problems with time windows,” Sixth International Conference on Genetics Algorithm, vol. 1, pp. 452–459, 1995., |

[94] | S.R. Thangiah, I.H. Osman, and T. Sun, ”Hybrid genetic algorithm, simulated annealing and Tabu search methods for vehicle routing problems with time windows,” Computer Science Department, Slippery Rock University, Technical Report SRU CpSc-TR-94-27, 1994., |

[95] | P. Toth and D. Vigo, The Vehicle Routing Problem, SIAM:Philadelphia, 2002., · Zbl 0979.00026 |

[96] | D. Van Veldhuizen and G.B. Lamont, ”Multiobjective evolutionary algorithm research: a history and analysis,” Department of Electrical and Computer Engineering, Air Force Institute of Technology, Ohio, Technical Report TR-98-03, 1998., |

[97] | D. Van Veldhuizen and G.B. Lamont, ”Multiobjective evolutionary algorithms: analyzing the state-of-art,” Evolutionary Computation, vol. 8, no. 2, pp. 125–147, 2000., · Zbl 05412910 |

[98] | D. Whitley, T. Starkweather, and D. Fuquay, ”Scheduling problems and traveling salesmen: The genetic edge recombination operator,” Third International Conference on Genetic Algorithms, San Mateo, CA, 1989, pp. 133–140., |

[99] | P. Yellow, ”A computational modification to the saving method of vehicle scheduling,” Operational Research Quart., vol. 21, no. 2, pp. 281–283, 1970., |

[100] | J. Czech Zbigniew and C. Piotr, ”Parallel simulated annealing for the vehicle routing problem with time windows,” Silesia University of Technology, Technical report, 2001., |

[101] | E. Zitzler and L. Thiele, ”Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach,” IEEE Transactions on Evolutionary Computation, vol. 3, no. 4, pp. 257–271, 1999., · Zbl 05452215 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.