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A multi-criteria vehicle routing problem with soft time windows by simulated annealing

This paper presents a multi-criteria vehicle routing problem with soft time windows (VRPSTW) to minimize fleet cost, routes cost, and violation of soft time windows penalty. In this case, the fleet is heterogeneous. The VRPSTW consists of a number of constraints in which vehicles are allowed to serve customers out of the desirable time window by a penalty. It is assumed that this relaxation affects customer satisfaction and penalty is equal to a degree of customer dissatisfaction. The VRP, which is an extension of traveling salesman problem (TSP), belongs to a class of NP-hard problems. Thus, it is necessary to use meta-heuristics for solving VRP in large-scale problems. This paper uses a simulated annealing (SA) approach with 1-Opt and 2Opt operators for solving the proposed mathematical model. The proposed model is then solved by the Lingo software and the associated solutions are compared with the computational results obtained by the SA approach for a number of instance problems. The obtained results are promising and indicating the efficiency of the proposed SA approach.