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 mini-mize fleet cost, routes cost, and violation of soft time windows penalty. in this case, the fleet is heterogene-ous. 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 sales-man 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 2-opt 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 ap-proach for a number of instance problems. the obtained results are promising and indicating the efficiency of the proposed sa approach.