Fuzzy multi-objective programming algorithm for vehicle routing problems with backhauls

The vehicle routing problem with backhauls (VRPB) is an extension of the standard vehicle routing problem. VRPB has two sets of customers: linehaul customers and backhaul customers. The aim of this study is to propose a new algorithm based on fuzzy multi-objective programming (FMOP-VRPB algorithm) to solve the VRPB. The FMOP-VRPB algorithm has three phases; clustering, routing and local search. In the clustering phase, customers are assigned to vehicles by the proposed multi-objective programming (MOP) model with two objective functions: minimizing the total distance and maximizing the total savings value. The proposed MOP model is solved by fuzzy operators. The weights of the objectives are also calculated by a fuzzy two-person zero-sum game with mixed strategies using membership functions in a fuzzy pay-off matrix. In the routing phase, each vehicle is routed as a traveling salesman problem with backhauls. The local search phase is used to improve the routes.