One of the most important and widely used problems in the logistics part of any supply chain is the location-routing problem (LRP) of vehicles. The purpose is to select distribution centers to supply goods for customers and create suitable travel routes for vehicles to serve customers. Studies conducted in the field of supply chain logistics systems have shown that if vehicle travel routing is neglected when locating supply centers, the costs of the logistics system may increase dramatically. Therefore, in the LRP problem, the location of supply centers and the routing of vehicles are considered simultaneously. In this paper, we will present a multi-objective model for vehicle location-routing problems with a flexible fuzzy approach. Its' goals are to make strategic decisions to deploy candidate supply centers at the beginning of the planning horizon, as well as form the vehicle travel at the tactical level to serve the customers in short-term periods of time. Therefore, in order to adapt the mathematical model to the real conditions, the constraints related to the capacity of the vehicles have been considered in a flexible fuzzy state, and also the problem has been modeled in a multi-period state along with the presence of the distance limit and the accessibility factor for each vehicle. The evaluation criterion is to minimize costs related to the establishment of candidate supply centers, the fixed cost of using vehicles and transportation costs, as well as maximizing customer satisfaction by reducing shortage costs and reducing harmful environmental effects. To solve the model, it is first converted into a single-objective model using the weight method and then solved using the proposed algorithm. Finally, using a numerical example in the field of waste management, the effectiveness of the proposed solution method is shown. It should be mentioned that the model was solved using GAMS software and the results are shown.
