Application of a hybrid simulated annealing-mutation operator to solve fuzzy capacitated location-routing problem

In the field of supply chain management and logistics, using vehicles to deliver products from depots to customers is one of the major operations. Before using vehicles, optimizing the location of depots is necessary in a location-routing problem (LRP). Also, before transportation products, optimizing the routing of vehicles is required so as to provide a low-cost and efficient service for customers. In this paper, the mathematical modelling of LRP is developed according to the existing condition and constraint in the real world. Maximum travelling time constraint is added, and we apply fuzzy numbers to determine customer demands, travelling time and drop time. The objective is to open a subset of depots to assign customers to these depots and to design vehicle routes, in order to minimize both the cost of open depots and the total cost of the routes. The proposed problem is modelled as a fuzzy linear programming (FLP), by applying the fuzzy ranking function method; the proposed FLP is converted to an exact linear programming (LP). A Lingo solver is used to solve this LP model in very small size. LRP is an non-deterministic polynomial-time hard (NP-Hard) problem, and because of the limitation of Lingo solver in solving medium, and large-size numerical examples, a hybrid algorithm including simulated annealing and mutation operator is proposed to solve these numerical examples. Also, a heuristic algorithm is proposed to find a suitable initial solution which is used in hybrid algorithm. At the end, a different analysis of the applied algorithm and a proposed model are introduced.