The sustainable two-echelon location-routing problem considering combined open and closed routes under uncertainty

Location-routing is an extremely important problem in supply chain management. In the location-routing problem, decisions are made about the location of facilities such as distribution centers as well as the set of vehicle routes. Today, organizations seek to reduce the transportation cost by outsourcing which leads to a specific type of transportation problems called open routing. On the other hand, the growing concerns of environmental impacts have led to paying more attention to environmental issues and reducing the environmental impacts of logistics activities. To this end, in this paper, both open and closed routes are simultaneously addressed by developing a multi-objective mixed integer linear programming model that included three economic, environmental, and social responsibility aspects. The three objective functions of the proposed model encompass the minimization of total costs and greenhouse gas emissions, and the maximization of employment rate and economic development. Also, in this study, a different type of routing is considered in each echelon. A small-sized problem instance is solved using the Augmented Epsilon Constraint (AEC) method with the CPLEX Optimizer Solver for the validation of the proposed model. Due to the NP-Hardness of the problem, two efficient metaheuristic algorithms of Non-dominated Sorting Genetic Algorithm (NSGA-II) and Multi-Objective Stochastic Fractal Search (MOSFS) are exploited to solve the medium and large size problems. The performance of the algorithms is compared in terms of time, MID, diversity, spacing, SNS, and RAS indexes. The results show that the MOSFS algorithm outperforms the NSGA-II based on several indexes.