A two-echelon location routing problem considering sustainability and hybrid 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 leading to a particular kind of transportation problems known as open routing. However, the increasing attention to environment 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 were 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 was considered in each echelon. A small-sized problem instance was solved using the Augmented Epsilon Constraint (AEC) method with the CPLEX Optimizer Solver for the validation of the proposed model. Moreover, the sensitivity analysis was performed to investigate the effect of changing main parameters on the values of the objective function. 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) were exploited to solve the medium and large size problems. The performance of the algorithms was compared on the basis of six different well-known indexes of Time, MID, RAS, Diversity, Spacing, and SNS. According to the obtained results, the performance of the MOSFS algorithm was %20, %9, %11.22, %10.03, and %19.06 higher than the performance of the NSGA-II on the basis of SNS, RAS, MID, Diversity, and Time indexes, respectively. On the other hand, the NSGA-II performance was %6.3 higher than the MOSFS performance in terms of Spacing index.