A Mathematical Model for Solving Location-Routing Problem with Simultaneous Pickup and Delivery Using a Robust Optimization Approach

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In this study, a robust optimization model is introduced, we propose a location-routing problem with simultaneous pickup and delivery under a hard time window that has a heterogeneous and limited depot and vehicle capacities and multi-variety of products and uncertain traveling time that considering all of these constraints together make the problem closer to real practical world’s problems, that not been studied in previous papers. For this purpose, a mixed-integer linear programming (MILP) model is proposed for locating depots and scheduling vehicle routing with multiple depots. Then, the robust counterpart of the proposed MILP model is proposed. The results show that GA performs much better than the exact algorithm concerning time. GAMS software fails to solve the large-size problem, and the time to find a solution grows exponentially with increasing the size of the problem. However, the GA quite efficient for problems of large sizes, and can nearly find the optimal solution in a much shorter amount of time. Also, results in the Robust model show that increasing the confidence level has led to an increase in the value of the objective function of the robust counterpart model, this increase does not exhibit linear behavior. At 80% confidence level, the minimum changes in the objective function are observed, if we want to obtain a 90% confidence level, it requires more cost, but increasing the confidence level from 70% to 80% does not need more cost, so an 80% confidence level can be considered as an ideal solution for decision-makers.