An exact algorithm for Two-Echelon Location-Routing problem with simultaneous pickup and delivery

This study considers an extension of the two-Echelon Location-Routing Problem (2E-LRP) namely two-Echelon Location-Routing Problem with Simultaneous Pickup and Delivery (2E-LRPSPD). A Mixed Integer Programming (MIP) formulation is developed and strengthened by the use of valid inequalities to solve the 2E-LRPSPD, and then a Branch and Cut-based exact algorithm (B & C) is presented. To improve the upper bounds, Iterated Local Search and Variable Neighborhood Search-based metaheuristic algorithms, namely, ILS-VNS and VNS, are developed and embedded into the B & C. Computational results, conducted on instances produced from the literature, demonstrate that the B & C yields tight lower and upper bounds, and also is capable of solving some instances optimally with up to 50 customers and 10 satellites in a reasonable time. Furthermore, sensitivity analysis reveals that the heuristic approaches and an increase in the runtime limit assist the algorithm to reach tight upper bounds, whereas the number of potential depots does not have any effect on the performance of the algorithm. Besides, further computational analysis on the 2E-LRPSPD with a single depot, which is studied in the literature, shows that the B & C improves some best-known solutions while proving the optimality of some instances. Finally, the B & C is applied to a two-echelon supermarket distribution system in Turkiye. The computational results show that simultaneous pickup and delivery operations, as well as depot location decisions are critical elements to consider in the design of cost-effective distribution networks.