A Branch-Cut-and-Price Approach for the Single-Trip and Multi-Trip Two-Echelon Vehicle Routing Problem with Time Windows

Abstract

The paper studies the two-echelon capacitated vehicle routing problem with time windows, in which delivery of freight from depots to customers is performed using intermediate facilities called satellites. We consider the variant of the problem with precedence constraints for unloading and loading freight at satellites. In this variant allows for storage and consolidation of freight at satellites. Thus, the total transportation cost may decrease in comparison with the alternative variant with exact freight synchronization at satellites. We suggest a mixed integer programming formulation for the problem with an exponential number of route variables and an exponential number of precedence constraints that link first-echelon and second-echelon routes. Routes at the second echelon connecting satellites and clients may consist of multiple trips and visit several satellites. A branch-cut-and-price algorithm is proposed to solve efficiently the problem. This is the first exact algorithm in the literature for the multi-trip variant of the problem. We also present a postprocessing procedure to check whether the solution can be transformed to avoid freight consolidation and storage without increasing its transportation cost. It is shown that all single-trip literature instances solved to optimality admit optimal solutions of the same cost for both variants of the problem either with precedence constraints or with exact synchronization constraints. Experimental results reveal that our algorithm can be used to solve these instances significantly faster than another recent approach proposed in the literature.

A matheuristic for a 2-echelon vehicle routing problem with capacitated satellites and reverse flows

“…The LNS and MIP components are evaluated in Section 5.2. In Section 5.3, we compare our algorithm on 2E-VRPTW instances of Marques et al (2020a). Finally, the sensitivity analyses of Section 5.4 provide some managerial insights.…”

“…The algorithm performs well on the instances generated by Dellaert et al (2019), computing 41 new optimal solutions in the 3-hour time limit. Independently, Marques et al (2020a) present BPC algorithms for solving the 2E-VRPTW and the 2E-MTVRPTW. The latter is a relaxed variant of the 2E-MTVRP-SS in which synchronization with precedences is required (instead of exact synchronization).…”

“…For the 2E-VRPTW benchmark, distances are rounded up to two digits as done by Grangier et al (2016). For the 2E-MTVRP-CSRF benchmark, travel times (in minutes) and distances (=routing costs) are computed as Euclidean distances rounded up to the next integer value (zero digits) as done for the instances described by Marques et al (2020a).…”

“…We cannot compare our results with those of Grangier et al (2016), who considered exact synchronization at satellites between first-echelon and second-echelon vehicles. Recently, Marques et al (2020a) extended this benchmark set by creating smaller instances with 25, 50, and 75 customers. They consider only the first customers in the respective 100-customer instance.…”

“…We can directly compare our method with the results of the branch-price-and-cut (BPC) algorithm of Marques et al (2020a) for the 2E-MTVRPTW. To do so, the capacity of satellites is assumed infinite and we minimize the vehicle-related cost that includes a fixed cost of 50 (25) per first(second)-echelon vehicle.…”