Abstract
The ongoing rise in e-commerce comes along with an increasing number of first-time delivery failures due to the absence of the customer at the delivery location. Failed deliveries result in rework which in turn has a large impact on the carriers’ delivery cost. In the classical vehicle routing problem (VRP) with time windows, each customer request has only one location and one time window describing where and when shipments need to be delivered. In contrast, we introduce and analyze the vehicle routing problem with delivery options (VRPDO), in which some requests can be shipped to alternative locations with possibly different time windows. Furthermore, customers may prefer some delivery options. The carrier must then select, for each request, one delivery option such that the carriers’ overall cost is minimized and a given service level regarding customer preferences is achieved. Moreover, when delivery options share a common location, e.g., a locker, capacities must be respected when assigning shipments. To solve the VRPDO exactly, we present a new branch-price-and-cut algorithm. The associated pricing subproblem is a shortest-path problem with resource constraints that we solve with a bidirectional labeling algorithm on an auxiliary network. We focus on the comparison of two alternative modeling approaches for the auxiliary network and present optimal solutions for instances with up to 100 delivery options. Moreover, we provide 17 new optimal solutions for the benchmark set for the VRP with roaming delivery locations.
Highlights
Over the past decade, mail-order trade has shown a strong compound annual growth rate, e.g., of 9.6 % in Germany with an overall revenue of 68.1 billion euros in 2018 (Furchheim et al 2020)
We stress that we have chosen this definition of the time windows because in the vehicle routing problem with delivery options (VRPDO) the total service time at a location is a variable
We show in the following that it is a variant of the shortest path problem with resource constraints (SPPRC, Irnich and Desaulniers 2005)
Summary
Mail-order trade has shown a strong compound annual growth rate, e.g., of 9.6 % in Germany with an overall revenue of 68.1 billion euros in 2018 (Furchheim et al 2020). Due to the increasing e-commerce, over 3.5 billion deliveries had to be handled in Germany in 2018 resulting in 12 million deliveries per operations day on average (BIEK 2019) These considerable numbers raise the question of how to cope with the strongly growing demand and general challenges in last-mile logistics. The main contributions of this work are the following: We introduce the VRPDO and present a branch-price-and-cut (BPC, Costa et al 2019) algorithm for its solution. We present a set-partitioning problem for the VRPDO, develop and analyze two different network structures for the solution of the pricing problem, and adapt cutting planes and branching rules. Our extensive computational study includes three parts: First, we evaluate the performance of our algorithm for the two different network structures on a newly introduced benchmark instance set.
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