The Vehicle Routing Problem with Partial Outsourcing

Abstract

This paper introduces the vehicle routing problem with partial outsourcing (VRPPO) in which a customer can be served by a single private vehicle, by a common carrier, or by both a single private vehicle and a common carrier. As such, it is a variant of the vehicle routing problem with private fleet and common carrier (VRPPC). The objective of the VRPPO is to minimize fixed and variable costs of the private fleet plus the outsourcing cost. We propose two different path-based formulations for the VRPPO and solve these with a branch-and-price-and-cut solution method. For each path-based formulation, two different pricing procedures are designed and used when solving the linear relaxations by column generation. To assess the quality of the solution methods and gain insight in potential cost improvements compared with the VRPPC, we perform tests on two instance sets with up to 100 customers from the literature.

The Vehicle Routing Problem with Ftl and Ltl Carriers

“…Table 5 compares our results to the exact results obtained by Baller et al (2020) on the VRPPC. Each line aggregates the results for one category of instances.…”

“…We benchmarked the results of the LNS-SPP algorithm against the most recent works on the HFF-VRP-TW-PC. Baller et al (2020) derive instances from those of Dabia et al (2019) and propose two categories of instances with high and low outsourcing costs, respectively. As in the original instances, they were adapted from Solomon's instances (Solomon, 1987).…”

“…Each category of instances is made up of 56 instances. The column entitled # shows the number of instances (out of 56) solved to optimality by the BCP algorithm of Baller et al (2020) within the one hour time limit. Note that these results ignore instances for which the optimal solution consists in shipping all orders in LTL.…”

“…Both models are solved with a Branch-and-Cut-and-Price algorithm, which is evaluated on instances of Liu and Shen (1999), that were adapted to handle the case of subcontracting costs and a fleet limited to three units of each vehicle type. Baller et al (2020) study the VRP with Partial Outsourcing (VRPPO), in which delivery to any customer can be split between the private fleet and external carriers. Two route-based SPP formulations are proposed as well as a Branch-and-Cut-and-Price to solve them.…”

“…Comparison to optimal solutions: results of LNS-SPP on VRP-PC instances compared to the branch-and-cutand-price ofBaller et al (2020) …”