An Exact Price-Cut-and-Enumerate Method for the Capacitated Multitrip Vehicle Routing Problem with Time Windows

Abstract

We consider the capacitated multitrip vehicle routing problem with time windows (CMTVRPTW), where vehicles are allowed to make multiple trips. The ability to perform multiple trips is necessary for some real-world applications where the vehicle capacity, the trip duration, or the number of drivers or vehicles is limited. However, it substantially increases the solution difficulty in view of the additional trip scheduling aspect. We propose an exact price-cut-and-enumerate method (EPCEM) that solves a novel superstructure-based formulation inspired by Paradiso et al. ( 2020 ). The EPCEM obtains a tight lower bound by an alternating column and row generation method and computes a valid upper bound in the early stage of the algorithm. It obtains an optimal solution and further proves its optimality by a new multiphase sift-and-cut method. Computationally, the EPCEM significantly outperforms the state-of-the-art exact method that only proves optimality for 9 of the 27 test instances with 50 customers. In particular, the EPCEM solves all test instances with up to 70 customers to optimality for the first time and obtains near-optimal solutions with an average optimality gap of no more than 0.3% for instances with 80 to 100 customers. From a practical point of view, solving the CMTVRPTW by the EPCEM yields a solution that, on average, uses at least 45% fewer vehicles and increases the travel cost by no more than 7% compared with the solution to the standard CVRPTW.

Solving vehicle routing problems with intermediate stops using VRPSolver models

“…To our knowledge, the most efficient current exact algorithms for the CMTVRPTW are proposed by [31] and [44]. The latter algorithm defines the current state-of-the-art for the exact solution of the CMTVRPTW.…”

“…Our approach is particularly efficient for instances in group C (clustered instances). However, for instances in groups R and RC, some instances can be solved by [44] and cannot be solved by our approach. Superiority of our algorithm can be explained by the fact that it incorporates the main state-of-the-art techniques known to be useful for many VRP variants.…”

“…There exist many vehicle routing problems which cannot be well described as VRPSolver models. These are for example the variants of the CMTVRPTW, considered by [31] and [44]. In this variants, there is a dependency between resources, which cannot be efficiently modelled in the current version of VRPSolver .…”

“…Recently, [31] and [44] proposed exact algorithms for the four variants: CMTVRPTW, CMTVRPTW-LT, CMTVRPTW-LD and CMTVRPTW-R. [31] introduced a novel structurebased formulation that involves fewer variables than the trip-based and route-based formulations. Based on this novel formulation, the authors propose an exact solution framework that relies on column generation, column enumeration, and generation of cutting planes.…”

“…They are based on variables representing parts of routes. Trip-based formulations are employed in [24,28,31,44] for solving multi-trip VRPs. [6] tackle the traveling salesperson problem with hotel selection using the formulation in which every variable represents a single-day part of the multi-day TSP tour.…”