Branch-and-cut-and-price for the Electric Vehicle Routing Problem with Time Windows, Piecewise-Linear Recharging and Capacitated Recharging Stations
Introduction
This paper proposes the Electric Vehicle Routing Problem with Time Windows, Piecewise-Linear Recharging and Capacitated Recharging Stations (EVRPTW-PLR-CRS). The EVRPTW-PLR-CRS adds two complications to the Electric Vehicle Routing Problem with Time Windows (EVRPTW) which is itself based on the classical Vehicle Routing Problem with Time Windows (VRPTW).
The VRPTW appoints a fleet of identical vehicles to deliver packages from a central depot to customers before returning to the depot at the end of their routes (see Vigo and Toth, 2014, Costa et al., 2019). Every package is associated with a quantity of load. Each vehicle can carry packages up to a maximum total load called the vehicle capacity. The customer of every package is given a time frame, called the time window, within which the package will be delivered. Vehicles can arrive at a customer prior to the time window but must wait until the window opens before commencing delivery. Performing a delivery requires an amount of time called its service duration. The goal of the VRPTW is to compute routes that deliver every package while minimizing the total travel distance.
The EVRPTW extends the VRPTW to electric vehicles, each equipped with a rechargeable battery (Desaulniers et al., 2016). The battery of a vehicle is initially fully charged at the depot and depletes while the vehicle travels along its route. The battery must maintain a minimum amount of energy, denoted as 0, at all times. Vehicles must detour to a recharging station to recharge before draining their battery. The recharge rate is assumed to be constant. Therefore, the energy replenished is a linear function of the time spent recharging.
In practice, the recharge rate progressively slows to avoid damaging the battery. It can be accurately modeled as a piecewise-constant function (Montoya et al., 2017), which leads to a piecewise-linear function for representing the energy restored. The EVRPTW-PLR-CRS includes this piecewise-linear recharging function absent in the EVRPTW.
Much of the literature on routing electric vehicles assumes that the recharging stations can simultaneously recharge an unlimited number of vehicles. In practice, the stations are often equipped with only a few chargers. Hence, a vehicle may arrive at a recharging station to find that all chargers are already in use by other vehicles and, consequently, has to wait until a charger becomes available. The issue of waiting is exacerbated by the slow recharging of batteries today. The EVRPTW-PLR-CRS supplements the routing structure of the EVRPTW with a novel scheduling structure at each recharging station that both determines when a vehicle should recharge and bounds the total number of vehicles simultaneously recharging. This type of scheduling is seen at privately-owned recharging facilities where the operator has full control of the chargers. Appendix A presents an example that illustrates the combined routing and scheduling.
The scheduling structure in the EVRPTW-PLR-CRS substantially complicates the VRPTW. Every vehicle in the VRPTW can be considered independent: as long as the packages delivered along a route fit within the vehicle capacity and can be delivered on-time, delaying the route does not impact the routes of the other vehicles. In the EVRPTW, adding the energy constraints retains the independence of the vehicles: as long as a vehicle does not completely drain its battery, the routes of the other vehicles remain unaffected. However, the recharging station scheduling of the EVRPTW-PLR-CRS makes the problem much more difficult. The vehicles are no longer independent as they interact at the charging stations. To deliver the on-board packages on time, the vehicles are required to self-organize at the stations to determine the order in which the vehicles recharge.
The main contributions of this paper are an exponential-size formulation of the EVRPTW-PLR-CRS and an accompanying exact optimization algorithm that elegantly decomposes the joint routing and scheduling structure to the best technologies available for exploiting these structures. It is well-known that integer programming and particularly branch-and-cut-and-price (BCP) power the state-of-the-art exact methods for many vehicle routing problems including the VRPTW (see Vigo and Toth, 2014, Costa et al., 2019) and the EVRPTW (Desaulniers et al., 2020). It is also well-known that constraint programming generally outperforms integer programming at scheduling problems (Schutt et al., 2010, Heinz, 2018). This paper introduces a hybrid BCP algorithm that uses integer programming for routing via a Dantzig–Wolfe decomposition (see Lübbecke and Desrosiers, 2005, Desaulniers et al., 2005) and defers the scheduling to a constraint programming subproblem using the generic form (Lam and Van Hentenryck, 2017, Davies et al., 2017, Lam et al., 2020) of logic-based Benders decomposition (Hooker and Ottosson, 2003). The paper also presents an ad-hoc technique to strengthen the Benders cuts and a polyhedral analysis to further lift the Benders cuts. Experimental results indicate that the BCP algorithm solves 34% of the instances with 100 customers.
The remainder of the paper is organized as follows. Section 2 reviews related problems and solution methods. Section 3 presents a mathematical formulation of the EVRPTW-PLR-CRS and describes the hybrid BCP algorithm. Section 4 compares the empirical performance of the algorithm on different variants of the problem. Section 5 concludes this paper.
Section snippets
Background and related work
This section reviews background material and surveys relevant studies.
The branch-and-cut-and-price algorithm
This section presents an exponential-size model of the EVRPTW-PLR-CRS and an accompanying BCP algorithm for solving it.
Experimental results
This section reports the performance of the BCP algorithm and analyzes the impacts of piecewise-linear recharging and the station capacity constraints.
Conclusion and future work
This paper proposes an exact method for solving the EVRPTW-PLR-CRS. The EVRPTW-PLR-CRS includes a more realistic non-linear recharging procedure and recharging stations with a limited number of chargers which must be shared among the vehicles. To ensure that the vehicles achieve their collective goal of delivering the packages to all customers on time, the vehicles are required to self-organize at the recharging stations. This interaction leads to a novel scheduling structure rarely seen in
CRediT authorship contribution statement
Edward Lam: Conceptualization, Methodology, Software, Investigation, Writing – original draft, Writing – review & editing. Guy Desaulniers: Investigation, Writing – review & editing, Supervision. Peter J. Stuckey: Supervision, Funding acquisition.
Funding sources
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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