Solving the battery swap station location-routing problem with a mixed fleet of electric and conventional vehicles using a heuristic branch-and-price algorithm with an adaptive selection scheme

Highlights

• Location routing problem, including time windows, and a mixed fleet, is investigated.

• A branch-and-price algorithm combining exact and heuristic policies is proposed.

• The proposed algorithm finds the optimal solution quickly to small-scale instances.

• The proposed algorithm performs better than existing ones for large-scale instances.

Abstract

In this paper, a battery swap station location and routing problem with time windows and a mixed fleet of electric and conventional vehicles (BSS–MF–LRPTW) is proposed. This problem is motivated by a real-life logistics application by extending the existing electric vehicle battery swap stations location routing problem (BSS–EV–LRP). The BSS–MF–LRPTW aims to simultaneously determine the locations of battery swap stations (BSSs) and the routing plan of a mixed fleet under the driving range, the load capacity limitation, and time windows. An integer programming (IP) model is developed for the proposed BSS–MF–LRPTW. As there are a large number of variables and complicating constraints of the IP model, we break it up into the master problem and the subproblem, based on Danzig–Wolfe decomposition. To enhance the tractability of the problem, we follow up with a heuristic branch-and-price algorithm with an adaptive selection scheme (HBP-ASS), which integrates the exact policy with a heuristic strategy. The HBP-ASS develops heuristic versions of the dynamic programming algorithm by designing seven operators for heuristic label extension and dominance. An adaptive selection scheme is presented to decide which operator is employed. The performance of the proposed HBP-ASS is evaluated based on an extensive computational study. The results show that the HBP-ASS can obtain the exact solution to small-scale instances much more quickly than commercial branch-and-bound/cut solvers such as CPLEX. Also, for all tested cases, the HBP-ASS can find better solutions to large-scale instances within a shorter time than the existing heuristics – adaptive large neighborhood search. Furthermore, sensitivity analyses are carried out to provide managerial insights.

Introduction

In China, the successful development of electronic commerce leads to the substantial growth of the transportation sector in recent years. From January 2020 to September 2020, the express delivery companies delivered 56.14 billion items, a year-on-year increase of 27.9% in China (State post bureau of the people’s republic of China, 2020). However, increasing delivery with conventional vehicles (CVs) causes environmental problems related to polluting emissions, noise, and congestion. At present, more and more laws and regulations are passed to control the emission of gases from CVs. Developing more environmentally friendly transportation modes, such as electric vehicles (EVs), is becoming necessary. EVs can provide an alternative to CVs. However, EVs face many limitations in real-world applications, such as the limited driving range, significantly considerable recharging time, and the availability of charging infrastructure. To tackle these issues, logistics companies open their battery swap stations (BSSs). The nearly depleted battery is replaced by a fully charged one at a BSS (Yang and Sun, 2015). The process is called battery swapping. The battery swapping process is auspicious. It usually takes less than 10 min and is much faster than the recharge with the fastest technology at a charging station (Kim, 2011). The battery swapping improves vehicles' productivity. Also, the depleted batteries can be recharged together at night at a discounted electricity price, decreasing charging costs (United Nations Environment Programme, 2009).

To render EVs more competitive, we should make simultaneous decisions, including (i) the BSS to be opened from a set of BSSs candidate locations and (ii) EVs' routes to be determined to minimize the total transportation cost and the cost for establishing the BSSs. These decisions combine the strategic level (location of the BSSs) and tactical level (routing of EVs). They integrate locations for electric vehicle battery swap stations and the routing problem (BSS–EV–LRP) introduced by Yang and Sun (2015).

In real-world applications, CVs are mainly used. A complete transition to an electric fleet is a very challenging task. Therefore, most logistics companies gradually introduce EVs into their existing CV fleet (Goeke and Schneider, 2015, Erdelić and Carić, 2019). A mixed fleet of EVs and CVs instead of a complete EV fleet can extend the existing BSS–EV–LRP. EVs and CVs have different characteristics in maximal driving range, operating cost, load capacity, and charging activities. Location and route planning for a mixed fleet should consider the tradeoff between EVs and CVs. Moreover, the existing BSS–EV–LRP models ignore time windows wherein the customer has to be visited. However, it is very common for customers to have their time window requirements in a real-world industrial application.

This research extends the existing work on the BSS–EV–LRP, including several crucial aspects such as a mixed fleet of EVs and CVs and time windows. Specifically, we consider the BBS locations (L), routing problem (RP) with time windows (TW), and a mixed fleet (MF), and that is called the BSS–MF–LRPTW. The proposed BSS–MF–LRPTW determines: (a) the location of the BSSs, (b) the assignment of customers to EVs and CVs, (c) the allocation of EVs to BSSs, and (d) the routes from the depot to customers considering the driving range and the load capacity limitation of EVs and CVs to minimize the total transportation cost and the cost for establishing the BSSs.

The BSS–EV–LRP can be classified as a location-routing problem (LRP), that is proven NP-hard (Hof et al., 2017). By adding the complexity of a mixed fleet and time windows requirements into the BSS–EV–LRP, the resulting BSS–MF–LRPTW is more challenging to be solved than the BSS–EV–LRP. Therefore, a proper and effective method must be developed for the proposed BSS–MF–LRPTW. The existing literature on BSS–EV–LRP proposes heuristic approaches, such as adaptive large neighborhood search (ALNS). However, none of them attempt to obtian the optimal solution based on the exact algorithm. Therefore, this study aims to introduce the exact algorithm, branch-and-price (B&P), to obtain a high-quality solution to the BSS–MF–LRPTW.

Given that the proposed BSS–MF–LRPTW is much more complicated than general LRPs, this research proposes a methodology. It embeds heuristic strategies and an adaptive selection scheme in the B&P framework to radically reduce computational effort while finding a high-quality solution, especially in large-sized instances. Thus, the proposed methodology can obtain the exact solution to small-scale instances with a significantly shorter run time than IBM CPLEX Optimizer. For all the tested cases, it can also find a better solution to large-scale instances more quickly than existing heuristics-ALNS. That makes the solution more practical in real-world logistics applications.

The main contributions of this research are threefold: (a) the BSS–EV–LRP is extended to the BSS–MF–LRPTW, which is motivated by a real-life logistics application. An integer program formulation is developed. It simultaneously determines the routing and locations of BSSs given the time window constraints and a mixed fleet of EVs and CVs; (b) A heuristic branch-and-price algorithm with an adaptive selection scheme (HBP-ASS), which combines exact and heuristic policies, is developed. The HBP-ASS introduces a heuristic version of the dynamic programming algorithm by developing seven heuristic operators in the label extension and dominance procedure. An adaptive selection scheme is also presented to choose the appropriate operator. The proposed heuristic version of dynamic programming can effectively generate new attractive columns and help achieve LP optimality or find a near-optimal solution within a short time. The proposed methodology also extends the elementary shortest path problem with resource constraints (ESPPRC) when solving the subproblem and (c) Effects of the operating cost of the BBS, the size of the mixed fleet, and the battery swapping time on the total cost are examined. This study may help logistics managers gain insight into applications in this problem.

The remainder of this paper is organized as follows. Section 2 provides an overview of the literature on the electric vehicle routing problem, electric vehicle routing problem with a mixed fleet, and electric location-routing problem. Section 3 presents an integer programming model for BSS–MF–LRPTW. Then the model is reformulated to the master problem and pricing problem. Section 4 provides a heuristic branch and price with an adaptive selection scheme (HBP-ASS) for the proposed model. Section 5 presents the performance of HBP-ASS on small-scale and large-scale instances. Managerial insights are also provided. Section 6 concludes the paper with a summary of the main findings.