A matheuristic algorithm for the vehicle routing problem with cross-docking

https://doi.org/10.1016/j.asoc.2021.107163Get rights and content

Highlights

  • The vehicle routing problem with cross-docking (VRPCD) is studied.

  • A matheuristic algorithm is proposed to solve the VRPCD.

  • Eighty new best known solutions are found.

  • New sets of large benchmark instances are introduced.

Abstract

This paper studies the integration of the vehicle routing problem with cross-docking (VRPCD). The aim is to find a set of routes to deliver products from a set of suppliers to a set of customers through a cross-dock facility, such that the operational and transportation costs are minimized, without violating the vehicle capacity and time horizon constraints. A two-phase matheuristic based on column generation is proposed. The first phase focuses on generating a set of feasible candidate routes in both pickup and delivery processes by implementing an adaptive large neighborhood search algorithm. A set of destroy and repair operators are used in order to explore a large neighborhood space. The second phase focuses on solving the set partitioning model to determine the final solution. The proposed matheuristic is tested on the available benchmark VRPCD instances and compared with the state-of-the-art algorithms. Experimental results show the competitiveness of the proposed matheuristic as it is able to improve the best known solutions for 80 instances and to obtain the same results for the remaining 10 instances, with an average improvement of 12.6%. On new and larger instances, our proposed matheuristic maintains its solution quality within acceptable CPU times and outperforms a pure ALNS algorithm. We also explicitly analyze the performance of the matheuristic considering the solution quality and CPU time.

Introduction

Cross-docking is an intermediate activity within a supply chain network for enabling a transshipment process. The two key points of the cross-docking are simultaneous arrival and consolidation [1]. Different shipments for a particular destination are consolidated in a full truckload, such that direct shipment with less than truckload can be avoided, and thus the transportation cost can be minimized [2]. Whenever an incoming (inbound) vehicle arrives at the cross-dock facility, its loads are sorted, moved, and loaded to the outgoing (outbound) vehicle for an immediate delivery elsewhere in the network [3].

The vehicle routing problem with cross-docking (VRPCD), as the integration of the vehicle routing problem (VRP) and cross-docking, is quite common in practice, however, only few papers consider both simultaneously [1]. Obviously, many papers consider the classical VRP, which involves the service of a set of customers with known demands by vehicles from a single distribution center or warehouse. The main objective is to minimize the total distance and the number of vehicles which start and end their tours at the central depot. The VRPCD was first introduced by Lee et al. [4]. It aims to construct a set of routes to deliver a single type of product from a set of suppliers to a set of customers through a cross-dock facility, such that the operational and transportation costs are minimized, with respect to vehicle capacity and time limitations.

In this study, we design a matheuristic based on a branch-and-price/column generation approach, which employs a restricted master heuristic scheme [5]. This approach is commonly used since it only requires a heuristic scheme to generate columns. In the implementation, the column generation is performed by an adaptive large neighborhood search (ALNS), due to its ability to explore a large neighborhood space. The proposed matheuristic is tested on benchmark VRPCD instances, and the results are compared against those of the state-of-the-art algorithms: tabu search (TS) [4], improved tabu search (imp-TS) [1], and simulated annealing (SA) [6]. Experimental results on the available benchmark VRPCD instances show that our proposed matheuristic outperforms the state-of-the-art algorithms. New sets of larger instances which are originally developed for the VRPCDTW [7] are also introduced. On those instances, our matheuristic outperforms an ALNS algorithm. An explicit analysis is included on the added value of solving the set partitioning formulation and implementing the adaptive scheme when selecting operators in the neighborhood search process.

The main contributions of this work are summarized as follows:

  • A matheuristic algorithm is developed in order to improve the solutions of the available benchmark VRPCD instances. The optimal solutions of one set of instances which were not available before are also provided.

  • New sets of larger instances are introduced in order to evaluate the performance of our matheuristic. These instances and our solutions can also be used for future research.

  • A comprehensive analysis of the properties of our algorithm, such as the set partitioning formulation, the adaptive scheme when selecting operators, and the performance of operators, is presented.

The rest of the paper is organized as follows. Section 2 summarizes related work to the VRPCD. Section 3 defines the VRPCD network. The proposed matheuristic is presented in Section 4, followed by the computational results and some analysis in Section 5. Finally, Section 6 concludes this study.

Section snippets

Related work

Lee et al. [4] first introduced the concept of VRPCD as the integration of vehicle routing problem (VRP) and cross-docking. A mathematical model for the VRPCD network is presented and a tabu search (TS) approach is proposed to solve the problem, since it is an NP-hard problem. The proposed TS is straightforward. It generates a new solution by selecting two adjacent arcs that have the highest transportation cost. The two selected arcs are then exchanged with other arcs on the same positions from

Problem description

The VRPCD network consists of a set of suppliers S={1,2,,|S|} delivering products to a set of customers C={1,2,,|C|} through a cross-dock facility, denoted as node 0. Two major processes in a VRPCD network are: the pickup process at the suppliers and the delivery process to the customers. However, the processes inside the cross-dock facility, such as loading, unloading and sorting processes, are assumed to be fast enough. Therefore, they are not taken into consideration. Pi products must be

Proposed algorithm

In this section, we present the two phases of our matheuristic. First, we briefly describe the overview of the matheuristic and the motivation behind it. Sections 4.2 Phase 1: Adaptive large neighborhood search algorithm, 4.4 Phase 2: Set partitioning formulation for the VRPCD then describe each phase respectively. Section 4.3 provides the list of destroy and repair operators used in ALNS.

Computational results

In this section, we first present the experimental setup of this study, which includes the details about the environment used, the selected parameter configurations of the proposed matheuristic, the benchmark instances and the metrics used. Section 5.2 presents the experimental results on the given benchmark instances and a thorough analysis. The results of the newly proposed larger instances solved by the matheuristic and a pure ALNS are presented in Section 5.3. Finally, the importance of

Conclusion

We study the integration of the vehicle routing problem with cross-docking (VRPCD). A set of homogeneous vehicles is used to deliver products from a set of suppliers to a set of customers through a cross-dock facility. The aim is to select a set of vehicles to be used and its corresponding routes, such that the operational and transportation costs are minimized.

A matheuristic approach is proposed. It consists of two phases: adaptive large neighborhood search (ALNS) and set partitioning. ALNS is

CRediT authorship contribution statement

Aldy Gunawan: Conceptualization, Methodology, Formal analysis, Investigation, Writing - review & editing. Audrey Tedja Widjaja: Software, Formal analysis, Investigation, Methodology, Writing - original draft. Pieter Vansteenwegen: Conceptualization, Writing - review & editing. Vincent F. Yu: Conceptualization, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This research is supported by the Singapore Ministry of Education (MOE) Academic Research Fund (AcRF) Tier 1 grant.

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