Logic-based Benders decomposition for the heterogeneous fixed fleet vehicle routing problem with time windows

https://doi.org/10.1016/j.cie.2020.106641Get rights and content

Highlights

  • We present the first exact algorithms for the standard HFFVRPTW.

  • Benders Cuts are devised to ensure the algorithmic convergence to a global optimum.

  • Several algorithmic enhancements are suggested.

  • Extensive computational experiments illustrate the effectiveness of the algorithms.

Abstract

This paper presents exact algorithms based on logic-based Benders decomposition and a variant, called branch-and-check, for the heterogeneous fixed fleet vehicle routing problem with time windows. The objective is to service, at the minimal cost, a set of geographically dispersed customers within their time windows by a limited and capacitated fleet of heterogeneous vehicles. The proposed algorithms decompose the problem into a generalized assignment master problem and independent traveling salesman subproblems with time windows. Valid optimality and feasibility cuts are devised to guarantee the convergence of the algorithms, which include enhancements to solve the master problem and the subproblems. Extensive computational experiments on 216 benchmark instances illustrate the effectiveness of the suggested approaches. Instances with up to 100 customers are solved to proven optimality and the results indicate that the best proposed algorithm is competitive with state-of-the-art methods.

Introduction

The vehicle routing problem (VRP) plays a key role in the optimization of logistic distribution management and is one of the main themes addressed in the field of operational research. In its classical version, introduced by Dantzig and Ramser (1959), we are given a set of geographically dispersed customers, each one with a demand, a single depot and a homogeneous vehicle fleet with fixed capacity. The objective is to determine minimum-cost routes, starting from and ending at the depot, such that the demand of each customer is satisfied, each customer is visited only once, and the total demand of each route does not exceed the capacity of a vehicle.

Over the past decades, research has been conducted on models and algorithms for a large number of practical VRP variants and, as a consequence, a vast literature has been developed as evidenced by the books edited by Golden et al., 2008, Toth and Vigo, 2014, see also the state of the art taxonomy and review provided by Eksioglu et al., 2009, Braekers et al., 2016.

Real-world VRP applications, together with complex objective functions and constraints, have been termed rich vehicle routing problems (RVRP) in the literature, see the surveys proposed by Drexl, 2012, Caceres-Cruz et al., 2015, Lahyani et al., 2015. The last provides a taxonomy for the RVRP literature with respect to real-life attributes such as load splitting constraints, time restrictions and vehicle characteristics.

This paper deals with two of the most common VRP attributes occurring in practice, namely time windows and the use of heterogeneous fleets. The resulting problem is called the heterogeneous fixed fleet vehicle routing problem with time windows (HFFVRPTW). Time windows restrict the time interval in which the service at each customer starts, and can be found in many applications of ground, maritime and air transportation problems. A recent survey on the VRP with time windows (VRPTW) is provided in Desaulniers, Madsen, and Ropke (2014).

A heterogeneous fleet consists of multiple types of vehicles with capacity constraints expressed in terms of weight, volume, number of pallets, stacks and compartments. This is a relevant VRP attribute since the assumption of homogeneous fleets is not realistic in most industrial applications (Hoff, Andersson, Christiansen, Hasle, & Løkketangen, 2010). To illustrate, the magazine Transport Topics specialized in logistics and trucking news presents the top 100 largest private carriers in North America for 2018 (Transport Topics, 2020). PepsiCo Inc. is placed in the first position with 11,100 tractors, 17,500 trailers, 3500 straight trucks, 20,600 pickup trucks and cargo vans. Another example is Amazon that is increasing its online grocery pickup and delivery in a 2-hour time window. For this purpose, they require a driver with a 4-door, mid-sized sedan or a larger vehicle, such as a truck with a covered bed, SUV or a van. Typically, the driver is hired for 3–6 h and the routes are determined by the Amazon software (Amazon, 2020).

Given the significance of such attributes, we propose exact algorithms based on logic-based Benders decomposition (LBBD) (Hooker, 2000, Hooker and Ottosson, 2003) and a variant, called branch-and-check (BAC) (Thorsteinsson, 2001), for the HFFVRPTW. Our algorithms are based on a mathematical formulation that suggests a two-phase method called cluster-first and route-second (Fisher & Jaikumar, 1981). Such approach is general enough to encompass both exact and heuristic methods, for further details see the comprehensive survey of Archetti and Speranza (2014) on decomposition matheuristics for VRP variants. In our application, the Benders master problem is a generalized assignment problem (GAP) with binary variables that indicate the assignment of each customer to a type of vehicle. When such variables are fixed, the Benders independent subproblems have the special structure of the traveling salesman problem with time windows (TSPTW). Optimality cuts and/or feasibility cuts are generated from the solutions of subproblems and are added to the master problem which is solved in the subsequent iteration.

The main contributions of this paper are threefold:

  • 1.

    To the best of our knowledge, we present the first exact algorithms for the standard HFFVRPTW.

  • 2.

    We suggest several enhancements for the proposed algorithms and provide a detailed analysis of their effectiveness.

  • 3.

    We conduct extensive computational experiments on benchmark instances and assess the performance of the best proposed algorithm against state-of-the-art methods.

The remainder of the article is organized as follows. Section 2 summarizes related work, while Section 3 introduces the problem description. Section 4 describes the exact algorithms based on LBBD and BAC. Section 5 details the suggested algorithmic enhancements. Computational results are reported in Section 6, and conclusions are outlined in Section 7.

Section snippets

Related work

There are two major VRP variants involving heterogeneous fleets, namely the fleet size and mix VRP (FSMVRP) and the heterogeneous fixed fleet VRP (HFFVRP). The first one was introduced in the seminal paper of Golden, Assad, Levy, and Gheysens (1984) and comprises both fleet composition and routing tasks, whereas the second considers a fleet with limited number of vehicles, as suggested by Taillard (1999). The objective of these problems includes vehicle fixed and/or variable costs, and their

Problem description

The HFFVRPTW is defined on a complete graph N,E where the node set N=0,1,,n consists of the depot 0 and the set of n customers C=N\{0}, while the set E=i,j:i,jN,ij represents the edges between the nodes. A heterogeneous fleet of vehicles is positioned at the depot in order to supply the customers. The set K=1,..., represents the distinct types of vehicles. For each type kK, the set Mk=1,,mk represents the mk vehicles available at the depot, each having capacity Qk and an associated

Logic-based Benders decomposition for the HFFVRPTW

Benders decomposition (Benders, 1962) was originally proposed for MIP problems and applies techniques of projection, outer linearization, and relaxation to reformulate a problem in a more amenable form. This approach partitions a MIP problem into two simpler ones: the master problem, a relaxed version of the original problem including its integer variables; and the subproblem, a linear program parametrized by the values of integer variables temporarily fixed in the master problem. At each

Algorithmic enhancements

Extensive research has been carried out to explore ways to accelerate the straightforward application of Benders decomposition, which may require too much computing time and memory. For a thorough review on this topic, see Rahmaniani, Crainic, Gendreau, and Rei (2017). In this section, we describe algorithmic enhancements that improve the convergence of LBBD and BAC.

Computational experiments

This section describes the computational results of the proposed algorithms, which were programmed in C# and run on an CPU Intel® Core™ i7-6500U 2.5 GHz with 16 GB RAM. As MIP solver, we used the Gurobi v.6.05 software. We start by presenting the benchmark instances and the parameters used within our algorithms in 6.1 Instance generation, 6.2 Parameter setting, respectively. Section 6.3 assesses the performance of LBBD versus BAC, and Section 6.4 analyses the impact of the algorithmic

Conclusions

We have proposed two exact algorithms under the cluster-first and route-second approach for the HFFVRPTW, namely LBBD and BAC. Both apply a Benders decomposition scheme that leads to a GAP master problem and independent TSPTW subproblems. Valid optimality and feasibility cuts are devised to guarantee the convergence of such methods to optimality in a finite number of iterations, and various algorithmic enhancements are presented. Furthermore, extensive computational tests were carried out on

CRediT authorship contribution statement

Ramon Faganello Fachini: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review & editing, Visualization, Project administration. Vinícius Amaral Armentano: Conceptualization, Methodology, Resources, Writing - review & editing, Supervision, Funding acquisition.

Acknowledgments

This research was funded by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq - Brazil, grants 141064/2015-3 and 305577/2014-0) and the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP - Brazil, grants 2016/01860-1 and 2016/06566-4). We are grateful for the valuable suggestions made by the editor and referees.

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