Heuristic methods for the fleet size and mix vehicle routing problem with time windows and split deliveries

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Abstract

This paper proposes a scatter-search (SS) approach to solve the Fleet Size and Mixed Vehicle Routing Problem with Time Windows and Split Deliveries (FSMVRPTWSD). In the Vehicle Routing Problem with Split Deliveries (VRPSD), each customer can be served by more than one vehicle, as opposed to the classical VRP in which each customer is served only once. In the FSMVRPTW, the customers must be serviced within their time windows with minimal costs using a heterogeneous fleet. Experimental testing and benchmark examples are used to assess the merit of our proposed procedure. The results show that the proposed heuristics are competitive with the best results found in the literature.

Highlights

► This paper studies the FSMVRPTWSD. ► The FSMVRPTW considers time windows and a heterogeneous fleet. ► In the VRPSD customers can be served by more than one vehicle. ► We propose a scatter search approach to solve the FSMVRPTWSD. ► Our algorithm is competitive with the best results found in the literature.

Introduction

The classical vehicle routing problem (VRP) aims to find a set of routes with minimal cost (by finding the shortest path, minimizing the number of vehicles, etc.) and beginning and ending the route at the depot such that the known demand of all nodes is fulfilled. Each node is visited only once by only one vehicle, and each vehicle has a limited capacity. Selected formulations also include constraints on the maximum travel time.

The VRPSD is a variation of the classical VRP in which each customer can be served by more than one vehicle. Thus, for the VRPSD, in addition to the delivery routes, the quantity to be delivered to each customer by each vehicle must also be determined. The option of splitting a given demand makes it possible to service a customer whose demand exceeds the vehicle capacity. Splitting may also result in decreased costs. The vehicle routing problem with time windows and split deliveries (VRPTWSD) presents an extension of the VRPSD in which the time window restraints are added.

In the literature, three variants of a VRP with a heterogeneous fleet have been studied. The first was introduced by Golden, Assad, Levy, and Gheysens (1984) in which the variable costs were uniformly spread over all vehicle types with the number of available vehicles assumed as unlimited for each type. This version is addressed in this paper and is known as the fleet size and mixed vehicle routing problem (FSMVRP), the vehicle fleet mix (VFM) or the fleet size and composition VRP. The second version considers the variable costs that depend on vehicle type, a component that is neglected in the first version. The third version, referred to as the heterogeneous fixed fleet vehicle routing problem (HFFVRP), generalizes the second version by limiting the number of available vehicles of each type.

Lenstra and Rinnoy Kan (1981) have analyzed the complexity of the vehicle routing problem and have concluded that practically all vehicle routing problems are NP-hard (including the classical vehicle routing problem) because they are not solved in polynomial time. Because the FSMVRPTWSD is a combination of the FSMVRP, VRPTW and VRPSD, it remains NP-hard (Archetti, Savelsbergh, et al., 2006, Dror and Trudeau, 1990, Dullaert et al., 2002, Taillard, 1999, Gendreau et al., 1999, Solomon and Desrosiers, 1988). Therefore, this observation makes a strong case for the application of heuristics and meta-heuristics to solve the problem.

This paper develops a scatter-search (SS) algorithm to solve the fleet size and mixed vehicle routing problem with time windows and split deliveries (FSMVRPTWSD). The algorithm proposed was adapted to solve three varieties of benchmark examples available in the literature: benchmark problems by Liu and Shen (1999) for the FSMVRP, benchmark problems by Ho and Haugland (2004) for the VRPSD, and benchmark problems by Solomon (1987) for the VRPTW.

The organization of this article is as follows. Section 2 reviews the literature on VRPSD and its extensions as well as FSMVRP and its extensions. Section 3 presents the problem definition, including the mathematical formulation, and Section 4 covers the scatter-search overview. Section 5 describes the constructive heuristics and the scatter-search approach proposed for solving the model. Section 6 presents the computational results, and conclusions are stated in the Section 7.

Section snippets

Literature review for the VRPSD and its extensions

The vehicle routing problem with split deliveries (VRPSD) was introduced in the literature by Dror and Trudeau, 1989, Dror and Trudeau, 1990, who presented the mathematical formulation of the problem and analyzed the savings that result when a customer is served by more than one vehicle; this situation creates economy with respect to both to the number of vehicles and the total distance traveled.

Dror, Laporte, and Trudeau (1994) have presented an integer programming formulation of the VRPSD,

Problem formulation and definitions

In this section, we define the problem under study and the notations used throughout the paper.

Scatter-search overview

According to Martí, Laguna, and Glover (2006), the scatter search is an evolutionary method that has been successfully applied to hard optimization problems. Glover, 1977, Glover, 1998 proposed the first description of this method and a scatter-search template. The SS operates on a set of reference solutions to generate new solutions via weighted linear combinations of structured subsets of solutions. This approach uses strategies for search diversification and intensification that have proven

Solution method

The initial reference set is composed of solutions generated by two different construction heuristics. An adaptation of Dullaert et al.’s sequential insertion heuristic (2002) is applied to generate quality solutions. To increase the diversity of the solution pool in the reference set, an adaptation of Golden et al.’s heuristic is also used. The solution methods are described below.

Experimental results

The algorithm was coded in Delphi 7 and run on an AMD Athlon computer with a 1-GHz processor and 512 MB of RAM memory. The algorithm proposed was adapted to solve three varieties of benchmark examples available in the literature: Solomon (1987) for the VRPTW, Ho and Haugland (2004) for the VRPSD, and Liu and Shen (1999) for the FSMVRP.

Conclusions and future research

We have developed a scatter-search algorithm for the Fleet Size and Mixed Vehicle Routing Problem with Time Windows and Split Deliveries (FSMVRPTWSD). Two constructive heuristics were proposed as initial solutions. The scatter-search framework provided a means to combine solutions as well as to diversify and intensify the meta-heuristic search process.

The model was adapted to solve three varieties of benchmark examples available in the literature: benchmark problems by Liu and Shen (1999) for

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