A hybrid evolution strategy for the open vehicle routing problem

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Abstract

This paper presents a hybrid evolution strategy (ES) for solving the open vehicle routing problem (OVRP), which is a well-known combinatorial optimization problem that addresses the service of a set of customers using a homogeneous fleet of non-depot returning capacitated vehicles. The objective is to minimize the fleet size and the distance traveled. The proposed solution method manipulates a population of μ individuals using a (μ+λ)-ES; at each generation, a new intermediate population of λ offspring is produced via mutation, using arcs extracted from parent individuals. The selection and combination of arcs is dictated by a vector of strategy parameters. A multi-parent recombination operator enables the self-adaptation of the mutation rates based on the frequency of appearance of each arc and the diversity of the population. Finally, each new offspring is further improved via a memory-based trajectory local search algorithm, while an elitist scheme guides the selection of survivors. Experimental results on well-known benchmark data sets demonstrate the competitiveness of the proposed population-based hybrid metaheuristic algorithm.

Introduction

The distribution of goods or the delivery of services is vital for both customers and modern business activities, since the relevant operating costs constitute a large proportion of the overall field expenses of a company. The problem intensifies from a practical perspective when the vehicle fleet is hired, that is, vehicles do not constitute company assets [1]). In such cases, effective planning is a critical success factor for the operational efficiency and the resulting service level, since non-company resources are responsible for the physical interface with the final customer [2].

The open vehicle routing operational framework is faced by a company which either does not own a vehicle fleet at all, or its fleet is inappropriate or inadequate to satisfy the demand of its customers [1]. Thus, the company has to contract all or part of its distribution activities to external carriers. These contractors have their own vehicles, they pay their own vehicle costs (e.g. capital, operating, maintenance and depreciation), and they usually consider a compensation model based on mileage [3]. Whenever the company does not need the contractor or the vehicle back at the depot, the paths followed by the vehicles must not include the vehicle trip after the last delivery (i.e. the return trip to the depot) that will add extra mileage to the compensation model.

The allocation of distribution functions to third party logistic (3PL) providers is a beneficial business practice for several companies. For example, when a company has its own fleet and customer demand varies over time, the solution to the open vehicle routing problem (OVRP) will provide the proper combination of owned and hired vehicles [2]. Similarly, companies that have a large number of deliveries encounter the same type of problem. Although hiring vehicles is more expensive per unit distance traveled (DT), a number of costs, such as capital, maintenance and depreciation costs, do not occur [4], [5]. Typical real-life OVRP examples are the home delivery of packages and newspapers [6]. Recently, Repoussis et al. [7] developed a web-based decision support system for a real-life OVRP application concerning the distribution of lubricant products. In all cases, the contractors who are not employees of the delivery company use their own vehicles and do not return to the depot, but their invoice is based on the total DT from the depot to the final customer. When all deliveries are completed, the travel distance and time associated with each vehicle is logged and drivers are free to return to their preferred location, since this portion of travel is not reimbursed. Furthermore, it is assumed that the cost of an additional vehicle will always outweigh any traveling costs that could be saved by its use. The latter reflects the tradeoff between the vehicle hiring cost and transportation cost expressed in terms of DT.

The OVRP can be described as the problem of designing least cost routes from a depot to a set of customers. Let an incomplete graph G=(V,A), where V={0,1,,n} is the node-customer set, A={ai,j:0i,jn, ij} is the arc set and the depot is represented by the node 0. Customers are geographically dispersed within a distance radius that allows their service to be performed via a homogeneous fleet of non-depot returning capacitated vehicles. Each customer i has a demand di and a service time ei. Furthermore, there is a non-negative distance cost cij, associated with the travel from i to j. Let m denote the necessary number of vehicles (NV) required to service all customers in a feasible manner, each traveling exactly one route and M={1,2,,m} be the set of vehicles. Each vehicle kM has a maximum capacity C and a maximum route length L that limits the maximum distance it can travel. A solution Ω consists of a set of feasible vehicle routes Ω={r1,r2,,rm} that correspond to Hamiltonian paths in G over the subset of customers that start from node 0. The routes must be designed such that each customer is visited only once by exactly one vehicle. Furthermore, feasible routes must satisfy both loading and route length constraints. The former state that the accumulated service up to any customer must not exceed the capacity of the vehicle, while the latter bounds to an upper limit the total traveling time of each vehicle route. The objective of the OVRP is first to find the minimum NV required and second to determine the sequence of customers visited by each vehicle such that the total DT is minimized.

This paper develops a population-based hybrid metaheuristic algorithm for solving the OVRP. The approach utilizes the basic solution framework of evolutionary algorithms (EA) combined with a memory-based trajectory local search algorithm. The main effort is to monotonically evolve a population of individuals on the basis of the (μ+λ)-evolution strategy (ES) [8], [35], following a strictly generational course of evolution and an elitist scheme for the selection of survivors. Initiating from a population of μ adequately diversified individuals, at each generation a new intermediate population of λ individuals is produced via mutation. For this purpose, a discrete arc-based representation of individuals is utilized, combined with a binary vector of strategy parameters that determines the corresponding mutation rates. A multi-parent recombination operator enables the self-adaptation of the strategy parameters, based on the frequency of appearance of each arc and the diversity of the population. Finally, each offspring is further modified–improved via a hybrid memory-based local search algorithm. The latter uses the basic tabu search (TS) [9] solution framework to drive the local search process, while the exploration of the solution space is performed on the basis of a guided local search (GLS) [10] augmented objective function. Computational experiments on small and large-scale data sets of the literature demonstrate the efficiency and effectiveness of the proposed solution method.

The remainder of the paper is organized as follows: Section 2 provides a comprehensive literature review on methods proposed for the OVRP. Section 3 elaborates the proposed hybrid ES and provides a detailed description of all algorithmic components. Computational experiments assessing the proof of concept and the quality of the proposed method, along with a comparative performance analysis, are presented in Section 4. Finally, in Section 5 conclusions are drawn and pointers for further research are suggested.

Section snippets

Literature review

Due to its wide applicability and high complexity, the OVRP has generated substantial research considering both its modeling and solution aspects. During the last eight years, tabu search, deterministic annealing, large neighborhood search and branch-and-cut, among other methods, have been successfully applied to the OVRP. Although optimal solutions can be obtained using exact methods, the computational time required to solve adequately large problem instances is still prohibitive. For this

Motivation and structure

The goal is to monotonically evolve a population of individuals on the basis of an (μ+λ)-ES. Let P(t) denote a population of size μ at generation t and P(t) a population of λ offspring. At each generation, the μ survivors from the union of parents P(t) and offspring P(t) are selected to form the next generation population P(t+1)=[P(t)P(t)]. Herein, a strictly generational course of evolution is adopted such that one offspring is produced per population individual (λ=μ). Recombination in

Benchmark data sets

The proposed method has been tested on the well-known benchmark data sets of Christofides et al. [31], Fisher [32] and Li et al. [3]. In particular, there are 14 problems denoted as C1–C14 from Christofides et al. [31] and 2 problems denoted as F11 and F12 from Fisher [32]. Both sets range in size from 50 to 199 customers, assume Cartesian coordinates and Euclidean distances. Among them seven problems, namely C6–C10, C13 and C14, consider route length restrictions L and a uniform service time e

Conclusions

This paper presented a hybrid ES for solving the OVRP. The suggested method evolves a population of individuals by means of a (μ+λ)-ES. Following a deterministic selection scheme of survivors, offsprings are produced exclusively via mutation out of arcs extracted from single parent individuals. The latter utilizes a discrete arc-based representation combined with a binary vector of strategy parameters that dictate the mutation rates. In addition, a multi-parent recombination operator is applied

Acknowledgments

The author is indebted to the anonymous reviewers for their accurate comments and suggestions that helped improve the work presented in this paper.

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