Production, Manufacturing and Logistics
An improved ant colony optimization for vehicle routing problem

https://doi.org/10.1016/j.ejor.2008.02.028Get rights and content

Abstract

The vehicle routing problem (VRP), a well-known combinatorial optimization problem, holds a central place in logistics management. This paper proposes an improved ant colony optimization (IACO), which possesses a new strategy to update the increased pheromone, called ant-weight strategy, and a mutation operation, to solve VRP. The computational results for fourteen benchmark problems are reported and compared to those of other metaheuristic approaches.

Introduction

Finding efficient vehicle routes is a representative logistics problem which has been studied for the last 40 years. A typical vehicle routing problem (VRP) aims to find a set of tours for several vehicles from a depot to a lot of customers and return to the depot without exceeding the capacity constraints of each vehicle at minimum cost. Since the customer combination is not restricted to the selection of vehicle routes, VRP is considered as a combinatorial optimization problem where the number of feasible solutions for the problem increases exponentially with the number of customers increasing (Bell and McMullen, 2004).

Heuristic algorithms such as simulated annealing (SA) (Chiang and Russell, 1996, Koulamas et al., 1994, Osman, 1993, Tavakkoli-Moghaddam et al., 2006), genetic algorithms (GAs) (Baker and Ayechew, 2003, Osman et al., 2005, Thangiah et al., 1994, Prins, 2004), tabu search (TS) (Gendreau et al., 1999, Semet and Taillard, 1993, Renaud et al., 1996, Brandao and Mercer, 1997, Osman, 1993) and ant colony optimization (Doerner et al., 2002, Reimann et al., 2002, Peng et al., 2005, Mazzeo and Loiseau, 2004, Bullnheimer et al., 1999, Doerner et al., 2004) are widely used for solving the VRP. Among these heuristic algorithms, ant colony optimizations (ACO) are new optimization methods proposed by Italian researchers Dorigo et al. (1996), which simulate the food-seeking behaviors of ant colonies in nature. It has been successfully applied as a solution to some classic compounding optimization problems, e.g. traveling salesman (Dorigo et al., 1996) quadratic assignment (Gambardella et al., 1997), job-shop scheduling (Colorni et al., 1994), telecommunication routing (Schoonderwoerd et al., 1997), etc.

If taking the central depot as the nest and customers as the food, the VRP is very similar to food-seeking behaviors of ant colonies in nature. This makes the coding of an ant colony optimization for the VRP is simple. Among the earliest studies was that of Bullnheimer et al. (1997) who presented a hybrid ant system algorithm with the 2-opt and the saving algorithm for the VRP. Other researches of the ACOs to the VRP included the work by Bullnheimer et al., 1999, Bell and McMullen, 2004, Chen and Ting, 2006. In the ACOs, the 2-opt exchange was used as an improvement heuristic within the routes found by individual vehicles and the pheromone updating rules mainly considered the global feature of the solution. This paper proposes an improved ant colony optimization with a new pheromone updating rule that can integrate the global feature and the local feature, a mutation operation and the 2-opt exchange for the VRP. The remainder of the paper is organized as follows. Section 2 presents the mathematical model for VRP. In Section 3, we present the IACO with ant-weight strategy and the mutation operation. Some computational results are discussed in Section 4 and lastly, the conclusions are provided in Section 5.

Section snippets

Vehicle routing problem

The VRP is described as a weighted graph G = (C, L) where the nodes are represented by C = {c0, c1, …, cN} and the arcs are represented by L = {(ci, cj): i  j}. In this graph model, c0 is the central depot and the other nodes are the N customers to be served. Each node is associated with a fixed quantity qi of goods to be delivered (a quantity q0 = 0 is associated to the depot c0). To each arc (ci, cj) is associated a value di,j representing the distance between ci and cj. Each tour starts from and

Generation of solutions

Using ACO whose colony scale is P, an individual ant simulates a vehicle, and its route is constructed by incrementally selecting customers until all customers have been visited. The customers, who were already visited by an ant or violated its capacity constraints, are stored in the infeasible customer list (tabu).

The decision making about combining customers is based on a probabilistic rule taking into account both the visibility and the pheromone information. Thus, to select the next

Numerical analysis

The heuristics described in the previous sections is applied to the 14 vehicle routing problems which can be downloaded from the OR-library (see Beasley, 1990), and which have been widely used as benchmarks, in order to compare its ability to find the solution to VRP. The information of the 14 problems is shown in columns 2–4 in Table 1, which consists of the problem size n, the vehicle capacity Q, and the well-known published results (Taillard, 1993, Rochat and Taillard, 1995). The IACO

Conclusions

The VRP has been an important problem in the field of distribution and logistics. Since the delivery routs consist of any combination of customers, this problem belongs to the class of NP-hard problems. This paper presents an IACO with ant-weight strategy and a mutation operation. The computational results of 14 benchmark problems reveal that the proposed IACO is effective and efficient. Further research on additional modifications of the IACO to extensions of the vehicle routing problem with

References (32)

  • B. Yu et al.

    Optimizing the distribution of shopping centers with parallel genetic algorithm

    Engineering Applications of Artificial Intelligence

    (2007)
  • J.E. Beasley

    OR-Library: distributing test problems by electronic mail

    Journal of the Operational Research Society

    (1990)
  • Bullnheimer, B., Hartl, R.F., Strauss, C., 1997. Applying the ant system to the vehicle routing problem. In: Second...
  • B. Bullnheimer et al.

    An improved ant system algorithm for the vehicle routing problem

    Annals of Operations Research

    (1999)
  • C.H. Chen et al.

    An improved ant colony system algorithm for the vehicle routing problem

    Journal of the Chinese Institute of Industrial Engineers

    (2006)
  • W.C. Chiang et al.

    Simulated annealing meta-heuristics for the vehicle routing problem with time windows

    Annals of Operations Research

    (1996)
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