An Ant System with Direct Communication for the Capacitated Vehicle Routing Problem Michalis Mavrovouniotis and Shengxiang Yang Abstract— Ant colony optimization (ACO) algorithms are population-based algorithms where ants communicate via their pheromone trails. Usually, this indirect communication leads the algorithm to a stagnation behaviour, where the ants follow the same path from early stages. This is because high levels of pheromone are generated into a single trail, where all the ants are influenced and follow it. As a result, the population gets trapped into a local optimum solution, which is difficult for the ACO algorithm to escape from it. In this paper, a direct communication scheme is proposed and applied to ACO for the capacitated vehicle routing problem (CVRP), which is a diffi- cult NP-hard optimization problem. The direct communication scheme allows the ants to exchange customers from different routes, with other ants that belong to their communication range. Experiments show that the direct communication scheme significantly improves the solution quality of a conventional ACO algorithm regarding CVRP with and without service time constraints. I. I NTRODUCTION Ant colony optimization (ACO) algorithms are inspired from the behaviour of real ant colonies, when ants search for food from their nest to food sources. Ants cooperate and communicate indirectly via their pheromone, where they leave a trail to the path they explore. The more pheromone on a specific trail, the higher the possibility of that trail to be followed by the ants. This behaviour inspired researchers to develop the first ACO algorithm, called the ant system (AS) [6], [8], which has been applied to the well-known travelling salesman problem (TSP). Moreover, AS has been applied to other combinatorial optimization problems, such as the quadratic assignment problem [10], the job scheduling problem [7], the vehicle routing problem (VRP) [2], [9], and many other optimization problems. In this paper, we focus on the VRP because it shares many similarities with real-world applications [21], where a population of ants begins from a depot and visits customers (or delivery points). When the demand of all the customers are satisfied, the ants will return back to the depot. Each ant represents several vehicle routes, due to the vehicle capacity constraint, which represent a complete capacitated VRP (CVRP) solution. The AS has been applied to the CVRP [2], and later on improved [3] using different heuristic information. However, it suffers from the stagnation be- haviour, where all ants generate the same solution from early M. Mavrovouniotis is with the Department of Computer Science, Uni- versity of Leicester, University Road, Leicester LE1 7RH, United Kingdom (email: mm251@mcs.le.ac.uk). S. Yang is with the Department of Information Systems and Computing, Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom (email: shengxiang.yang@brunel.ac.uk). iterations. This is because a high intensity of pheromones may be generated into a single trail, and the ants may stuck on a local optimum solution. In nature, ants do not only communicate indirectly by pheromone trails, but also directly with other ants and gather important information [15]. A direct communication (DC) scheme has been found beneficial to the TSP, where ants communicate and exchange cities [16]. In this paper, a sim- ilar DC scheme is proposed for ACO algorithms to address the CVRP. The ants will be able to communicate with other ants within their neighbourhood (or communication range), which is based on a similarity metric. Ants within their communication range are allowed to exchange customers with each other, only if there is an improvement and the CVRP constraints are not violated. Additionally, a small amount of pheromone is added to the exchanged customers in order to influence ants towards new promising paths generated from DC. In order to investigate the performance of the proposed DC scheme for ACO algorithms, we consider the AS for the CVRP [3], and experiments are carried out to compare the conventional AS algorithm and the AS with the proposed DC scheme, denoted as AS+DC, on a set of benchmark CVRP instances. Experimental results show that the proposed scheme improves the solution quality of the conventional AS since it enables ants to avoid local optima and leads the population towards the global optimum. The rest of the paper is organized as follows. Section II, defines the framework of two CVRP variations, i.e., the basic CVRP and CVRP with service time constraints. In Section III, we describe the AS algorithm for the CVRP, which will be used in the experiments. In Section IV, we describe the proposed DC scheme, giving details on how it can be applied to AS, and to any ACO algorithm. In addition, we discuss possible advantages and disadvantages of using this scheme. Section V presents the experimental results with the corresponding statistical tests of the proposed AS+DC in comparison with the conventional AS on different CVRP problem instances. Finally, Section VI provides concluding remarks and several directions for future work. II. THE VEHICLE ROUTING PROBLEM The VRP became one of the most popular combinatorial optimization problems, due to its similarities with many real- world applications. The VRP is classified as NP-hard [13]. The basic VRP is the CVRP, where a number of vehicles with a fixed capacity need to satisfy the demand of all the customers, starting from and finishing to the depot. A VRP