An Ant System Algorithm for the Vehicle Routing Problem with Backhauls

The ant system is a metaheuristic developed for the solution of hard combinatorial optimization problems. It was first proposed by Colorni et al. [4] and Dorigo [6]. The method was inspired by the observation of the behaviour of real life ant colonies, in particular the way in which real ants find the shortest path between food sources and their nest. While walking ants deposit a substance called pheromone on to the ground which forms a pheromone trail. Ants can detect the pheromone and choose their way according to the level of the pheromone trail. The greater the concentration of pheromone on the ground the higher the probability that an ant will choose that path. Where there is a shorter path from a food source to the nest, ants will reach the end of the path in a quicker time compared to ants on a longer path. This means that the trail will build up at a faster rate on the shorter path which in turn causes more ants to choose the shorter path which also causes a greater level of pheromone. In time all ants will have the tendency to choose the shorter path. This real life behaviour of ants has been adapted to solve combinatorial optimization problems using simulation. A number of artificial ants build solutions in parallel using a form of indirect communication. The artificial ants co-operate the artificial pheromone level deposited on arcs which is calculated as a function of the quality of the solution found. The artificial ants construct solutions iteratively by adding a new node to a partial solution using information gained from past performance and also a greedy heuristic. The greedy heuristic, known as the visibility, is introduced in an attempt to guide the search. Ant systems were first developed to solve the Travelling Salesman Problem (TSP), a number of ant systems have been proposed for the TSP. Ant systems have also been proposed for many other combinatorial problems, such as the sequential ordering problem, the quadratic assignment problem and the vehicle routing problem. For an overview see Bonabaeu et al. [1]. In this paper we propose an ant system for the vehicle routing problem with backhauls (VRPB). First we introduce the VRPB, then we present our ant system algorithm followed by preliminary results and conclusions.