Pheromone Modification Strategy for the Dynamic Travelling Salesman Problem with Weight Changes Michalis Mavrovouniotis School of Science and Technology Nottingham Trent University Nottingham NG11 8NS, U.K. email: michalis.mavrovouniotis@ntu.ac.uk Mien Van School of Science and Technology Nottingham Trent University Nottingham NG11 8NS, U.K. email: mien.van@ntu.ac.uk Shengxiang Yang School of Computer Science and Informatics De Montfort University Leicester LE1 9BH, U.K. email: syang@dmu.ac.uk Abstract—Ant colony optimization (ACO) algorithms have proved to be able to adapt in problems that change dynamically. One of the key issues for ACO when a change occurs is that the pheromone trails generated in the previous environment will not be compatible with the new environment. Therefore, the optimization process may be biased from the pheromone trails of the previous environment and fail to search for the newly generated global optimum. In this paper, we consider the dynamic travelling salesman problem (DTSP) in which the weights of the arcs are modified. A pheromone strategy that utilizes change-related information and regulates heuristically the pheromone trails of the affected arcs is proposed. From the experimental results the heuristic-based pheromone strategy performs statistically significant better in most DTSP test cases than other peer ACO algorithms. I. I NTRODUCTION Ant colony optimization (ACO) has proved that is a pow- erful metaheuristic to provide optimal (or near-optimal) solu- tions for solving different combinatorial optimization problems (e.g., the travelling salesman problem (TSP) [3]). Traditionally, researchers have drawn their attention on static optimization problems, where the environment remains fixed during the execution of ACO. However, many real-world applications are subject to dynamic changes. Such problems are known as dynamic optimization problems (DOPs). DOPs are challenging since the aim of an optimization algorithm is not only to find the optimum of the problem quickly, but also to efficiently track the moving optimum during the changing environments [8]. A change in a DOP may involve factors like the objective function, input variables, problem instance and constraints. ACO algorithms have been originally designed to address static optimization problems [2] (e.g., to converge fast into an optimum or near-optimum solution) and may face a serious challenge when addressing DOPs. This is because, after a change, the pheromone trails of the previous environment may bias the colony to search into an old optimum, making it difficult to track the moving optimum. As a result, ACO will not adapt well to dynamic changes once the colony converges into an optimum. Considering that DOPs can be taken as a series of static problem instances, a simple way to tackle them is to reinitialize all the pheromone trails with an equal amount and consider every dynamic change as the arrival of a new problem instance which needs to be solved from scratch [5], [11]. However, this restart strategy is generally not very efficient because all the information gained from previously optimized environments is removed. In contrast, once ACO algorithms are enhanced properly they are able to adapt to dynamic changes [1], [8]. Several pheromone strategies have been proposed and integrated with ACO to shorten the re-optimization time and maintain a high quality of the output efficiently, simultaneously. These strate- gies can be categorized as: increasing diversity after a change [5], [7], [11]; maintaining diversity during the execution [4], [12]; multi-colony schemes [13], [15]; memetic algorithms [9], [10] and memory-based schemes [6] (refer to [14] for a recent comprehensive survey). From the existing pheromone strategies only few of them utilize change-related information such as the location of dynamic changes [5], [7]. For example, in [5] the pheromone trails associated with the arcs incident to a newly added city are regulated accordingly for the dynamic TSP (DTSP) where the topology changes. However, pheromone strategies that utilize change-related information for DTSPs with weight changes (e.g., the weight of the arcs may increase or decrease but the topology remains the same) do no exist. In this paper, a pheromone strategy that regulates heuristically the pheromone trails for the DTSP with weight change is proposed. The key idea of the strategy is to utilize the existing heuristic information of the arcs affected by the dynamic change and regulate their pheromone trails accordingly. For example, an arc may have very high concentration of pheromone trails before the change but its weight may increase after a change. However, due to its high intensity of pheromone trails, it may be still very attractive and may result in poor solution quality. Therefore, penalizing the pheromone trails of the particular arc will make it less attractive for the colony. The rest of the paper is organized as follows. Section II describes the generation of a dynamic environment using the TSP as the base problem. Specifically, the DTSP with weight changes is described. Section III describes the ACO meta- heuristic and how to respond to dynamic changes. Section IV describes the proposed pheromone strategy which utilizes change-related and heuristic information. Section V presents the experimental results and analysis. Finally, Section VI concludes this paper with discussions on future work.