Empirical Study on the Effect of Population Size on MAX -MIN Ant System in Dynamic Environments Michalis Mavrovouniotis Centre for Computational Intelligence (CCI), School of Computer Science and Informatics, De Montfort University, The Gateway, Leicester, LE1 9BH, U.K. Email: mmavrovouniotis@dmu.ac.uk Shengxiang Yang Centre for Computational Intelligence (CCI), School of Computer Science and Informatics, De Montfort University, The Gateway, Leicester, LE1 9BH, U.K. Email: syang@dmu.ac.uk Abstract—In this paper, the effect of the population size on the performance of the MAX -MIN ant system for dynamic optimization problems (DOPs) is investigated. DOPs are gener- ated with the dynamic benchmark generator for permutation- encoded problems. In particular, the empirical study investigates: a) possible dependencies of the population size parameter with the dynamic properties of DOPs; b) the effect of the population size with the problem size of the DOP; and c) whether a larger population size with less algorithmic iterations performs better than a smaller population size with more algorithmic iterations given the same computational budget for each environmental change. Our study shows that the population size is sensitive to the magnitude of change of the DOP and less sensitive to the frequency of change and the problem size. It also shows that a longer duration in terms of algorithmic iterations results in a better performance. I. I NTRODUCTION Ant colony optimization (ACO) is a metaheuristic inspired by the foraging behaviour of real ant colonies [2], [4]. ACO algorithms have been successfully applied to many NP - hard combinatorial problems such as the travelling salesman problem (TSP) [3] and vehicle routing problem (VRP) [8]. Although, there are many existing ACO variations in this paper, we consider one of the state-of-the-art variations, i.e., the MAX -MIN Ant System (MMAS) [20]. The construction of solutions from ants is biased by ex- isting pheromone trails and heuristic information. Pheromone trails are updated according to the search experience and towards solution with good quality. This is similar to a learning reinforcement scheme. The behaviour and performance of MMAS algorithm depends strongly on the number of ants used [5], [22]. When a given computational budget is available, e.g., the maximum number of function evaluations, a smaller number of ants will produce more algorithmic iterations whereas a larger number of ants less. Hence, the population size affects the duration of the learning reinforcement. In [22], it was investigated that when fewer ants are used, the algorithm may converge quickly at early stages of the optimization but get stuck in the stagnation behaviour later on. When more ants are used, the algorithm performs broader search but may waste computational resources. For the TSP, it was found that a higher number of ants performs better at later stages of the optimization process. However, the effect of the population size parameter on the performance of MMAS algorithm was only investigated for stationary optimization problems. In this paper, we investigate the effect of the population size parameter on the MMAS algorithm for dynamic optimization problems (DOPs), e.g., the dynamic TSP (DTSP) and dynamic VRP (DVRP). In particular, we are interested to investigate: a) the dependency of the population size with the dynamic prop- erties of a DOP, i.e., magnitude and frequency; b) the effect of the population size parameter with the problem size; and c) whether a broader search with less learning reinforcement time leads to better performance than a limited search with more learning reinforcement time given the same computation budget between environmental changes in DOPs. Several dy- namic test cases are generated using the dynamic benchmark generator for permutation-encoded problems (DBGP) [16] for our study. The rest of the paper is organized as follows. Section II and Section III describe the DOPs generated and ACO algorithm used for this study, respectively. Section IV discusses the importance of the population size parameter. Section V presents the experimental study and gives a discussion. Finally, Section VI concludes this paper. II. DYNAMIC ENVIRONMENTS A. Dynamic Optimization Problems 1) Dynamic Travelling Salesman Problem (DTSP): The DTSP is modelled by a fully connected weighted graph G =(N,A), where N = {v 1 ,...,v n } is a set of n nodes (e.g., cities) and A = {(v i ,v j ) | v i ,v j ∈ N,i = j } is a set of arcs (i.e., links), where n represents the size of a problem instance. Each arc (v i ,v j ) ∈ A is associated with a non-negative value d ij ∈ R + , which represents the distance between cities v i and v j . The objective of the problem is to find the shortest Hamiltonian cycle that starts from one node and visits each of the other cities once before returning to the starting city. The distance matrix of the DTSP is subject to changes, which is defined as follows: D(t)= {d ij (t)} n×n , where t is