Future Generation Computer Systems 16 (2000) 889–914 MAX –MIN Ant System Thomas Stützle a,∗,1 , Holger H. Hoos b,2 a IRIDIA, Université Libre de Bruxelles, Avenue Franklin Roosevelt 50, CP 194/6, 1050 Brussels, Belgium b Computer Science Department, University of British Columbia, 2366 Main Mall, Vancouver, BC, Canada V6T 1Z4 Abstract Ant System, the first Ant Colony Optimization algorithm, showed to be a viable method for attacking hard combinatorial optimization problems. Yet, its performance, when compared to more fine-tuned algorithms, was rather poor for large instances of traditional benchmark problems like the Traveling Salesman Problem. To show that Ant Colony Optimization algorithms could be good alternatives to existing algorithms for hard combinatorial optimization problems, recent research in this area has mainly focused on the development of algorithmic variants which achieve better performance than Ant System. In this paper, we present MAX –MIN Ant System (MMAS), an Ant Colony Optimization algorithm derived from Ant System. MMAS differs from Ant System in several important aspects, whose usefulness we demonstrate by means of an experimental study. Additionally, we relate one of the characteristics specific to MMAS — that of using a greedier search than Ant System — to results from the search space analysis of the combinatorial optimization problems attacked in this paper. Our computational results on the Traveling Salesman Problem and the Quadratic Assignment Problem show that MMAS is currently among the best performing algorithms for these problems. ©2000 Elsevier Science B.V. All rights reserved. Keywords: Ant Colony Optimization; Search space analysis; Traveling Salesman Problem; Quadratic Assignment Problem; Combinatorial optimization 1. Introduction Ant Colony Optimization (ACO) [8,11,13,14] is a recently developed, population-based approach which has been successfully applied to several NP -hard combinatorial optimization problems [5,7,12,19,20,27,32,41] (see [10,11] for an overview). As the name suggests, ACO has been inspired by the behavior of real ant colonies, in particular, by their foraging behavior. One of its main ideas is ∗ Corresponding author. Tel.: +32-2-650-3167; fax: +32-2-650-2715. E-mail addresses: tstutzle@ulb.ac.be (T. Stützle), hoos@cs.ubc.ca (H.H. Hoos) 1 On leave from FG Intellektik, TU Darmstadt, Germany. 2 Tel.: +1-604-822-5109; fax: +1-604-822-5485. the indirect communication among the individuals of a colony of agents, called (artificial) ants, based on an analogy with trails of a chemical substance, called pheromone, which real ants use for commu- nication. The (artificial) pheromone trails are a kind of distributed numeric information (called stigmergic information in [9]) which is modified by the ants to reflect their experience accumulated while solv- ing a particular problem. Recently, the ACO meta- heuristic has been proposed to provide a unifying framework for most applications of ant algorithms [10,11] to combinatorial optimization problems. Al- gorithms which actually are instantiations of the ACO metaheuristic will be called ACO algorithms in the following. The first ACO algorithm, called Ant System (AS) [8,13,14], was applied to the Traveling Salesman Prob- 0167-739X/00/$ – see front matter ©2000 Elsevier Science B.V. All rights reserved. PII:S0167-739X(00)00043-1