Extending ACO R to Solve Multi-Objective Problems Abel Garcia-Najera School of Computer Science University of Birmingham Birmingham, B15 2TT UK A.G.Najera@cs.bham.ac.uk John A. Bullinaria School of Computer Science University of Birmingham Birmignahm, B15 2TT UK J.A.Bullinaria@cs.bham.ac.uk Abstract Ant Colony Optimization (ACO) was first proposed to solve the Traveling Sales- man Problem, and later applied to solve more problems of a combinatorial na- ture. Some research based on ACO to tackle continuous problems has been pub- lished, but this has not followed the origi- nal ACO metaheuristic exactly. Recently, ACO R has been proposed to solve contin- uous function optimization problems. We have taken this work and extended it to solve multi-objective optimization prob- lems. After an analysis of the results ob- tained, including comparisons with two other well-known methods, we conclude that ACO R is a promising new technique for solving multi-objective problems. 1 Introduction The Ant Colony Optimization (ACO) meta- heuristic was first proposed to solve combina- torial problems like the traveling salesman prob- lem [8, 6, 7], vehicle routing [1, 2, 9] and schedul- ing [15], among others. Even though ACO has been used widely to solve combinatorial prob- lems efficiently, its use on continuous function problems has been limited due to the fact that there is not a straightforward extension. Nevertheless, some methods based on ACO have been proposed to tackle continuous prob- lems [14, 12, 13], but these have not followed the original metaheuristic exactly [11]. Recently, a new state-of-the-art technique has been proposed, extending ACO to continu- ous domains without the need to make any ma- jor conceptual change to its structure. This has been called ACO R [11]. In that work, the au- thors deal with single-objective problems, com- paring their solutions with other previously pub- lished results. These solutions suggest that ACO R might usefully be extended to perform well on multi-objective problems too. Multi-objective problems can be found in most engineering domains in the real world, and most of the time the various objectives are in- consistent. This means that, when we try to optimize one objective, we will probably not op- timize them all. So, instead of looking for op- timized parameter values to give an optimal so- lution, we must search for values that provide appropriate trade-offs. In this paper we describe how we have ex- tended the ACO R approach to deal with multi- objective problems, and provide a visual analysis of comparative results obtained on some stan- dard benchmark problems. The remainder of this paper is organized as follows. In section 2 we review the standard ACO metaheuristic. A brief description of the ACO R approach is given in section 3. In section 4 we define what a multi-objective optimization problem is. Our proposal for extending ACO R to tackle multi-objective problems is described in section 5. The experimental setup and results are described in section 6. Finally, in section 7, we give our conclusions about this work and a few ideas for future work in this area. 2 The ACO metaheuristic Ant Colony Optimization makes use of agents, called ants, which mimic the behavior of real ants in how they manage to establish shortest- route paths from their colony to feeding sources and back [8]. Ants communicate information through pheromone trails, which influence which routes the ants follow, and eventually lead to a solution route. ACO was initially designed to solve the Trav- eling Salesman Problem (TSP) and works as fol- lows. The salesman must visit a number of cities exactly once each by the shortest total path pos- sible. The cities and routes between them can be represented as a connected graph, and the ants move from one city to another following