Analyzing the impact of MOACO components: An algorithmic study on the multi-objective shortest path problem

Abstract

Multi-objective Ant Colony Optimization (MOACO) algorithms have been successfully applied to several multi-objective combinatorial optimization problems (MCOP) over the past decade. Recently, we proposed a MOACO algorithm named GRACE for the multi-objective shortest path (MSP) problem, confirming the efficiency of such metaheuristic for this MCOP. In this paper, we investigate several extensions of GRACE, proposing several single and multi-colony variants of the original algorithm. All variants are compared on the original set of instances used for proposing GRACE. The best-performing variants are also assessed using a new benchmark containing 300 larger instances with three different underlying graph structures. Experimental evaluation shows one of the variants to produce better results than the others, including the original GRACE, thus improving the state-of-the-art of MSP.

Introduction

Multi-objective Ant Colony Optimization (MOACO) algorithms have been studied for a decade now, and have been applied to several important multi-objective combinatorial optimization problems (MCOP) (Chica et al., 2011, Doerner et al., 2001, Doerner et al., 2006, Iredi et al., 2001, Ke et al., 2010, Mora et al., 2009). Given that a large amount of design possibilities can be devised, a number of experimental studies have been conducted aiming at analyzing individual algorithmic components and their contribution (García-Martínez et al., 2007, López-Ibáñez et al., 2004, López-Ibáñez and Stützle, 2010). The influence of multiple colonies, multiple heuristics and multiple pheromone structures are some of the most investigated topics.

In a previous work addressing the multi-objective shortest path problem (MSP), we have proposed a two-phase MOACO algorithm named GRACE (Bezerra, Goldbarg, Goldbarg, & Buriol, 2011) and showed its efficiency in a comparison to a well-known evolutionary multi-objective (EMO) algorithm, named NSGA-II (Deb, Pratap, Agarwal, & Meyarivan, 2002), and to another MOACO algorithm from the literature of MSP (Häckel, Fischer, Zechel, & Teich, 2008). Given the observed quality of the results, we continue such investigation by extending GRACE. Several single and multi-colony variants are proposed, using many of the efficient algorithmic components recently identified in the MOACO literature. To compare the variants, two sets of tri-criteria instances are used: (i) the original set of 18 instances with two different underlying graph structures used in the original GRACE (MSPP, 2010), ranging from 100 to 1000 nodes, and; (ii) a new instance benchmark proposed in this work containing 300 larger instances, ranging from 1000 to 8000 nodes, with three different underlying graph structures. All comparisons use solid assessment methodology: dominance rankings (Knowles, Thiele, & Zitzler, 2006), unary Pareto-compliant quality indicators (Zitzler, Thiele, Laumanns, Fonseca, & da Fonseca, 2003), and non-parametrical statistical tests (Conover, 1999, Holland, 1975). Results show the performance of the variants highly depends on the underlying graph structure for smaller instances. For larger instances, one of the variants outperforms the others as well as the original GRACE algorithm, thus establishing a new state-of-the-art for the problem.

The contribution of this work, however, is not limited to an experimental study of existing algorithmic components or to the improvement of the state-of-the-art of an important MCOP. Over the literature, one of the key aspects that has not yet been investigated is how to generate and assign scalarization vectors to colonies in a MOACO algorithm. Scalarization vectors play a central role in multi-objective optimization, allowing algorithms to deal with single objective versions of the problem, which are generally less difficult to solve. Efficient multi-objective algorithms make use of such approach, regardless of the metaheuristic employed (Paquete and Stützle, 2003, Vianna and Arroyo, 2004, Zhang and Li, 2007). Traditionally, scalarization vectors have only been generated randomly in the [0, 1] space, or systematically, parameterized by a number of divisions. In this work, all the variants proposed use a different method to generate or assign such vectors, many of them novel. In fact, the variant that outperforms the others differs from the original GRACE only on the way scalarization vectors are generated and assigned to colonies.

This paper is organized as follows. Section 2 describes the MSP and reviews its state-of-the-art. In Section 3, the ACO metaheuristic is revised and the different approaches found over the literature are detailed. In Section 4, GRACE and its single-species variants are described and compared. In Section 5, the experimental setup for the comparison of the variants on the original set of instances is described, and results are presented and discussed. In Section 6, the new instance benchmark is proposed, listing results from an exact algorithm. The comparison between the two best performing variants are also presented in this section. Finally, conclusions and future work possibilities are discussed in Section 7.