Decision Support g-dominance: Reference point based dominance for multiobjective metaheuristics Julián Molina c, * , Luis V. Santana a , Alfredo G. Hernández-Díaz b , Carlos A. Coello Coello a , Rafael Caballero c a CINVESTAV-IPN, Computer Science Department, Mexico b Pablo de Olavide University, Seville, Spain c University of Málaga, Applied Economics (Mathematics), Campus El Ejido s./n., 29071, Málaga, Spain article info Article history: Received 1 February 2007 Accepted 7 July 2008 Available online 25 July 2008 Keywords: Multiple-criteria decision making Interactive methods Preference information Reference point abstract One of the main tools for including decision maker (DM) preferences in the multiobjective optimization (MO) literature is the use of reference points and achievement scalarizing functions [A.P. Wierzbicki, The use of reference objectives in multiobjective optimization, in: G. Fandel, T. Gal (Eds.), Multiple-Criteria Decision Making Theory and Application, Springer-Verlag, New York, 1980, pp. 469–486.]. The core idea in these approaches is converting the original MO problem into a single-objective optimization problem through the use of a scalarizing function based on a reference point. As a result, a single efficient point adapted to the DM’s preferences is obtained. However, a single solution can be less interesting than an approximation of the efficient set around this area, as stated for example by Deb in [K. Deb, J. Sundar, N. Udaya Bhaskara Rao, S. Chaudhuri, Reference point based multiobjective optimization using evolution- ary algorithms, International Journal of Computational Intelligence Research, 2(3) (2006) 273–286]. In this paper, we propose a variation of the concept of Pareto dominance, called g-dominance, which is based on the information included in a reference point and designed to be used with any MO evolution- ary method or any MO metaheuristic. This concept will let us approximate the efficient set around the area of the most preferred point without using any scalarizing function. On the other hand, we will show how it can be easily used with any MO evolutionary method or any MO metaheuristic (just changing the dominance concept) and, to exemplify its use, we will show some results with some state-of-the-art- methods and some test problems. Ó 2008 Elsevier B.V. All rights reserved. 1. Introduction Multiple-criteria optimization naturally appears in most real- world applications, and the term MultiObjective Programming (MOP) problem refers to such problems. The first difficulty that we face when dealing with Multiobjective Optimization (MO) is that the notion of ‘‘optimum” changes. In this case, rather than aiming to find the global optimum, we look for good trade-offs among the objectives, which are obtained by using the definition of Pareto efficiency. Such a definition will lead us to obtain not one, but a set of (Pareto) efficient solutions (the Pareto front, PF). The idea of solving a multiobjective optimization problem is understood as helping a human Decision Maker (DM) in consider- ing the multiple criteria simultaneously and in finding a Pareto efficient solution that pleases him/her the most. More details about the resolution of a MOP can be found in Ref. [31]. The common element in all MOP techniques is the need to find a sufficiently wide and representative set of efficient points where the DM is able to find an alternative adjusted to his/her prefer- ences. A commonly adopted approach to find this type of solutions are the so-called Interactive MultiObjective methods (see Miettinen [25]), which assume that the DM is able to provide consistent feedback regarding which preferences to include in the resolution process. This interaction can guide a search towards the most preferred areas of the Pareto front obtained and avoids exploring non-interesting solutions. These methods are very useful in real-world cases, as they help the DM to find the most preferred solutions in a consistent and reliable way. The main problem when solving a real application is that some of the existing methods generate the entire Pareto set (most of the MO metaheuristics) whilst others produce a single point (most of the Interactive MultiObjective methods). Our aim in this paper is producing something in-between. Thus, we will show how the use of g-dominance within a MO metaheuristic will let us produce a (reduced) set of efficient points adapted to the DM’s preferences instead of the entire Pateto Set or a single efficient solution. One of the main tools for expressing preference information is the use of reference points [33]. Reference points consist of aspira- tion levels reflecting desirable values for the objective functions. This is a natural way of expressing preference information and lets the DM express hopes about his/her most preferred solutions. The reference point is projected onto the Pareto front by minimizing a so-called achievement scalarizing function [33] outlined in Section 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.07.015 * Corresponding author. Tel.: +34 952 13 11 71; fax: +34 952 13 20 61. E-mail address: jumolina@uma.es (J. Molina). European Journal of Operational Research 197 (2009) 685–692 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor