International Journal of Computer Applications (0975 – 8887) Volume 87 – No.19, February 2014 22 Decision-Making in Complicated Geometrical Problems Amir Mosavi University of Debrecen Egyetem ter 1. Debrecen, Hungary ABSTRACT Due to increasing the number of decision-making criteria in today’s ever complicated geometrical optimization problems, the traditional multiobjective optimization approaches, whether a priori, a posteriori or interactive's, found to be insufficient and ineffective. In this paper the drawbacks of the current algorithms are reviewed and the urgent need for inserting a learning component in the optimization loop is discussed. In the following the methodology of reactive optimization for evolutionary interactive multiobjective optimization for solving complicated geometrical decision- making problems is adopted. The proposed brain-computer optimization follows to the paradigm of learning while optimizing, through the use of online machine learning techniques as an integral part of a self-tuning optimization scheme. At the end the effectiveness of the approach to geometrical problems is emphasized by providing the study case of optimal design problem of curves and surfaces. Keywords Decision-making, geometry, optimization 1. INTRODUCTION We According to [3] the general form of the Multiobjective optimization (MOO) problems is stated as Minimize           Subjected to   where   ℝ n is a vector of  decision variables;   ℝ n is the feasible region and is specified as a set of constraints on the decision variables;    ℝ m is made of  objective functions subjected to be minimization. Objective vectors are images of decision vectors written as             An objective vector is considered optimal if none of its components can be improved without worsening at least one of the others. An objective vector  is said to dominate  , denoted as     , if       for all  and there exist at least one  that       . A point   is Pareto optimal if there is no other   such that  dominates   The set of Pareto optimal points is called Pareto set (PS). The corresponding set of Pareto optimal objective vectors is called Pareto front (PF). Solving a MOO problem would be done by providing the decision-maker (DM) with the optimal solution according to some certain utility criteria allowing to choose among competing PF. Such utility criteria are often inconsistent, difficult to formalize and subjected to revision. Approaches to MOO are divided into the two broad categories of non-interactive and interactive ones [13]. According to [14], through interactive MOO the DM is building a conviction of what is possible and confronting this knowledge with the preferences that also evolve where learning task and the computer-supported solution processes are involved. On the other hand the non-interactive approaches are divided into a priori approaches, where the preferences are formulated in advance, and a posteriori approaches, where a PS is created. A priori methods have the drawback of requiring the user to accurately pre-specify the preferences which is actually hard for a DM. A posteriori methods, on the other hand, imply a confusing selection task among a large and complicated set of candidate solutions. 2. FROM EVOLUTIONARY ALGORITHMS TO BRAIN-COMPUTER OPTIMIZATION All Evolutionary algorithms (EAs) are among the most popular a posteriori methods to generating PS to a MOO problem. The evolutionary algorithms of MOO for solving MCDM problems have been around for almost twenty years now [15]. EA are ideally suited to search for a set of PS to be presented to the DM. In this paradigm, evolutionary multiobjective optimization algorithms (EMOAs) aim at building a set of points near the PF. Currently, most evolutionary EMOAs apply Pareto-based ranking schemes. Some of the most successful EMOAs [16] rely on Pareto dominance classification as a fitness measure to guide selection of the new population. The work [17] indicates that resorting to Pareto dominance classification to assign fitness becomes ineffective for increasing number of objectives and proposes a refined preference ordering based on the notion of order of efficiency [18]. MOO of curve and surfaces [5] would be a good example for such ineffective attempt and increasing complexity. The reviewed and applied approaches for solving the MOO of the curve and surfaces [5] whether a priori or a posteriori, in particular EA, involve plenty of various complications. The reason is that the proportion of PF in a set grows very rapidly with the dimension . In fact the reality of applied DM has to consider plenty of priorities and drawbacks to both interactive and non-interactive approaches. Although the mathematical representative set of the DM model is often created however presenting a human DM with numerous representative solutions on a multi-dimensional PF is very far from reality. This is because the typical DM cannot deal with more than a very limited number of information items at a time [19]. Therefore a satisficing decision procedures should be developed according to human memory and data processing capabilities. Moreover often DMs cannot formulate their objectives and preferences at the beginning. Instead they would rather learn on the job. This is already recognized in the MOO formulation, where a combination of the individual objectives