EngOpt 2008 - International Conference on Engineering Optimization Rio de Janeiro, Brazil, 01 - 05 June 2008. Multicriteria Analysis Based on Constructing Payoff Matrices and Methods of Decision Making in Fuzzy Environment Patricia Bernardes Pontifícia Universidade Católica de Minas Gerais Belo Horizonte, MG, Brasil patib@pucminas.br Petr Ekel Pontifícia Universidade Católica de Minas Gerais Belo Horizonte, MG, Brasil ekel@pucminas.br Jorge Kotlarewski FURNAS Centrais Elétricas Rio de Janeiro, RJ, Brasil jkotlar@furnas.com.br Reinaldo Palhares Universidade Federal de Minas Gerais Belo Horizonte, MG, Brazil palhares@cpdee.ufmg.br Roberta Parreiras Pontifícia Universidade Católica de Minas Gerais Belo Horizonte, MG, Brasil roberta.parreiras@terra.com.br 1. Abstract There exist two major classes of problems, which need the use of a multicriteria approach: problems whose solution consequences cannot be estimated with a single criterion and problems that, initially, may require a single criterion, but their unique solutions are unachievable, due to the existence of decision uncertainty regions, which can be contracted using additional criteria. According to this, two classes of multicriteria models ( > < M X , and > < R X , models) can be constructed. The Bellman-Zadeh approach to decision making in a fuzzy environment is utilized for analyzing > < M X , models. Its application conforms to the principle of guaranteed result and provides constructive lines in obtaining harmonious solutions on the basis of analyzing associated maxmin problems. The analysis of > < R X , models (which contain fuzzy preference relations as criteria of optimality) is based on fourth techniques for fuzzy preference modeling. They permit the evaluation, comparison, selection, prioritization, and/or ordering of alternatives with the use of quantitative as well as qualitative estimates based on knowledge, experience, and intuition of professionals. With the availability of different techniques, the most appropriate one can be chosen, considering possible sources of information and its uncertainty. The analysis of > < M X , and > < R X , models serves as parts of a general scheme for multicriteria decision making under information uncertainty. This scheme is also associated with a generalization of the classic approach to considering the uncertainty of information (based on analyzing payoff matrices constructed for different combinations of solution alternatives and states of nature) in monocriteria decision making to multicriteria problems. 2. Keywords: Multicriteria decision making, Information uncertainty, Bellman-Zadeh approach, Fuzzy preference relations, Payoff matrices 3. Introduction Diverse types of uncertainty are often encountered in a wide range of problems related to the design, planning, and control of complex systems [1]. Taking into account the uncertainty factor in constructing mathematical models serves as a means for increasing their adequacy and, as a result, the credibility and factual efficiency of decisions based on their analysis. Investigations of recent years show the benefits of applying fuzzy set theory [2,3] to deal with diverse types of uncertainty. Its use in problems of optimization character offers advantages of both fundamental nature (the possibility of validly obtaining more effective, less "cautious" solutions) and of a computational character [4,5]. The uncertainty of goals is an important kind of uncertainty related to a multicriteria character of many problems of optimization nature. It is possible to classify two major classes of problems, which need the use of a multicriteria approach: • problems whose solution consequences cannot be estimated on the basis of a single criterion: these problems are associated with the analysis of economic as well as natural indices (when alternatives cannot be reduced to comparable form) and also with the need to consider indices whose cost estimations are hampered; • problems that, from the substantial point of view, may require the use of a single criterion, but their unique solutions are unachievable, due to the existence of the uncertainty of information, generating the decision uncertainty regions. The convincing means to contract these regions is the introduction of additional criteria, including criteria of qualitative character (based on knowledge, experience, and intuition of involved experts) [4,5]. In this context, two classes of multicriteria models, so-called > < M X , and > < R X , models, may be constructed [4-6]. Their analysis is associated with the use of the Bellman-Zadeh approach to decision making in a fuzzy environment and with the use of techniques involving fuzzy preference modeling, respectively.