Cybernetics and Systems Analysis, Vol. 55, No . 6, November, 2019 MAXIMIZING ALTERNATIVES IN A DECISION-MAKING PROBLEM WITH A GOAL TYPE-2 FUZZY SET S. O. Mashchenko UDC 519.8 Abstract. The Bellman and Zadeh approach is applied to a decision-making problem posed on type-2 fuzzy sets (T2FSs). The concept of a solution T2FS is defined. The natural order relation was extended to the class of fuzzy sets to compare fuzzy sets of membership degrees of alternatives. Based on this preference relation, a fuzzy set of nondominated alternatives is constructed. The concept of a nondominated alternative of level a is introduced. It is shown that such an alternative can be obtained from the optimization problem in which the primary membership degree of a T2FS of solutions is maximized under a constrained secondary degree. The existence of nondominated alternatives of level a = 1 is investigated. The problem of choosing alternatives by two criteria (the primary and secondary degrees of membership in the T2FS of solutions) is formulated. Keywords: fuzzy set, type-2 fuzzy set, fuzzy mathematical programming, decision-making. INTRODUCTION The Bellman and Zadeh approach [1] to the solution of a problem with a fuzzy goal is that the decision-making goal and the set of feasible alternatives are considered as equal fuzzy sets of some universal set of alternatives. This allows to sufficiently simply find a problem solution. The practical use of this approach has shown its high efficiency and wide range of applications. It gave an impetus to developing methods for solving optimization problems under conditions of fuzzy information of different nature. Models and methods of fuzzy mathematical programming are well developed. Achievements in this area are described in more detail in [2–4]. In the Bellman and Zadeh approach, fuzzy sets are the main tool for formalizing uncertainty in the description of a decision-making problem, but they can also be undefined. Mendel, John, and Liu noted in [5] that there are at least four sources of uncertainties in classical fuzzy sets. These are, for example, meanings of words used in describing fuzzy sets (words have different meanings for different persons). The cause of uncertainty can also be the ambiguity of opinions of experts. Moreover, it is stipulated by both measurement noises and data noises. All this leads to the uncertainty of the membership function of a fuzzy set. Such uncertainties can be modeled by type-2 fuzzy sets (T2FSs) since their membership functions are fuzzy in themselves. At the same time, T2FSs are difficult to interpret and their use is computationally more complicated. Despite these problems, T2FSs are widely used. PROBLEM STATEMENT Type-2 fuzzy sets were introduced by Zadeh in 1971 [6] as an extension of classical fuzzy sets (of type-1). The membership degree of elements of fuzzy sets is determined by the value in the interval [0, 1], and the membership 933 1060-0396/19/5506-0933 © 2019 Springer Science+Business Media, LLC Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, s.o.mashchenko@gmail.com. Translated from Kibernetika i Sistemnyi Analiz, No. 6, November–December, 2019, pp. 62–72. Original article submitted November 8, 2018. DOI 10.1007/s10559-019-00203-x