Fuzzy Aggregators - an Overview

Interdisciplinary Description of Complex Systems 21(4), 356-364, 2023 FUZZY AGGREGATORS – AN OVERVIEW Dragan Z. Šaletić* University “Donja Gorica” Podgorica, Republic of Montenegro DOI: 10.7906/indecs.21.4.5 Regular article Received: 15 June 2023. Accepted: 23 July 2023. ABSTRACT The article deals with mathematical formalism of the process of combining several inputs into a single output in fuzzy inteligent systems, the process known as aggregation. We are interested in logic aggregation operators. Such aggregators are present in most decision problems and in fuzzy expert systems. Fuzzy inteligent systems are equipped with aggregation operators (aggregators) with which reasoning models adapt well to human reasoning. A brief overview of the field of fuzzy aggregators is given. Attention is devoted to so called graded logic aggregators.. The role of fuzzy agregators in modelling reasoning and the way they are chosen in modelling are pointed out. The conclusions are given and research in the field is pointed out. KEY WORDS aggregation, fuzzy intelligent systems, conjunction, disjunction, compensatory operators CLASSIFICATION ACM: D.1.1. JEL: O31 PACS: 89.70.H *Corresponding author : ; +382 20 410 777; *Oktoih 1, Podgorica 81000, Montenegro Fuzzy aggregators – an overview INTRODUCTION In order to achieve an intelligent system, we need intelligence and a device – a computer. In order to implement intelligence with a computer, we need to model intelligence (knowledge representation), we need the automation of the process of (intelligent) reasoning to get new ideas about the world, and we need to implement the process of intelligent action based on new ideas [1]. Logic is one of tools for modelling the observable properties of human reasoning. We use logic to implement decision-making process or knowledge representation and automatic reasoning. In the last century, it has been noticed that the classical two-value logic is a limited framework for modelling the representation of knowledge and human reasoning. The ways to expand the possibilities of representation by logic have been proposed. One of the most fruitful of these attempts was initiated by Lotfi Zadeh [2]. Zadeh has expanded the idea of the degree to which an element belongs to a set from two values, 0 (for non-belonging), and 1 (for belonging), to a range between 0 and 1, which allows the development of models in which key elements are not precise numbers but vague sets, i.e. a class of objects in which the transition from non-belonging to belonging is gradual, not abrupt. Zadeh described the mathematical theory of fuzzy sets and the corresponding fuzzy logic (a kind of a continuous logic with truth value from [0, 1], instead as in standard logic where each sentences have truth value from {0, 1}, there is no “in between”). Zadeh, also, proposed appropriate set and logical operations, which improved the expressiveness of the model, i.e. enabled dealing with uncertain and vague information common in human reasoning. Operations on fuzzy sets of unions, intersection and complement are defined using max, min and 1 −(x) operations, (where  is degree of membership of element x in a fuzzy set), which correspond to fuzzy logic functions disjunction, conjunction, and negation. In fuzzy intelligent systems [3], one of the key issues is the problem of aggregation of fuzzy information represented by membership functions (whose values are in [0, 1]). Fuzzy membership can be interpreted as a degree of truth, so we have fuzzy logic aggregation. Aggregation operators combine multiple input values into one output value, which represents all input values. In this article, the aggregation operator (aggregator), present in fuzzy intelligent systems, is considered. In Section 2, the considered problem is formulated. In Section 3 a formal definition of aggregator is given, as well as main classes of that operator. Section 4 deals with compensatory aggregators. Special attention is devoted to the aggregator called graded conjunction/disjunction. The selection of an aggregator is discussed in Section 5. Section 6 contains the conclusions. A list of references is given. AGGREGATION In fuzzy intelligent systems, one of the key problems is the problem of agrregating fuzzy information represented by membership functions (whose values are in [0, 1]). Aggregators combine multiple input values into one output value, which represents all input values. For example, the general form of a fuzzy multicriteria decision-making system is shown in the Figure 1. 357 D.Z. Šaletić Figure 1. Aggregation in a type of fuzzy multicriteria decision-making system. In Figure 1 meanings of symbols are as follows: • xi, i = 1, 2, ..., m, are vectors of object properties, which are considered in decision-making process; • Cj, j = 1, 2, ..., n, are decision-making criteria; • ji(xj), j = 1, 2, ..., m, i = 1,2, ...,n, ji  [0, 1], are scores – degrees in which an object xi (or its property) satisfies the criteria Cj, ji is the degree of fuzzy membership in a fuzzy set of object property that completely satisfies criterion Cj; • Di, i = 1, 2, ..., m, Di  [0, 1], are decisions (performance indices) of an object xi with respect to all the criteria Cj; decisions Di are obtained by aggregation of information ji(xj), using appropriate aggregation operation. • The decision D*, on object xi that best satisfies all the criteria Cj, j = 1, 2, ..., m, is obtained by aggregation of decisions Di – using suitable aggregation operation, appropriate for the considered problem. The procedure used to combine the scores by which the object xi, or one of its characteristics, satisfies the criteria Ci into one decision Dj, i.e. D*, is: Dj = A1(j1(xj), ..., jn(xj)), D* = A2(D1, ..., Dm). (1) The symbol A in the above expressions indicates aggregators. In the more general case, expressions (1) can be given in the form a = A(a1, ..., ar), (2) where aj, j = 1, ... , r, r  {n, m}, and a are values from interval of degrees of membership [0, 1]. Fuzzy operators, min for conjunction and max for disjunction, for A1 or A2 in (1), are to restrictive in practice and do not coincide with how people perform this operations. This lead to studies of other aggregators. In the huge majority of applications, primarily in decision-support systems, aggregators are developed as models of observable human reasoning. So, we are interested in graded logic aggregators, i.e., aggregators that aggregate degrees of truth. Such aggregators are present in most decision problems. We assume that decision-making commonly includes evaluation of alternatives and selection of the most suitable alternative, Figure 1. 358 Fuzzy aggregators – an overview Some other examples of applications of fuzzy set theory, for modelling complex and perhaps incompletely defined systems, use knowledge bases in which knowledge is represented by a base of fuzzy rules. These applications include fuzzy rule-based s (...truncated)