FUZZY AGGREGATORS – AN OVERVIEW

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.


INTRODUCTION
In order to achieve an intelligent system, we need intelligence and a devicea 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.
• The decision D*, on object xi that best satisfies all the criteria Cj, j = 1, 2, ..., m, is obtained by aggregation of decisions Diusing 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:
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.
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 systems (and fuzzy logic control).What is typical for these situations is the set of rules, which emphasizes the aggregation components, also.

DEFINITION AND CLASSES OF AGGREGATORS
Let us aggregate n degrees of truth x = ( 1 ,...,  ),  > 1,   ∈  = [0, 1], =1, ..., .A general logic aggregator :   →  is defined as a continuous function that is nondecreasing in all components of x : x  y implies A(x)  A(y) for every x, y  [0, 1] n , (nondecreasing monotonicity); and satisfies the boundary conditions (idempotency in extreme points): It is assumed that the vector inequality is componentwise.
Typical examples of aggregators are: weighted means, medians, OWA operators and t-norms / t-conorms.But there are many other aggregators and an infinite number of aggregator members in most families.Not all aggregators have the same properties, so they are grouped into separate classes according to the properties they satisfy.

Conjunctive/Disjunctive Aggregators
For this class of aggregators holds duality: for strong negation N, In a special case of standard negation: Duals of conjunctive operators are disjunctive operators, and vice versa, duals of disjunctive operators are conjunctive operators Among conjunctive/disjunctive aggregators are t (triangular) normsconorms, copulas and their duals, and others, [3].
Other Aggregators, Not Conjunctive/Disjunctive Or Averaging In that class of aggregating operators are uninorms, nulnorms, T-S operators, symmetric sums, and others operators.

COMPENSATORY AGGREGATORS
Fuzzy logic theory offers a multitude of connectives that can be used as aggregators to aggregate membership values representing uncertain information.These operators can be classified, as we have seen, into the following three general classes: conjunction, disjunction (Section 3.2.1),and compensation operators (Section 3.2.2).In the case of Zadeh's min for conjunction and max for disjunction, used as aggregators, only inputs with extreme values affect the value of the output fuzzy set.However, both intuitive and formal criteria of human reasoning contain numerous requirements that are combined using models of simultaneity and substitutability (partial conjunction and partial disjunction), which set requirements for further development of fuzzy aggregators.In [5], logic operators based on continuous transition from conjunction to disjunction, were introduced, see also [6].Results from [5] were strong contribution to development of aggregation as part of a soft computing.Those results, [6], improve Zadeh's approach in dealing with uncertain and vague information common in human reasoning.So, any operator A, that, for example, applies to two arguments a1 and a2 from [0, 1], is compensatory operator if it satisfies the following: min(a1, a2)  A(a1, a2)  max(a1, a2).
After [5], others also dealt with this issue of compensatory operators, the review is given in [3; p.183].
The disjunction (union) operator provides full compensation, and the conjunction (intersection) operator does not allow compensation.The arithmetic mean is neutral in terms of disjunction and conjunction.It represents the midpoint between them and represents a special case of weighted averaging.
In [5] andness and orness were defined by Dujmović as the level of simultaneity and substitutability, respectively, of the aggregation.They are defined in terms of the similarity to minimum and maximum, respectively.Andness was introduced as a degree of conjunction, Orness was introduced as a degree of disjunction.A high orness permits that a bad criteria be compensated by a good one.On the other hand, a high andness requires both criteria to be satisfied to a great degree.Andness and orness are related and add up to one.So, andness-directed transition from conjunction to disjunction (introduced in 1973 to its current status [6]), is the history of an effort to interpret aggregation as a soft computing propositional calculus.
In some cases, we need to consider stronger functions in the sense that the outcome of an aggregation is less than the minimum or it is larger than the maximum.Fuzzy logic provides these type of operators, they are called tnorms and t-conorms, (xy  min(x, y), product t-norm is still more conjunctive than minimum).Because of this relationship, while minimum has an andness equal to one, product t-norm has an andness that is larger than one.When operators are between minimum and maximum, andness is for any number of inputs in the range [0, 1].Operators that can return values smaller than the minimum (as t-norms) or larger than the maximum (as tconorms) will provide andness outside [0, 1], reaching the minimum and the maximum of the interval with drastic disjunction and drastic conjunction [6].
The resulting analytic framework is a graded logic [6], based on analytic models of graded simultaneity (various forms of conjunction), graded substitutability (various forms of disjunction) and complementing (negation).
Basic graded logic functions can be conjunctive, disjunctive, or neutral.Conjunctive functions have andness  greater than orness ,  > .Similarly, disjunctive functions have orness greater than andness,  < , and neutral is only the arithmetic mean where  =  = ½.Between the drastic conjunction and the drastic disjunction, we have andness-directed logic aggregators that are special cases of a fundamental logic function called graded conjunction/disjunction (GCD) [6].GCD has the status of a logic aggregator, and it can be idempotent or nonidempotent, as well as hard (supporting annihilators) or soft (not supporting annihilators).The annihilator of hard conjunctive aggregators is 0, and the annihilator of hard disjunctive aggregators is 1.
The whole range of conjunctive aggregators is presented in Figure 2   A detailed classification of GCD aggregators, based on combinations conjunctive/ disjunctive, idempotent/nonidempotent, and hard/soft aggregators is presented in Table 1 [6].