Aggregation functions: Construction methods, conjunctive, disjunctive and mixed classes

doi:10.1016/j.ins.2010.08.040

Information Sciences

Volume 181, Issue 1, 1 January 2011, Pages 23-43

https://doi.org/10.1016/j.ins.2010.08.040 Get rights and content

Abstract

In this second part of our state-of-the-art overview on aggregation theory, based again on our recent monograph on aggregation functions, we focus on several construction methods for aggregation functions and on special classes of aggregation functions, covering the well-known conjunctive, disjunctive, and mixed aggregation functions. Some fields of applications are included.

Introduction

This paper is a continuation of [29] and it is based on our monograph [28], as explained in the first part of this paper, i.e., in [29]. This two-part overview paper is an invited state-of-art overview for Information Sciences, and as all necessary details are in [28] (as mentioned in introduction), we do not give here proofs, detailed related references etc. Using the same notation and terminology as in [29], this paper is organized as follows. The next section deals with construction methods for aggregation functions. Section 3 is devoted to conjunctive aggregation functions, with a special stress on distinguished classes of triangular norms and copulas. In Section 4, disjunctive aggregation functions are discussed, though due to their duality with conjunctive aggregation functions, many results are omitted. Mixed aggregation functions, mixing in some sense conjunctive and disjunctive aggregation functions, are introduced in Section 5. In concluding remarks, several applied fields where aggregation functions play an important role are summarized.

Section snippets

Some construction methods for aggregation functions

There is a strong demand for an ample variety of aggregation functions having predictable and tailored properties to be used in modeling processes. Several construction methods have been introduced and developed for extending the known classes of aggregation functions (defined either on [0, 1] or, possibly, on some other domains). In this paper we present some well-established construction methods as well as some new ones.

The first group of construction methods can be characterized “from simple

Conjunctive aggregation functions

As already mentioned in the introduction in [29], a conjunctive aggregation function is any aggregation function which is bounded from above by the Min, independently of the interval $I$ we are dealing with. For the sake of transparency, and because of the fact that the majority of applications dealing with conjunctive aggregation functions is linked to the unit interval $I = [0, 1]$ , we restrict our considerations in this section to the unit interval. Directly from the definition it follows that Min

Disjunctive aggregation functions

As already mentioned, disjunctive aggregation functions are those which are bounded from below by Max. Hence Max is the smallest disjunctive aggregation function. When restricting our consideration to the unit interval $I = [0, 1]$ , there is a one-to-one connection between conjunctive and disjunctive aggregation functions (this is not true, for example, if $I =] 0, 1]$ or $I = [0, 1 [$ ).

Lemma 1

Let φ : [0, 1] → [0, 1] be a decreasing bijection. Then the (extended) aggregation function A is disjunctive if and only if its

Mixed aggregation functions

There are several aggregation functions which are neither averaging, nor conjunctive or disjunctive. As a typical example recall the standard summation on the real line, or the standard product on [0,∞]. Aggregation functions of this type will be called mixed. On some sub-domains, mixed aggregation functions behave like averaging, or conjunctive, or disjunctive aggregation functions. For example, considering the product Π:[0,∞]² → [0,∞], it is conjunctive on [0,1]², disjunctive on [1,∞]² and

Concluding remarks

In our two-part contribution we have introduced and discussed the basics of the theory of aggregation functions. Besides several properties and construction methods, also several kinds of aggregation functions have been introduced and examined. The application of aggregation functions can be found in any domain where the observed pieces of information are merged into a single value. We indicate some of the domains where aggregation functions play a substantial role.

We do not pretend to be

Acknowledgments

The authors gratefully acknowledge the support of following projects: bilateral project between Slovakia and Serbia SK-SRB-19-06, the internal research project supported by the University of Luxembourg “Mathematics Research in Decision Making and Operations Research”, F1R-MTH-PUL-09MRDO, project APVV-0012–07 supported by the Slovak grant agency, Grant MSMVZ 6198898701, the national grants Ministry of Sciences of Serbia 174009 (Mathematical models of nonlinearity, uncertainty and decision),