Fuzzy centroids

As probably in most fields of application of mathematics, the construction of fuzzy methods has not been developed in a natural language. For this reason it is convenient to start with doing a translation of the basic concepts and principles of fuzzy logic in a colloquial language.

To model any type of social phenomenon, the definition of its domains is an aspect of fundamental importance. In general terms, the domains of the models are expressed as sets; sets understood in the manner in which they are commonly discussed in elementary set theory, represented by Venn diagrams.

In the "normal" sets, the elements belong, or do not belong to the set; there are no intermediate states. But there is another type of sets, which follows the rules of another type of mathematical logic: fuzzy sets. The main characteristic that distinguishes this other type of sets is that it is accepted that the membership of their elements can be given to a certain degree, not necessarily totally or categorically. That is, under the logic of fuzzy sets, an element can belong to a set, for example, in a degree of 0.75 (or 75%). This contrasts with the categorical rules of belonging that are held for traditional sets, where an element can only belong (that is, 100% belonging), or not belong (that is, 0% belonging).

In mathematical terms, we write that a fuzzy set X in a space F associates with each element x ⸦ X a degree of belonging F (x) ⸦ [0, 1], indicating the degree to which the element x satisfies the concept that F represents.

If, for example, the criterion "relevance of the company" is being modeled and if x is a company, then in fuzzy logic F (x) represents the degree to which x satisfies the concept "relevance of the company".

There is a wide range of applications of fuzzy logic in all fields of knowledge. Particularly, it is very useful to represent valuations in qualitative scales, given that this type of measurements is generally based on nominal variables; that is, modalities of sizing a variable with words: large, small, wide, deep, etc. The possibility of analyzing fuzzy terms represented by linguistic expressions is one of the fields in which fuzzy logic contributes more significantly to systems theory.

This app focuses exclusively on the application of fuzzy logic to the methodology of defining centroids. Thus, we will limit ourselves to the purpose of describing analytically the form taken by centroid models under the fuzzy logic, that is, recognizing the fact that research in economic sciences is done by human beings, individuals with complex ideas and perceptions, generally not categorical.

The type of expression of the decision function will express the profile of the decision maker. It is intended that this be reflected in the structure of the model.

A simple way, perhaps the most elementary, to model the decision function is to consider the minimum operator, and apply it to the degree of membership of x within each of the fuzzy sets; that is:

D (x) = minJ [Cj (x)].

which can result in a certain number of values of x, since the minimum can be not unique. However, we can take as x* the value of x that has the largest D (x).

With mathematical resources like the above, a vast combination of criteria for decisions can now be considered. You can configure very different ways of modeling the problem of determination of centroids. The screens show how to introduce the parameters for determining fuzzy centroids and their neighborhoods.