МІНІСТЕРСТВО ТРАНСПОРТУ ТА ЗВ’ЯЗКУ УКРАЇНИ

ДЕРЖАВНА АДМІНІСТРАЦІЯ ЗВ’ЯЗКУ

МЕТОДИЧНІ ВКАЗІВКИ

до виконання лабораторної роботи

“Обработка нечетких множеств”

METHODICAL INSTRUCTIONS

for laboratory lesson

“PROCESSING OF FUZZY SETS”

Odessa 2011

Laboratory work № 2

Fuzzy sets processing

Objectives: to study the basic methods of software for fuzzy sets modeling and processing, crisp sets и systems

1. Key position

1.1. Fuzzy sets.

More often than not, the classes of objects encountered in the real physical world do not have precisely defined criteria of membership. For example, the classes of animals clearly exclude such objects as rocks, fluids, plants, etc. However, such objects as starfish, bacteria, etc. have an ambiguity arise in the case of a number such as 10 in relation to the “class” of all real numbers which are much greater than 1.

Clearly, the “class of all real numbers which are much greater than 1”, or “the class of beautiful women”, or “the class of tall men”, do not constitute classes or sets in the usual mathematical sense of these terms. Yet, the fact remains that such imprecisely in the domains of pattern recognition, communication of information, and abstraction. The concept in question is that of a fuzzy set, that is, a “class” with a continuum of grades of membership. The notion of fussy sets provides a convenient point of departure for the construction of a conceptual framework which parallels in many general than the latter and, potentially, may prove to have a much wider scope of applicability, particularly in the field of pattern classification and information processing. Essentially, such a framework provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply definite criteria of class membership rather than the presence of random variables.

The notion of fuzzy sets provides a convenient tool for representing vague concepts by allowing partial memberships. Among many formulations of fuzzy sets, we choose two systems that are related to many-valued logic and modal logic.In both systems, a fuzzy set can be interpreted by a family of crisp sets, and fuzzy set operators can be defined using standard set operators.

Using either of these scales (or their average median scores) and dividing by 100 gives a mapping between these subjective terms and membership in the fuzzy set “ALWAYS”.

Similar surveys can put other descriptive terms on a quantitative scale. For instance, we are interested in learning how many trials would be successful (out of 100) if the likelihood of success was described by the words probable, very probable, likely, certain, improbable, possible, impossible.

Fuzzy sets and fuzzy logic were introduced by Lotfi A.Zadehin 1965 in the article “Fuzzy sets”. Zadeh was almost single-handedly responsible for the early development in this field. The article contains new definition of notion fuzzy set, used to describe and research complex fuzzy systems.

Let X be a space of points (objects), with a generic element of X denoted by x. Thus

A fuzzy set (class) A in X is characterized by a membership (characteristic) function which associates with each point in X a real number in interval [0, 1], with the value of at x representing the “grade of membership” of x in A. Thus, the nearer the value of to unity, the higher the grade of membership of x in A. When A is asset in the ordinary sense of the term, its membership function can take on only two values 0 and 1, with =1 or 0 according to the familiar characteristic function of a set A. (When there is a need to differentiate between such sets and fuzzy sets, the sets with two-valued characteristic function will be referred to as ordinary sets or simply sets).

Fuzziness is the property of objects and events, if, an object may belong to the given set and not belong to it. The uncertainty of this type is described with membership functions. The value of this function presents the degree of certainty whether this object belongs to the given set. The set is not definite and is called a fuzzy set.

Generic element is the variable which value is defined by the set of verbal characteristics of some property.

The value of generic element is defined due to fuzzy set notion as the basic type of variables used in natural languages used to construct models of systems (generic models).

The generic element

,

where is the value of generic element, is the sequence of its generic values, is the carrier, is the syntax rule that continue the sequence of , is the semantic rule that unites the value of every generic element w with its sense , defines fuzzy subset of carrier . Set is known as basic scale of generic element.

The generic element called “employee age” has values: =”optimal” и =”not optimal”. The syntax rule is “employee age optimal for industry”. Then sequence of generic values contains two elements: =”optimal employee age” and =”not optimal employee age”. Carrier is the range , contains all the possible ages of employees. This carrier defines all functions of membership: for - fuzzy set with membership function and for - fuzzy set with membership function . If the optimal employee age for industry is about 40 years, then semantic rule = “if employee age is about 40, then it is optimal employee age for industry” and = “if employee age differs from 40, then it is not optimal employee age for industry”. Fig.1. illustrates the membership function for sets и

and

Figure 1. Membership function graph for sets А and В.

We have a domain X. Now examine a set A with objects .

The fuzzy set A for the set is the set of pairs (i.e. ), where x is the linguistic variable, is the membership function.

The membership function is defined as

So, an object x is either fully part of A or not at all part of A. We call such a set A a crisp set.

However, in fuzzy logic, things are different. Now an object x can also be partially in A. In other words, can take values between 0 and 1 as well. We call such a set A a fuzzy set. Also, the value of is called the membership degree or membership grade.

The membership function can be presented analytically or as a table.

Tabular membership function. The fuzzy set “several” corresponds to some subset A of arbitrarily numbers taken from the set of natural numbers . Table below illustrates the membership function of the set A:

	1	2	3	4	5	6	7	8	9	10
	0	0,1	0,6	0,8	1	1	0,9	0,7	0,2	0

The value of membership function is represented verbally as follows: number 2 belongs to the set A on 10%, or corresponds to the set “several” on 10%. The same way fuzzy sets “many”, “few”, “about 50”, “almost 20”.

Analytical membership function. The fuzzy set that corresponds to the notion “about 50”: the subset B of the set of real numbers D. Sets D and B are continuous, so the membership function is to be analytical.

So to define the probability density of normal distribution

Here is the average value of arbitrarily value, is the dispersion. The notion “about 50” corresponds to number range (45,55). To build the membership = 50, а = 4,5. So

Fig.2 illustrates .

The fuzzy set of “old” people. The notion “age” contains value “baby”, “child”, “teenager”, “young”, ”mature” and “old”. The basic scale X for this notion is the set of real (0, 130), . The age of 50 is the intermediate between maturity and old age. So the membership function :

Here is the parameter that takes into account the subjectivity of age evaluation. Fig.3. shows the fuzzy set “old” age:

The fussy set is empty fussy set if and only if its membership function is identically zero on X, or .

Equality of two fuzzy sets. Two fuzzy sets A and B are equal, written as A = B, if and only if their functions fot all .

Containment. A is contained in B (or, equivalently, A is a subset of B, or A is smaller than or equal to B) if and only if

As an example, a set of long – livers is a subset of old people.

The height of a fuzzy set hgt(A) is the supremum (maximum) of the membership grades of A. So,

A fuzzy set A is normal if . Any set that is not normal is called subnormal. Such a set A can be normalized using the normalization function norm(A). It is defined such that, for all , we have

The fuzzy set of old people C is nominal one, because the exact high edge of its membership function equals to 1:

The basic operations on fuzzy sets are the same as the operations on rough sets and have linguistic meanings.