Construction of the membership function of a fuzzy set. Fuzzy sets

The membership function μ A (x) ∈ assigns each number

x ∈ X is a number from the interval characterizing the degree of membership of the solution to subset A.

Those. this is some non-probabilistic subjective measure of fuzziness, determined as a result of a survey of experts about the degree of correspondence of element x to the concept formalized by the fuzzy set A. In contrast to the probabilistic measure, which is an assessment of stochastic uncertainty, dealing with the ambiguity of the occurrence of some event in various moments time, the fuzzy measure is a numerical assessment of linguistic uncertainty associated with the ambiguity and vagueness of the categories of human thinking. When constructing a membership function μ A (x), each fuzzy set A is associated with a certain property, sign or attribute that characterizes a certain set of objects X. The more a specific object x ∈ X has this property, the closer to 1 the corresponding value μ A(x). If an element x ∈ X definitely has this property, then μ A (x)=1, but if x ∈ X definitely does not have this property, then μ A (x)=0.

Main types of membership functions

In practice, it is convenient to use those membership functions that allow analytical representation in the form of some simple mathematical function.

1. Piecewise linear,

used to specify uncertainties of the type: “approximately equal”, “average value”, “located in the interval”, “similar to an object”, “similar to an object”, etc.

Triangular trimf

Trapezoidal trapmf

2. S-shaped,

used to specify uncertainties like: " a large number of", "great value", "significant value", "high level", etc.

Quadratic S-spline smf

3. Z-shaped,

used to specify uncertainties such as “small quantity”, “small value of e”, “insignificant value”, “low level”, etc.

QuadraticZ-spline zmf

4. U-shaped,

used to specify uncertainties of the type: “approximately within the range from and to”, “approximately equal”, “about”, etc.

This type of membership function includes whole class curves that resemble a bell, a flattened trapezoid, or the letter “P” in shape.

Bell-shaped gbellmf

a is the concentration coefficient of the membership function; b – coefficient of slope of the membership function; c – coordinate of the maximum of the membership function.

Gaussian gaussmf

a – coordinate of the maximum of the membership function; b – concentration coefficient of the membership function.

Methods for constructing membership functions

Direct and indirect

Depending on the number of experts involved in the survey, both direct and indirect methods are divided into single And group.

Direct

In direct methods, an expert or a group of experts simply sets for each

x ∈ X is the value of the membership function μ A (x).

As a rule, direct methods for constructing membership functions are used for properties that can be measured on a certain quantitative scale. For example, such physical quantities as speed, time, distance, pressure, temperature and others have corresponding units and standards for their measurement.

When directly constructing membership functions, it should be taken into account that the theory of fuzzy sets does not require an absolutely accurate specification of membership functions. Often it is enough to record only the most characteristic values and the type of membership function.

So, for example, if it is necessary to construct a fuzzy set that represents the property “car speed is approximately 50 km/h”, at the initial stage it may be sufficient to represent the corresponding fuzzy set as a triangular membership function with parameters a = 40 km/h, b = 60 km /h and s = 50 km/h. Subsequently, the membership function can be refined experimentally based on an analysis of the results of solving specific problems.

The process of constructing or specifying a fuzzy set based on some known quantitative value of a measurable attribute has even received a special name - fuzzification or reduction to fuzziness. The point is that although sometimes we know some value of a measurable quantity, we recognize the fact that this value is known inaccurately, possibly with an error or random mistake. Moreover, the less confident we are in the accuracy of the measurement of a feature, the larger the interval of the carrier of the corresponding fuzzy set will be. It should be remembered that in most practical cases, absolute measurement accuracy is only a convenient abstraction for constructing mathematical models. It is for this reason that fuzzification allows us to more adequately represent the objectively present inaccuracy of the results of physical measurements.

Relative frequency method (direct group)

Let there be m experts, n 1 of which answer positively to the question about whether an element x ∈ X belongs to a fuzzy set A. Another part of the experts n 2 = m – n 1 answers this question in the negative. Then it is accepted μ A (x) = n 1 / (n 1 + n 2) = n 1/m.

Example. Let us consider the fuzzy set A corresponding to the concept of “positive average rate of temperature change.” Object x – rate of temperature change. Experts are presented with different values ​​for the rate of change of temperature x, and each of them is asked whether the expert believes that this rate of change of temperature x is a positive average. The survey results are summarized in table.

As a continuous representation of this fuzzy variable, you can use the Gaussian FP gaussmf with the maximum of the membership function a=5 and the concentration coefficient of the membership function b=1.7:

μ(x) = exp [ – (x–5) 2 / 2*1.7 2 ]

Indirect

They are used to solve problems for which the properties of physical quantities cannot be measured. Most common among indirect methods received the method of paired comparisons.

Paired comparison method

The intensity of membership is determined based on pairwise comparisons of the elements under consideration.

For each pair of elements of the universal set, the expert evaluates the advantage of one element over the other in relation to the property fuzzy set. It is convenient to represent pairwise comparisons with the following matrix:

,

where is the elementadvantage level(), determined on the nine-point Saaty scale:

1 - if there is no advantage of the element over the element;

3 - if there is a weak advantage over;

5 - if there is a significant advantage over;

7 - if there is a clear advantage over;

9 - if there is an absolute advantage over;

2, 4, 6, 8 - intermediate comparative estimates.

Example.Construct the membership function of a fuzzy set " a tall man" on the universal set (170, 175, 180, 185, 190, 195), if the following expert pairwise comparisons are known:

absolute advantage 195 over 170;

clear advantage of 195 over 175;

significant advantage of 195 over 180;

slight advantage 195 over 185;

there is no advantage of 195 over 190.

The given expert statements correspond to the following matrix of paired comparisons:

If expert opinions are consistent, the paired comparison matrix has the following properties:

it is diagonal, i.e. a ii =1 ‚ i=1..n ;

it is inversely symmetrical, that is, elements symmetrical with respect to the main diagonal are connected by the dependence a ij =1/a ji , i,j=1..n ;

it is transitive‚ i.e. a ik a kj =a ij , i,j,k=1..n .

The presence of these properties allows us to determine all elements of the pairwise comparison matrix:

After determining all elements of the pairwise comparison matrix, the degrees of membership of the fuzzy set are calculated using the formula:

To normalize a fuzzy set, we divide all degrees of membership into maximum value, i.e. at 0.3588.

When solving problems, we encounter situations where an element to some extent belongs to a given set. For example, many small quantities are determined. Who can say exactly from what value a quantity can be considered small? There is no clear answer to this question. Therefore, one of the ways mathematical description of a fuzzy set is to determine the degree of membership of an element in a fuzzy set. The degree of membership is specified by a number from the interval. The boundaries of the interval - 0, 1, mean, respectively, “does not belong” and “belongs”. In Sect. 1 element affiliation x many A written down in formal form xÎA. This entry can be represented as a characteristic function:

Membership in a set can be represented graphically. For example, in one-dimensional arithmetic space R two sets are given AÎ R And BÎ R . Affiliation xÎA can be represented as a rectangle P A, shown in Fig. 2.1, and affiliation xÎB- in the form of a rectangle P V, shown in Fig. 2.2. Affiliation x union of sets xОАХВ represented by a rectangle P A Ç V, shown in Fig. 2.3. Belonging to a two-dimensional set will be represented by a parallelepiped in three-dimensional space, and belonging n–dimensional set – ( n+1)-dimensional parallelepiped.

Rice. 2.1 Fig. 2.2

Fuzzy subset A sets X called the set of twos. Function m A, which is a reflection of the elements xÎX into the elements of the set (m a:X®), is called the fuzzy set membership function, and X- basic set.

Specific meaning mA(x), specified for the element x, is called the degree of membership of an element x to a fuzzy set. The carrier of a fuzzy set is a subset ÎX, containing those elements xÎX, for which the value of the membership function Above zero.

Example. Let X- a bunch of natural numbers X=(1,2,3, ...,x max ), intended to determine the price of the product. The fuzzy subset “small price” can be specified in the following form:

={<1/1>,<0,9/2>,<0,8/3>,<0,7/4>,<0,6/5>,<0,5/6>,<0,4/7>,<0,3/8>,

<0,2/9>,<0,1/10>,<0/11>,...,<0/x max >}.

The belonging of price values ​​to the fuzzy subset “small price” is shown in Fig. 2.4.

If we consider the set X How continuous set natural numbers, then the price values ​​belonging to the fuzzy subset “small price” will have the form of a continuous function, as shown in Fig. 2.5. Let's look at the properties fuzzy sets.

Height (height - hgt) of a fuzzy set: .

Fuzzy set with hgtA=1 is called normal, and when hgtA<1 - subnormal. Core (core, kernal, nucleus) or center of a fuzzy set: core =(xÎX/m A (x)=1). Foundation (support – supp) of a fuzzy set: supp =(xÎX/m A (x)>1). The crossover point of a fuzzy set is a collection core (xÎX/m A (x)=0.5). Level a, or a– cut (section) of a fuzzy set: a=(xÎX/m A (x)³a). a– a cut of a fuzzy set is also denoted by: a-cut. Strict a– cut of a fuzzy set: a=(xÎX/m A (x)>a). Convex fuzzy set: "x 1 ,x 2 ,x 3 ОX:x 1 £x 2 £x 3 ®m A (x 2)³min(m A (x 1),m A (x 3)). If the inequality does not hold, the fuzzy set is called non-convex. In Fig. Figure 2.6 provides an illustration of the above properties.

Separate view fuzzy set A is a fuzzy number (fuzzy singleton) if the following conditions are met: A is convex, height is normal ( hgt A=1), m A (x) is piecewise continuous function, core or center of a set A (core A) contains one point. Example of affiliation x The fuzzy number “approximately 5” is shown in Fig. 2.7.

Another type of fuzzy set is the specification of some variables in the form of a fuzzy interval. The definition is known.

A fuzzy interval is a convex fuzzy quantity A, whose membership function is quasi-concave, so that

"u,v, "wÎ, m A (w)³min(m A (u), m A (v)), u,v,wÎX.

Then the fuzzy number is an upper semi-continuous fuzzy interval with compact support and a single modal value. Specifying the parameters of a problem in the form of a fuzzy interval is a very convenient form for formalizing imprecise quantities. The usual interval is often an unsatisfactory representation because... its boundaries must be fixed. Estimates may be overestimated or underestimated, which will cast doubt on the calculation results. Setting the task parameters in the form of a fuzzy interval will be both overestimated and underestimated, and the carrier ( base set) of the fuzzy interval will be chosen so that the kernel contains the most plausible values ​​and it will be guaranteed that the parameter in question is within the required limits.

Defining fuzzy intervals can be done by experts as follows. The fuzzy interval is specified by four parameters M=() (see Fig. 2.8), where and are the lower and upper respectively modal values fuzzy interval, and a And b represent the left and right fuzziness coefficient. Setting a fuzzy interval can be done in the following ways.

Option 1. The lower and upper modal values ​​of the interval are the same, and a and b are equal to zero. The value of x is determined with uncertainty equal to zero. To define a fuzzy input variable on the set X, we formally define a fuzzy interval =(x min =x, x m ax =x,0,0), where x imin is the lower modal value, and x m ax is the upper modal value.

A clear assignment of x on a set of values ​​of X, as shown in Fig. 2.9 is a special case of specifying a fuzzy interval, and m A (x) is the value of the degree of membership in the interval.

Option 2. Assignment x is determined with uncertainty other than zero. An example is shown in Fig. 2.10. The fuzzy interval is defined as =(x min , x m ax =x min ,0,b), those. the upper and lower modal values ​​of the interval coincide.

Rice. 2.9 Fig. 2.10

Option 3. Assignment x can be obtained from the interval [A,B]. An example is shown in Fig. 2.11. The degree of membership is equal to one, and =(A=x min ,B=x m ax ,0.0), Where A– lower modal value (minimum possible value of the input variable x), IN– upper modal value (maximum value of the input variable x.

Option 4. Value of the input variable x i can be obtained from a range of values [A,C] [A,B] (A£B£C). Formally, the fuzzy interval is defined as = (A=x min ,B=x max ,0,b). An example of a task is shown in Fig. 2.12, where b=N-E.

Option 5. Value of the input variable qi can be determined by experts from the range of values [A,D] in such a way that in the interval [B,C] the uncertainty of receipt is equal to one (A£B£C£D). Formally, the fuzzy interval in this case is defined as = (B=x min ,C=x max ,a,b). An example of specifying a fuzzy interval is shown in Fig. 2.13, where a=B-A, b=D-C.

Let us consider operations on fuzzy intervals.

Rice. 2.11 Fig. 2.12

The fuzzy summation operation for fuzzy intervals is defined as follows. Sum of two fuzzy intervals M i =() and M j =(), written in the form M i M j, there is also a fuzzy interval M i M j =, Where a=a i + a j ; b=b i + b j ;, . Sum n fuzzy intervals is determined by the formulas:

.

If , a , where and are convex intervals, then , and is a set of intervals, which is determined by the previous formulas.

The operation of difference of fuzzy intervals is defined as follows. The fuzzy difference of two fuzzy intervals is a trapezoidal interval for which c=|a-h|, d=|b-l|,, , where are, respectively, the lower modal values ​​of fuzzy intervals, and are the upper modal values ​​of fuzzy intervals.

Decision making is associated with comparisons of the resulting fuzzy interval either by experts or according to modeling data with a real number. The operation of comparing a fuzzy interval and a real number is performed as follows.

Real number A represent it as an interval (A,A,0,0). Definition of less or greater value fuzzy interval with respect to a real number A produced according to the formulas:

A, If |A-()|£|A-()| And A£;

A, If |A-()|³|A-()| And A³ .

For fuzzy intervals there is an operation of product and division. The product of two fuzzy intervals is determined in the form of a trapezoidal interval, the parameters of which are determined by the formulas:

c=ah, d=bl, ; .

These rules for multiplying two fuzzy intervals depending on the signs of numbers , , , take the form:

If , then ;

If , then ;

If , then ;

If , then ;

If , then ;

If , then ;

If , then ;

If , then ;

If , That .

Let's consider the division operation. Dividing two fuzzy intervals will give a trapezoidal interval, the parameters of which are determined as follows:

c=ah, d=bl, ; ,

and depending on the signs of the numbers , , , this rule to divide two fuzzy intervals will look like this:

If , then ;

If , then ;

If , then ;

If , then ;

If , then ;

If , then ;

If , then ;

If , then ;

If , That .

Membership functions

Membership functions are a subjective concept, because they are determined by people (experts) and each person gives his own assessment. Exist various methods assignment of membership functions.

We will assume that the membership function is some incredible subjective measurement of fuzziness and that it differs from the probability measure, i.e. degree of membership mA(x) element x fuzzy set is a subjective measure of how much an element xÎX corresponds to a concept whose meaning is formalized by a fuzzy set.

Element Match Level x concept, formalized by a fuzzy set, is determined by a survey of experts and represents a subjective measure.

There are two classes of methods for constructing set membership functions: direct and indirect.

2.2.1. Direct construction methods. Direct methods for constructing membership functions are those methods in which the membership degrees of elements x sets X are directly set either by one expert or by a team of experts. Direct methods are divided into direct methods for one expert and for a group of experts, depending on the number of experts.

The direct method for one expert is that an expert for each element xÎX matches a certain degree accessories mA(x), which, in his opinion, the best way is consistent with the semantic interpretation of the set.

Application simple methods for a group of experts allows you to integrate the opinions of all experts and build a graph of correspondence between elements from a set X. The following procedure for constructing the membership function is possible mA(x).

The experts who make up the group of m man asking a question about the ownership of an element xÎX fuzzy set. Let the part of experts consisting of n 1 person answered the question positively, and the other part of the experts n 2 =m-n 1 answered negatively. Then it is decided that m A (x)=n 1 /m.

In more general case expert assessments are compared with weighting coefficients a i О. Odds a i reflect the degree of competence of experts. The degree of membership of an element x to a fuzzy set will be determined

Where p i =1 if the answer is positive and p i =0 if the expert's answer is negative.

The disadvantages of direct methods are their inherent subjectivity because It is human nature to make mistakes.

2.2.2. Indirect methods for constructing membership functions. Indirect methods for constructing membership functions are those methods in which a reduction in subjective influence is achieved due to partitioning common task determining the degree of membership mA(x), xÎX into a number of simpler subtasks. One of the indirect methods is the method of paired comparisons. Let's consider its essence.

Based on the experts’ answers, a matrix of pairwise comparisons is constructed M=½½m ij ½½, in which the elements m ij represent estimates of the intensity of membership of elements x i ОX subset versus elements x j ОX. Membership function m a (x) determined from the matrix M. Let us assume that the values ​​of the membership function are known mA(x) for all values xÎХ. Let m A (x)=r i , Then pairwise comparisons are defined m ij =r i /r j. If the ratios are exact, then the ratio is matrix form MR=n*R, Where R=(r 1 ,r 2 ,...,r n), n- eigenvalue of the matrix M, from which the vector is reconstructed R taking into account the condition Empirical vector R has a solution to the eigenvalue problem M*R=l max, Where lmax- most proper meaning. The problem comes down to finding the vector R, which satisfies the equation

M*R=l max *R. (2.1)

This equation has only decision. Coordinate values eigenvector, corresponding to the maximum eigenvalue lmax, divided by their sum, will be the required degrees of membership. The concepts that are proposed to experts, as well as the correspondence of these concepts to the quantities m ij, are given in Table 2.1.

Table 2.1

Experts are surveyed regarding how much, in their opinion, the value m A (x i) exceeds the value m A (x i), i.e. how much element x i more significant for the concept described by the fuzzy set than the element x j. The survey will allow you to build a matrix of pairwise comparisons, which has the form

Element Definition r i ОR happens as follows. The sum of each j th matrix column M. From constructing the matrix M follows that It follows that r i =1/k i .

Having determined all the quantities k j, we get the values ​​of the vector elements R. Based on the fact that the matrix M, as a rule, is constructed inaccurately, the found vector R used as initial in iterative method solutions to equation (2.1).

2.2.3. Types of membership functions. It was determined above that membership functions can have a trapezoidal form (see Fig. 2.7), triangular view(see Fig. 2.7). Membership functions can also have a bell-shaped form (Fig. 2.14).

For the bell-shaped form, the membership function is defined by

,

Where m- a given number, d- indicator of fuzziness.

For a trapezoidal view, the membership function is defined by the expression: m A (x)=min(max(a-k|x-b|;0);1), Where a, b - given numbers, k- indicator of fuzziness.

When solving fuzzy control problems, other functions can be used:

m A (x)=e -kx , x>0; m A (x)=1-a x , 0£x£a -1/k ; m A (x)=(1+kx 2) -1, k>1.

Fuzzy set with one-dimensional membership function mA(x) usually called fuzzy set of the first kind.

Exist fuzzy sets of the second kind, for which the membership function is: .

Two-dimensional fuzzy set A defined as follows: A=(A 1 ´A 2: m A (x 1 ,x 2)), Where A 1 ´A 2- Cartesian product, m A (x 1 ,x 2)=min(a-k 1 |x 1 -b| - k 2 |x 2 -c|; (x 1 =0, x 2 =0));- two-dimensional trapezoidal membership function, in which: a, b, c - given numbers, k 1, k 2- indicators of fuzziness. An example of specifying a two-dimensional trapezoidal membership function is shown in Fig. 2.15.

2D function The bell-shaped accessory is determined by the formula:

Where m 1, m 2- given numbers, d 1, d 2- indicators of fuzziness.

Fuzzy Logic Toolbox includes 11 built-in accessory functions that use the following basic functions:

• piecewise linear;
• Gaussian distribution;
• sigmoid curve;
• quadratic and cubic curves.

For convenience, the names of all built-in membership functions end in mf. The membership function is called as follows:

namemf(x, params),

Where namemf– name of the membership function;
x– vector for whose coordinates it is necessary to calculate the values ​​of the membership function;
params– vector of parameters of the membership function.

The simplest membership functions are triangular ( trimf) and trapezoidal ( trapmf) is formed using piecewise linear approximation. The trapezoidal membership function is a generalization of the triangular one; it allows you to specify the core of a fuzzy set in the form of an interval. In the case of a trapezoidal membership function, the following convenient interpretation is possible: the kernel of a fuzzy set is an optimistic estimate; the carrier of a fuzzy set is a pessimistic assessment.

Two membership functions – symmetric Gaussian ( gaussmf) and two-sided Gaussian ( gaussmf) is formed using a Gaussian distribution. gaussmf Function gbellmf allows you to specify asymmetric membership functions. Generalized bell-shaped membership function () are similar in shape to Gaussian ones. These membership functions are often used in

fuzzy systems , since throughout the entire domain of definition they are smooth and take non-zero values.,Membership functions, sigmf dsigmf

psigmf are based on the use of a sigmoid curve. These functions allow you to generate membership functions whose values ​​starting from a certain argument value and up to + (-) are equal to 1. Such functions are convenient for specifying linguistic terms of the “high” or “low” type. And smf, Polynomial approximation is used when generating functions zmf, pimf , since throughout the entire domain of definition they are smooth and take non-zero values.,graphic images, sigmf which are similar to functions

dsigmf , respectively. Basic information about built-in membership functions is summarized in Table. 6.1. In Fig. 6.1 shows graphical representations of the membership functions obtained using the demo script

mfdemo . As can be seen from the figure, the built-in membership functions allow you to specify a variety of fuzzy sets. IN Fuzzy Logic Toolbox it is possible for the user to create m own function m accessories. To do this you need to create :

-function containing two input arguments - a vector for the coordinates of which it is necessary to calculate the values ​​of the membership function and a vector of parameters of the membership function. The output argument of the function must be a vector of degrees of membership. Below is
-function that implements the bell-shaped membership function
function mu=bellmf(x, params)
%bellmf – bell membership function;
%x – input vector;
%params(1) – concentration coefficient (>0);
%params(2) – coordinate of maximuma.
a=params(1);

b=params(2);

mu=1./(1+ ((x-b)/a).^2);

Fuzzy set- key concept fuzzy logic. Let E- universal set, X- element E, a R is some property. Regular (crisp) subset A universal set E, whose elements satisfy the property R is defined as the set of ordered pairs

A = ( μA(x) / x},

Where μ A (x) —characteristic function, taking the value 1 if X satisfies property R, and 0 otherwise.

Fuzzy subset is different from regular topics, which is for elements X from E there is no clear yes-no answer regarding property R. In this regard, the fuzzy subset A universal set E is defined as the set of ordered pairs

A = ( μA(x) / x},

Where μ A (x) — characteristic membership function(or simply membership function), taking values ​​in some completely ordered set M(For example, M = ).

The membership function indicates the degree (or level) of membership of an element X subset A. A bunch of M called a set of accessories. If M= (0, 1), then the fuzzy subset A can be considered as an ordinary or crisp set.

Examples of writing a fuzzy set

Let E = {x 1 , x 2 , x z,x 4 , x 5 ), M = ; A is a fuzzy set for which μ A ( x 1 )= 0.3; μ A ( x 2)= 0; μ A ( X 3) = 1; μ A (x 4) = 0.5; μ A ( x 5)= 0,9.

Then A can be represented in the form

A ={0,3/x 1 ; 0/X 2 ; 1/X 3 ; 0,5/X 4 ; 0,9/X 5 } ,

or

A={0,3/x 1 +0/X 2 +1/X 3 +0,5/X 4 +0,9/X 5 },

or

Comment. Here the “+” sign does not denote the operation of addition, but has the meaning of union.

Basic characteristics of fuzzy sets

Let M= and A— fuzzy set with elements from the universal set E and many accessories M.

The quantity is called height fuzzy set A. Fuzzy set It's okay if its height is 1, i.e. upper limit its membership function is 1 (= 1). At< 1нечеткое множество называется subnormal.

Fuzzy set empty if ∀ xϵ E μ A ( x) = 0. A non-empty subnormal set can be normalized using the formula

Fuzzy set unimodal, If μ A ( x) = 1 on only one X from E.

. Carrier fuzzy set A is an ordinary subset with the property μ A ( x)>0, i.e. carrier A = {x/x ϵ E, μ A ( x)>0}.

Elements xϵ E, for which μ A ( x) = 0,5 , are called transition points sets A.

Examples of fuzzy sets

1. Let E = {0, 1, 2, . . ., 10}, M =. Fuzzy set"Several" can be defined as follows:

“Several” = 0.5/3 + 0.8/4 + 1/5 + 1/6 + 0.8/7 + 0.5/8; its characteristics:height = 1, carrier = {3, 4, 5, 6, 7, 8}, transition points — {3, 8}.

2. Let E = {0, 1, 2, 3,…, n,… ). The fuzzy set “Small” can be defined:

3. Let E= (1, 2, 3,..., 100) and corresponds to the concept “Age”, then the fuzzy set “Young” can be defined using

Fuzzy set “Young” on the universal set E"= (IVANOV, PETROV, SIDOROV,...) is specified using the membership function μ Young ( x) on E =(1, 2, 3, ..., 100) (age), called in relation to E" compatibility function, while:

Where X— SIDOROV’s age.

4. Let E= (ZAPOROZHETS, ZHIGULI, MERCEDES,...) - a set of car brands, and E"= is the universal set “Cost”, then on E" we can define fuzzy sets of the type:

Rice. 1.1. Examples of membership functions

“For the poor”, “For the middle class”, “Prestigious”, with affiliation functions like Fig. 1.1.

Having these functions and knowing the cost of cars from E V this moment time, we will thereby determine E" fuzzy sets with the same names.

So, for example, the fuzzy set “For the poor”, defined on the universal set E =(ZAPOROZHETZ, ZHIGULI, MERCEDES,...), looks as shown in Fig. 1.2.

Rice. 1.2. An example of specifying a fuzzy set

Similarly, you can define the fuzzy set “High-speed”, “Medium”, “Slow-speed”, etc.

5. Let E- set of integers:

E= {-8, -5, -3, 0, 1, 2, 4, 6, 9}.

Then the fuzzy subset of numbers, according to absolute value close to zero, can be defined, for example, like this:

A ={0/-8 + 0,5/-5 + 0,6/-3 +1/0 + 0,9/1 + 0,8/2 + 0,6/4 + 0,3/6 + 0/9}.

On methods for constructing membership functions of fuzzy sets

The above examples used straight methods when an expert either simply sets for each X ϵ E meaning μ A (x), or defines a compatibility function. As a rule, direct methods for specifying the membership function are used for measurable concepts such as speed, time, distance, pressure, temperature, etc., or when polar values ​​are distinguished.

In many problems, when characterizing an object, it is possible to select a set of features and for each of them determine polar values ​​corresponding to the values ​​of the membership function, 0 or 1.

For example, in the task of face recognition, we can distinguish the scales given in table. 1.1.

Table 1.1. Scales in the face recognition task

For a specific personAthe expert, based on the given scale, setsμ A(x)ϵ, forming the vector membership function (μ A(x 1) , μ A(x 2),…, μ A(x 9)}.

With direct methods, group direct methods are also used, when, for example, a group of experts is presented with a specific person and everyone must give one of two answers: “this person is bald” or “this person is not bald”, then the number of affirmative answers divided on total number experts, gives meaning μ bald ( of this person). (In this example, you can act through the compatibility function, but then you will have to count the number of hairs on the head of each person presented to the expert.)

Indirect methods for determining the values ​​of the membership function are used in cases where there are no elementary measurable properties through which the fuzzy set of interest to us is determined. As a rule, these are pairwise comparison methods. If the values ​​of the membership functions were known to us, for example, μ A(X-i) = ω i , i= 1, 2, ..., n, then pairwise comparisons can be represented by a matrix of relations A= ( a ij ), where a ij= ωi/ ω j(division operation).

In practice, the expert himself forms the matrix A, in this case it is assumed that the diagonal elements are equal to 1, and for elements that are symmetrical with respect to the diagonal a ij = 1/a ij , i.e. if one element evaluates to α times stronger than the other, then this latter must be 1/α times stronger than the first. In the general case, the problem reduces to finding a vector ω that satisfies an equation of the form Aw= λmax w, where λ max is the largest eigenvalue of the matrix A. Since the matrix A is positive by construction, a solution to this problem exists and is positive.

Two more approaches can be noted:

• use of standard forms curves for specifying membership functions (in the form of (L-R)-Type - see below) with clarification of their parameters in accordance with experimental data;
• use of relative frequenciesaccording to the experiment as membership values.

Let X be a universal (basic) set, x be an element of X, and R be some property. An ordinary (crisp) subset A of a universal set X, whose elements satisfy the property R, is defined as the set of ordered pairs
A = μ A x / x, where μ A x is a characteristic function that takes the value 1 if x satisfies property R, and 0 otherwise.

A fuzzy subset differs from a regular subset in that for elements x of X there is no clear yes-no answer regarding the property R. In this regard, a fuzzy subset A of a universal set X is defined as a set of ordered pairs A = μ A x / x , where μ A x – characteristic membership function(or simply membership function), taking values ​​in some completely ordered set M = 0 ;

Carrier 1 . The membership function indicates the degree (or level) of membership of an element x to a subset A. The set M is called the membership set. If M = 0 ;

1, then the fuzzy subset A can be considered as an ordinary or crisp set. The degree of membership μ A x is a subjective measure of how much an element x ∈ X corresponds to the concept, the meaning of which is formalized by the fuzzy set A. fuzzy set A is a clear subset S A of the universal set X with the property μ A x > 0, i.e. S A = x ∣ x ∈ X ∧ μ A x > 0 . In other words, the carrier of a fuzzy set A is a subset S A of the universal set X, for whose elements the membership function μ A x > 0 is greater than zero. Sometimes the support of a fuzzy set is denoted by support A.< x 2 < x 3 < … < x n .

If the carrier of a fuzzy set A is a discrete subset S A , then the fuzzy subset A of a universal set X consisting of n elements can be represented as a union

finite number

Example. single-point sets μ A x / x using the symbol ∑ : A = ∑ i = 1 n μ A x i / x i . This implies that the elements x i are ordered in ascending order in accordance with their indices, i.e.

A = 1 / 10 ;

0.9/11;

0.8/12; 0.7/13; 0.5 / 14 ;

0.3 / 15 ;

0.1 / 16 ; 0 / 17 ;... ; 0 / 40 A = 1 / 10 + 0.9 / 11 + 0.8 / 12 + 0.7 / 13 + 0.5 / 14 + 0.3 / 15 + 0.1 / 16 + 0 / 17 + … + 0 / 40,

where the summation sign denotes a non-operation arithmetic addition, but combining elements into one set. The carrier of the fuzzy set A will be a finite subset (discrete carrier):

S A = 10 ;

Example. eleven ; 12 ; 13 ; 14 ; 15 ; 16 .

If the universal set X is a set real numbers from 10 to 40, i.e. the thickness of the product can accept anything

possible values within these limits, then the carrier of the fuzzy set A is the segment S A = 10 ; 16 . A fuzzy set with a discrete carrier can be represented as individual points on a plane, a fuzzy set with a continuous carrier can be represented as a curve, which corresponds to a discrete and continuous functions belonging to μ A x defined on the universal set X (Fig. 2.1). Fig.2.1. Membership functions of fuzzy sets with (a)-discrete and (b)-continuous carriers Let X = 0; 1 ; 2 ; ... is a set of non-negative integers. The fuzzy set ital small can be defined as μ ital small x = x 1 + 0.1 x 2 − 1 . Fig.2.2. Graphical representation fuzzy set small The fuzzy set A is called final

Example. The fuzzy concept of “a very small number of parts” can be represented as a finite fuzzy set A = 1 / 0 + 0.9 / 1 + 0.8 / 2 + 0.7 / 3 + 0.5 / 4 + 0.1 / 5 + 0 / 6 + … with card power (A) = 6 and carrier S A = 0 ;

Example. 1 ; 2 ;

3; height 4 ; 5, which is a finite crisp set. The fuzzy concept of “a very large number of parts” can be represented as A = 0 / 0 + … + 0.1 / 1 0 + 0.4 / 11 + 0.7 / 12 + 0.9 / 13 + 1 / 14 + 1 / 15 + … + 1 / n + … , n ∈ N – a fuzzy set with infinite countable support S A ≡ N (set of natural numbers), which has countable power in the usual sense.

An uncountable fuzzy set A corresponding to the fuzzy concept “very hot” is defined on the universal set of temperature values ​​(in Kelvin) by temperature x ∈ [ 0 ; ∞) and the membership function μ A = 1 − e − x , with support S A ≡ R + (the set of non-negative real numbers), which has an uncountable continuum power. The quantity sup x ∈ X μ A x is called< 1 subnormal.

fuzzy set. Fuzzy set A Fine

, if its height is 1, i.e. the upper bound of its membership function is sup x ∈ X μ A x = 1 . For sup x ∈ X μ A x

fuzzy set. A fuzzy set is called empty , if ∀ x ∈ X μ A x = 0 . A non-empty subnormal set can always be normalized by dividing all values ​​of the membership function by its maximum value μ A x sup x ∈ X μ A x .

fuzzy set. unimodal, if μ A x = 1 for only one point x (

fashion α ) of the universal set X. point

, if μ A x > 0 only for one point x of the universal set X .

Example. Many

-level fuzzy set A defined on a universal set X is called a clear subset A α of the universal set X, defined as: A α = x ∈ X ∣ μ A x ≥ α, where α ∈ 0;

Elements x ∈ X for which μ A x = 0.5 are called transition points fuzzy set A.

Core a fuzzy set A defined on a universal set X is called a clear set core A, the elements of which satisfy the condition core A = x ∈ X ∣ μ A x = 1.

Border of a fuzzy set A defined on a universal set X is called a clear set front A whose elements satisfy the condition front A = x ∈ X ∣ 0< μ A x < 1 .

Example. Let X = 0;

1 ; 2 ; ... ; 10, M = 0;< x < b ; x , a , b ∈ X (рис.2.3).

1 . The fuzzy set several can be defined on the universal set of natural numbers as follows: several = 0.5 / 3 + 0.8 / 4 + 1 / 5 + 1 / 6 + 0.8 / 7 + 0.5 / 8 ; its characteristics: height = 1, media = 3;