From natural or artificial language. For example, the linguistic variable “speed” can have the values “high”, “medium”, “very low”, etc. The phrases whose value the variable takes are, in turn, names of fuzzy variables and are described by a fuzzy set.
Mathematical definition
The linguistic variable is called five , Where - variable name; - a certain set of values of a linguistic variable , each of which is a fuzzy variable on the set ; there is a syntactic rule for the formation of names of new values ; there is a semantic procedure that allows you to convert a new name formed by the procedure , into a fuzzy variable (specify the type of membership function), associates a name with its value, concept.
also called the basic term set, since it specifies the minimum number of values on the basis of which, using rules, And you can generate the remaining valid values of the linguistic variable. A bunch of and new ones educated with the help And the values of a linguistic variable form an extended term set.
Example: fuzzy age
Consider a linguistic variable describing a person's age, then:
- : "age";
- : set of integers from the interval;
- : meanings “young”, “mature”, “old”. a bunch of - a set of fuzzy variables, for each value: “young”, “mature”, “old”, it is necessary to set a membership function, which specifies information about what age people should be considered young, mature, old;
- : “very”, “not very”. Such additions allow the formation of new meanings: “very young”, “not very old”, etc.
- : mathematical rule, which determines the type of membership function for each value formed using the rule .
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Excerpt characterizing the Linguistic variable
The Count again went behind the partition and lay down. The Countess approached Natasha, touched her head with her inverted hand, as she did when her daughter was sick, then touched her forehead with her lips, as if to find out if there was a fever, and kissed her.-You're cold. You're shaking all over. You should go to bed,” she said.
- Go to bed? Yes, okay, I'll go to bed. “I’ll go to bed now,” Natasha said.
Since Natasha was told this morning that Prince Andrei was seriously wounded and was going with them, only in the first minute she asked a lot about where? How? Is he dangerously injured? and is she allowed to see him? But after she was told that she could not see him, that he was seriously wounded, but that his life was not in danger, she, obviously, did not believe what she was told, but was convinced that no matter how much she said, she would be answer the same thing, stopped asking and talking. All the way with big eyes, which the Countess knew so well and whose expressions the Countess was so afraid of, Natasha sat motionless in the corner of the carriage and now sat in the same way on the bench on which she sat down. She was thinking about something, something she was deciding or had already decided in her mind now - the countess knew this, but what it was, she did not know, and this frightened and tormented her.
- Natasha, undress, my dear, lie down on my bed. (Only the countess alone had a bed made on the bed; m me Schoss and both young ladies had to sleep on the floor on the hay.)
“No, mom, I’ll lie here on the floor,” Natasha said angrily, went to the window and opened it. The adjutant groans from open window was heard more clearly. She stuck her head out into the damp air of the night, and the countess saw how her thin shoulders were shaking with sobs and beating against the frame. Natasha knew that it was not Prince Andrei who was moaning. She knew that Prince Andrei was lying in the same connection where they were, in another hut across the hallway; but this terrible incessant groan made her sob. The Countess exchanged glances with Sonya.
“Lie down, my dear, lie down, my friend,” said the countess, lightly touching Natasha’s shoulder with her hand. - Well, go to bed.
“Oh, yes... I’ll go to bed now,” said Natasha, hastily undressing and tearing off the strings of her skirts. Having taken off her dress and put on a jacket, she tucked her legs in, sat down on the bed prepared on the floor and, throwing her short thin braid over her shoulder, began to braid it. Thin, long, familiar fingers quickly, deftly took apart, braided, and tied the braid. Natasha's head turned with a habitual gesture, first in one direction, then in the other, but her eyes, feverishly open, looked straight and motionless. When the night suit was finished, Natasha quietly sank down onto the sheet laid on the hay on the edge of the door.
Linguistic Variables( LP ) are a way of describing complex systems, the parameters of which are considered not from quantitative positions, but as qualitative ones. At the same time, linguistic variables make it possible to correlate quality characteristics some quantitative interpretation with a given degree of confidence, which makes it possible to process qualitative data on a computer. Another area of application of linguistic variables is fuzzy logical inference, the difference from the usual one is that the truth logical statements is determined not by two values 0 and 1, but by a set of values in the interval .
The concept of a linguistic variable is based on the concept of an odd variable.
fuzzy variable is a combination of three elements:
< X, U, µ A(u) >,
Where X– name of the fuzzy variable; U– universal set; µ A(u) – fuzzy subset A universal set U. In other words, a fuzzy variable is a named fuzzy set.
Linguistic variable is called a set of five elements:
< L, T(X), U, G, M >,
Where L– name of the linguistic variable;
T(X) – a set of basic terms of a linguistic variable, consisting of a set of names of values of linguistic variables ( T 1 , T 2 , …, Tn), each of which corresponds to a fuzzy variable X universal set U;
U– a universal set on which a linguistic variable is defined;
G- syntactic rule that generates names X variable values;
M– a semantic rule that assigns each fuzzy variable X its meaning M(X), i.e. fuzzy subset of the universal set U.
The terms of a linguistic variable are subject to the ordering requirement: T 1 < T 2 < … < Tn.
Membership functions fuzzy sets, constituting the quantitative meaning of the basic terms of a linguistic variable, must satisfy the following conditions:
2. : 
;
4. : 
.
Here n– the number of basic terms of the linguistic variable; umin, u max– boundaries of the universal set U, on which the linguistic variable is defined. If U R (R is the set of real numbers, then U = [umin, u max].
Syntax rule G is a combination of four elements: G = < V T, V N, T, P >,
Where V T– a set of terminal symbols or words; V N– a set of non-terminal symbols or phrases; T– a set of basic terms; R– a set of substitution rules that determine the equivalence of phrases.
Semantic rule M assigns each phrase a new non-
a clear set defined on the basis of membership functions of basic terms and a set of operations with fuzzy sets.
As an example, consider the numerical linguistic variable “person height”. Let the variable values be specified using three basic terms: “low”, “medium”, “high”. The terms are ordered. Universal number set U V in this case is the interval U = .
The membership functions of terms are shown in Fig. 7.6 and satisfy the requirements discussed above.
Rice. 7.6 Linguistic variable “Human height”
As a syntactic rule, we define that the set of non-terminal symbols includes the words “and”, “or”, “more or less”, “not”, “very”, which can be combined with the basic terms “low”, “medium”, “ high”, and the following rules must be followed:
The symbols “and” and “or” can only connect two phrases or basic terms, and the remaining non-terminal symbols are unary, i.e. may precede a phrase or base term; for example, “not high”, “very low”, “low or average”;
The simultaneous negation of two basic terms, for example, “not low and not high,” is equivalent to the remaining basic term, i.e. "average".
By applying these rules, you can construct many phrases and substitution rules. If the syntactic rule cannot be specified algorithmically, then all possible phrases are simply listed.
As a semantic rule, we define the correspondence between non-terminal symbols and operations on fuzzy sets:
“not” – addition;
“and” - intersection;
“or” - union;
“very” - concentration;
"more or less" is an extension.
Using the considered linguistic variable, we can estimate
determine the height of people without resorting to precise measurements.
Thus, using linguistic variables you can describe objects precise measurement characteristics of which is either extremely labor-intensive or completely impossible.
The formation of a linguistic variable, as a rule, is carried out on the basis of a survey of experts - specialists in the field for which the LP is being built. Wherein Special attention is paid to the formation of membership functions of fuzzy sets, which are the basic terms of a linguistic variable, since the definition of syntactic and semantic rules for most linguistic variables is standard and in practice comes down to listing all possible phrases and interpreting non-terminal symbols, as shown above.
The process of forming a linguistic variable includes the following stages:
1. Definition of the set of LP terms and its ordering.
2. Construction of the numerical domain of definition of the LP.
3. Determining the scheme for interviewing experts and conducting the survey.
4. Construction of membership functions for each LP term.
Stage 1 involves the expert specifying the number of LP terms and the names of the corresponding fuzzy variables. The number of terms is selected from the range n= 7±2.
At stage 2, the universal set is described U, which can be numeric or non-numeric. The type of universal set depends on the objects being described and determines the method of forming membership functions of LP terms.
Stage 3 is key in the formation of the LP. There are two types
expert survey: direct and indirect. Each of these methods can be individual or group. The simplest from the point of view of organization and
software implementation is an individual way of interviewing experts.
During a direct survey of experts, all parameters of the membership functions are directly indicated. The disadvantage here is the manifestation of subjectivity in judgments, as well as the need for an expert to know the basics of fuzzy logic. In an indirect survey, membership functions are formed based on the expert’s answer to “leading” questions. This increases the objectivity of the assessment and does not require knowledge of fuzzy logic, but increases the risk of inconsistency in the expert’s judgments.
With group survey methods, the result is formed based on combining the opinions of several experts. In practice, individual indirect interviews are most often used.
Lecture. Fuzzy calculations
Concept of fuzzy number
One of the areas of application of fuzzy logic is the performance of arithmetic operations with fuzzy sets. To reduce the complexity of such operations, a special type of fuzzy sets is used - fuzzy numbers.
Fuzzy number(NF) is a fuzzy variable that has following properties: 
; .
In other words, a fuzzy number is a named fuzzy set for which the universal set U represents the real axis interval R.
IN real problems piecewise linear fuzzy numbers are used. To simplify arithmetic operations, piecewise linear membership functions are additionally approximated to obtain a special type of fuzzy numbers - parametric fuzzy numbers or fuzzy numbers
(L–R)-type, which are characterized by compactness of representation and simple-
that implementation of arithmetic operations.
fuzzy number A called fuzzy number (L–R)–type, if its membership function has the following form (Fig. 7.8):
0, 
where are the parameters of the fuzzy number; L(x), R(x) – some functions.
A fuzzy parametric number is denoted by ( a, b, c, d)LR.
Thus, the fuzzy number ( L–R)-type is described by six parameters: four numbers indicating its boundaries, and two functions determining the form of its membership function.
 |
Fig.7.8 Parametric fuzzy numbers
Fuzzy number is called unimodal, if it has only one point at which the membership function is equal to one, i.e. its parameters b And c are equal, otherwise the fuzzy number is called tolerant(see Fig. 7.8). Unimodal fuzzy numbers are denoted by five parameters ( a, b, d)LR.
As LR–functions most often used linear dependencies, given by the following relations:
LR– functions can also be specified by quadratic, exponential and other dependencies.
In case of use linear functions unimodal and tolerant fuzzy numbers are called triangular and trapezoidal, respectively, and denoted by ( a, b, d) And ( a, b, c, d).
For fuzzy numbers, the concept of sign and zero value is defined in a special way.
fuzzy number A called positive, if its base lies in the positive real semi-axis or
fuzzy number A called negative, if its base lies in the negative real semi-axis or
For parametric fuzzy numbers, the sign is determined by the values of the parameters: a positive fuzzy number if a> 0; negative if d < 0; нечеткий ноль, если .
Let us recall that a linguistic variable is a variable that takes values from a set of words or phrases of some natural or artificial language. A bunch of acceptable values a linguistic variable is called a term set. Setting the value of a variable in words, without using numbers, is more natural for humans. Every day we make decisions based on linguistic information like: “very heat"; "long trip"; "quick response"; " beautiful bouquet"; "harmonious taste", etc. Psychologists have found that in human brain almost all numerical information verbally recoded and stored in the form of linguistic terms. The concept of linguistic variable plays important role in fuzzy logical inference and in decision making based on approximate reasoning. Formally, a linguistic variable is defined as follows.
Definition 44.Linguistic variable is given by five, where -; variable name; - ; term-set, each element of which (term) is represented as a fuzzy set on the universal set; - ; syntactic rules, often in the form of a grammar, giving rise to the names of terms; - ; semantic rules that specify the membership functions of fuzzy terms generated by syntactic rules.
Example 9. Consider a linguistic variable called "room temperature". Then the remaining four can be defined as follows:
Table 4 - Rules for calculating membership functions
Graphs of the membership functions of the terms “cold”, “not very cold”, “comfortable”, “more or less comfortable”, “hot” and “very hot” of the linguistic variable “room temperature” are shown in Fig. 13.
Figure 13 - Linguistic variable "room temperature"
Fuzzy truth
A special place in fuzzy logic is occupied by the linguistic variable “truth”. In classical logic, truth can only take two meanings: true and false. In fuzzy logic, truth is "fuzzy". Fuzzy truth is defined axiomatically, and different authors do it in different ways. The interval is used as a universal set to define the linguistic variable "truth". Ordinary, clear truth can be represented by fuzzy singleton sets. In this case, a clear concept will truly correspond to the membership function 
, and the clear concept is false -; 
, .
To define fuzzy truth, Zadeh proposed the following membership functions for the terms “true” and “false”:
;
Where - ; a parameter that determines the carriers of the fuzzy sets “true” and “false”. For a fuzzy set “true” the carrier will be the interval , and for a fuzzy set “false” - ; .
The membership functions of the fuzzy terms “true” and “false” are shown in Fig. 14. They are built with the parameter value . As you can see, the graphs of the membership functions of the terms “true” and “false” are mirror images.
Figure 14 - Linguistic variable “truth” according to Zadeh
To define fuzzy truth, Baldwin proposed the following membership functions for fuzzy “true” and “false”:
The quantifiers “more or less” and “very” are often applied to the fuzzy sets “true” and “false”, thus obtaining the terms “very false”, “more or less false”, “more or less true”, “very true”. , “very, very true”, “very, very false”, etc. The membership functions of new terms are obtained by performing the operations of concentration and stretching of the fuzzy sets “true” and “false”. The operation of concentration corresponds to squaring the membership function, and the operation of stretching corresponds to raising it to the power of ½. Consequently, the membership functions of the terms “very, very false”, “very false”, “more or less false”, “more or less true”, “true”, “very true” and “very, very true” are given as follows.
In our informal discussion of the concept of a linguistic variable in §1, we stated that a linguistic variable differs from a numerical variable in that its values are not numbers, but words or sentences in a natural or formal language. Because words are generally less precise than numbers, the concept of a linguistic variable makes it possible to approximate phenomena that are so complex that they cannot be described in conventional quantitative terms. In particular, a fuzzy set, which is a constraint associated with the values of a linguistic variable, can be considered as a collective characteristic of various subclasses of elements of a universal set. In this sense, the role of fuzzy sets is similar to the role played by words and sentences in natural language. For example, adjective Beautiful reflects a complex of characteristics of an individual’s appearance. This adjective can also be considered as the name of a fuzzy set, which is a constraint imposed by a fuzzy variable Beautiful. From this point of view, the terms very beautiful, ugly, extremely beautiful, quite beautiful etc. - names of fuzzy sets formed by the action of modifiers Very, Not, extremely, quite etc. on a fuzzy set Beautiful. In essence, these fuzzy sets, together with the beautiful fuzzy set, play the role of values of a linguistic variable Appearance.
An important aspect of the concept of a linguistic variable is that it is a higher order variable than a fuzzy variable in the sense that the values of a linguistic variable are fuzzy variables. For example, the values of the linguistic variable Age can be: young, middle-aged, old, very old, middle-aged and not old, quite old etc. Each of these values is the name of a fuzzy variable. If is the name of a fuzzy variable, then the constraint imposed by this name can be interpreted as the meaning of the fuzzy variable. So, if the constraint due to the fuzzy variable old, is a fuzzy subset of a set of the form
, , (5.1)
Another important aspect of the concept of a linguistic variable is that a linguistic variable corresponds to two rules: (1) a syntactic rule, which can be given in the form of a grammar that generates the names of the values of the variable; (2) a semantic rule that specifies an algorithmic procedure for calculating the meaning of each value. These rules form an essential part of the description of the structure of a linguistic variable.
Rice. 5.1. Compatibility functions for values and 
.
Since a linguistic variable is a variable of a higher order than a fuzzy variable, its description should be more complex than the description of a fuzzy variable given in Definition 4.1.
Definition 5.1. A linguistic variable is characterized by a set 
, in which is the name of the variable; (or simply) denotes a term-set of a variable, i.e., a set of names of linguistic values of a variable, each of such values being a fuzzy variable with values from a universal set with a base variable; - a syntactic rule (usually having the form of a grammar) that generates the names of the values of the variable, and - a semantic rule that associates each fuzzy variable with its meaning, i.e., a fuzzy subset of the universal set. Specific name generated by a syntactic rule is called a term. A term consisting of one word or several words that always appear together with each other is called an atomic term. A term consisting of one or more atomic terms is called a compound term. The concatenation of some components of a compound term is a subterm. If are terms in , then they can be represented as a union
(5.2)
If it is necessary to explicitly indicate what was generated by the grammar, we will write .
The meaning of a term is defined as a constraint on a base variable conditioned by a fuzzy variable:
, (5.3)
keeping in mind that and, therefore, can be considered as a fuzzy subset of the set having the name . The relationship between its linguistic meaning and the underlying variable is illustrated in Fig. 1.3.
Remark 5.2. In order to avoid large quantity symbols, it is appropriate to assign multiple meanings to some of the symbols appearing in Definition 5.1, relying on context to resolve possible ambiguities. In particular:
a) We will often use the symbol to denote both the name of the variable itself and the general name of its values. Likewise, will denote both the general name of the values of the variable and the name of the variable itself.
b) We will use the same symbol to denote the set and its name. Thus, the symbols , and will be interchangeable, although, strictly speaking, as a name (or ) is not the same as a fuzzy set. In other words, when we say that a term (for example, young) there is a variable value (for example, Age), then we mean that real value is , a is just the name of this value.
Example 5.3. Age, i.e. , let it go . Linguistic meaning of the variable Age maybe, for example, old, and the value old is an atomic term. Another meaning could be very old, i.e. a compound term in which old - atomic term, and Very And old- subterms.
Meaning more or less young variable Age - a compound term in which the term young - atomic, and More or less- subterm. Term set of variable Age can be written as follows:
(5.4)
Here, each term is the name of a fuzzy variable in the universal set. The limitation imposed by the term, say, is the meaning of the linguistic meaning old. Thus, if determined according to (5.1), then the meaning of the linguistic meaning old is determined by the expression
, (5.5)
or simpler (see remark 5.2)
. (5.6)
Likewise, the meaning of such linguistic meaning as very old, can be expressed as follows (see Fig. 5.1):
The assignment equation in the case of a linguistic variable takes the form
whence it follows that the meaning assigned to the term is expressed by the equality
In other words, the meaning of a term is obtained by applying a semantic rule to the meaning of the term assigned according to the right-hand side of equation (5.8). Moreover, from definition (5.3) it follows that it is identical to the limitation due to the term .
Remark 5.4. In accordance with Remark 5.2(a), the assignment equation will usually be written as
, (5.10)
understanding this in such a way that old- restriction on the values of the base variable, defined by (5.1), - assigned to the linguistic variable Age. It is important to note that the equal sign in (5.10) does not denote a symmetric relation, as in the case of arithmetic equality. Thus, it makes no sense to write (5.11) in the form
To illustrate the concept of a linguistic variable, we will first consider a very simple example in which only small number terms, and the syntactic and semantic rules are trivial.
Example 5.5. Consider the linguistic variable Number, whose finite term set has the form
where each term represents a constraint on the values of the base variable in the universal set
It is assumed that these restrictions are fuzzy subsets of the set and they are defined as follows:
, (5.15) with binary constraint approximately equal.
To assign a value, let's say approximately equal linguistic variable, we write
where, as in (5.18), it is meant that a binary fuzzy relation is assigned as the value of the variable approximately equal, which is a binary constraint on the values of the base variable in the universal set (5.20).
Rice. 5.2. The carpetbag analogy for a linguistic variable
Remark 5.7. Using the traveling bag analogy (see Remark 4.3), a linguistic variable in the sense of Definition 5.1 can be likened to a hard traveling bag into which soft traveling bags can be placed, as shown in Fig. 5.2. The soft bag corresponds to a fuzzy variable, which is the linguistic value of the variable , and plays the role of a label on the soft bag.
The concept of fuzzy and linguistic variables is used to describe objects and phenomena using fuzzy sets.
Fuzzy variable characterized by three (α, X, A), Where
α — name of the variable;
X— universal set (domain α);
A- fuzzy set on X, describing the restrictions (i.e. μ A(x) ) to the values of the fuzzy variable α.
Linguistic variable (LP) is the set ( β , T, X, G, M), where
β — name of the linguistic variable;
T— a set of its values (term set), which are the names of fuzzy variables, the domain of definition of each of which is a set X. A bunch of T called basic term-set linguistic variable;
G is a syntactic procedure that allows you to operate with elements of the term set T, in particular, to generate new terms (values). The set T∪G(T), where G(T) is the set of generated terms, is called the extended term set of a linguistic variable;
M— a semantic procedure that allows you to turn each new value of a linguistic variable generated by procedure G into a fuzzy variable, i.e. form the corresponding fuzzy set.
Comment. To avoid too many characters:
1) symbol β used both for the name of the variable itself and for all its values;
2) use the same symbol to denote a fuzzy set and its name, for example the term “Young”, which is the value of a linguistic variable β = “age”, at the same time there is a fuzzy set M("Young").
Assigning multiple meanings to symbols assumes that context allows possible ambiguities to be resolved.
Example. Let the expert determine the thickness of the manufactured product using the concepts “Small thickness”, “Medium thickness” and “Large thickness”, while minimum thickness is equal to 10 mm, and the maximum is 80 mm.
Formalization of such a description can be carried out using the following linguistic variable ( β , T, X, G, M ), Where
β — thickness of the product;
T— (“Small thickness”, “Medium thickness”, “Large thickness”);
X— ;
G - the procedure for the formation of new terms using connectives “and”, “or” and modifiers such as “very”, “not”, “slightly”, etc. For example: “Small or medium thickness”, “Very small thickness”, etc.;
M- task procedure for X = fuzzy subsets A 1 = “Small thickness”, A 2 = "Medium thickness", A 3 = “Large thickness”, as well as fuzzy sets for terms from G (T) in accordance with the rules of translation of fuzzy connectives and modifiers “and”, “or”, “not”, “very”, “slightly” and other operations on fuzzy sets of the form: A⋂IN,A∪INA, CON A =A 2 , DIL A = A 0.5 and so on.
Comment. Along with the basic values of the linguistic variable “Thickness” discussed above (T =(“Small thickness”, “Medium thickness”, “Large thickness”)) possible values depend on the domain of definition X. In this case, the values of the linguistic variable “Product thickness” can be defined as “about 20 mm”, “about 50 mm”, “about 70 mm”, i.e. in the form of fuzzy numbers.
The term set and extended term set in the example conditions can be characterized by the membership functions shown in Fig. 1.5 and 1.6.
Rice. 1.5. Fuzzy set membership functions: “Small thickness” = A 1,"Medium thickness" = A 2, "Large thickness" = A 3
Rice. 1.6. Fuzzy set membership function “Small or medium thickness” = A 1 ∪ A 2
Fuzzy numbers
Fuzzy numbers- fuzzy variables defined on the number axis, i.e. a fuzzy number is defined as a fuzzy set A on the set of real numbers ℝwith membership function μ A(X) ϵ , where X — real number, i.e. X ϵ ℝ.
fuzzy number It's okay if tah μ A(x) = 1; convex, if for any X ≤ at ≤ z performed
μ A (x)≥ μ A(at) ˄ μ A(z).
A bunch of α -fuzzy number level A defined as
Aα= {x/μ α (x) ≥ α } .
Subset S A⊂ ℝ is called the support of the fuzzy number A, If
S A= { x/μ A(x) > 0 }.
fuzzy number And unimodally, if condition μ A(X) = 1 is valid only for one point of the real axis.
Convex fuzzy number A called fuzzy zero, If
μ A(0) = sup ( μ A(x)).
fuzzy number And positively, if ∀ xϵ S A, X> 0 and negative, if ∀ X ϵ S A, X< 0.
Operations on fuzzy numbers
Extended binary arithmetic operations(addition, multiplication, etc.) for fuzzy numbers are determined through the corresponding operations for clear numbers using the generalization principle as follows.
Let A And IN- fuzzy numbers, and - fuzzy operation corresponding to an arbitrary algebraic operation * on ordinary numbers. Then (using here and henceforth the notation instead instead of ) we can write
Fuzzy Numbers (L-R)-Type
Fuzzy numbers (L-R)-type are a type of fuzzy numbers special type, i.e. specified by certain rules in order to reduce the amount of calculations when performing operations on them.
Membership functions of (L-R)-type fuzzy numbers are specified using functions of the real variable L(, non-increasing on the set of non-negative real numbers x) and R( x), satisfying the following properties:
a) L(- x) = L( x), R(- x) = R( x);
b) L(0) = R(0).
Obviously, the class of (L-R)-functions includes functions whose graphs look like those shown in Fig. 1.7.
Rice. 1.7. Possible view(L-R)-functions
Examples of analytical tasks of (L-R) functions can be
Let L( at) and R( at)—(L-R)-type (concrete) functions. Unimodal fuzzy number A With fashion a(i.e. μ A(A) = 1) using L( at) and R( at) is given as follows:
where a is the mode; α > 0, β > 0 - left and right fuzziness coefficients.
Thus, for given L( at) and R( at) fuzzy number (uni-modal) is given by a triple A = (A, α, β ).
The tolerant fuzzy number is specified, respectively, by four parameters A = (a 1 , A 2 , α, β ), Where A 1 and A 2 - limits of tolerance, i.e. in the interim [ a 1 , A 2 ] the value of the membership function is 1.
Examples of graphs of membership functions of (L-R)-type fuzzy numbers are shown in Fig. 1.8.
Rice. 1.8. Examples of graphs of membership functions of fuzzy numbers (L-R)-type
Note that in specific situations the functions L (y), R (y), as well as parameters A, β fuzzy numbers (A, α, β ) And ( a 1 , A 2 , α, β ) must be selected in such a way that the result of the operation (addition, subtraction, division, etc.) is exactly or approximately equal to a fuzzy number with the same L (y) and R (y), and the parameters α" And β" the results did not go beyond the restrictions on these parameters for the original fuzzy numbers, especially if the result will subsequently participate in operations.
Comment. Solving problems of mathematical modeling of complex systems using the apparatus of fuzzy sets requires performing a large volume of operations on various kinds linguistic and other fuzzy variables. For ease of execution of operations, as well as for input-output and data storage, it is advisable to work with standard-type membership functions.
Fuzzy sets, which have to be operated in most problems, are, as a rule, unimodal and normal. One of possible methods approximation of unimodal fuzzy sets is approximation using (L-R)-type functions.
Examples of (L-R)-representations of some linguistic variables are given in Table. 1.2.
Table 1.2. Possible (L- R)-representation of some linguistic variables
