An approach to constructing fuzzy numbers and fuzzy intervals based on fuzzy linguistic statements has been proposed. Mechanisms are proposed that allow for the direct construction of fuzzy numbers of the L-R type from fuzzy linguistic statements. This allows setting model parameters, fuzzy time series, forming databases in fuzzy inference systems, as well as information queries with fuzzy linguistic statements under conditions of significant uncertainty in the information environment. An example of the application of such an approach in modeling random variables of predictive parameters is given. fuzzy numbers and intervals, membership function, fuzzy linguistic statements, soft computing
Lately, fuzzy modelling has become one of the most active and promising areas of applied research
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Arithmetic operations for fuzzy numbers and intervals can be defined using the Zadeh's extension principle. However, carrying out such operations is quite labour-intensive. Therefore, in practice, operations over (L-R)-type fuzzy numbers and intervals, which reduce the volume of computations, have gained widespread acceptance. In addition, (L-R) numbers can be used to define intervals of parameter values, the precise boundaries of which are difficult to specify in conditions of uncertainty.
The issues of (L-R)-approximation have been examined by many researchers [
Despite a wide range of research on various aspects of (L-R)-approximation, there remains a need for the development of powerful, yet simple, ways of approximating the (L-R)-numbers themselves.
When it is necessary to evaluate a certain parameter of the model by a fuzzy magnitude or to construct a fuzzy time series, fuzzy numbers and intervals could be directly given by fuzzy linguistic
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by statements with the quantifiers "approximately/about" [
or "The value is approximately in the range from c to d" (1)
These estimates can approximate (L-R)-type fuzzy numbers. Triangular numbers (c,α,β) are approximated by estimates of the first type, and trapezoidal numbers (c,d,α,β) are approximated by estimates of the second type. Since the values of c and d in these numbers are given by linguistic estimates, therefore, the task of constructing such numbers is only to determine the fuzziness coefficients α and β.
The fuzziness coefficients determine the boundaries of the support of the fuzzy set A, that is, points at which, typically,
̃( ) = 0.01. There are several ways to determine these coefficients. The simplest, and at the same time, the most unreliable is direct expert evaluation. Since in this case the task of psychological measurements is complicated by the fact that a person, as a rule, has an uncertainty about the correctness of the estimates of these values. This difficulty is to some extent eliminated by a group expert evaluation. However, this approach has its known challenges.
Considering this, it is proposed to calculate the fuzziness coefficients based on membership functions. Fuzzy (L-R)-numbers are represented by triangular (trapezoidal) membership functions or Gaussian functions, which are most widely used in solving practical problems. Consider first triangular and trapezoidal functions (Fig. 1) 1 0.5 µ(x) c
a) 0 1 2 1 c d 2 x = −1(0.01), where the line = ( ) passes through points (с, 1) and ( 2, 0.5).
= −1(0.01). Similarly, the coefficient
Now it is necessary to determine the transition points 1 and 2. These points can also be determined by experts. However, this way leads to known problems. Given this, a more constructive approach is one that allows one to calculate transition points based on the distance between them. Let d is the 2 distance between points 1 and 2. Then 1 = − and x2 = c + .
To determine such a distance, an algorithm discussed in [
Let a fuzzy set be defined as "the number is around ", where is a natural number. If ∈ [1,99], . Possible values of are divided into residue classes modulo 3. As a result, three classes , ∈ {0,1,2} are obtained, where = to which the number belongs.
(3). In this case, the value ( ) also depends on the class Number
Let be the digit in the place of the number . Then: 1. if ∈ 0, then ( ) = ( )⋅ 10 −2, where = ⋅ 10, and ( ) is taken from Table 1.
a) if +1 = 0, then ( ) = ( )⋅ 10 −1, where = ; b) if +1 ≠ 0, then ( ) = ( )⋅ 10 −1, where = +1 ⋅ 10 + .
a) if +1 = 0, then = ⋅ 10; ( ) = ( )⋅ 10 −2;; b) if +1 ≠ 0, then = +1 ⋅ 10 + ; ( ) = ( )⋅ 10 −1..
As a result, the value ( ) of will be obtained. This algorithm can also be used in cases where the number is expressed as a decimal fraction. In this case, the algorithm is applied to the mantissa of the fraction, and then its order is considered.
2
2
For trapezoidal functions, the distances (с), ( ) are calculated, and the points 1 and 2 (Fig. 1b) are determined: 1 = − (с), 2 = +
( ). Then, the equations of the lines = 1( )and = 2( ) are constructed, passing respectively through the points ( 1, 0.5),(с, 1), ( , 1), ( 2, 0.5)and similarly, the coefficients , are calculated.
The standard Gaussian function is used to define fuzzy sets ̃ ≜ "the number is approximately equal to
c". The form of the Gaussian function used is as follows [
(− ( − )2),
2( )
where = − 4 0.5, and ( ) is the distance between the transition points. Gaussian function has an
unbounded support, as it asymptotically tends to zero on both the left and right. However, in practice,
the support of this function can be considered to be limited by points, at which its value is approximately
equal to 0.01, which corresponds to the complete non-membership of an element in the fuzzy set ̃.
Therefore, the coefficients and are found from the equation ̃ ( ) = 0.01. These coefficients can
also be calculated approximately as follows: = с − ⋅ (с) and = + ⋅ ( ), where ≈ 2.5 is a
scaling factor [
2 where ̃ ( ) is the membership function of the fuzzy set ̃ ≜ "the number is around c", and ̃( ) is the membership function of the fuzzy set ̃ ≜ "the number is around d". In this case, the fuzziness (2) (3) 0.5 0 c a) fuzzy interval ( , , , ) is obtained.
can be found from equations μB̃(x) = 0.01 and ̃( ) = 0.01. As a result, a
the value of the parameter under consideration [
uncertainty about the accuracy of their estimates under conditions of uncertainty. However, they are subjective. Therefore, when increased requirements are placed on the accuracy of the results, an obvious condition for reducing the degree of subjectivity is the conduct of a group expertise. The results of such an examination are generally considered reliable when the assessments of the experts are is good consistency.
Issues of consistency in group expertise evaluations have been discussed in many studies [
Suppose it is necessary to give a fuzzy linguistic assessment of the form "the parameter value is approximately in the interval from a to b". Also, let's suppose [ 1, 1],…,[ , ] are the interval [а, ], evaluations given by experts. Then, the coefficients of variation for the boundaries a and b are ̄ , where s is the sample standard determined as follows:
for the left boundaries, the formula is where where for the right boundaries, the formula is = ,
̄ = ,
̄ = √
∑ = ( − ) , ̄ = ∑ =1 .
(7) Here, , , ̄ and , , ̄ are the coefficients of variation, sample standard deviations, and mean values for the estimates and ,, respectively. And is the weight coefficient of the j-th expert, where ∑
=1 =1.
The practice of applying expert assessment methods shows that the results of the expertise can be weighted average values of the boundaries of the interval [a, b] are calculated. considered satisfactory if 0,2 ≤ ≤ 0,3, and good if < 0,2. These conditions can be used as a criterion for the consistency of assessments and as a basis for their refinement. At the same time, the aim of the clarification is to reduce the spread of estimates , . After the estimates are refined, the
approach when simulating random variables using the Monte Carlo method.
≜ "project implementationtime". needs to be played out. First, its value range is defined. Since this random variable has a predictive nature, there is significant uncertainty about the boundaries of its values at these stages. Hence, to obtain more reliable values for the boundaries of this area, a group of experts is engaged who provide fuzzy linguistic evaluations for these boundaries (table 2). According to formulas (4)-(7) for intervals [87,123], [90,134], and [93,145], we have: ̄ = 90, ̄ = 134, = 3, =11, =0.03, = 0.08. Therefore, assuming the experts are balanced, the average values of their estimates are taken as the boundaries of the value range of the random variable. As a result, a group of agreed estimate that the "project implementation time is approximately in the interval from 90 to 134" is obtained.
= ( , , , ), where с = 90 and = 134.
To determine the coefficients α and β, we first calculate (90) and (134). The distance b(90) is determined according to table 1 and equals (90) ≈ 19, and the value of (134) is calculated using the above algorithm.
The least significant digit of the number 134 is in the ones place (q=1), therefore = 1 = 7, +1 = 2 = 3 is the digit, the order of which is one higher than the order of the least significant digit of the number 134. When dividing q by 3 we get a remainder of 1, therefore, the number 134 belongs to the equivalence class 1, so = 1.
Since +1 ≠ 0, then according to item 2b of this algorithm we have = +1 ⋅ 10 + = 2 ⋅ 10 + 1 = 34 and (134) = (34), and (34) is calculated by the formula (34) = calculated using both the trapezoidal and the combined Gaussian function. 1 2 in which (35) and (4) are found from table 1: (35) = 6.63 and (4) = 1.84. Then (134) = (6.63 + 1.84) ≈ 4. Knowing the distances (90) and (134), the coefficients α and β can be
First, the trapezoidal function is used. For this function, the transition points 1 and 2 are calculated by the formulas: 1 = 90 − = 80.5 and x2 = 134 + =136. Equations of straight lines = 1( ) and = 2( ), passing through the points (80.5,0.5), (90,1) and (134,1), (136,0.5). are then b(134) 2 constructed. These equations take the form = 19
4 −71 and = 138− , which at μ = 0.01 have roots x ≈ 71 = и ≈ 138 = respectively. As a result, a fuzzy number = (90,134,71,138) is obtained, implying the value range of the random variable lies within the interval [71, 138].
Now, a fuzzy number
= ( , , , ) is constructed using the combined Gaussian function (3), which describes the fuzzy set ̃ ≜ "the number lies approximately in the interval from 90 to 134" and has the form: ̃ ( ) = {90 ≤ ≤ 134,
(x) and µC̃(x) = − "the number is around 90" and ̃ ≜ "the number is around 134". In this case, coefficients α and β are derived from the equations µB̃(x) = 0.01 and ̃( ) = 0.01 and equal ≈ 66, of the random variable ≜ "project realization time" lies within the interval [66, 139]. ≈139. Hence, the range
It's worth noting that the interval of values for the random variable , obtained using the Gaussian function, exceeds the corresponding interval of the trapezoidal function. Therefore, such intervals will assuredly contain the values of forecast parameters, with more distant forecast horizons having increasingly blurred boundaries of their values. Consequently, the use of the Gaussian function in the construction of fuzzy numbers is preferable.
To describe random variables, the values of which are limited by a finite interval, a beta distribution
is primarily used [
The standard beta distribution on the interval ∈ [0,1] is given by the density function: ( ) =
1 ( , )
−1(1 − ) −1, where ( , ) = ∫1 −1 ⋅ (1 − ) −1 0
is Euler's beta function.
Meanwhile, the distribution function is expressed through the incomplete beta function: ( ) = 1
( , ) 0
∫ −1(1 − ) −1 ,
whereas this function is tabulated [
Considering that the function ( ) is tabulated, the simulation of the random variable in the
interval [66, 139] will be carried out using the Neumann exclusion method [
Let the random variable be defined on the interval [ , ] and has a bounded density function ( ) from above. Also, let 1, 2 be independent realizations of the random variable and = + ( − ) 1 , = 2, where =
( ).
≤ ≤
< ( ), then the value x is a realization of the random variable . The efficiency of the exclusion method is directly proportional to the probability of the condition < ( ) being met, i.e.
{ < ( )} = [ ( − )]−1.
This probability allows for the desired number of realizations of the random variable to determine the number of necessary model’s runs. The main advantage of this method is its universality, i.e., its applicability for generating random variables that have any computable or tabularly given probability density.
To model the values of the random variable , beta distribution will be used with parameters = 1 12 2, = 3. In this case, (2,3) =
. Since the quantity is defined on the interval [66, 139], it is necessary to scale the density function (8). In general, for the interval [ , ], this function has the form: (8) (9) 0, ℎ In our case, on the interval [66, 139], the density function has the form:
12 ( ) = {( − )4 ⋅ ( − )( − )2, ≤ ≤ .
12 ( ) = { 734 ⋅ ( − 66)(139 − )2, 66 ≤ ≤ 139
0, ℎ and takes the maximum value of ≈ 0.024.
Suppose that random numbers 1 = 0.4 and 2 = 0.7are obtained by different generators (under the condition of independence). These numbers are scaled respectively into the interval [66, 139] and [0, 0.024]: = 66 + 73 ⋅ 0.4 = 95 and = 0.024 ⋅ 0.7 = 0.016. Then, (95) = 0.024 is calculated. Since the condition is met, the value = 95 is accepted as a realization of the random variable . Otherwise, this value is discarded. In this case, the efficiency of modelling by the exclusion method, according to (9), is directly proportional to the probability of 0.57. This means that to obtain, for example, 1000 realizations of the random variable, approximately 1750 model runs must be carried out.
The considered approach can also be used in constructing, for example, linear S- and Z-shaped membership functions, which look like:
Z-shaped function
In these formulas, the parameters α and β define the bounds of the support of the fuzzy set à and can be determined based on the fuzzy linguistic estimation of the bounds of the corresponding interval. Let's say à ≜ "high pressure in the tank." Then the interval of values "high pressure" a fuzzy linguistic score may be given, such as "high pressure is approximately in the range of 70 to 90".
An approach to the construction of fuzzy numbers and intervals based on fuzzy linguistic statements has been proposed. Depending on the type of statements, triangular or trapezoidal fuzzy numbers of the L-R type are constructed. Such numbers are constructed using triangular, trapezoidal, and Gaussian membership functions. It has been demonstrated that the use of Gaussian function in the construction of fuzzy numbers is preferable, as their intervals will assuredly contain the values of forecast parameters, especially in cases when the forecasting horizon is more distant in time.
When constructing membership functions, it is proposed to use the distances between transition points, which are experimental data reflecting a person's perception of the boundaries of number classes, approximately equal to some number. This allows the construction of fuzzy numbers to be automated and provides the possibility to use this approach when setting the parameters of models, modelling random variables, presenting fuzzy time series in a verbal form, forming databases in fuzzy inference systems, information requests, as well as in many other applied aspects under conditions of significant uncertainty in the information environment.
[1] Fernando Matía, G. Nicolás Marichal, Emilio Jiménez. Fuzzy Modeling and Control: Methods,
[2] Myrto Konstandinidou, Zoe Nivolianitou, Chris Kiranoudis, Nikolaos Markatos.