=Paper=
{{Paper
|id=Vol-3018/Paper_16.pdf
|storemode=property
|title=Modeling of Random Variables on Fuzzy Intervals of their Values
|pdfUrl=https://ceur-ws.org/Vol-3018/Paper_16.pdf
|volume=Vol-3018
|authors=Yuri Samokhvalov
|dblpUrl=https://dblp.org/rec/conf/intsol/Samokhvalov21
}}
==Modeling of Random Variables on Fuzzy Intervals of their Values==
https://ceur-ws.org/Vol-3018/Paper_16.pdf
Modeling of Random Variables on Fuzzy Intervals of Their
Values
Yuri Samokhvalov
Taras Shevchenko National University of Kyiv, Volodymyrs’ka str. 64/13, Kyiv, 01601, Ukraine
Abstract
An approach to modeling random variables on fuzzy intervals of their values is proposed.
The approach includes two stages. At the first stage, a triangular or trapezoidal fuzzy number
is built on the basis of a fuzzy linguistic evaluation of the boundaries of the values of a
random variable, the fuzzy coefficients of which determine the boundaries of the interval of
values of a fuzzy variable. Such numbers are constructed using the Gaussian membership
function. At the second stage, Monte Carlo simulation is carried out using Gaussian
membership functions and beta distribution. In addition, declaring a random parameter by a
fuzzy number makes it possible not only to determine the interval of its possible values when
modeling a random variable, but also to use this parameter in fuzzy arithmetic.
Keywords 1
random variable, fuzzy set, membership function, linguistic assessment, fuzzy interval,
modeling, beta distribution, Monte Carlo method.
1. Introduction
Modern real systems and processes to one degree or another have development in time, therefore,
they are stochastic. This means that the characteristics that describe their functioning are probabilistic
and are random variables. The values of these quantities, as a rule, are in a certain interval, which
sometimes has clearly defined boundaries, and more often - the boundaries are indefinite, vague. For
example, such boundaries are inherent in parameters that are predictive in nature. Moreover, the more
distant in time the forecasting horizon, the less its accurate, i.e. accuracy of estimates of the
boundaries of possible values of such parameters. Therefore, in such conditions, the use of fuzzy
intervals is preferable. Declaring model parameters in the form of a fuzzy interval is a convenient
form for formalizing imprecise values. It is psychologically easy to give a fuzzy interval estimation,
and the carrier of a fuzzy interval is guaranteed to contain the value of the parameter under
consideration. Recently, fuzzy modeling has become one of the most active and promising areas of
applied research in various fields [1-4]. In fuzzy modeling, to represent fuzzy sets, fuzzy values are
most often used, which are the basis for constructing mathematical models using linguistic variables.
Fuzzy Monte Carlo Simulation (FMCS) [5-9] is widely used in stochastic fuzzy models for modeling
random variables. The main point of the FMCS considered in these works is the representation of
parameters and variables only by triangular fuzzy numbers. However, in practice, the intervals of
possible values of a random variable are often known. In this case, such parameters are given by
trapezoidal fuzzy numbers. In article [10], a mechanism for fuzzy modeling of random variables by
the Monte Carlo method based on the Gaussian membership function is proposed. This article is a
development of these studies. It discloses a method for modeling random variables, the value intervals
of of which are given in a fuzzy linguistic form. In this case, both the Gaussian membership function
and the beta distribution are used.
II International Scientific Symposium «Intelligent Solutions» IntSol-2021, September 28–30, 2021, Kyiv-Uzhhorod, Ukraine
EMAIL: yu1953@ukr.net
ORCID: 0000-0001-5123-1288
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
167
2. Representation of fuzzy linguistic evalutions by fuzzy values
As noted, under conditions of uncertainty, it is easier to specify the range of values of a random
variable by a fuzzy linguistic evaluation. Fuzzy linguistic evaluation is understood as a numerical
score, which is expressed using the modalities " approximately / near".
The intervals of possible values of random variables are determined by their physical content. If
the random variable is, for example, the diameter of the machined part, then its deviation from the
specified value under the influence of various factors may be insignificant. That is, the values of this
random variable are near the norm. If we consider the random variable project implementation time,
then in this case there is a large uncertainty and it is most plausible to assert that the possible values of
the project implementation time are approximately in a certain interval. With this in mind, we will use
two types of fuzzy linguistic evaluations " the value is near c " or " the value is approximately in the
range from c to d " and represent them as fuzzy values. From a linguistic point of view, a fuzzy value
is an imprecise, indefinite numerical value of a certain parameter of a model, which is the result of its
evaluation in the absence of complete and accurate information. Fuzzy variables include fuzzy
numbers and fuzzy intervals. To declare such values when solving practical problems, several
approaches can be used [11,13]. In the case of fuzzy modeling, an approach using the standard and
combined (double) Gaussian membership function (MF) [14] (Fig. 1) was applied.
µ(x)
1
b 𝑏1 𝑏2
0.5
0 c c d x
a) б)
Figure 1: Gaussian membership functions: a) - standard; b) - combined (double)
The standard Gaussian function is used to define fuzzy sets 𝐶̃ ≜ "the number is near 𝑐". We will
use the Gaussian function of the form:
𝜇𝐶̃ (𝑥) = exp(−𝑎(𝑥 − 𝑐)2 ), (1)
4𝑙𝑛0.5
where 𝛼 = − 𝑏2(𝑐) , and 𝑏(𝑐) is the distance between the transition points.
The combined function describes fuzzy set 𝐴̃ ≜"the number is approximately in the range from 𝑐
to 𝑑". This function has the form:
𝜇𝐶̃ (𝑥), 𝑥<𝑐
𝜇𝐴̃ (𝑥) = { 1, 𝑐≤𝑥≤𝑑 (2)
𝜇𝐷̃ (𝑥), 𝑥>𝑑
where 𝜇𝐶̃ (𝑥) is the membership function of the fuzzy set 𝐶̃ ≜“the number is near 𝑐”, and 𝜇𝐷̃ (𝑥) is
the membership function of the fuzzy set 𝐷 ̃ ≜ “the number is near 𝑑”. These functions are built in a
similar way.
The Gaussian function has an unbounded support, since it tends to zero asymptotically on the left
and right. However, in practice, the carrier of this function can be considered limited by points 𝑥 =
𝑐 ± 3𝜎, at which its value is approximately equal to 0.01. Therefore, it can be assumed that the value
of the function equal to 0.01 corresponds to the complete non-belonging of the element to the fuzzy
168
𝑘⋅𝑏(𝑐)
set 𝐴̃. If we go from 𝜎 to 𝑏(𝑐) then the boundaries of this interval will be equal to 𝑐 ± , where
2
𝑘 ≈ 2.5 is the scaling factor [10]. These boundaries are the coefficients 𝛼 and 𝛽 of the triangular
fuzzy number 𝑀1 = (𝑐, 𝛼, 𝛽).
The fuzzy interval, described by this function, is constructed in a similar way. In this case, the
𝑘⋅𝑏(𝑐) 𝑘⋅𝑏(𝑑)
indistinctness coefficients will be equal to 𝛼 = 𝑐 − 2 and 𝛽 = 𝑑 + 2 . As a result, we get
trapezoidal fuzzy interval 𝑀2 = (𝑐, 𝑑, 𝛼, 𝛽).
Thus, depending on the type of fuzzy linguistic evaluation of the interval of possible values of a
random variable, its boundaries will be the fuzzy coefficients 𝛼 and 𝛽, accordingly, a fuzzy triangular
number 𝑀1 = (𝑐, 𝛼, 𝛽) or a trapezoidal number 𝑀2 = (𝑐, 𝑑, 𝛼, 𝛽).
3. Harmonization of linguistic evaluations
When determining the range of values of random parameters, the problem of objectivity
(reliability) of their evaluations may arise. This is especially true for parameters that are predictive in
nature. In this case, in order to increase the reliability of the evaluations of such parameters, a group
examination is carried out. The results of the examination are considered to be reliable if there is good
agreement in the evaluations of experts. The issues of harmonization of evaluations of group expertise
were considered in many studies [15-18], among which the article [18] can be highlighted. In this
paper, a mechanism for harmonization interval evaluations is presented. In this case, the coefficient of
variation is used as a measure of the consistency of evaluations. This coefficient is determined
separately for the left and right boundaries of the intervals by the formula 𝑉 = 𝑠/𝑥, where 𝑠 is the
sample standard deviation of the evaluations; 𝑥 - their average value.
Let (𝑐1 , 𝑑1 ),…, (𝑐𝑘 , 𝑑𝑘 ) be evaluations of values c and d of interval linguistic evaluation of some
random parameter, which are given by k experts. Then the coefficients of variation of the boundaries
of the corresponding intervals are determined as follows:
for left borders by the formula
𝑉𝐿 = 𝑠𝐿 /𝑥𝐿 , (3)
where
1
𝑠𝐿 = √𝑘−1 ∑𝑘𝑗=1(𝑐𝑗 − 𝑥𝐿 )2 𝑟𝑗 , 𝑥𝐿 = ∑𝑘𝑗=1 𝑐𝑗 𝑟𝑗 , (4)
for right borders by the formula
𝑉𝑅 = 𝑠𝑅 /𝑥𝑅 , (5)
where
(6)
1
𝑠𝑅 = √𝑘−1 ∑𝑘𝑗=1(𝑑𝑗 − 𝑥𝑅 )2 𝑟𝑗 , 𝑥𝑅 = ∑𝑘𝑗=1 𝑑𝑗 𝑟𝑗 ,
Here 𝑟𝑖𝑗 is the weighting coefficient of the j-ith expert, moreover ∑𝑘𝑗=1 𝑟𝑗 = 1.
The practice of applying the methods of expert evaluations shows that the results of the
examination can be considered satisfactory, if 0,2 ≤ 𝑉 ≤ 0,3, and good, if 𝑉 < 0,2. These conditions
can be used as a criterion for the consistency of estimates and the basis for their specification. This
approach can also be used for fuzzy point evaluation. In this case, the coefficient of variation of the
point value is used.
4. Calculation of the distance between the transition points
When constructing a Gaussian function, the distance between the transition points is determined
mainly by an expert. At the same time, the task of measuring such a distance is complicated by the
fact that a person, as a rule, has a lack of confidence in the accuracy of his evaluation. Therefore, a
more constructive approach is that excludes the conduct of such examinations [14]. This algorithm is
based on experimental data, which, according to experts, reflect the transition points for numbers
169
approximately equal to 𝑇. Based on this data, formulas were obtained to calculate the distance
between the transition points for each number 𝑇 ∈ [1, 99]. The results are shown in Table 1.
Table 1
The distance between the transition points
Number The distance between the transition points
𝑥 𝑏(𝑥)
1, 2, 3, 4, 6, 7, 8, 9 0,46 𝑥
10, 20, 30, 40, 60, 70, 80, 90 (0,357 − 0,00163𝑥 )𝑥
35, 45, 55, 65, 75, 85, 95 (0,213 − 0,00067𝑥)𝑥
5 2,8
15 6,48
25 6,75
50 24
1 𝑥 𝑥
Other two-digit numbers (𝑏 ([ ] ⋅ 10 + 5) + 𝑏(𝑥 − [ ] ⋅ 10))
2 10 10
For the purpose of a holistic perception of the material, we present the main provisions of this
approach. Let a fuzzy linguistic quantity “the number is near the number 𝑇” be given, where 𝑇 is a
natural number. If 𝑇 ∈ [1, 99], then 𝑏(𝑇) could be found according to the Table 1. Otherwise, the
following algorithm is used. Let the least significant digit of T has an order of 𝑞. We divide the
possible values of q into the residue classes modulo 3. As a result, we obtain three classes 𝑀𝑑 , 𝑑 ∈
{0, 1, 2}, where 𝑑 = 𝑞 𝑚𝑜𝑑 3. In this case the value 𝑏(𝑇) also depends on the class 𝑀𝑑 , to which the
number 𝑇 belongs. Let 𝑟𝑞 be the numeral that is in the 𝑞 th place of the number 𝑇. Then:
1. If 𝑇 ∈ 𝑀0 (for example, 300, 300000 etc.), then 𝑏(𝑇) = 𝑏(𝑥) ⋅ 10𝑞−2 , where 𝑥 = 𝑟𝑞 ⋅ 10 and
𝑏(𝑥) is taken from Table 1.
2. If 𝑇 ∈ 𝑀1 (for example, 101, 202000, 15000 etc.), then two options are possible:
a) if 𝑟𝑞+1 = 0, then 𝑏(𝑇) = 𝑏(𝑥) ⋅ 10𝑞−1 , where 𝑥 = 𝑟𝑞 ;
b) if 𝑟𝑞+1 ≠ 0, then 𝑏(𝑇) = 𝑏(𝑥) ⋅ 10𝑞−1 where 𝑥 = 𝑟𝑞+1 ⋅ 10 + 𝑟𝑞 .
3. If 𝑇 ∈ 𝑀2 (for example, 2030, 2140 etc.), then two options are also possible:
a) if 𝑟𝑞+1 = 0, then 𝑥 = 𝑟𝑞 ⋅ 10; 𝑏(𝑇) = 𝑏(𝑥) ⋅ 10𝑞−2 ;
b) if 𝑟𝑞+1 ≠ 0, then 𝑥 = 𝑟𝑞+1 ⋅ 10 + 𝑟𝑞 ; 𝑏(𝑇) = 𝑏(𝑥) ⋅ 10𝑞−1 ;
As a result, the value 𝑏(𝑇) will be obtained.
This algorithm can be used in the case when T is expressed as a decimal fraction. In this case, the
algorithm is applied to the mantissa of the fraction, and then its order is taken into account.
5. Modeling of random variables
Modeling of a random variable using the Monte Carlo method involves drawing a specific value of
the random variable. Consider two ways of drawing: based on the Gaussian membership function (1)
and in accordance with a given distribution law.
Draw based on the Gaussian function. Consider two cases.
Case 1. Let a random variable X be given a fuzzy linguistic estimate “value is near c”. The
membership function 𝜇𝐶̃ (𝑥) of the fuzzy set 𝐶̃ ≜“the number is near 𝑐” and the corresponding fuzzy
number 𝑀 = (𝑐, 𝛼, 𝛽) are constructed, which sets the boundaries 𝛼 and 𝛽 of the interval of possible
values of the X.
Further, the evaluation "value near c" assumes that the random variable X in the interval [𝛼, 𝛽] is
distributed according to the normal law. Therefore, the membership function 𝜇𝐶̃ (𝑥) can be considered
as a density function with mathematical expectation c and variance 𝑏2 (𝑐). And since max 𝜇𝐶̃ (𝑥) = 1,
then this function will be used in the drawing of a random variable X.
170
Let r is a random number. If 𝑟𝜖[0.01,0.5], then the value of X is the root of the equation 𝜇𝐶̃ (𝑥) =
2𝑟 that is in the interval[𝛼, 𝑐]. If 𝑟𝜖(0.5,0.99], then the root of the equation 𝜇𝐶̃ (𝑥) = 2(1 − 𝑟) is taken
that belongs to the interval [𝑐, 𝛽].
Case 2. Let a random variable X be given a fuzzy linguistic estimate “the value is approximately in
the range from c to d”. First, the membership functions 𝜇𝐶̃ (𝑥) and 𝜇𝐷̃ (𝑥) of the fuzzy sets 𝐶̃ ≜“the
number is near 𝑐” and 𝐷 ̃ ≜ “the number is near 𝑑” are constructed, as well as the corresponding fuzzy
numbers 𝑀1 = (𝑐, 𝛼1 , 𝛽1 ) and 𝑀2 = (𝑑, 𝛼2 , 𝛽2 ). As a result, we get the following fuzzy trapezoidal
number 𝑀 = (𝑐, 𝑑, 𝛼1 , 𝛽2 ). In this case, the possible values of X will belong to the interval [𝛼1 , 𝛽2 ].
𝑐+𝑑
Then the value of the middle of the tolerance interval 𝑏 = 2 is calculated and the membership
function 𝜇𝐵̃ (𝑥) is constructed (Fig. 2).
µ(x)
1
𝜎1 𝜎2
0 α1 b β2 x
Figure 2: Membership function 𝜇𝐵̃ (𝑥).
This function is defined by the expression:
𝜇1̃ , 𝑥𝜖[𝛼1 , 𝑏]
𝜇𝐵̃ (𝑥) = { 𝐵2 , (7)
𝜇𝐵̃ , 𝑥𝜖{𝑏, 𝛽2 ]
(𝑥−𝑏)2 (𝑥−𝑏)2
− −
2𝜎2 2𝜎2
where 𝜇1𝐵̃ = 𝑒 1 and 𝜇𝐵2̃ = 𝑒 2 are Gaussian functions. In these functions, the
(𝛼 −𝑏) 2 (𝛽 −𝑏) 2
− 1 2 − 2 2
2𝜎
parameters 𝜎1 and 𝜎2 are found from the equations 𝑒 2𝜎
1 = 0.01 and 𝑒 2 = 0.01.
The function 𝜇𝐵̃ (𝑥) constructed in this way will be used to model the random variable X on an
interval [𝛼1 , 𝛽2 ]. Then, as in case 1, if 𝑟𝜖[0.01,0.5], then the value of X is the root of the equation
𝜇1𝐵̃ (𝑥) = 2𝑟, which is in the interval [𝛼1 , 𝑏], and if 𝑟𝜖(0.5,0.99], then the root of the equation
𝜇2𝐵̃ (𝑥) = 2(1 − 𝑟), which belongs to the interval [𝑏, 𝛽2 ].
The drawing in accordance with distribution law. As noted, the result of a fuzzy linguistic
evaluation of variable X is approximated by the interval of its possible values. To describe random
variables, the values of which are limited to a finite interval, beta distribution is mainly used [19]. The
beta distribution is parameterized by two positive parameters 𝛼 and 𝛽, which determine its shape. Due
to the fact that the beta distribution can have a different shape, practically all the applied probability
distributions can be expressed in terms of this distribution.
The standard beta distribution over the interval 𝑥𝜖[0,1] is given by the density function:
1 (8)
𝑓 (𝑥) = 𝐵(𝛼,𝛽) 𝑥 𝛼−1 (1 − 𝑥)𝛽−1 ,
1
where 𝐵(𝛼, 𝛽) = ∫0 𝑦 𝛼−1 ∗ (1 − 𝑦)𝛽−1 𝑑𝑦 - Euler's beta function.
In this case, the distribution function is expressed through the incomplete beta function:
1 𝑥
(𝑥) = ∫ 𝑦 𝛼−1 ∗ (1 − 𝑦)𝛽−1 𝑑𝑦,
𝐵(𝛼,𝛽) 0
moreover this function is tabulated [20].
171
In practice, of greater interest are, as a rule, beta values determined in an arbitrary interval [𝑎, 𝑏].
Taking into account that the function 𝐹 (𝑥) is tabulated, the drawing of random variables on the
interval [𝑎, 𝑏] will be carried out by the Neumann elimination method [21]. This method is based on
the following theorem.
Let the random variable X be defined on an interval [𝑎, 𝑏] and has an upper bounded density
function f(x). Let also 𝑟1 , 𝑟2 𝜖[0,1] be independent implementations of the base random variable ξ, also
𝑥 = 𝑎 + (𝑏 − 𝑎)𝑟1 and 𝑦 = 𝑀𝑟2
where 𝑀 = max 𝑓(𝑥).
𝑎≤𝑥≤𝑏
Then if 𝑦 < 𝑓(𝑥), then the value x is the realization of the random variable X. At the same time,
the effectiveness of the elimination method is directly proportional to the probability of fulfilling the
condition 𝑦 < 𝑓(𝑥), i.e.
𝑃{𝑦 < 𝑓 (𝑥)} = [𝑀(𝑏 − 𝑎)]−1 . (9)
This probability allows for the desired number of realizations of a random variable X to determine
the number of necessary model runs. The main advantage of this method is its versatility, i.e.
applicability for generating random variables having any computable or tabular probability density.To
better understand such calculations, consider an example. Let us draw a random value 𝑋 ≜“project
implementation time”. First, the range of its values is determined. Since the random variable has a
predictive nature, therefore, in order to obtain more reliable values of the boundaries of the region, a
group of, say, three equivalent experts is involved, whose evaluations are given in Table 2.
Table 2
Evaluations of intervals in days
The project implementation time is approximately in the
interval
expert 1 from 87 to 123
expert 2 from 90 to 134
expert 3 from 93 to 145
Using formulas (3) - (6), for the boundaries of the given intervals, we obtain the coefficients of
variation, respectively, 0.03 and 0.08. Since the experts' assessments are reasonably well agreed, there
is no need to refine them. Therefore, the average values of 90 and 134 are taken as the boundaries of
the range of values of the random variable X. As a result, we obtain the collective estimate "the time
of the project implementation is approximately in the range from 90 to 134".
Then, the membership functions μC̃ (x) and μD̃ (x) of both fuzzy sets 𝐶̃ ≜“the number is near the
number 90” and 𝐷 ̃ ≜ “the number is near the number 134” are constructed. To construct the function
μC̃ (x), it is necessary to calculate the distance 𝑏(90). This value is found according to Table 1 and is
4𝑙𝑛0.5((𝑥−90)2
−
equal to 𝑏(90) ≈ 32. As a result, we get a function 𝜇C̃ (𝑥) = 𝑒 322 and a fuzzy
number 𝑀1 = (90,50,130). The value 𝑏 (134) for the function μD̃ (x) is calculated by the above
algorithm. The least significant digit of number 134 is in the ones place (q = 1), therefore 𝑟𝑞 = 𝑟1 = 7,
𝑟𝑞+1 = 𝑟2 = 3 is a digit whose order is one higher than the order of the least significant digit of
number 134. When dividing q by 3 in the remainder, we get 1, therefore, the number 134 belongs to
the equivalence class M1, so d = 1.
Since 𝑟𝑞+1 ≠ 0, then, according to clause 2b of this algorhythm, we have 𝑥 = 𝑟𝑞+1 ∗ 10 + 𝑟𝑞 =
𝑟2 ∗ 10 + 𝑟1 = 34 and 𝑏(134) = 𝑏(34), and 𝑏(34) is calculated by the formula
1 34 34 1
𝑏(37) = 2 (𝑏 ([10] ∗ 10 + 5) + 𝑏 (34 − [10] ∗ 10)) = 2 (𝑏(35) + 𝑏(4)),
172
in which 𝑏(35) and 𝑏(35) are found in Table 1: 𝑏(35) = 6.63 and 𝑏 (4) = 1.84. Then 𝑏(134) =
4𝑙𝑛0.5((𝑥−134)2
1 −
2
(6.63 + 1.84) ≈ 4. As a result, we get 𝜇D̃ (𝑥 ) = 𝑒 42 the corresponding fuzzy
number 𝑀2 = (134,129,139). Then the interval of possible values of the random variable X will be
the interval [50,139]. We will model the values of a random variable X using two methods: based on
the Gaussian function and beta distribution. According to the first method, we determine the value
90+134
𝑏 = 2 = 112 and build the membership function (7):
𝜇1̃ , 𝑥𝜖[50,112]
𝜇𝐵̃ (𝑥) = { 2𝐵 ,
𝜇𝐵̃ , 𝑥𝜖{112,139]
(𝑥−112)2 (𝑥−112)2
− −
Where 𝜇1𝐵̃ = 𝑒 2∗212 and 𝜇𝐵2̃ = 𝑒 2∗92 . Then, in the process of modeling,
if 𝑟𝜖[0.01,0.5], then the value of X is the root of the equation 𝜇1𝐵̃ (𝑥) = 2𝑟, which is in the interval
[50,112], and if 𝑟𝜖(0.5,0.99], then the root of the equation 𝜇2𝐵̃ (𝑥) = 2(1 − 𝑟), which belongs to
the interval [112,139]. For example, if 𝑟 = 0.4, then the value x is equal to 98, if 𝑟 = 0.7, then it is
equal to 118. Let us now consider modeling a random variable X using the beta distribution, for
1
example, with parameters 𝛼 = 2, 𝛽 = 3. In this case 𝐵(2,3) = . Since the value X is determined on
12
interval [50, 139], the density function (8) must be scaled. In the general case, for the interval [a,b]
this function has the form
12
∗ (𝑡 − 𝑎)(𝑏 − 𝑡)2 , 𝑖𝑓 𝑎 ≤ 𝑡 ≤ 𝑏
𝑓 (𝑡) = {(𝑏−𝑎)4 .
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
For the interval [50, 139] density function has the form
12
( )( )2
𝑓(𝑡) = {894 ∗ 𝑡 − 50 139 − 𝑡 , 𝑖𝑓 50 ≤ 𝑡 ≤ 139
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
and takes maximum value 𝑀 ≈ 0.02.
Let the random numbers 𝑟1𝑖 = 0.4 and 𝑟2𝑖 = 0.7 be obtained by different generators
(independence condition). These numbers are scaled to the interval [50, 139] and [0, 0.02]: 𝑥𝑖 = 50 +
89 ∗ 0.4 = 85.6 and 𝑦𝑖 = 1.78 ∗ 0.7 = 0.014. Then 𝑓(𝑥𝑖 ) = 𝑓(85.6) ≈ 0.019 is calculated. Since
the condition 𝑦𝑖 ≤ 𝑓(𝑥𝑖 ) is met, the value 𝑥𝑖 = 85.6 is taken as a realization of the random variable X.
Otherwise, this value is discarded. In this case, the efficiency of modeling by the elimination method
on the interval [50, 139], according to (9), is directly proportional to the probability 0.56. That is, to
obtain, for example, 1000 realizations of a random variable, it is necessary to carry out approximately
1800 runs of the model.
6. Conclusion
An approach to modeling random variables on fuzzy intervals of their values is proposed. This
approach includes two stages. At the first stage, on the basis of a fuzzy linguistic evaluation of a
random parameter, a fuzzy number is constructed, which declares a fuzzy interval of its possible
values. Fuzzy linguistic evaluations can be either point or interval. Depending on the type of
evaluation, a triangular or trapezoidal fuzzy number is constructed, the fuzzy coefficients of which
determine the range boundaries of values of the fuzzy variable. Such numbers are constructed using
the Gaussian membership function. At the second stage, a random variable is modeled on the
constructed interval of its values. The drawing is performed by the Monte Carlo method using
Gaussian membership functions and beta distribution. In this case, the drawing of a random variable
by the beta distribution function is carried out by the Neumann method. Note that the representation
of a random parameter as a fuzzy number allows not only to determine the interval of its possible
values, but also to use this parameter in calculations in the process of fuzzy modeling.
173
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