=Paper=
{{Paper
|id=Vol-3018/Paper_5
|storemode=property
|title=On Fuzzy Similarity Relations for Heterogeneous Fuzzy Sets
|pdfUrl=https://ceur-ws.org/Vol-3018/Paper_5.pdf
|volume=Vol-3018
|authors=Leonid Hulianytskyi,Iryna Riasna
|dblpUrl=https://dblp.org/rec/conf/intsol/HulianytskyiR21
}}
==On Fuzzy Similarity Relations for Heterogeneous Fuzzy Sets==
https://ceur-ws.org/Vol-3018/Paper_5.pdf
On Fuzzy Similarity Relations for Heterogeneous Fuzzy Sets
Leonid Hulianytskyi and Iryna Riasna
V.M. Glushkov Institute of Cybernetics of National Academy of Sciences of Ukraine, Academician Glushkov
Avenue, 40, Kyiv, 03187, Ukraine
Abstract
The fundamental problem of development of formal models and methods of ill-structured
problems of combinatorial optimization under uncertainty requires utilizing of fuzzy concepts
of informal models and different types of scales for measurement of quality and quantity
characteristics. In the paper, we introduce a concept of a fuzzy similarity scale. An
inadequacy of traditional building a fuzzy similarity scale based on operators of fuzzy logic
is shown. A concept of a linguistic correlation coefficient is offered, and conditions of its
adequacy in scales of measurement of empirical objects properties on Stevens' classification
are derived. For determining fuzzy similarity relations on heterogeneous fuzzy sets, we use
the concept of the linguistic correlation coefficient.
Keywords 11
Fuzzy relation, similarity, fuzzy similarity measure, fuzzy set, fuzzy logical operator, fuzzy
measurement scale, empirical system with relations, mathematical system with relations.
1. Introduction
Issues of formalization of ill-structured problems under quantitative and qualitative information
arise in various areas of human activity. Often, in problems under uncertainty we obtain the
information about data from verbal expert assessments, conclusions, judgments, and generalizations.
Developing adequate mathematical models and methods for solving this type of problems and
approaches providing a meaningful interpretation of their modeling results are relevant; see [1]. The
examples are classification and clustering, other ones reducible to combinatorial optimization
problems. To formalize such ill-structured problems, methods of representative measurement theory
and fuzzy sets theory are normally used. The paper considers two ways of building a fuzzy similarity
measure on homogeneous and heterogeneous fuzzy sets under uncertainty in measurements of
membership functions values caused by not statistical in nature and subjective expert assessments.
In clustering and classification, much attention is devoted to the identification of fuzzy similarities
and differences, as well as to metrics of fuzzy sets; see [2, 3, 4, 5, 6, 7, 8, 9]. In papers [10] and [11],
fuzzy logic is used in methodologies where objects under consideration as words and sentences of
natural languages – computing with words (CWW). They provide an overview of approaches to
constructing similarity measures for interval fuzzy sets of type-1 and type-2. Non-similarity measures
on graphs are proposed in [12]. These questions are discussed in [13] for fuzzy intuitionistic sets.
Fuzzy nominal scales are applied in [14] and [15] for generating signals on fuzzy subsets of
natural language words. The application of cluster analysis in intelligent systems is considered in [16],
thoroughly outlining the application of crisp and fuzzy algorithms. In [9], an overview of approaches
to formalization of fuzzy cluster analysis problems is given. There the comprehensive list of methods
and algorithms for their solution is also given.
Many scientific studies are devoted to formalizing and solving combinatorial optimization
problems on fuzzy sets; see [17].
II International Scientific Symposium «Intelligent Solutions» IntSol-2021, September 28–30, 2021, Kyiv-Uzhhorod, Ukraine
EMAIL: leonhul.icyb@gmail.com (L. Hulianytskyi); riasnaia@gmail.com (I. Riasna)
ORCID: 0000-0002-1379-4132 (L. Hulianytskyi); 0000-0003-1370-3066 (I. Riasna)
©️ 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)
48
The approach to formalization of Euclidean combinatorial configurations dealing with finite sets
embedded in Euclid space is developed in [18].
2. Way 1 to define a fuzzy similarity measure
Consider the case where a fuzzy scale defines a fuzzy similarity relation or measure. The scale will
further referred to as a fuzzy similarity scale. The problem of building a scale is important, for
example, in fuzzy clustering under uncertainty due to a verbal description of properties of empirical
objects.
2.1. Basic concepts of representative measurement theory
The subject of measurements is empirical objects properties (features, characteristics) and relations
between them. The basic concepts of representative measurement theory are a system with relations
and a measurement scale, however, fuzzy sets are not involved in the main works of representative
measurement theory [19], [20], and [21].
The study [21] presents three main problems of the theory, i.e., presenting the results of measuring
properties or relations in numerical form, the problem of uniqueness of measurement results, and the
one of adequacy (meaningfulness [19], [22]).
A subset of the Cartesian product A A is the binary relation on a set A, and a subset of the
Cartesian product or degree of the set Ak is the k -ary relation.
Definition 1 A tuple M A; R1,..., Rn is called a system with relations, where A is a non-
empty set called a domain (support) of a system with relations, and R1,..., Rn are the relations given
on A .
Let Ri is a ki - ary relation given on A , і 1,..., n.
n
Definition 2 An ordered sequence of positive integers ki i 1 we call the type of a system M
with relations.
Specifying the system type singles out significant relations in the domain of defining a given
system and is determining by the structure of data or its theoretical-set model. We call two systems
with relations are similar if they possesses of the same type.
A k -ary relation R is characterized by set of ordered collections a1,..., ak Ak such that
R a1,..., ak Ak | R a1,..., ak 1 .
If a set A consists of empirical objects and relations on A empirically defined, then we call the
system an empirical system with relations (ESR). It is not important how these empirical relations are
determined in practice. They can be defined by physical objects (for example, consecutive connection
of measuring apparatuses), or by human answers stated verbally or derived from qualitative and
quantitative expert conclusions.
If A R1 , where R1 is the set of real numbers, then the system M is called a numerical system
with relations (NSR) [20]. If A is a set of non-numeric mathematical objects (for example, symbols,
vectors, functions), then M is called a mathematical system with relations (MSR) [22].
Let two similar systems are given:
j is a set of ESR objects (elements), S1,..., Sn are
an ESR M 1 E; S1,..., Sn , where E e j
relations given on E; and M 2 A; R1,..., Rn is a NSR (or MSR), where A a j is a set of
j
NSR elements (or MSR), R1,..., Rn are relations given on A .
49
Definition 3 A mapping f : E A is called a scale of measurements (in short, a scale) if f is
homomorphism, that is,
Si e1,..., e j ,..., eki Ri a1,..., a j ,..., aki ,
where e1,..., e j ,..., ek E , a1,..., a j ,..., ak A, a j f e j are the scale values, and
i i
i 1,..., n .
If the mapping f is bijection, then f is isomorphism.
Let us introduce the definition of a "fuzzy scale" of measurements [23]. Let U be a non-empty set
of empirical objects, Si i 1,..., n is a set of relations on U , L is a subset of real numbers, T is a
set of fuzzy subsets on L , Ri ( i 1,..., n ) are ki -ary relations.
Definition 4 A mapping :U T is called a fuzzy scale, if i 1,..., n and
u1,..., uki U ki ,
Si u1 ,..., uki Ri u1 ,..., uki ,
are the corresponding scale values from T .
where u1 ,..., uk
i
The questions about permissible transformations, adequate and invariant functions and other
relevant issues are not investigated yet [23] for fuzzy scales.
2.2. Using fuzzy operators for building a fuzzy similarity scale
If there are several properties of empirical objects, a formal ESR model, i.e., an MSR, for which
measurements of properties of empirical objects represented by a membership function of a fuzzy set,
is determined, using concept of "fuzzy measurement scale". The scale we define as a homomorphic
mapping of an ESR onto MSR if relations in the MSR we define on a set of fuzzy subsets. Defining
the scale in representative measurement starts with building an ESR model.
Let X be a finite set of objects (elements) of the empirical system and W be a finite set of fuzzy
properties of elements defined verbally. The result of measurements of values of membership
functions for a set of properties W w1, ..., wn , where n is the number of properties, can be a
homogeneous or heterogeneous fuzzy set. Membership function is defined as follows:
W : X L1 ... Ln , and Li is some lattice, i.e., W x 1 x ,..., n x , i : X Li , and
i 1,..., n. Based on heterogeneous fuzzy sets, it is possible to build models under different types
of properties, which are measured by both quantitative and qualitative scales.
Let the property (for convenience, an index i will be omitted below) takes a finite set
of verbal values (features, gradations, linguistic terms), where m w is the
Tw t1,..., tm w
number of these values. If values t ,..., t of property w W are measured on a certain scale,
1 m w
the result of the property measurement is a homogeneous fuzzy subset of Tw . If, in verbal
measurements, different scale types or scales of same type are applied, while the valid transformations
are independent, the measurements results t1,..., tm w of membership function for any wW is a
heterogeneous fuzzy set. Let for any w W be defined R : X Tw 0,1 – a fuzzy binary relation
on X Tw , then R ( x, t ) be a membership function defining this fuzzy relation and
R ( x, t ) 0,1 , x X , t Tw .
50
Denote by FT a set of all fuzzy subsets of Tw and let : X FT be a mapping such that for it
w w
t j Tw x X x (t j ) R ( x, t j ) is met. Here and beyond, in similar cases, j 1,..., m w .
Let for any x X the set Dx t , t t T be a fuzzy subset from T , that we
j x j j w w
call a measurement description of w W . The meaning of x (t j ) will be considered as degree of
truth of statement "x has the meaning t j of a fuzzy property w ". Let х Х D x be a
homogeneous fuzzy set.
Let D = Dx x X be a set of fuzzy measurement descriptions. We consider D as a
vector-valued fuzzy set [24] or as a type-2 fuzzy set:
D x, x x X x, x t1 ,..., x tm w x X ,
where x t j 0,1 , j 1,..., m w , and m w is the number of property values.
For any x X , measurements of a certain property w W are not a real numbers, but are a fuzzy
subset of set of values from the corresponding Tw , to which the studying of problems of
representation, uniqueness and adequacy is carried out taking into account the structure of fuzzy
measurement scale. However, in this case, we change known definition of fuzzy scale: we define
fuzzy subsets on a set of verbal values Tw t1,..., tm w of a certain property w, and not on a
subset of real numbers, as is certain in [23].
A fuzzy similarity measure on X X is a function ( x, y ), satisfying following conditions:
1) ( x, y ) X X ( x, y) 0,1;
2) x X ( x, x ) 1 (reflexivity);
3) ( x, y ) X X ( x, y ) ( y, x) (symmetry).
A function ( x, y ) defines a fuzzy set in sense of L. Zadeh. A fuzzy similarity measure specifies
on the finite set X a fuzzy similarity relation R : X X 0,1 .
A fuzzy similarity measure ( x, y ) should be determined taking into account the results of
measuring for all values of fuzzy property w W for any x X , that is, by aggregating fuzzy
logical operators.
A method of calculating ( x, y ) values is determined both by type of scale of measuring values of
w W for objects of an empirical system, and by theoretical-set interpretation of semantic link AND
and OR in form of fuzzy logical operators.
A fuzzy analogue of semantic link AND is triangular norm and fuzzy analogue of OR is an
operator of triangular conorm; see [24]. In fuzzy sets theory, there are many variants for choosing of
norm and conorm operators.
We use fuzzy operators of Zadeh norms and conorms:
T (a, b) min(a, b) , S a, b max a, b , a, b 0,1 ,
and fuzzy operators of norms and conorms (Ia. Lukasiewicz)
T (a, b) max(0, a b 1) , S a, b min 1, a b , a, b0,1 .
Let (, ) be a fuzzy similarity measure defined on X X and *(, ) be a fuzzy similarity
measure defined on FT FT . The Homomorphic mapping : X FT for w W determines a
w w w
fuzzy measurement scale. However, the existence of a scale, i.e., a homomorphism of mapping we
must prove [20].
51
Denote by FX a set of all subsets of Х. Let : Tw FX be the mapping such that
x X t j Tw t j ( x) R ( x, t j ) and let E j x, x x X be a fuzzy subset of Х,
tj
called a meaning of t j Tw .
Obviously, a condition for similarity (( x, y ) 0) of x, y X is an existence of at least one
common property, i.e.:
x, y 0 t j Tw : x Supp E j y Supp E j . (1)
Since t ( x) x (t j ), then it is from (1) follows that intersection of supports, i.e.,
j
Supp Dx I Supp Dy is also a condition of similarity of empirical objects. If
Supp Dx I Supp Dy case is valid, then ( x, y ) 0 we obtain. Thus,
x, y 0 t j Tw : x t j 0 y t j 0 . (2)
Considering (2), we define ( Dx , Dy ) as follows:
Dx , Dy x t1 0 y t1 0 ... x tm 0 y tm 0 . (3)
Replacing in (3) inequalities with corresponding values of membership function, and logical
operators with triangular norms T and conorms S, we get:
t T T x t j , y t j , Dx , Dy D .
Dx , Dy S
j w
(4)
Obviously, ( Dx , Dy ) 0,1. Symmetry ( Dx , Dy ) follows from the symmetry of operators of
norm and conorm operators.
If (4) satisfies reflexivity condition, i.e., if the condition, ( Dx , Dy ) 1 is met, then we define a
fuzzy similarity measure on FT FT as
w w
* Dx , Dy Dx , Dy .
Therefore,
t T T x t j , y t j , Dx , Dy D .
* Dx , Dy S
j w
(5)
Such a method of defining a fuzzy similarity measure, which is an analogue of method of
calculating a similarity measure for crisp properties based on direct replacement of logical operators
with their fuzzy analogues, we call as "direct" method.
We suppose, that x, y * Dx , Dy is met. If reflexivity condition * Dx , Dx =1 is met,
then formula (5) defines a fuzzy similarity measure.
Obviously, mapping : X D gives rise to an equivalence relation , i.e.:
xy Dx Dy ,
that, under the reflexivity of * Dx , Dy is a congruence relation:
xx yy x, y x, y .
Thus, we define two systems with relations: an ESR, M 1 X , , and a MSR,
M 2 D , , * . A tuple M 1, M 2 , we call a fuzzy scale for measuring a partial fuzzy
similarity measure (further – a fuzzy similarity scale).
52
2.3. Representation theorem
We formulate a representation theorem determining conditions for existence of measurement scale
of a fuzzy similarity measure (5).
When proving representation theorem, we believe that an absolute measurement scale we use,
while only identical transformations [20] are permissible for it.
Theorem 1 A reflexivity condition for fuzzy similarity measure (5) is in generally not satisfied.
Proof. If we use in (5) Zadeh norm and conorm operators, we get
t T
* Dx , Dy max min x t j , y t j
j w
, Dx , Dy D .
Because of idempotency of min operator, we get
t T .
* Dx , Dx max x t j
j w
In turn,
t T 1
* Dx , Dx max x t j
j w
will be the reflexivity condition of fuzzy similarity measure.
That is, we can write
x X t j : x (t j ) 1 . (6)
Therefore, measurement results must consist of normal [24] fuzzy sets. In general, this condition
may not meet.
The Lukasevich norm operator cannot used in (5), since in case where the sum of values of
membership functions is less than one, the value of this norm is zero, then
( Dx , Dx ) 0,
and the reflexivity condition is not fulfilled. If in (5) we use Zadeh norm and Lukasevich conorm,
then we get
( Dx , Dx ) min 1,
t jTw
x (t j ) .
In this case, the reflexivity condition is as follows:
x (t j ) 1 x X , (7)
t j Tw
which in practice would not always be fulfilled.
Theorem proved.
2.4. Investigating of adequacy of fuzzy similarity measure
Let us study an adequacy of fuzzy similarity measure.
Theorem 2 Fuzzy similarity measure (5) is not adequate while measuring values of membership
function of a fuzzy property w W in ratio, interval, and order scales.
Proof. Truthiness of theorem falls out of absence of invariance of conditions (6) and (7) for
measurements in ratio and interval scales.
Condition (7) we cannot use in order scale because sum operation is invalid. It is obvious that
condition (6) violates in monotonic transformations permissible for order scale.
Theorem proved.
Thus, fuzzy similarity measure in "direct" method of determining (5), used, for example, in [25], is
generally unsuitable for constructing an adequate formal model, while component of it is a fuzzy
similarity relation.
53
3. Way 2 to define a fuzzy similarity measure
Let X be a finite set of empirical system objects (elements), W w1,..., wn is a finite set of
fuzzy properties of X defined verbally. Let property wi W takes a finite set of verbal values
Twi t1i ,..., tmi w , where m( wi ) is the number of values of wi W .
i
Definition 5 A linguistic correlation coefficient (LCC, Klingv ) and a partial linguistic correlation
coefficient ( ki ) we call, respectively,
n
1
K lingv ( x, y ) ki ( x, y ), (8)
n i 1
m( wi ) m ( wi )
ki x, y *i Dxi , Diy
j 1
min x (t ij ), y (t ij )
max x (t ij ), y (t ij ) ,
(9)
j 1
where *i Dxi , Diy is a partial similarity measure in MSR of w W , and D , Diу are i
i
x
measurements of wi W for elements x, y X , respectively, n is the total number of properties;
x (t ij ) determines a measure of belonging of value t ij to wi of x Х,
j 1, ..., m(wi ), i 1,..., n.
Therefore, a partial LCC determines a value of a partial similarity measure on a set of empirical
objects. Obviously, when measuring values of membership functions in an absolute scale, we get,
ki x, x 1, ki x, y ki y, x ; Klingv ( x, x) 1, Klingv ( x, y) Klingv ( y, x) ;
ki x, y 0,1 , Klingv ( x, y) 0,1 .
That is, according to formulas (8) and (9), a fuzzy similarity measure: x, y Klingv ( x, y) on
X X is determined. Unlike "direct" method of determining of similarity measure (5) in the case of
measurements of values of membership functions on an absolute scale, in order to ensure reflexivity
condition of fuzzy similarity measure with help of LCC, no restrictions on value of membership
functions of fuzzy objects properties are necessary. Since min and max operators are valid for order,
interval, ratio, and absolute scale [23], such theorems are valid.
Theorem 3 Partial fuzzy similarity measure of (9) is invariant when measuring values of
membership function of a fuzzy qualitative property wi W in ratio scales.
Proof. A valid transformation of membership function values in ratio scale is a similarity
transformation, i.e., y х , where 0 1 . It is not hard to see that
m( wi ) m ( wi )
ki x, y
*i Dxi , Diy
j 1
min x (t ij ), y (t ij ) max x (t ij ), y (t ij )
j 1
m( wi ) m ( wi )
j 1
min x (t ij ), y (t ij )
max x (t ij ), y (t ij ) .
j 1
Theorem proved.
Theorem 4 When measuring values of membership functions of a fuzzy qualitative property in an
order scale and an interval one, there exist a permissible monotonic transformation x (t ij ) ,
54
t ij Twi , leading to invariance of values of membership function of a partial fuzzy similarity measure
(9), i 1,..., n .
Proof. For a set of empirical objects X x1,..., xN , result t ij of measurements of term values
1 k N
of property wi we represent as a sequence A x (t ij ),..., x (t ij ),..., x (t ij ) . For simplicity, we
denote by аk x (t ij ) and аk 0,1. If we order the elements of sequence in ascending order,
k
1
then the collection A аk ... аk ... аk
r N is a ranked sequence, where the number
1 r N we call the rank of аk А . If there are no identical elements in sequence A, then
akr akr 1 ; if there are equal elements in sequence A ( аk p аk p 1 ... аk p m ), then rank value in
the interval p, p m is ( p ... ( p m)) / ( m 1) (fractional ranks). In both cases, the sum of
ranks is equal to N N 1 / 2. Denote by r ak , r ak 1, N , rank value of ak . In order scale,
it is allowing a monotone transformation of that does not change ratio relation, i.e.,
аk аm аk аm , аk аm аk аm . Then obviously, in such
transformations, rank values of r ak in the sequence A do not change. We define transformation
ak r ak N ; because r ak 1, N , then 0 ak 1. Value ak does not change
under any valid monotone transformation. Therefore, membership function value of partial fuzzy
similarity measure
m( wi )
m ( wi )
ki x, y
j 1
min x (t j ) , y (t j )
i i
max x (t ij ) , y (t ij )
j 1
is invariant under transformation. This transformation is also valid in interval scales.
Theorem proved.
Invariance of partial similarity measures leads to invariance of LCC, using which as a similarity
measure provides an adequacy of formal model of ESR as example in fuzzy clustering.
4. Examples of LCC Calculation
Example 1 Let us give a method of calculating of LCC for the case where elements of a MSR are
fuzzy combinatorial configurations (objects) of 1st order of the first type [26]
1(1) Y , , X (1) , ,
where a base set X (1) coincides with fuzzy generating set Z z1,..., zn , X (1) Z , :Y X 1 ,
and Y {1,..., p}. Let be p n, and :Y X 1 – some crisp mapping. Then ordered fuzzy set
A (1),..., (l ),..., ( p) a1,..., al ,..., a p is an arrangement with the repetitions of fuzzy
elements of Z where (l ) al and al zi Z.
Let A a1,..., a p and B b1,..., bp be two arrangements. A LCC ( Klingv ) and a partial
LCC ( ki ) by the property wi W we calculate as follows:
p
n
1 1
K lingv ( A, B) ki ( A, B), ki ( A, B) *i Dai l , Dbil ,
n i 1 p l 1
55
m ( wi ) m ( wi )
where
*i Dai l , Dbil
min al (t ij ), bl (t ij ) max a (t ij ), b (t ij )
l l
is partial
j 1 j 1
similarity measure of al , bl for wi W ; Twi t1i ,..., tm
i
w i is a set of verbal values of w W , i
and Dai , Dbi are measurements of wi W for elements al А , bl B , respectively, n is the total
l l
number of properties, and a (t ij ), b (t ij ) determine the measure of membership value t ij of wi
l l
for elements al A, bl B, respectively, a (t ij ) 0,1 , b (t ij ) 0,1 , j 1, ..., m(wi ) ,
l l
and i 1,..., n. Obviously, when measuring values a (t ij ) and b (t ij ) of membership functions
l l
in an absolute scale then ki ( A, A) 1, ki ( A, B) ki ( B, A); Klingv ( A, A) 1,
Klingv ( A, B) Klingv ( B, A); and ki ( A, B) 0,1 , Klingv ( A, B) 0,1 , that is Klingv ( A, B)
determines a fuzzy similarity in an ESR. Proof of invariance of a Klingv ( A, B) under permissible
transformations in applied measurement scales is the same with proof of theorems 3 and 4.
Example 2 Let the manager of the enterprise personnel department be face with the problem of
selecting several of the best representatives from the candidates for vacant positions of engineers of
the department producing equipment, for example, for a new model of the aircraft.
To determine the best applicants, it is necessary to evaluate their competitiveness according to the
totality of two fuzzy qualitative characteristics, i.e., w1 , w2 , where w1 – the work experience, w2 –
the level of qualification. Characteristics w1 , w2 are fuzzy multidimensional expert assessments
measured in order scales.
We believe that the description of the work experience has two gradations, i.e., w1 t11, t12 ,
where t11 – the work experience according to the profile of the enterprise, t12 – the work experience
on a computer with programmable logic matrices. We assume that the qualification level has three
gradations, i.e., w2 t12 , t22 , t32 , where by t12 denote knowledge of foreign languages, by t22 denote
knowledge of a specify set of programming languages, and by t32 denote duration of work in leading
companies in the industry.
Let m ( m 2 ) be the number of vacancies, X xi i 1 be the set of applicants, and p ( p 4 )
p
be the number of applicants.
Let the manager give the following gradations of t11 , t12 for characteristic w1 for applicants:
x1 0,6;0,9 , x2 0,5;0, 4 , x3 0,3;0,7 , x4 0, 4;0,5 . The ideal applicant has the
maximum value of gradations of characteristics, i.e., xid 1;1 . The variation series of manager
assessments of gradation of t11 , t12 for the characteristic w1 are respectively:
t11 x1 , t11 x2 , t11 x3 , t11 x4 , t11 xid 0,6;0,5;0,3;0, 4;1 ,
t12 x1 , t12 x2 , t12 x3 , t12 x4 , t12 xid 0,9;0, 4;0,7;0,5;1 .
1
Obviously, the corresponding rank vector of the assessments of t1 (ordered in ascending order) for
the sequence x1, x2 , x3, x4 , xid has the form: r11 x1 , r11 x2 , r11 x3 , r11 x4 , r11 xid
56
4;3;1;2;5 . Similarly, the rank vector of the assessments of t12 is
r21 x1 , r21 x2 , r21 x3 , r21 x4 , r21 xid 4;1;3;2;5.
Since rank values are invariant to permissible (monotonic) transformations in the order scale, the
value of the membership functions of fuzzy characteristics w1 and w2 is calculated by dividing the
rank assessments by a maximum rank value of 5.
Thus, we obtain the values of the membership functions to the assessments t11 and t12 for the
ordered totality x1 , x2 , x3 , x4 , xid , respectively,
x (t11), x (t11), x (t11), x (t11), x (t11) 0,8;0,6;0, 2;0, 4;1 ,
1 2 3 4 id
x (t12 ), x (t12 ), x (t12 ), x (t12 ), x (t12 ) 0,8;0, 2;0,6;0, 4;1 .
1 2 3 4 id
Let the manager give the following gradations of t12 , t22 , t32 for w2 characteristic for applicants:
x1 0,8;0,7;0,3 , x2 0,3;0,6;0,8 , x3 0,5;0,9;0, 2 , and x4 0,6;0,8;0,6 .
The ideal applicant has the maximum value of gradations of w2 characteristics, i.e., xid 1;1;1 .
The variation series of gradation manager assessments of t12 , t22 , t32 for w2 have the form:
t12 x1 , t12 x2 , t12 x3 , t12 x4 , t12 xid 0,8;0,3;0,5;0,6;1 ,
t22 x1 , t22 x2 , t22 x3 , t22 x4 , t22 xid 0,7;0,6;0,9;0,8;1 ,
t32 x1 , t32 x2 , t32 x3 , t32 x4 , t32 xid 0,3;0,8;0, 2;0,6;1 .
It is not difficult to show that the corresponding rank vector of assessments of t12 for the sequence
x1, x2 , x3, x4 , xid is r12 x1 , r12 x2 , r12 x3 , r12 x4 , r12 xid 4;1;2;3;5 .
Similarly, the rank assessment vector for t22 is r22 x1 , r22 x2 , r22 x3 , r22 x4 , r22 xid
2;1;4;3;5 , and r32 x1 , r32 x2 , r32 x3 , r32 x4 , r32 xid 2;4;1;3;5 is the ranking
vector for t32 .
The value of the membership functions of fuzzy gradations of t12 , t22 , t32 we calculate by dividing
the rank assessments by the maximum rank value (5).
Thus, we get the values of the membership functions of the assessments for t12 , t22 , and t32 of
ordered sequence x1 , x2 , x3 , x4 , xid :
x (t12 ), x (t12 ), x (t12 ), x (t12 ), x (t12 ) 0,8;0, 2;0, 4;0,6;1 ,
1 2 3 4 id
x (t22 ), x (t22 ), x (t22 ), x (t22 ), x (t22 ) 0, 4;0, 2;0,8;0,6;1 ,
1 2 3 4 id
x (t32 ), x (t32 ), x (t32 ), x (t32 ), x (t32 ) 0, 4;0,8;0, 2;0,6;1 .
1 2 3 4 id
We use Klingv ( x p , xid ) to evaluate the measure of similarity of the applicant x p X
p 1, 2,3, 4 with the ideal element xid calculated as follows
57
n
K lingv ( x p , xid )
1
n i 1
ki ( x p , xid ) , ki x p , xid *i Dxi p , Dxi id ,
m( wi )
m ( wi )
*i Dxi p , Dxi id min x p (t ij ), xid (t ij ) max x p (t ij ), xid (t ij ) ,
j 1
j 1
where ki ( x p , xid ) is a partial LCC which defines the value of the similarity measure on a set Х of
empirical objects by property wi W ; *i Dx , Dx p id is a partial similarity measure in MSR by
wi W , Dxi p and Dxi id are measurements of wi W for elements x p and xid , respectively, n is
the total number of properties; x (t j ) and x (t ij ) determine a measure of belonging of value t ij
i
p id
to wi , respectively, of x p , xid ; m( w1 ) 2 , m( w2 ) 3.
After simple calculations according to the above formulas, we get
Klingv ( x1, xid ) 0,67 ; Klingv ( x2 , xid ) 0, 40 ; Klingv ( x3 , xid ) 0, 44 ; Klingv ( x4 , xid ) 0,50 .
According to calculations,
Klingv ( x1, xid ) Klingv ( x4 , xid ) Klingv ( x3 , xid ) Klingv ( x2 , xid ) .
Thus, LCC assessments of the competitiveness of applicants, which use expert assessments of
multidimensional fuzzy characteristics measured in order scales, show that one of the two vacancies
should be offered to the applicant x1 , and the second vacancy should be offered to the applicant x4 .
5. Conclusions
The paper concerns the features of combinatorial optimization problems on fuzzy sets under
multidimensional qualitative and quantitative information. Uncertainty caused by measurements of
membership functions values of fuzzy sets in different scales leads to accounting the base problems
studied in representative measurement theory, i.e., a presentation problem, uniqueness one and an
adequacy. We offer a concept of fuzzy similarity scale. An inadequacy of traditional building a fuzzy
similarity scale based on fuzzy logic operators we prove considering the representative measurement
theory positions. A concept of a linguistic correlation coefficient is introduced. Conditions of its
adequacy in different scales of measuring of empirical objects properties, i.e., order, ratio, intervals,
and absolute scales (according to Stevens’ Classification) are derived. As further step of research,
based on linguistic correlation coefficient, a fuzzy binary relation of difference on homogeneous and
heterogeneous fuzzy sets will be determined.
6. References
[1] L. F. Hulianytskyi, I. I. Riasna, Automatic classification method based on a fuzzy similarity
relation, Cybernetics and Systems Analysis 52.1 (2016) 30–37. doi:10.1007/s10559-016-9796-3.
[2] V. Balopoulos, A. G. Hatzimichailidis, B. K. Papadoupoulos, Difference and similarity measures
for fuzzy operators, Information Sciences (2007) 2336–2348. doi:10.1016/j.ins.2007.01.005.
[3] B. B. Chaudur, A. Rosenfeld, On a metric distance between fuzzy sets, Pattern Recognition
Letters 17.11 (1996) 1157–1160. doi:10.1016/0167-8655(96)00077-3.
[4] Andrew Gardner, et al., Measuring distance between unordered sets of different sizes, in:
Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2014,
pp. 137–143. doi:10.1109/CVPR.2014.25.
58
[5] S. Kosub, A note on the triangle inequality for the Jaccard distance, Pattern Recognition Letters
(2019) 36–38. URL: https://doi.org/10.1016/j.patrec.2018.12.007.
[6] J. S. Williams, N. Steele, Difference, distance and similarity as a basis for fuzzy decision support
based on prototypical decision classes, Fuzzy Sets and Systems (2002) 35–46. URL:
https//doi.org/10.1016/S0165-0114(01)00253-6.
[7] A. Bogiatzis, B. Papadopoulos, Producing fuzzy inclusion and entropy measures and their
application on global image thresholding, Computer science. Evolving Systems (2018) 331–353.
doi:10.1007/s12530-017-9200-1.
[8] A. Bogiatzis, B. Papadopoulos, Global image thresholding adaptive neuro-fuzzy inference
system trained with fuzzy inclusion and entropy measures, Symmetry 11.2 (2019) 286. URL:
https://doi.org/10.3390/sym11020286.
[9] P. D’Urso, M. A. Gil, Fuzzy data analysis and classification, Adv. Data Anal. Classif. (2017)
645–657. URL: https://doi.org/10.1007/s11634-017-0304-z.
[10] D. Wu, J. M. Mendel, A comparative study of ranking methods, similarity measures and
uncertainty measures for interval type-2 fuzzy sets, Information Sciences 179.8 (2009) 1169–
1192. doi:10.1016/j.ins.2008.12.010.
[11] D. Wu, J. M. Mendel, A vector similarity measure for linguistic approximation: Interval type-2
and type-1 fuzzy sets, Information Sciences 178.2 (2008) 381–402. doi:10.1007/978-3-540-
72950-1_57.
[12] A. V. Kuznetsov, Nonsimilarity measure on graph set and its application, Vestnik VGU, Seria:
System analysis and Information Technology (2017) 125–131. URL:
http://www.vestnik.vsu.ru/pdf/analiz/2017/01/2017-01-17.pdf. (In Russian).
[13] E. Szmidt, J. Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems 114
(2000) 505–518. doi:10.1016/S0165-0114(98)00244-9.
[14] T. Allevard, E. Benoit, L. Foulloy, The transportation distance for fuzzy descriptions of
measurement, Metrol. Meas. Syst. XIV.1 (2007) 25–27. URL: https://hal.archives-
ouvertes.fr/hal-00581046/document.
[15] E. Benoit, L. Foulloy, Towards fuzzy nominal scales, Measurement (2003) 49–55. URL:
https://doi.org/10.1016/S0263-2241(03)00019-8.
[16] M. Z. Zgurovsky, Y. P. Zaychenko, The Fundamentals of Computational Intelligence: System
Approach, Springer, Switzerland, 2016.
[17] P. M. Pardalos, et al, Fuzzy combinatorial optimization problems, Handbook of Combinatorial
Optimization, Springer New York, 2013, pp. 1357–1413. URL: https://doi.org/10.1007/978-1-
4419-7997-1_68.
[18] O. S. Pichugina, S. V. Yakovlev, Continuous functional representations in combinatorial
optimization problems, Zolotaya milya, Kharkov, 2018. (In Russian).
[19] R. Luce, D. Krantz, P. Suppes, and A. Tversky, Vol. III: Foundations of Measurement, Dover
Publications, Inc., New York, 2007.
[20] I. Pfanzagl, Measurement Theory, Мir, Moscow, 1976. (In Russian).
[21] P. Suppes, J. Zinnes, The fundamentals of measurement theory. Psychological measurements,
Мir, Moscow, 1967, pp. 9–110. URL: http://www.libex.ru/detail/book525723.html. (In Russian).
[22] Yu. N. Tolstova, Measurement in sociology: course of lectures, Infra-M, Moscow, 1998. URL:
https://www.isras.ru/publ.html?id=6544. (In Russian).
[23] A. F. Blishun, Comparative analysis of fuzzy measurement methods, Izv. of Academy of
Sciences of the USSR. Technical cybernetics, volume 5, 1988, pp. 152–175. (In Russian).
[24] A. Kaufmann, Fuzzy sets theory, Radio and svyaz, Moscow, 1982. (In Russian).
[25] S. V. Ovchinnikov, T. R’era, Fuzzy classifications, in: R. R. Yager (Ed.), Fuzzy sets and
possibility theory. Recent developments, Radio and svyaz, Moscow, 1986, pp. 100–113. (In
Russian).
[26] L. F. Hulianytskyi, I. I. Riasna, Formalization and Сlassification of Сombinatorial Optimization
Problems, in: S. Butenko, P. M. Pardalos, V. Shylo (eds.), Optimization Methods and
Applications, Cham, Springer International Publishing AG, 2017, pp. 239–250. doi:10.1007/978-
3-319-68640-0_11.
59