RETRIEVING THE MISSING DATA FROM DIFFERENT
INCOMPLETE SOFT SETS
Julee Srivastava
Assistant Professor, Department of Mathematics and Statistics, Deen Dayal Upadhyaya Gorakhpur University.
Gorakhpur (India).
E-mail: mathjulee@gmail.com
ORCID: https://orcid.org/0000-0002-1764-7797
Sudhir Maddheshiya
Research Scholar, Department of Mathematics and Statistics, Deen Dayal Upadhyaya Gorakhpur University.
Gorakhpur (India).
E-mail: sudhirmaddy149@gmail.com
ORCID: https://orcid.org/0000-0001-6027-7245
Reception: 13/09/2022 Acceptance: 28/09/2022 Publication: 29/12/2022
Suggested citation:
Julee Srivastava and Sudhir Maddheshiya (2022). Retrieving the Missing Data From Different Incomplete Soft Sets. 3C
Empresa. Investigación y pensamiento crítico,11 (2), 104-114. https://doi.org/10.17993/3cemp.2022.110250.104-114
https://doi.org/10.17993/3cemp.2022.110250.104-114
ABSTRACT
Dealing an incomplete information has been a major issue in the theory of soft sets. In this paper, we
have presented an approach to deal with incomplete soft set, incomplete fuzzy soft set and incomplete
intuitionistic fuzzy soft set. For this purpose we have discussed about the notion of distance between
two objects (parameters) which will be used to compute the degree of interdependence between them.
This approach will use the full information of known data and the relationships between them. Data
filling converts an incomplete soft set into complete one which makes the soft sets applicable not only to
decision making but also to other fields.
KEYWORDS
Soft set, incomplete information system, fuzzy soft set, object parameter approach.
https://doi.org/10.17993/3cemp.2022.110250.104-114
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
104
RETRIEVING THE MISSING DATA FROM DIFFERENT
INCOMPLETE SOFT SETS
Julee Srivastava
Assistant Professor, Department of Mathematics and Statistics, Deen Dayal Upadhyaya Gorakhpur University.
Gorakhpur (India).
E-mail: mathjulee@gmail.com
ORCID: https://orcid.org/0000-0002-1764-7797
Sudhir Maddheshiya
Research Scholar, Department of Mathematics and Statistics, Deen Dayal Upadhyaya Gorakhpur University.
Gorakhpur (India).
E-mail: sudhirmaddy149@gmail.com
ORCID: https://orcid.org/0000-0001-6027-7245
Reception: 13/09/2022 Acceptance: 28/09/2022 Publication: 29/12/2022
Suggested citation:
Julee Srivastava and Sudhir Maddheshiya (2022). Retrieving the Missing Data From Different Incomplete Soft Sets. 3C
Empresa. Investigación y pensamiento crítico,11 (2), 104-114. https://doi.org/10.17993/3cemp.2022.110250.104-114
https://doi.org/10.17993/3cemp.2022.110250.104-114
ABSTRACT
Dealing an incomplete information has been a major issue in the theory of soft sets. In this paper, we
have presented an approach to deal with incomplete soft set, incomplete fuzzy soft set and incomplete
intuitionistic fuzzy soft set. For this purpose we have discussed about the notion of distance between
two objects (parameters) which will be used to compute the degree of interdependence between them.
This approach will use the full information of known data and the relationships between them. Data
filling converts an incomplete soft set into complete one which makes the soft sets applicable not only to
decision making but also to other fields.
KEYWORDS
Soft set, incomplete information system, fuzzy soft set, object parameter approach.
https://doi.org/10.17993/3cemp.2022.110250.104-114
105
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
1 INTRODUCTION
There are various real life problems involving uncertainties and classical mathematical tools are not
sufficient for handling them. There are many theories developed recently for dealing with them. Some
of them are probability theory, theory of fuzzy sets [16], theory of intuitionistic fuzzy sets [1,2], theory
of vague sets [4], theory of interval mathematics [11] and theory of rough sets [13]. All of these theories
have their own advantages and some limitations as well. For example, in the theory of fuzzy sets and
intuitionistic fuzzy sets, it is very difficult to choose the membership and non membership functions
that give us the desired result; in the theory of probability the outcomes of an event must be unbiased;
in the theory of rough sets, the indiscernibility relation may create a situation where two completely
different objects are same. One major common drawback of these theories is probably the inadequacy
of parameterization tools which was observed by Molodtsov in 1999. Consequently he introduced the
concept of soft set theory [10] that is free from the difficulties that have troubled the usual theoretical
approaches. The absence of any restriction on the approximate description in soft set theory makes
it easily applicable in practice. A soft set model requires no prior knowledge of data sets. Molodtsov
provided several applications of soft set theory in his work. Maji et al. [9] introduced fuzzy soft set
by allowing the parameters to be mapped to the fuzzy sets. Further allowing the parameters to be
mapped to the intuitionistic fuzzy sets, Maji et al. [8] introduced the concept of intuitionistic fuzzy soft
set which is a generalization of standard soft set and fuzzy soft set in the sense that it is a soft set
whose approximate values are the intuitionistic fuzzy sets.
Lots of research work are currently active in the field of theoretical and practical soft sets. The
major portion of these works is based on complete information. However, incomplete information widely
exists in real life due to mishandling data, mistakes in processing or transferring data, mistakes in
measuring and collecting data or any other factor. Soft set under incomplete information is referred to
as an incomplete soft set. Similarly fuzzy soft set and intuitionistic fuzzy soft set under incomplete
information are referred to as incomplete fuzzy soft set and incomplete intuitionistic fuzzy soft set
respectively.
The simplest approach to transform an incomplete data set to a complete one is to delete all objects
related to missing information. But in this process we may deduce wrong information from it. On the
other hand, predicting the unknown information gives more fruitful results. Zou et al. [17] initiated
the study of incomplete soft sets. For incomplete soft set, they computed decision values rather than
filling the empty cells in the corresponding incomplete information system. The decision values are
calculated by the weighted average of all the choice values and the weight of each choice value is decided
by the distribution of other available objects. Incomplete fuzzy soft set is completed by the method
of average probability. Zou’s method is too complicated and it does not fill the empty cells of the
corresponding information system. So the soft set obtained by this method is only useful in decision
making. Using average probability method we can predict individual unknown value of fuzzy soft set
but all the predicted values of a parameter for different objects are equal, so this method is also of low
accuracy. Kong et al. [6] proposed a simple method equivalent to that of Zou which fills the empty cells.
To fill the empty cells in the incomplete information system, Kong’s method uses the values of target
parameter (the parameter for which the cell is empty) on the objects other than target object (the
object for which the cell is empty). Qin et al. [14] presented a method called DFIS (data filling approach
for incomplete soft set). In that paper, empty cells are filled in terms of the association degree between
the parameters, when a strong association exists between the parameters, otherwise they are filled
in terms of probability of other available objects. Khan et al. [5] proposed an alternative data filling
approach for incomplete soft set (ADFIS) to predict the missing data in soft sets. In ADFIS, the value
of the empty cell whose corresponding parameter has strongest association is computed first. Unlike
the DFIS, before filling second empty cell, the value of the first is inserted in the information table.
But the drawback of DFIS and ADFIS is that a parameter can have strongest association or maximal
association with more than one parameters having opposite type of association. In that case empty cell
can’t be filled. This method considers only the relation between parameters and does not take the effect
of objects into account. However there may be some relationship between objects too. For example
the houses in same locality have nearly same price. Deng et al. [3] introduced an object-parameter
https://doi.org/10.17993/3cemp.2022.110250.104-114
approach which uses the full information between object and between parameters. Deng’s approach
has some drawbacks as: (i) the estimated value may not be in the interval [0,1]; (ii) the information
between objects and between parameters is not comprehensive. To overcome these drawbacks Liu et
al. [7] improved Deng’s approach by redefining the notion of distance and dominant degree.
It is a review paper. In paper [15] we have put forward an algorithm to predict missing data in
an incomplete soft set and incomplete fuzzy soft set. For this we have defined the notion of distance
(emerged from the concept of Euclidean distance in
Rn
) between two objects (parameters) and defined
the degree of interdependence between two objects (parameters). And thus we have taken account
of the effect of other objects (parameters) on the target object (parameter). This algorithm uses the
full available data to reveal the hidden relationship between objects (parameters). Moreover we have
introduced an approach to predict missing data in an incomplete intuitionistic fuzzy soft set with the
help of algorithm for incomplete soft set and incomplete fuzzy soft set.
Rest of the paper has been organized as follows. Section 2 recalls the basic definitions and concepts
of soft set theory and information system. In section 3 we have introduced an algorithm to predict
missing data in an incomplete soft set and incomplete fuzzy soft set and given an application through
an example. In section 4, we have given an algorithm to predict the missing data in an incomplete
intuitionistic fuzzy soft set and given an application through an example. Finally we have concluded
this paper in section 5.
2 PRELIMINARIES
Let U={u1,u
2,...,u
m}be a universe set of objects and E={e1,e
2,...,e
n}be a set of parameters.
Definition 1 (Fuzzy set).[16] A fuzzy set Aover Uis given by
A={⟨u, µA(u)⟩|u∈U}
where
µA
:
U→
[0
,
1] is called the membership function of the fuzzy set
A
.
µA
(
u
)is said to be the degree
of membership of uin A.
Definition 2 (Intuitionistic fuzzy set).[1] An intuitionistic fuzzy set (IFS) Aover Uis given by
A={⟨u, µA(u),ν
A(u)⟩|u∈U;µA(u),ν
A(u)∈[0,1] and µA(u)+νA(u)≤1}}
where
µA
:
U→
[0
,
1] and
νA
:
U→
[0
,
1] are said to be the membership and non membership functions
of the intuitionistic fuzzy set Arespectively.
Definition 3 (Soft set).[10] A pair
A
=(
F, E
)is said to be a soft set over
U
, where
F
is a mapping
from
E
to
P
(
U
)(set of all crisp subsets of
U
). Sometimes it is also called a crisp soft set to emphasize
the fact that F(e)is a crisp set for every e∈E.
Alternatively, a soft set Ais given by
A={F(e)|e∈E}
where Fis a mapping from Eto P(U).
Definition 4 (Fuzzy soft set).[9] A pair
A
=(
F, E
)is said to be a fuzzy soft set over
U
, where
F
is
a mapping from Eto F(U)(set of all fuzzy sets over U).
Alternatively, a fuzzy soft set Ais given by
A={F(e)|e∈E}
where Fis a mapping from Eto F(U).
https://doi.org/10.17993/3cemp.2022.110250.104-114
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
106
1 INTRODUCTION
There are various real life problems involving uncertainties and classical mathematical tools are not
sufficient for handling them. There are many theories developed recently for dealing with them. Some
of them are probability theory, theory of fuzzy sets [16], theory of intuitionistic fuzzy sets [1,2], theory
of vague sets [4], theory of interval mathematics [11] and theory of rough sets [13]. All of these theories
have their own advantages and some limitations as well. For example, in the theory of fuzzy sets and
intuitionistic fuzzy sets, it is very difficult to choose the membership and non membership functions
that give us the desired result; in the theory of probability the outcomes of an event must be unbiased;
in the theory of rough sets, the indiscernibility relation may create a situation where two completely
different objects are same. One major common drawback of these theories is probably the inadequacy
of parameterization tools which was observed by Molodtsov in 1999. Consequently he introduced the
concept of soft set theory [10] that is free from the difficulties that have troubled the usual theoretical
approaches. The absence of any restriction on the approximate description in soft set theory makes
it easily applicable in practice. A soft set model requires no prior knowledge of data sets. Molodtsov
provided several applications of soft set theory in his work. Maji et al. [9] introduced fuzzy soft set
by allowing the parameters to be mapped to the fuzzy sets. Further allowing the parameters to be
mapped to the intuitionistic fuzzy sets, Maji et al. [8] introduced the concept of intuitionistic fuzzy soft
set which is a generalization of standard soft set and fuzzy soft set in the sense that it is a soft set
whose approximate values are the intuitionistic fuzzy sets.
Lots of research work are currently active in the field of theoretical and practical soft sets. The
major portion of these works is based on complete information. However, incomplete information widely
exists in real life due to mishandling data, mistakes in processing or transferring data, mistakes in
measuring and collecting data or any other factor. Soft set under incomplete information is referred to
as an incomplete soft set. Similarly fuzzy soft set and intuitionistic fuzzy soft set under incomplete
information are referred to as incomplete fuzzy soft set and incomplete intuitionistic fuzzy soft set
respectively.
The simplest approach to transform an incomplete data set to a complete one is to delete all objects
related to missing information. But in this process we may deduce wrong information from it. On the
other hand, predicting the unknown information gives more fruitful results. Zou et al. [17] initiated
the study of incomplete soft sets. For incomplete soft set, they computed decision values rather than
filling the empty cells in the corresponding incomplete information system. The decision values are
calculated by the weighted average of all the choice values and the weight of each choice value is decided
by the distribution of other available objects. Incomplete fuzzy soft set is completed by the method
of average probability. Zou’s method is too complicated and it does not fill the empty cells of the
corresponding information system. So the soft set obtained by this method is only useful in decision
making. Using average probability method we can predict individual unknown value of fuzzy soft set
but all the predicted values of a parameter for different objects are equal, so this method is also of low
accuracy. Kong et al. [6] proposed a simple method equivalent to that of Zou which fills the empty cells.
To fill the empty cells in the incomplete information system, Kong’s method uses the values of target
parameter (the parameter for which the cell is empty) on the objects other than target object (the
object for which the cell is empty). Qin et al. [14] presented a method called DFIS (data filling approach
for incomplete soft set). In that paper, empty cells are filled in terms of the association degree between
the parameters, when a strong association exists between the parameters, otherwise they are filled
in terms of probability of other available objects. Khan et al. [5] proposed an alternative data filling
approach for incomplete soft set (ADFIS) to predict the missing data in soft sets. In ADFIS, the value
of the empty cell whose corresponding parameter has strongest association is computed first. Unlike
the DFIS, before filling second empty cell, the value of the first is inserted in the information table.
But the drawback of DFIS and ADFIS is that a parameter can have strongest association or maximal
association with more than one parameters having opposite type of association. In that case empty cell
can’t be filled. This method considers only the relation between parameters and does not take the effect
of objects into account. However there may be some relationship between objects too. For example
the houses in same locality have nearly same price. Deng et al. [3] introduced an object-parameter
https://doi.org/10.17993/3cemp.2022.110250.104-114
approach which uses the full information between object and between parameters. Deng’s approach
has some drawbacks as: (i) the estimated value may not be in the interval [0,1]; (ii) the information
between objects and between parameters is not comprehensive. To overcome these drawbacks Liu et
al. [7] improved Deng’s approach by redefining the notion of distance and dominant degree.
It is a review paper. In paper [15] we have put forward an algorithm to predict missing data in
an incomplete soft set and incomplete fuzzy soft set. For this we have defined the notion of distance
(emerged from the concept of Euclidean distance in
Rn
) between two objects (parameters) and defined
the degree of interdependence between two objects (parameters). And thus we have taken account
of the effect of other objects (parameters) on the target object (parameter). This algorithm uses the
full available data to reveal the hidden relationship between objects (parameters). Moreover we have
introduced an approach to predict missing data in an incomplete intuitionistic fuzzy soft set with the
help of algorithm for incomplete soft set and incomplete fuzzy soft set.
Rest of the paper has been organized as follows. Section 2 recalls the basic definitions and concepts
of soft set theory and information system. In section 3 we have introduced an algorithm to predict
missing data in an incomplete soft set and incomplete fuzzy soft set and given an application through
an example. In section 4, we have given an algorithm to predict the missing data in an incomplete
intuitionistic fuzzy soft set and given an application through an example. Finally we have concluded
this paper in section 5.
2 PRELIMINARIES
Let U={u1,u
2,...,u
m}be a universe set of objects and E={e1,e
2,...,e
n}be a set of parameters.
Definition 1 (Fuzzy set).[16] A fuzzy set Aover Uis given by
A={⟨u, µA(u)⟩|u∈U}
where
µA
:
U→
[0
,
1] is called the membership function of the fuzzy set
A
.
µA
(
u
)is said to be the degree
of membership of uin A.
Definition 2 (Intuitionistic fuzzy set).[1] An intuitionistic fuzzy set (IFS) Aover Uis given by
A={⟨u, µA(u),ν
A(u)⟩|u∈U;µA(u),ν
A(u)∈[0,1] and µA(u)+νA(u)≤1}}
where
µA
:
U→
[0
,
1] and
νA
:
U→
[0
,
1] are said to be the membership and non membership functions
of the intuitionistic fuzzy set Arespectively.
Definition 3 (Soft set).[10] A pair
A
=(
F, E
)is said to be a soft set over
U
, where
F
is a mapping
from
E
to
P
(
U
)(set of all crisp subsets of
U
). Sometimes it is also called a crisp soft set to emphasize
the fact that F(e)is a crisp set for every e∈E.
Alternatively, a soft set Ais given by
A={F(e)|e∈E}
where Fis a mapping from Eto P(U).
Definition 4 (Fuzzy soft set).[9] A pair
A
=(
F, E
)is said to be a fuzzy soft set over
U
, where
F
is
a mapping from Eto F(U)(set of all fuzzy sets over U).
Alternatively, a fuzzy soft set Ais given by
A={F(e)|e∈E}
where Fis a mapping from Eto F(U).
https://doi.org/10.17993/3cemp.2022.110250.104-114
107
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
Definition 5 (Intuitionistic fuzzy soft set).[8] A pair
A
=(
F, E
)is said to be an intuitionistic fuzzy
soft set (IFSS) over
U
, where
F
is a mapping from
E
to
IF
(
U
)(set of all intuitionistic fuzzy sets over
U).
Alternatively, an intuitionistic fuzzy soft set Ais given by
A={F(e)|e∈E}
where Fis mapping from Eto IF(U).
Definition 6 (Information system).[12] A quadruple
S
=(
U,A,F,V
)is called an information system,
where
U
=
{u1,u
2,...,u
m}
is a universe of discourse,
A
=
{a1,...,a
n}
is a set of attributes and
V
=
n
j=1 Vj
, where each
Vj
is the value set of the attribute
aj
and
F
=
{f1,...,f
n}
where
fj
:
U→Vj
for every j.
If
Vj
=
{
0
,
1
}
for every 1
≤j≤n
then the corresponding information system is called Boolean
valued information system and if
Vj
= [0
,
1] for every 1
≤j≤n
then the corresponding information
system is called fuzzy information system. In an information system
uik
=
fk
(
ui
)denotes the value of
the attribute akon the object ui. An information system is often represented by an information table.
Remark: (i) Every soft set can be considered as a Boolean valued information system with each
entry filled by 1 or 0 depending on whether an object belongs to range of the parameter or not.
(ii) Every fuzzy soft set can be considered as a fuzzy information system with each entry filled by a
quantity in [0
,
1] which represents the membership degree of object in the range of the related parameter.
(iii) Every intuitionistic fuzzy soft set can be considered as an information system with each entry filled
by an element of [0
,
1]
×
[0
,
1] where the first and second coordinates represent the membership degree
and non membership degree of the object in the range of the related parameter respectively.
Example 2.1. Every incomplete soft set can be considered as an incomplete information system.
Examples of incomplete soft set, incomplete fuzzy soft set and incomplete intuitionistic fuzzy soft set
are given in table 1, 2 and 3 respectively. The unknown value in incomplete information system is
denoted by ‘∗’.
Table 1 – Incomplete soft set
U e1e2e3e4e5e6
u11 0 1 0 1 0
u21 0 0 1 0 0
u30 1 0 0 1 0
u40 1 ∗1 0 ∗
u51 0 1 1 0 0
u60 1 0 0 ∗0
u71∗1 0 1 0
u80 0 1 1 0 0
Table 2 – Incomplete fuzzy soft set
U e1e2e3e4e5e6e7
u10.9 0.4 0.1 0.9 0.6 0.3 0.4
u20.8 0.6 0.5 ∗0.5 0.3 0.3
u3∗0.8 0.9 ∗0.9 0.9 0.9
u40.9 0.8 0.9 0.8 ∗0.8 0.9
u50.9 0.2 0.2 0.6 0.3 0.4 ∗
u60.9 0.2 0.4 0.4 0.4 0.3 0.3
https://doi.org/10.17993/3cemp.2022.110250.104-114
Table 3 – Incomplete Intuitionistic fuzzy soft set
U e1e2e3e4e5e6
u1(0.8,0.1) (0.2,0.1) (0.8,0.1) (0.4,0.5) (0.4,0.5) (0.6,0.2)
u2(0.8,0.2) (0.8,0.1) (0.7,0.2) (0.6,0.4) (0.5,0.5) (0.6,0.2)
u3(0.7,0.2) (0.3,0.1) (0.8,0.2) ∗(0.6,0.1) (0.4,0.2)
u4(0.6,0.2) (0.7,0.2) (0.7,0.3) (0.4,0.3) (0.7,0.1) (0.6,0.1)
u5(0.5,0.3) (0.6,0.3) (0.4,0.5) (0.7,0.3) (0.8,0.1) ∗
u6(0.2,0.4) (0.4,0.4) (0.5,0.5) (0.4,0.3) (0.4,0.3) (0.5,0.1)
u7(0.7,0.2) (0.8,0.1) (0.5,0.4) (0.9,0.1) (0.5,0.3) (0.4,0.1)
3
ALGORITHM TO PREDICT MISSING DATA IN AN INCOMPLETE SOFT SET
AND INCOMPLETE FUZZY SOFT SET AND ITS APPLICATION
3.1 A PREPARATORY STEP
There is always a direct or indirect relationship between objects (parameters). To measure this
relationship, we will define the ‘degree of interdependence’ between objects (parameters). To determine
the unknown value in the incomplete soft set we will examine the remaining known values and
interdependence between target object (parameter) and other objects (parameters).
Let
U
=
{u1,u
2,...,u
m}
be universe set of objects and
E
=
{e1,e
2,...,e
n}
be set of parameters.
Suppose that
µF(ek)
(
ui
)=
uik
. For every 1
≤i≤m
; denote
E(i)
=
{k|uik
=
∗}
and for every 1
≤k≤n
;
U(k)={i|uik =∗}.
Now we will define distance and degree of interdependence between two objects and between two
parameters.
Definition 7 (Distance).For uiand ujin U, the distance between uiand ujis defined by
d(ui,u
j)=
k∈E(i)∩E(j)
(uik −ujk)2
1/2
(1)
where E(i)∩E(j)={k|uik =∗and ujk =∗}.
Similarly, for ekand elin E, distance between ekand elis defined by
d(ek,e
l)=
i∈U(k)∩U(l)
(uik −uil)2
1/2
(2)
where U(k)∩U(l)={i|uik =∗and uil =∗}.
Definition 8 (Degree of Interdependence).For
ui
and
uj
in
U
, the degree of interdependence between
uiand ujis denoted by αij and is defined as αij =1
1+d(ui,uj).
Similarly, for
ek
and
el
in
E
, the degree of interdependence between
ek
and
el
is denoted by
βkl
and
is defined as βkl =1
1+d(ek,el).
Suppose that the value
uik
is missing, then we will call
ui
as target object and
ek
as target parameter.
The prediction of
uik
will contain two parts: (i) object part
uobj
ik
and (ii) parameter part
upar
ik
. As the
distance between two objects (parameters) increases, the interdependence between them decreases.
So the objects (parameters) which are nearer to target object (parameter) will be more reliable to
determine the object (parameter) part of the unknown value. Object part of an unknown value is
determined using the values of the target parameter on the objects other than target object and the
parameter part is determined using the values of all parameters other than target parameter on target
object.
https://doi.org/10.17993/3cemp.2022.110250.104-114
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
108
Definition 5 (Intuitionistic fuzzy soft set).[8] A pair
A
=(
F, E
)is said to be an intuitionistic fuzzy
soft set (IFSS) over
U
, where
F
is a mapping from
E
to
IF
(
U
)(set of all intuitionistic fuzzy sets over
U).
Alternatively, an intuitionistic fuzzy soft set Ais given by
A={F(e)|e∈E}
where Fis mapping from Eto IF(U).
Definition 6 (Information system).[12] A quadruple
S
=(
U,A,F,V
)is called an information system,
where
U
=
{u1,u
2,...,u
m}
is a universe of discourse,
A
=
{a1,...,a
n}
is a set of attributes and
V
=
n
j=1 Vj
, where each
Vj
is the value set of the attribute
aj
and
F
=
{f1,...,f
n}
where
fj
:
U→Vj
for every j.
If
Vj
=
{
0
,
1
}
for every 1
≤j≤n
then the corresponding information system is called Boolean
valued information system and if
Vj
= [0
,
1] for every 1
≤j≤n
then the corresponding information
system is called fuzzy information system. In an information system
uik
=
fk
(
ui
)denotes the value of
the attribute akon the object ui. An information system is often represented by an information table.
Remark: (i) Every soft set can be considered as a Boolean valued information system with each
entry filled by 1 or 0 depending on whether an object belongs to range of the parameter or not.
(ii) Every fuzzy soft set can be considered as a fuzzy information system with each entry filled by a
quantity in [0
,
1] which represents the membership degree of object in the range of the related parameter.
(iii) Every intuitionistic fuzzy soft set can be considered as an information system with each entry filled
by an element of [0
,
1]
×
[0
,
1] where the first and second coordinates represent the membership degree
and non membership degree of the object in the range of the related parameter respectively.
Example 2.1. Every incomplete soft set can be considered as an incomplete information system.
Examples of incomplete soft set, incomplete fuzzy soft set and incomplete intuitionistic fuzzy soft set
are given in table 1, 2 and 3 respectively. The unknown value in incomplete information system is
denoted by ‘∗’.
Table 1 – Incomplete soft set
U e1e2e3e4e5e6
u1101010
u2100100
u3010010
u40 1 ∗1 0 ∗
u5101100
u60100∗0
u71∗1010
u8001100
Table 2 – Incomplete fuzzy soft set
U e1e2e3e4e5e6e7
u10.9 0.4 0.1 0.9 0.6 0.3 0.4
u20.8 0.6 0.5 ∗0.5 0.3 0.3
u3∗0.8 0.9 ∗0.9 0.9 0.9
u40.9 0.8 0.9 0.8 ∗0.8 0.9
u50.9 0.2 0.2 0.6 0.3 0.4 ∗
u60.9 0.2 0.4 0.4 0.4 0.3 0.3
https://doi.org/10.17993/3cemp.2022.110250.104-114
Table 3 – Incomplete Intuitionistic fuzzy soft set
U e1e2e3e4e5e6
u1(0.8,0.1) (0.2,0.1) (0.8,0.1) (0.4,0.5) (0.4,0.5) (0.6,0.2)
u2(0.8,0.2) (0.8,0.1) (0.7,0.2) (0.6,0.4) (0.5,0.5) (0.6,0.2)
u3(0.7,0.2) (0.3,0.1) (0.8,0.2) ∗(0.6,0.1) (0.4,0.2)
u4(0.6,0.2) (0.7,0.2) (0.7,0.3) (0.4,0.3) (0.7,0.1) (0.6,0.1)
u5(0.5,0.3) (0.6,0.3) (0.4,0.5) (0.7,0.3) (0.8,0.1) ∗
u6(0.2,0.4) (0.4,0.4) (0.5,0.5) (0.4,0.3) (0.4,0.3) (0.5,0.1)
u7(0.7,0.2) (0.8,0.1) (0.5,0.4) (0.9,0.1) (0.5,0.3) (0.4,0.1)
3
ALGORITHM TO PREDICT MISSING DATA IN AN INCOMPLETE SOFT SET
AND INCOMPLETE FUZZY SOFT SET AND ITS APPLICATION
3.1 A PREPARATORY STEP
There is always a direct or indirect relationship between objects (parameters). To measure this
relationship, we will define the ‘degree of interdependence’ between objects (parameters). To determine
the unknown value in the incomplete soft set we will examine the remaining known values and
interdependence between target object (parameter) and other objects (parameters).
Let
U
=
{u1,u
2,...,u
m}
be universe set of objects and
E
=
{e1,e
2,...,e
n}
be set of parameters.
Suppose that
µF(ek)
(
ui
)=
uik
. For every 1
≤i≤m
; denote
E(i)
=
{k|uik
=
∗}
and for every 1
≤k≤n
;
U(k)={i|uik =∗}.
Now we will define distance and degree of interdependence between two objects and between two
parameters.
Definition 7 (Distance).For uiand ujin U, the distance between uiand ujis defined by
d(ui,u
j)=
k∈E(i)∩E(j)
(uik −ujk)2
1/2
(1)
where E(i)∩E(j)={k|uik =∗and ujk =∗}.
Similarly, for ekand elin E, distance between ekand elis defined by
d(ek,e
l)=
i∈U(k)∩U(l)
(uik −uil)2
1/2
(2)
where U(k)∩U(l)={i|uik =∗and uil =∗}.
Definition 8 (Degree of Interdependence).For
ui
and
uj
in
U
, the degree of interdependence between
uiand ujis denoted by αij and is defined as αij =1
1+d(ui,uj).
Similarly, for
ek
and
el
in
E
, the degree of interdependence between
ek
and
el
is denoted by
βkl
and
is defined as βkl =1
1+d(ek,el).
Suppose that the value
uik
is missing, then we will call
ui
as target object and
ek
as target parameter.
The prediction of
uik
will contain two parts: (i) object part
uobj
ik
and (ii) parameter part
upar
ik
. As the
distance between two objects (parameters) increases, the interdependence between them decreases.
So the objects (parameters) which are nearer to target object (parameter) will be more reliable to
determine the object (parameter) part of the unknown value. Object part of an unknown value is
determined using the values of the target parameter on the objects other than target object and the
parameter part is determined using the values of all parameters other than target parameter on target
object.
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3.2 ALGORITHM
Suppose we have to predict the value of uik. Before giving Algorithm we define some notations here:
U∗
i={p|up∈U−{ui}and upk =∗} and
E∗
k={q|eq∈E−{ek}and uiq =∗}
And we define Urand Errecursively as
Ur={jr|d(ui,u
jr) = min
j∈U∗
i−(U0∪U1∪···∪Ur−1)d(ui,u
j)}; where U0=∅.
Er={lr|d(ek,e
lr) = min
l∈E∗
k−(E0∪E1∪···∪Er−1)d(ek,e
l)}; where E0=∅.
First we compute the object part:
1. Input the incomplete soft set (F, E).
2. Find uisuch that uik is unknown.
3. Compute d(ui,u
j)for all j∈U∗
i.
4. Let ¯u1,obj
ik =
j1∈U1
uj1k
|U1|.
5.
Compute degree of interdependence between
ui
and
uj1
, which is given by
αij1
=
1
1+d(ui,uj1)
where
j1∈U1.
6. Define u1,obj
ik =¯u1,obj
ik ×αij1.
7. Let ¯u2,obj
ik =
j2∈U2
uj2k
|U2|.
8.
Compute degree of interdependence between
ui
and
uj2
, which is given by
αij2
=
1
1+d(ui,uj2)
where
j2∈U2.
9. Define u2,obj
ik =¯u2,obj
ik ×αij2.
10. Continue in this way until U1∪U2∪···∪Ut=U∗
i.
11. Hence object part of unknown value uobj
ik =
t
r=1
ur,obj
ik
t
r=1
αijr
.
Now we compute the parameter part:
1. Input the incomplete soft set (F, E).
2. Find eksuch that uik is unknown.
3. Compute d(ek,e
l)for all l∈E∗
k.
4. Let ¯u1,par
ik =
l1∈E1
uil1
|E1|.
5. Compute degree of interdependence between ekand el1:βkl1=1
1+d(ek,el1)for l1∈E1.
6. Define u1,par
ik =¯u1,par
ik ×βkl1.
7. Let ¯u2,par
ik =
l2∈E2
uil2
|E2|.
8. Compute degree of interdependence between ukand ul2:βkl2=1
1+d(ek,el2)for l2∈E2.
https://doi.org/10.17993/3cemp.2022.110250.104-114
9. Define u2,par
ik =¯u2,par
ik ×βkl2.
10. Continue in this way until E1∪E2∪···∪Et=E∗
k.
11. Hence parameter part of unknown value is upar
ik =
t
r=1
ur,par
ik
t
r=1
βklr
.
Now the unknown value uik of a fuzzy soft set can be predicted by the equation
uik =w1.uobj
ik +w2.upar
ik (3)
where
w1
and
w2
are weights of the objects and parameters measuring the impact on unknown data,
respectively. The weights can be assigned according to the given problem. If the objects and parameters
are treated equally, the weights can be set as w1=w2=1
2.
In case
uik
is an unknown value of a soft set then we compute
hik
=
w1.uobj
ik
+
w2.upar
ik
as above. If
hik <1
2, put uik =0and if hik ≥1
2, put uik =1.
3.3 APPLICATION OF ALGORITHM FOR INCOMPLETE SOFT SET
Consider the incomplete soft set represented in tabel 1. In this table there are eight objects, six parameters
and four unknown values to be predicted. Suppose that the weights of objects and parameters be equal,
i.e.,
w1
=
w2
=
1
2
. By using our algorithm we compute
h43
=0
.
5628,
h46
=0
.
2682,
h65
=0
.
4903,
h72
=0
.
4334. Since
h43 >1
2
,
h46 <1
2
,
h65 <1
2
and
h72 <1
2
, therefore we obtain
h43
=1,
h46
=0,
h65 =0,h72 =0.
3.4 APPLICATION OF ALGORITHM FOR INCOMPLETE FUZZY SOFT SET
Consider the incomplete fuzzy soft set represented in table 2. In this table there are six objects, seven
parameters and five unknown values to be predicted. Here also we suppose that the weights of objects
and parameters are equal, i.e.,
w1
=
w2
=
1
2
. By using our algorithm we obtain the unknown values as
u24 =0.5825,u31 =0.8815,u34 =0.7910,u45 =0.7202,u57 =0.4575.
4
ALGORITHM TO PREDICT MISSING DATA IN AN INCOMPLETE INTUITIO-
NISTIC FUZZY SOFT SET AND ITS APPLICATION
4.1 A PREPARATORY STEP
To predict the unknown values of incomplete intuitionistic fuzzy sets we will construct four fuzzy soft
sets from given intuitionistic fuzzy soft set as follows. We take an example given in table 3 to make
it more clear. For this incomplete intuitionistic fuzzy soft set we construct four tables; first by using
membership degrees (table 4), second by using non membership degrees (table 5), third by using the
sum of membership and non membership degrees (table 6) and fourth by their differences (table 7).
Table 4 – Membership degrees
U e1e2e3e4e5e6
u10.8 0.2 0.8 0.4 0.4 0.6
u20.8 0.8 0.5 0.6 0.5 0.6
u30.7 0.3 0.8 ∗0.6 0.4
u40.6 0.7 0.7 0.4 0.7 0.6
u50.5 0.6 0.4 0.7 0.8 ∗
u60.2 0.4 0.5 0.4 0.4 0.5
u70.7 0.8 0.5 0.9 0.5 0.4
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Ed. 50 Vol. 11 N.º 2 August - December 2022
110
3.2 ALGORITHM
Suppose we have to predict the value of uik. Before giving Algorithm we define some notations here:
U∗
i={p|up∈U−{ui}and upk =∗} and
E∗
k={q|eq∈E−{ek}and uiq =∗}
And we define Urand Errecursively as
Ur={jr|d(ui,u
jr) = min
j∈U∗
i−(U0∪U1∪···∪Ur−1)d(ui,u
j)}; where U0=∅.
Er={lr|d(ek,e
lr) = min
l∈E∗
k−(E0∪E1∪···∪Er−1)d(ek,e
l)}; where E0=∅.
First we compute the object part:
1. Input the incomplete soft set (F, E).
2. Find uisuch that uik is unknown.
3. Compute d(ui,u
j)for all j∈U∗
i.
4. Let ¯u1,obj
ik =
j1∈U1
uj1k
|U1|.
5.
Compute degree of interdependence between
ui
and
uj1
, which is given by
αij1
=
1
1+d(ui,uj1)
where
j1∈U1.
6. Define u1,obj
ik =¯u1,obj
ik ×αij1.
7. Let ¯u2,obj
ik =
j2∈U2
uj2k
|U2|.
8.
Compute degree of interdependence between
ui
and
uj2
, which is given by
αij2
=
1
1+d(ui,uj2)
where
j2∈U2.
9. Define u2,obj
ik =¯u2,obj
ik ×αij2.
10. Continue in this way until U1∪U2∪···∪Ut=U∗
i.
11. Hence object part of unknown value uobj
ik =
t
r=1
ur,obj
ik
t
r=1
αijr
.
Now we compute the parameter part:
1. Input the incomplete soft set (F, E).
2. Find eksuch that uik is unknown.
3. Compute d(ek,e
l)for all l∈E∗
k.
4. Let ¯u1,par
ik =
l1∈E1
uil1
|E1|.
5. Compute degree of interdependence between ekand el1:βkl1=1
1+d(ek,el1)for l1∈E1.
6. Define u1,par
ik =¯u1,par
ik ×βkl1.
7. Let ¯u2,par
ik =
l2∈E2
uil2
|E2|.
8. Compute degree of interdependence between ukand ul2:βkl2=1
1+d(ek,el2)for l2∈E2.
https://doi.org/10.17993/3cemp.2022.110250.104-114
9. Define u2,par
ik =¯u2,par
ik ×βkl2.
10. Continue in this way until E1∪E2∪···∪Et=E∗
k.
11. Hence parameter part of unknown value is upar
ik =
t
r=1
ur,par
ik
t
r=1
βklr
.
Now the unknown value uik of a fuzzy soft set can be predicted by the equation
uik =w1.uobj
ik +w2.upar
ik (3)
where
w1
and
w2
are weights of the objects and parameters measuring the impact on unknown data,
respectively. The weights can be assigned according to the given problem. If the objects and parameters
are treated equally, the weights can be set as w1=w2=1
2.
In case
uik
is an unknown value of a soft set then we compute
hik
=
w1.uobj
ik
+
w2.upar
ik
as above. If
hik <1
2, put uik =0and if hik ≥1
2, put uik =1.
3.3 APPLICATION OF ALGORITHM FOR INCOMPLETE SOFT SET
Consider the incomplete soft set represented in tabel 1. In this table there are eight objects, six parameters
and four unknown values to be predicted. Suppose that the weights of objects and parameters be equal,
i.e.,
w1
=
w2
=
1
2
. By using our algorithm we compute
h43
=0
.
5628,
h46
=0
.
2682,
h65
=0
.
4903,
h72
=0
.
4334. Since
h43 >1
2
,
h46 <1
2
,
h65 <1
2
and
h72 <1
2
, therefore we obtain
h43
=1,
h46
=0,
h65 =0,h72 =0.
3.4 APPLICATION OF ALGORITHM FOR INCOMPLETE FUZZY SOFT SET
Consider the incomplete fuzzy soft set represented in table 2. In this table there are six objects, seven
parameters and five unknown values to be predicted. Here also we suppose that the weights of objects
and parameters are equal, i.e.,
w1
=
w2
=
1
2
. By using our algorithm we obtain the unknown values as
u24 =0.5825,u31 =0.8815,u34 =0.7910,u45 =0.7202,u57 =0.4575.
4
ALGORITHM TO PREDICT MISSING DATA IN AN INCOMPLETE INTUITIO-
NISTIC FUZZY SOFT SET AND ITS APPLICATION
4.1 A PREPARATORY STEP
To predict the unknown values of incomplete intuitionistic fuzzy sets we will construct four fuzzy soft
sets from given intuitionistic fuzzy soft set as follows. We take an example given in table 3 to make
it more clear. For this incomplete intuitionistic fuzzy soft set we construct four tables; first by using
membership degrees (table 4), second by using non membership degrees (table 5), third by using the
sum of membership and non membership degrees (table 6) and fourth by their differences (table 7).
Table 4 – Membership degrees
U e1e2e3e4e5e6
u10.8 0.2 0.8 0.4 0.4 0.6
u20.8 0.8 0.5 0.6 0.5 0.6
u30.7 0.3 0.8 ∗0.6 0.4
u40.6 0.7 0.7 0.4 0.7 0.6
u50.5 0.6 0.4 0.7 0.8 ∗
u60.2 0.4 0.5 0.4 0.4 0.5
u70.7 0.8 0.5 0.9 0.5 0.4
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Table 5 – Non membership degrees
U e1e2e3e4e5e6
u10.1 0.1 0.1 0.5 0.5 0.2
u20.2 0.1 0.2 0.4 0.5 0.2
u30.2 0.1 0.2 ∗0.1 0.2
u40.2 0.2 0.3 0.3 0.1 0.1
u50.3 0.3 0.5 0.3 0.1 ∗
u60.4 0.4 0.5 0.3 0.3 0.1
u70.2 0.1 0.4 0.1 0.3 0.1
Table 6 – Sum of membership and non membership degrees
U e1e2e3e4e5e6
u10.9 0.3 0.9 0.9 0.9 0.8
u21 0.9 0.7 1 1 0.8
u30.9 0.4 1 ∗0.7 0.6
u40.8 0.9 1 0.7 0.8 0.7
u50.8 0.9 0.9 1 0.9 ∗
u60.6 0.8 1 0.7 0.7 0.6
u70.9 0.9 0.9 1 0.8 0.5
Table 7 – Difference of membership and non membership degrees
U e1e2e3e4e5e6
u10.7 0.1 0.7 0.1 0.1 0.4
u20.6 0.7 0.3 0.2 0 0.4
u30.5 0.2 0.6 ∗0.5 0.2
u40.4 0.5 0.4 0.1 0.6 0.5
u50.2 0.3 0.1 0.4 0.7 ∗
u60.2 0 0 0.1 0.1 0.4
u70.5 0.7 0.1 0.8 0.2 0.3
4.2 ALGORITHM
Suppose we have to predict the unknown value of (
µF(ek)
(
ui
)
,ν
F(ek)
(
ui
)) = (
uik,v
ik
). Let
mik
,
nik
,
sik
and
tik
denote the corresponding unknown values of fuzzy soft set of membership degrees, fuzzy soft
set of non membership degrees, fuzzy soft set of sum of membership and non membership degrees and
fuzzy soft set of difference of membership and non membership degrees respectively.
1. Input the incomplete intuitionistic fuzzy soft set.
2. Compute sik =uik +vik using algorithm 3.2.
3. Compute tik =|uik −vik|using algorithm 3.2.
4. Compute mik and nik using algorithm 3.2.
5.
If
mik >n
ik
, put
|uik −vik|
=
uik −vik
otherwise put
|uik −vik|
=
vik −uik
. Accordingly we get
tik =uik −vik or tik =vik −uik.
6. Solve equations obtained from step (ii) and step (v) to get the values of uik and vik.
https://doi.org/10.17993/3cemp.2022.110250.104-114
4.3 APPLICATION
Consider the incomplete intuitionistic fuzzy soft set given in table 3. In this table there are seven objects,
six parameters and two unknown values to be predicted. Here also we suppose that the weights of
objects and parameters are equal, i.e.,
w1
=
w2
=
1
2
. Now we predict the two unknown values (
u34,v
34
)
and (u56,v
56)as follows:
For (
u34,v
34
)we obtain
m34
=0
.
5609,
n34
=0
.
2360,
s34
=0
.
8086 and
t34
=0
.
3331. Using algorithm
4.2 we get u34 =0.5709 and v34 =0.2377.
For (
u56,v
56
)we obtain
m56
=0
.
5598,
n56
=0
.
2273,
s56
=0
.
7804 and
t56
=0
.
3523. Using algorithm
4.2 we get u56 =0.5664 and v56 =0.2140.
5 CONCLUSION
This paper analyzes the effect of known data on unknown ones in an incomplete data set and proposes
algorithms to predict unknown values. The concept of Euclidean distance on
Rn
is used to measure
the distance between objects (parameters). This distance is further used in measuring the degree
of interdependence between objects (parameters). An approach to predict the missing data in an
intuitionistic fuzzy set is also given.
Our proposed methodology has the following advantages:
1.
There is only one basic algorithm (algorithm 3.2) given in this paper which is used to predict the
missing data in each of incomplete soft set, incomplete fuzzy soft set and incomplete intuitionistic
fuzzy soft set.
2.
Algorithms given in this paper makes full use of known data so that the predicted values have
higher accuracy.
3.
The basic algorithm 3.2 produces a finite sequence of predictions based on the distance and degree
of interdependence between objects (parameters).
4.
In this paper the relation between objects (parameters) is determined using degree of interde-
pendence. If the degree of interdependence between an object (parameter) and the target object
(parameter) is less, then the missing values corresponding to the target object (parameter) is less
expected to be same as the corresponding values of former object (parameter).
5. The algorithm 3.2 predicts the unknown values of incomplete soft set to be in {0,1}precisely.
6. Algorithm 4.2 predicts the unknown values of incomplete intuitionistic fuzzy soft set.
ACKNOWLEDGMENT
This work is supported by Council of Scientific & Industrial Research (CSIR), India, under Junior
Research Fellowship with Ref. No. 17/06/2018(i)EU-V.
REFERENCES
[1] Atanassov, K. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20, 87–96 pp.
[2]
Atanassov, K. (1994). Operators over interval valued intuitionistic fuzzy sets. Fuzzy Sets and
Systems, 64, 159–174 pp.
[3]
Deng, T. and Wang, X. (2013). An object-parameter approach to predicting unknown data in
incomplete fuzzy soft sets. Applied Mathematical Modelling, 37, 4139–4146 pp.
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3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
112
Table 5 – Non membership degrees
U e1e2e3e4e5e6
u10.1 0.1 0.1 0.5 0.5 0.2
u20.2 0.1 0.2 0.4 0.5 0.2
u30.2 0.1 0.2 ∗0.1 0.2
u40.2 0.2 0.3 0.3 0.1 0.1
u50.3 0.3 0.5 0.3 0.1 ∗
u60.4 0.4 0.5 0.3 0.3 0.1
u70.2 0.1 0.4 0.1 0.3 0.1
Table 6 – Sum of membership and non membership degrees
U e1e2e3e4e5e6
u10.9 0.3 0.9 0.9 0.9 0.8
u21 0.9 0.7 1 1 0.8
u30.9 0.4 1 ∗0.7 0.6
u40.8 0.9 1 0.7 0.8 0.7
u50.8 0.9 0.9 1 0.9 ∗
u60.6 0.8 1 0.7 0.7 0.6
u70.9 0.9 0.9 1 0.8 0.5
Table 7 – Difference of membership and non membership degrees
U e1e2e3e4e5e6
u10.7 0.1 0.7 0.1 0.1 0.4
u20.6 0.7 0.3 0.2 0 0.4
u30.5 0.2 0.6 ∗0.5 0.2
u40.4 0.5 0.4 0.1 0.6 0.5
u50.2 0.3 0.1 0.4 0.7 ∗
u60.2 0 0 0.1 0.1 0.4
u70.5 0.7 0.1 0.8 0.2 0.3
4.2 ALGORITHM
Suppose we have to predict the unknown value of (
µF(ek)
(
ui
)
,ν
F(ek)
(
ui
)) = (
uik,v
ik
). Let
mik
,
nik
,
sik
and
tik
denote the corresponding unknown values of fuzzy soft set of membership degrees, fuzzy soft
set of non membership degrees, fuzzy soft set of sum of membership and non membership degrees and
fuzzy soft set of difference of membership and non membership degrees respectively.
1. Input the incomplete intuitionistic fuzzy soft set.
2. Compute sik =uik +vik using algorithm 3.2.
3. Compute tik =|uik −vik|using algorithm 3.2.
4. Compute mik and nik using algorithm 3.2.
5.
If
mik >n
ik
, put
|uik −vik|
=
uik −vik
otherwise put
|uik −vik|
=
vik −uik
. Accordingly we get
tik =uik −vik or tik =vik −uik.
6. Solve equations obtained from step (ii) and step (v) to get the values of uik and vik.
https://doi.org/10.17993/3cemp.2022.110250.104-114
4.3 APPLICATION
Consider the incomplete intuitionistic fuzzy soft set given in table 3. In this table there are seven objects,
six parameters and two unknown values to be predicted. Here also we suppose that the weights of
objects and parameters are equal, i.e.,
w1
=
w2
=
1
2
. Now we predict the two unknown values (
u34,v
34
)
and (u56,v
56)as follows:
For (
u34,v
34
)we obtain
m34
=0
.
5609,
n34
=0
.
2360,
s34
=0
.
8086 and
t34
=0
.
3331. Using algorithm
4.2 we get u34 =0.5709 and v34 =0.2377.
For (
u56,v
56
)we obtain
m56
=0
.
5598,
n56
=0
.
2273,
s56
=0
.
7804 and
t56
=0
.
3523. Using algorithm
4.2 we get u56 =0.5664 and v56 =0.2140.
5 CONCLUSION
This paper analyzes the effect of known data on unknown ones in an incomplete data set and proposes
algorithms to predict unknown values. The concept of Euclidean distance on
Rn
is used to measure
the distance between objects (parameters). This distance is further used in measuring the degree
of interdependence between objects (parameters). An approach to predict the missing data in an
intuitionistic fuzzy set is also given.
Our proposed methodology has the following advantages:
1.
There is only one basic algorithm (algorithm 3.2) given in this paper which is used to predict the
missing data in each of incomplete soft set, incomplete fuzzy soft set and incomplete intuitionistic
fuzzy soft set.
2.
Algorithms given in this paper makes full use of known data so that the predicted values have
higher accuracy.
3.
The basic algorithm 3.2 produces a finite sequence of predictions based on the distance and degree
of interdependence between objects (parameters).
4.
In this paper the relation between objects (parameters) is determined using degree of interde-
pendence. If the degree of interdependence between an object (parameter) and the target object
(parameter) is less, then the missing values corresponding to the target object (parameter) is less
expected to be same as the corresponding values of former object (parameter).
5. The algorithm 3.2 predicts the unknown values of incomplete soft set to be in {0,1}precisely.
6. Algorithm 4.2 predicts the unknown values of incomplete intuitionistic fuzzy soft set.
ACKNOWLEDGMENT
This work is supported by Council of Scientific & Industrial Research (CSIR), India, under Junior
Research Fellowship with Ref. No. 17/06/2018(i)EU-V.
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