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Generalized intuitionistic fuzzy soft rough set and its application in decision making

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Generalized intuitionistic fuzzy soft rough set and its application in decision making

Abstract

Intuitionistic fuzzy set theory, soft set theory and rough set theory are mathematical tools for dealing with uncertainties and are closely related. This paper is devoted to the discussion of the combinations of intuitionistic fuzzy set, rough set and soft set. The concept of generalized intuitionistic fuzzy soft rough sets is proposed, and its properties are investigated. Furthermore, classical representations of generalized intuitionistic fuzzy soft rough approximation operators are presented. Finally, we develop an approach to generalized intuitionistic fuzzy soft rough sets based decision making and a practical example is provided to illustrate the developed approach.
Volume 20, Number 4 April 2016
ISSN:1521-1398 PRINT,1572-9206 ONLINE
Journal of
Computational
Analysis and
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EUDOXUS PRESS,LLC
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Journal of Computational Analysis and Applications
ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE
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602
TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL
ANALYSIS AND APPLICATIONS, VOL. 20, NO. 4, 2016
Modelling By Shepard-Type Curves and Surfaces, Umberto Amato, and Biancamaria Della
Vecchia,……………………………………………………………………………………..…611
Hesitant Fuzzy Set Theory Applied to BCK/BCI-Algebras, Young Bae Jun, and Sun Shin
Ahn,……………………………………………………………………………………………635
Fractional q-Integrodifference Equations and Inclusions With Nonlocal Fractional q-Integral
Conditions, Sotiris K. Ntouyas, and Jessada Tariboon,………………………………………..647
On the Solvability of a System of Multi-Point Second Order Boundary Value Problem, Tugba
Senlik Cerdik, Ilkay Yaslan Karaca, Aycan Sinanoglu, and Fatma Tokmak Fen,……………..666
Cascadic Multigrid Method for the Elliptic Monge-Ampère Equation, Zhiyong Liu,…………674
Iterative Algorithms for Zeros of Accretive Operators and Fixed Points of Nonexpansive
Mappings in Banach Spaces, Jong Soo Jung,…………………………………………………..688
Fixed Points by Some Iterative Algorithms in Banach and Hilbert Spaces with Some
Applications, S. A. Ahmed, A. El-Sayed Ahmed, Abdulfattah K. A. Bukhari, and Vesna
Cojbasic Rajic,…………………………………………………………………………………707
A Variant of Second-Order Arnoldi Method for Solving the Quadratic Eigenvalue Problem, Peng
Zhou, Xiang Wang, Ming He, and Liang-Zhi Mao,……………………………………………718
Solvability for Fractional Differential Inclusions with Fractional Nonseparated Boundary
Conditions, Xianghu Liu, Xiaoyou Liu, and Yanmin Liu,……………………………………..734
Generalized Intuitionistic Fuzzy Soft Rough Set and Its Application in Decision Making,
Haidong Zhang, Lianglin Xiong, and Weiyuan Ma,……………………………………….…..750
Global Exponential Dissipativity of Static Neural Networks with Time Delay and Impulses,
Liping Zhang, Shu-Lin Wu, and Kelin Li,………………………………………………….….767
Generalized intuitionistic fuzzy soft rough set and its
application in decision making
Haidong Zhang1
, Lianglin Xiong2, Weiyuan Ma1
1. School of Mathematics and Computer Science,
Northwest University for Nationalities,
Lanzhou, Gansu, 730030, P. R. China
2. School of Mathematics and Computer Science,
Yunnan Minzu University,
Kunming, Yunnan, 650500, P. R. China
Abstract
Intuitionistic fuzzy set theory, soft set theory and rough set theory are mathemat-
ical tools for dealing with uncertainties and are closely related. This paper is devoted
to the discussion of the combinations of intuitionistic fuzzy set, rough set and soft
set. The concept of generalized intuitionistic fuzzy soft rough sets is proposed, and its
properties are investigated. Furthermore, classical representations of generalized intu-
itionistic fuzzy soft rough approximation operators are presented. Finally, we develop
an approach to generalized intuitionistic fuzzy soft rough sets based decision making
and a practical example is provided to illustrate the developed approach.
Key words: Soft set; Rough set; Generalized intuitionistic fuzzy soft rough set;
Decision making
1 Introduction
To solve complicated problems in economics, engineering, environmental science and
social science, methods in classical mathematics are not always successful because of var-
ious types of uncertainties presented in these problems. There are several well-known
theories to describe uncertainty. For instance, fuzzy set theory [1], intuitionistic fuzzy set
theory [2,3], rough set theory [4,5] and other mathematical tools. But each of these theories
has its inherent difficulties as pointed out in [6]. Perhaps above mentioned these theories
are due to lack of parametrization tools. Theory of soft sets presented by Molodtsov [6]
has enough parameters, so that it is free from inherent difficulties of above mentioned
Corresponding author. Address: School of Mathematics and Computer Science Northwest University
for Nationalities, Lanzhou, Gansu, 730030, P. R. China. E-mail:lingdianstar@163.com
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these theories. Soft set theory deals with uncertainty and vagueness on the one hand
while on the other it has enough parametrization tools. These qualities of soft set theory
make it popular among researchers and experts working in diverse areas. Soft set theory
has potential applications in many different fields including the smoothness of functions,
game theory, operational research, Perron integration, probability theory, and measure-
ment theory [6, 7]. Presently, works on soft set theory are progressing rapidly. Maji et
al. [8] defined several operations on soft sets and made a theoretical study on the theory
of soft sets. Furthermore, based on [8], Ali et al. [9] introduced some new operations on
soft sets and improved the notion of complement of soft set. They proved that certain
De Morgans laws hold in soft set theory. Park et al [10] discussed some properties of
equivalence soft set relations. The study of hybrid models combining soft sets with other
mathematical structures is also emerging as an active research topic of soft set theory.
Maji et al. [11] initiated the study on hybrid structures involving fuzzy sets and soft sets.
They introduced the notion of fuzzy soft sets, which can be seen as a fuzzy generalization
of soft sets. Furthermore, based on [11], Maji et al [12] modified definition of fuzzy soft
sets, and presented the notion of generalized fuzzy soft sets theory. Yang et al. [13] pre-
sented the concept of the interval-valued fuzzy soft sets by combining interval-valued fuzzy
set [14,15] and soft set models. By combining the concept of trapezoidal fuzzy set and soft
set models, Xiao et al. [16] presented the concept of the trapezoidal fuzzy soft set which
can deal with certain linguistic assessments. Yang et al. [17] presented the concept of the
multi-fuzzy soft set by combining the multi-fuzzy set and soft set models, and provided
its application in decision making under an imprecise environment.
The concept of rough sets, proposed by Pawlak [4, 5] as a framework for the con-
struction of approximations of concepts, is a formal tool for modeling and processing
insufficient and incomplete information. In order to handle vagueness and imprecision in
the data equivalence relations play an important role in this theory. This theory has been
applied successfully to solve many problems, but in daily life, it is very difficult to find
an equivalence relation among the elements of a set under consideration. Therefore many
authors have generalized the notion of Pawlak rough set by using non-equivalence binary
relations. This has led to various other generalized rough set models [18–27].
Soft set theory, fuzzy set theory and rough set theory are all mathematical tools to
deal with uncertainty. It has been found that soft set, fuzzy set and rough set are closely
related concepts [28]. Feng et al. [29] provided a framework to combine fuzzy sets, rough
sets and soft sets all together, which gives rise to several interesting new concepts such as
rough soft sets, soft rough sets and soft rough fuzzy sets. The combination of soft set and
rough set models was also discussed by some researchers [30–32].
In this paper, we devote to the discussion of the combinations of intuitionistic fuzzy
set, rough set and soft set. The traditional intuitionistic fuzzy rough set [33–35] and soft
set theory are two different tools to deal with uncertainty. Apparently there is no direct
connection between these two theories. The major criticism on rough set theory is that
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it lacks parametrization tools. In order to make parametrization tools available in rough
sets, we construct an intuitionistic fuzzy soft relation between the universe set Uand the
parameter set Ein intuitionistic fuzzy soft set so that it is natural to combine intuitionistic
fuzzy rough set and soft set theory. So the concept of generalized intuitionistic fuzzy
soft rough sets is propose, and its some properties are discussed. Then the relationships
between generalized intuitionistic fuzzy soft rough sets and the existing generalized soft
rough sets are also established. We finally present an illustrative example which show
that the decision making method of generalized intuitionistic fuzzy soft rough sets can be
successfully applied to many problems that contain uncertainties.
The rest of this paper is organized as follows. Section 2 briefly reviews some preliminar-
ies. In section 3, we construct the crisp soft rough approximation operators, and discuss
their some interesting properties. In Section 4, an intuitionistic fuzzy soft relation is first
defined by us. By combining the intuitionistic fuzzy soft relation with intuitionistic fuzzy
rough sets, then the concept of generalized intuitionistic fuzzy soft rough approximation
operators is presented and the properties of the generalized upper and lower intuitionistic
fuzzy soft rough approximation operators are examined. Furthermore, classical representa-
tions of generalized intuitionistic fuzzy soft rough approximation operators are presented.
Section 5 is devoted to studying the application of generalized intuitionistic fuzzy soft
rough sets. Some conclusions and outlooks for further research are given in Section 6.
2 Preliminaries
In this section, we shall briefly recall some basic notions being used in the study.
Definition 2.1 ( [36]) Let L={(µ, ν)[0,1]×[0,1]|µ+ν1}and denote (µ1, ν1)L
(µ2, ν2)µ1µ2and ν1ν2,(µ1, ν1),(µ2, ν2)L.Then the pair (L,L)is
called a complete lattice. The operators and on (L,L)are defined as follows:for
(µ1, ν1),(µ2, ν2)L,
(µ1, ν1)(µ2, ν2) = (min{µ1, µ2}, max{ν1, ν2}),
(µ1, ν1)(µ2, ν2) = (max{µ1, µ2}, min{ν1, ν2}),
Obviously, a complete lattice on Lhas the smallest element 0L= (0,1) and the
greatest element 1L= (1,0). The definitions of fuzzy logical operators can be straight-
forwardly extended to the intuitionistic fuzzy case. The strict partial order <Lis defined
by
(µ1, ν1)<L(µ2, ν2)(µ1, ν1)L(µ2, ν2)and (µ1, ν1)L(µ2, ν2).
In the following, we review the concept of the intuitionistic fuzzy set introduced by
Atanassov [2, 3].
Definition 2.2 ( [2, 3]) Let a set Ube fixed. An intuitionistic fuzzy (IF, for short) set
Ain Uis an object having the form
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A={< x, µA(x), γA(x)>|xU},
where µA:U[0,1],and γA:U[0,1],satisfy 0µA(x) + γA(x)1for all xU,
and µA(x)and γA(x)are, respectively, called the degree of membership and the degree of
non-membership of the element xUto A.
The family of all intuitionistic fuzzy subsets in Uis denoted by IF (U). The comple-
ment of an IF set Ais denoted by A={< x, γA(x), µA(x)>|xU}.
Obviously, every fuzzy set A={< x, A(x)>|xU}={< x, µA(x)>|xU}can be
identified with the IF set of the form {< x, µA(x),1µA(x)>|xU}and is thus an IF
set.
The basic operations on I F (U) are defined as follows [2,3,37–39]: for all A, B I F (U),
(1) ABiff µA(x)µB(x) and γA(x)γB(x) for all xU,
(2) A=Biff ABand BA,
(3) AB={< x, min{µA(x), µB(x)}, max{γA(x), γB(x)>|xU},
(4) AB={< x, max{µA(x), µB(x)}, min{γA(x), γB(x)>|xU}.
For (α, β )L,
\
(α, β) denotes a constant IF set:
\
(α, β)(x) = {< x, α, β > |xU}
,where α, β [0,1] and α+β1; The IF universe set is U= 1U=
[
(1,0) = {< x, 1,0>
|xU}and the IF empty set is = 0U=
[
(0,1) = {< x, 0,1>|xU}.
Definition 2.3 ( [33, 35]) Let A={< x, µA(x), γA(x)>|xU} ∈ IF (U), and (α, β )
L. The (α, β )-level cut set of A, denoted by Aβ
α, is defined as follows:
Aβ
α={xU|µA(x)α, γA(x)β}.
Aα={xU|µA(x)α},and Aα+={xU|µA(x)> α},are, respectively, called
the α-level cut set and the strong α-level cut set of membership generated by A. And
Aβ={xU|γA(x)β}and Aβ+={xU|γA(x)< β}are, respectively, referred to as
the β-level cut set and the strong β-level cut set of non-membership generated by A.
At the same time, other types of cut sets of the IF set Aare denoted as follows:
Aβ
α+={xU|µA(x)> α, γA(x)β},which is called the (α+, β)-level cut set of A;
Aβ+
α={xU|µA(x)α, γA(x)< β},which is called the (α, β+)-level cut set of A;
Aβ+
α+={xU|µA(x)> α, γA(x)< β},which is called the (α+, β+)-level cut set of A.
Theorem 2.4 ( [33, 35]) The cut sets of IF sets satisfy the following properties: A
I F (U), α, β [0,1] with α+β1,
(1) Aβ
α=AαAβ,
(2) (A)α=Aα+,(A)β=Aβ+.
Definition 2.5 ( [22, 24]) Let Ube a nonempty and finite universe of discourse and R
U×Uan arbitrary crisp relation on U. We define a set-valued function Rs:UP(U)
by Rs(x) = {yU|(x, y)R}, x U.
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The pair (U, R)is called a crisp approximation space. For any AU, the upper and
lower approximations of Awith respect to (U, R), denoted by R(A)and R(A), are defined,
respectively, as follows:
R(A) = {xU|Rs(x)A6=∅}, R(A) = {xU|Rs(x)A}.
The pair (R(A), R(A)) is referred to as a crisp rough set, and R, R :P(U)P(U)
are, respectively, referred to as upper and lower crisp approximation operators induced
from (U, R).
3 Construction of crisp soft rough sets
In this section, we will introduce the concept of crisp soft rough sets by combining the
crisp soft relation from Uto Ewith crisp rough sets.
Definition 3.1 ( [6]) Let Ube an initial universe set and Ebe a universe set of param-
eters. A pair (F, E)is called a soft set over Uif F:EP(U),where P(U)is the set of
all subsets of U.
Definition 3.2 ( [11]) Let Ube an initial universe set and Ebe a universe set of pa-
rameters. A pair (F, E)is called a fuzzy soft set over Uif F:EF(U),where F(U)is
the set of all fuzzy subsets of U.
By using the concepts of the above soft set and fuzzy soft set, Cagman et al. [40,41]
introduce the definitions of crisp soft relation and fuzzy soft relation, respectively.
Definition 3.3 ( [40]) Let (F, E)be a soft set over U. Then a subset of U×Ecalled a
crisp soft relation from Uto Eis uniquely defined by
R={<(u, x), µR(u, x)>|(u, x)U×E},
where µR:U×E→ {0,1}, µR(u, x) = (1,(u, x)R
0,(u, x)/R.
Definition 3.4 ( [41]) Let (F, E)be a fuzzy soft set over U. Then a fuzzy subset of U×E
called a fuzzy soft relation from Uto Eis uniquely defined by
R={<(u, x), µR(u, x)>|(u, x)U×E},
where µR:U×E[0,1], µR(u, x) = µF(x)(u).
Based the crisp soft relation proposed by Cagman, we can construct the following crisp
soft rough sets.
Definition 3.5 Let Ube an initial universe set and Ebe a universe set of parameters.
For an arbitrary crisp soft relation Rover U×E, we can define a set-valued function
Rs:UP(E)by Rs(u) = {xE|(u, x)R}, u U.
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Ris referred to as serial if for all uU, Rs(u)6=. The pair (U, E, R)is called a crisp
soft approximation space. For any AE, the upper and lower soft approximations of A
with respect to (U, E, R), denoted by R(A)and R(A), are defined, respectively, as follows:
R(A) = {uU|Rs(u)A6=∅},(1)
R(A) = {uU|Rs(u)A}.(2)
The pair (R(A), R(A)) is referred to as a crisp soft rough set, and R, R :P(E)P(U)
are, referred to as upper and lower crisp soft rough approximation operators, respectively.
Example 3.6 Let Ube a universal set, which is denoted by U={u1, u2, u3, u4, u5}. Let
Ebe a set of parameters, where E={e1, e2, e3, e4}. Suppose that a soft set over Uis
defined as follows:
F(e1) = {u1, u3, u4}, F (e2) = {u2, u4}, F (e3) = , F (e4) = U.
Then the crisp soft relation on U×Eis written by
R={(u1, e1),(u3, e1),(u4, e1),(u2, e2),(u4, e2),(u1, e4),(u2, e4),(u3, e4),(u4, e4),(u5, e4)}.
If the set of parameter A={e2, e3, e4},by Equations (1) and (2), we have R(A) =
{u2, u5},and R(A) = U.
Theorem 3.7 Let (U, E, R)be a crisp soft approximation space. Then upper and lower
crisp soft approximation operators R(A)and R(A)in Definition 3.5 satisfy the following
properties: for all A, B P(E)
(CSL1) R(A) =R(A),(CSU1) R(A) =R(A);
(CSL2) R(AB) = R(A)R(B),(CSU2) R(AB) = R(A)R(B);
(CSL3) ABR(A)R(B),(CSU3) ABR(A)R(B);
(CSL4) R(AB)R(A)R(B),(CSU4) R(AB) = R(A)R(B).
Proof. The proof can be directly followed from Definition 3.5. 2
From Definition 3.5, the following theorem can be easily derived.
Theorem 3.8 Let (U, E, R)be a crisp soft approximation space, and R(A)and R(A)be
the upper and lower soft approximations operators in Definition 3.5. Then
Ris serial R(A)R(A),AER() = ∅ ⇔ R(E) = U.
4 Construction of generalized intuitionistic fuzzy soft rough
sets
In this section, inspired by the constructive method regard to generalized intuitionistic
fuzzy rough sets in [35], we will present the concept of generalized intuitionistic fuzzy soft
rough sets by combining the intuitionistic fuzzy soft relation from Uto Ewith generalized
intuitionistic fuzzy rough sets.
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Definition 4.1 ( [42]) Let Ube an initial universe set and Ebe a universe set of pa-
rameters. A pair (F, E)is called an intuitionistic fuzzy soft set over Uif F:EIF (U),
where IF (U)is the set of all intuitionistic fuzzy subsets of U. That is, for xE, F (x) =
{< u, µF(x)(u), γF(x)(u)>|uU} ∈ I F (U), where µF(x)(u)[0,1] and γF(x)(u)[0,1]
denote membership and non-membership degrees of an element uregard to intuitionistic
fuzzy set F(x)respectively, which satisfy the condition µF(x)(u) + γF(x)(u)1.
In the following, an intuitionistic fuzzy soft relation will be presented by us which is
important for us to construct generalized intuitionistic fuzzy soft rough sets.
Definition 4.2 Let (F, E)be an intuitionistic fuzzy soft set over U. Then an intuitionistic
fuzzy subset of U×Ecalled an intuitionistic fuzzy soft relation from Uto Eis uniquely
defined by
R={<(u, x), µR(u, x), γR(u, x)>|(u, x)U×E},
where µR:U×E[0,1] and γR:U×E[0,1] satisfy the condition 0µR(u, x) +
γR(u, x)1for all (u, x)U×E.
If U={u1, u2,· · · , um}, E ={x1, x2,· · · , xn}then the intuitionistic fuzzy soft relation
Rfrom Uto Ecan be presented by a table as in the following form
R x1x2· · · xn
u1(µR(u1, x1), γR(u1, x1)) (µR(u1, x2), γR(u1, x2)) · · · (µR(u1, xn), γR(u1, xn))
u2(µR(u2, x1), γR(u2, x1)) (µR(u2, x2), γR(u2, x2)) · · · (µR(u2, xn), γR(u2, xn))
.
.
..
.
..
.
.....
.
.
um(µR(um, x1), γR(um, x1)) (µR(um, x2), γR(um, x2)) · · · (µR(um, xn), γR(um, xn))
From the above form and according to the definition of IF soft relation, we could find
that every IF soft set (F, E) is uniquely characterized by the IF soft relation, namely they
are mutual determined. It means that an IF soft set (F, E) is formally equal to its IF soft
relation. Therefore, we shall identify any IF soft set with its IF soft relation and view
these two concepts as interchangeable. Now, any discussion regard to IF soft set could be
converted into analysis about IF soft relation, which will bring great convenience for our
future researches.
In this case, according to the definition IF soft relation, we can construct generalized
intuitionistic fuzzy soft rough sets.
Definition 4.3 Let Ube an initial universe set and Ebe a universe set of parameters.
For an arbitrary intuitionistic fuzzy soft relation Rover U×E, the pair (U, E , R)is called
an intuitionistic fuzzy soft approximation space. For any AIF (E),we define the upper
and lower soft approximations of Awith respect to (U, E, R), denoted by R(A)and R(A),
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respectively, as follows:
R(A) = {< u, µR(A)(u), γR(A)(u)>|uU},(3)
R(A) = {< u, µR(A)(u), γR(A)(u)>|uU}.(4)
where
µR(A)(u) = W
xE
[µR(u, x)µA(x)], γR(A)(u) = V
xE
[γR(u, x)) γA(x)];
µR(A)(u) = V
xE
[γR(u, x)µA(x)], γR(A)(u) = W
xE
[µR(u, x)γA(x)].
The pair (R(A), R(A)) is referred to as a generalized IF soft rough set of Awith respect
to (U, E, R).
By µR(u, x) + γR(u, x)1 and µA(x) + γA(x)1, it can be easily verified that R(A)
and R(A)I F (U).In fact,
µR(A)(u) + γR(A)(u) = _
xE
[µR(u, x)µA(x)] + ^
xE
[(γR(u, x)) γA(x)]
_
xE
[(1 γR(u, x)) (1 γA(x))] + ^
xE
[(γR(u, x)) γA(x)]
= 1 ^
xE
[(γR(u, x)) γA(x)] + ^
xE
[(γR(u, x)) γA(x)]
= 1.
Hence, R(A)I F (U).Similarly, we can obtain R(A)IF (U).So we call R, R :I F (E)
I F (U) generalized upper and lower IF soft rough approximation operators, respectively.
Remark 4.4 If γR(u, x) = 1µR(u, x)in Definition 4.2, then Ris a fuzzy soft relation on
U×E(see Definition 3.4), that is, R={<(u, x), µR(u, x),1µR(u, x)>|(u, x)U×E}.
In this case, we call (U, E, R)a fuzzy soft approximation space. Let (U, E, R)be the fuzzy
soft approximation space and AIF (E), then generalized IF soft rough approximation
operators R(A)and R(A)in Definition 4.3 degenerate to the following forms:
R(A) = {< u, µR(A)(u), γR(A)(u)>|uU},
R(A) = {< u, µR(A)(u), γR(A)(u)>|uU}.
where
µR(A)(u) = W
xE
[µR(u, x)µA(x)], γR(A)(u) = V
xE
[(1 µR(u, x)) γA(x)];
µR(A)(u) = V
xE
[(1 µR(u, x)) µA(x)], γR(A)(u) = W
xE
[µR(u, x)γA(x)].
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In this case, the pair (R(A), R(A)) is referred to as an IF soft rough set of Awith
respect to (U, E, R). That is, generalized IF soft rough set in Definition 4.3 has included
IF soft rough set.
Remark 4.5 Let (U, E, R)be a fuzzy soft approximation space. If AF(E),then gen-
eralized IF soft rough approximation operators R(A)and R(A)degenerate to the following
forms:
R(A) = {< u, µR(A)(u)>|uU}, R(A) = {< u, µR(A)(u)>|uU}.
where µR(A)(u) = W
xE
[µR(u, x)µA(x)], µR(A)(u) = V
xE
[(1 µR(u, x)) µA(x).
In this case, generalized IF soft rough approximation operators R(A)and R(A)are
identical with the soft fuzzy rough approximation operators defined by Sun [32]. That is,
generalized IF soft rough approximation operators in Definition 4.3 are an extension of
the soft fuzzy rough approximation operators defined by Sun [32].
In order to better understand the concept of generalized IF soft rough approximation
operators, let us consider the following example.
Example 4.6 Suppose that U={u1, u2, u3, u4, u5}is the set of five houses under con-
sideration of a decision maker to purchase. Let Ebe a parameter set, where E=
{e1, e2, e3, e4}={expensive; beautiful; size; location}. Mr. X wants to buy the house which
qualifies with the parameters of Eto the utmost extent from available houses in U. Assume
that Mr. X describes the “attractiveness of the houses” by constructing an IF soft relation
Rfrom Uto E. And it is presented by a table as in the following form.
R e1e2e3e4
u1(0.7,0.2) (0.6,0.3) (0.1,0.9) (0.1,0.6)
u2(0.2,0.6) (0.6,0.4) (0.5,0.4) (0.7,0.3)
u3(0.4,0.6) (0.8,0.1) (0.2,0.7) (0.6,0.3)
u4(0.2,0.5) (0.3,0.6) (0.8,0.1) (0.1,0.7)
u5(0.6,0.4) (0.5,0.2) (0.3,0.5) (0.2,0.5)
As a generalization of Zadeh’s fuzzy set, intuitionistic fuzzy set is characterized by
a membership function and a nonmembership function, and thus can depict the fuzzy
character of data more detailedly and comprehensively than Zadehs fuzzy set which is only
characterized by a membership function. Therefore, the characteristics of the five houses
with respect to the four parameters can be represented by the intuitionistic fuzzy sets. For
example, the characteristics of the house u1under the parameter e1is (0.7,0.2). The
values of 0.7 and 0.2 are the degrees of membership and non-membership of the house u1
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with respect to the parameter e1, respectively. In other words, house u1is expensive on
the membership degree of 0.7 and it is not expensive on the non-membership degree of 0.2.
Now suppose that Mr X gives the optimum normal decision object Awhich an IF subset
over the parameter set Eas follows:
A={< e1,0.7,0.1>, < e2,0.3,0.6>, < e3,0.8,0.2>, < e4,0.3,0.5>}
By Equations (3) and (4), we have
µR(A)(u1)=0.7, γR(A)(u1)=0.2, µR(A)(u2) = 0.5, γR(A)(u2)=0.4,
µR(A)(u3)=0.4, γR(A)(u3)=0.5, µR(A)(u4) = 0.8, γR(A)(u4)=0.2,
µR(A)(u5)=0.6, γR(A)(u5)=0.4; µR(A)(u1) = 0.3, γR(A)(u1)=0.6,
µR(A)(u2)=0.3, γR(A)(u2)=0.6, µR(A)(u3) = 0.3, γR(A)(u3)=0.6,
µR(A)(u4)=0.3, γR(A)(u4)=0.3, µR(A)(u5) = 0.3, γR(A)(u5)=0.5.
Thus
R(A) = {< u1,0.7,0.2>, < u2,0.5,0.4>, < u3,0.4,0.5>,
< u4,0.8,0.2>, < u5,0.6,0.4>}
and
R(A) = {< u1,0.3,0.6>, < u2,0.3,0.6>, < u3,0.3,0.6>,
< u4,0.3,0.3>, < u5,0.3,0.5>}.
Theorem 4.7 Let (U, E, R)be an intuitionistic fuzzy soft approximation space. Then
the generalized upper and lower IF soft rough approximation operators R(A)and R(A)in
Definition 4.3 satisfy the following properties: A, B I F (E),(α, β)L,
(GIFSL1) R(A) =R(A),(GIFSU1) R(A) =R(A);
(GIFSL2) R(A
\
(α, β)) = R(A)
\
(α, β),(GIFSU2) R(A
\
(α, β)) = R(A)
\
(α, β);
(GIFSL3) R(AB) = R(A)R(B),(GIFSU3) R(AB) = R(A)R(B);
(GIFSL4) ABR(A)R(B),(GIFSU4) ABR(A)R(B);
(GIFSL5) R(AB)R(A)R(B),(GIFSU5) R(AB) = R(A)R(B);
Proof. It can be easily followed from Definition 4.3. 2
In Theorem 4.7, properties (GIFSL1) and (GIFSU1) show that the generalized upper
lower IF soft rough approximation operators Rand Rare dual to each other.
Assume that Ris an intuitionistic fuzzy soft relation from Uto E, denote
Rα={(u, x)U×E|µR(u, x)α}, Rα(u) = {xE|µR(u, x)α}, α [0,1],
Rα+={(u, x)U×E|µR(u, x)> α}, Rα+(u) = {xE|µR(u, x)> α}, α [0,1),
Rα={(u, x)U×E|γR(u, x)α}, Rα(u) = {xE|γR(u, x)α}, α [0,1],
Rα+={(u, x)U×E|γR(u, x)< α}, Rα+(u) = {xE|γR(u, x)< α}, α (0,1].
Then Rα,Rα+,Rα, and Rα+are crisp soft relations on U×E.
The following Theorems 4.8 and 4.9 show that the generalized IF soft rough approxi-
mation operators can be represented by crisp soft rough approximation operators.
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Theorem 4.8 Let (U, E, R)be an intuitionistic fuzzy soft approximation space, and A
I F (E). Then the generalized upper IF soft rough approximation operator can be repre-
sented as follows: uU
(1)
µR(A)(u) = _
α[0,1]
[αRα(Aα)(u)] = _
α[0,1]
[αRα(Aα+)(u)]
=_
α[0,1]
[αRα+(Aα)(u)] = _
α[0,1]
[αRα+(Aα+)(u)],
(2)
γR(A)(u) = ^
α[0,1]
[αRα(Aα)(u)] = ^
α[0,1]
[αRα(Aα+)(u)]
=^
α[0,1]
[αRα+(Aα)(u)] = ^
α[0,1]
[αRα+(Aα+)(u)]
and moreover, for any α[0,1],
(3) [R(A)]α+Rα+(Aα+)Rα+(Aα)Rα(Aα)[R(A)]α,
(4) [R(A)]α+Rα+(Aα+)Rα+(Aα)Rα(Aα)[R(A)]α.
Proof. (1) For any uU, we have that
_
α[0,1]
[αRα(Aα)(u)] = sup{α[0,1]|uRα(Aα)}
=sup{α[0,1]|Rα(u)Aα6=∅}
=sup{α[0,1]|∃xE[xRα(u), x Aα]}
=sup{α[0,1]|∃xE[µR(u, x)α, µA(x)α]}
=_
xE
[µR(u, x)µA(x)] = µR(A)(u)
Similarly, we can prove
µR(A)(u) = W
α[0,1]
[αRα(Aα+)(u)] = W
α[0,1]
[αRα+(Aα)(u)] = W
α[0,1]
[αRα+(Aα+)(u)].
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(2) By the definition of upper crisp soft rough approximation operator in Definition
3.5, we have that
^
α[0,1]
[αRα(Aα)(u))] = inf{α[0,1]|uRα(Aα)}
=inf{α[0,1]|Rα(u)Aα6=∅}
=inf{α[0,1]|∃xE[xRα(u), x Aα]}
=inf{α[0,1]|∃xE[γR(u, x)α, γA(x)α]}
=^
xE
[γR(u, x)γA(x)] = γR(A)(u).
Likewise, we can prove that
γR(A)(u) = ^
α[0,1]
[αRα(Aα+)(u)] = ^
α[0,1]
[αRα+(Aα)(u)]
=^
α[0,1]
[αRα+(Aα+)(u)].
(3) It is easily verified that Rα+(Aα+)Rα+(Aα)Rα(Aα). We only need to prove
that [R(A)]α+Rα+(Aα+) and Rα(Aα)[R(A)]α.
In fact, u[R(A)]α+, we have µR(A)(u)> α. According to Definition 4.3, W
xE
[µR(u, x)
µA(x)] > α holds. Then x0E, such that µR(u, x0)µA(x0)> α, that is, µR(u, x0)> α
and µA(x0)> α. Thus x0Rα+(u) and x0Aα+. Consequently, Rα+(u)Aα+6=. By
Definition 3.5, we have uRα+(Aα+).Hence [R(A)]α+Rα+(Aα+).
On the other hand, for any uRα(Aα),we have Rα(Aα)(u) = 1. Since µR(A)(u) =
W
β[0,1]
[βRβ(Aβ)(u)] αRα(Aα)(u) = α, we obtain u[R(A)]α.Hence, Rα(Aα)
[R(A)]α.
(4) Similar to the proof of (3), it can be easily verified. 2
Theorem 4.9 Let (U, E, R)be an intuitionistic fuzzy soft approximation space, and A
I F (E). Then the generalized lower IF soft rough approximation operator can be repre-
sented as follows: uU
(1)
µR(A)(u) = ^
α[0,1]
[α(1 Rα(Aα+)(u))] = ^
α[0,1]
[α(1 Rα(Aα)(u))]
=^
α[0,1]
[α(1 Rα+(Aα+)(u))] = ^
α[0,1]
[α(1 Rα+(Aα)(u))],
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(2)
γR(A)(u) = _
α[0,1]
[α(1 Rα(Aα+)(u))] = _
α[0,1]
[α(1 Rα(Aα)(u))]
=_
α[0,1]
[α(1 Rα+(Aα+)(u))] = _
α[0,1]
[α(1 Rα+(Aα)(u))]
and moreover, for any α[0,1],
(3) [R(A)]α+Rα(Aα+)Rα+(Aα+)Rα+(Aα)[R(A)]α,
(4) [R(A)]α+Rα(Aα+)Rα+(Aα+)Rα+(Aα)[R(A)]α.
Proof. (1) and (2) According to Theorem 4.8 and 2.4, for all uUwe have
µR(A)(u) = _
α[0,1]
[αRα(A)α(u)] = _
α[0,1]
[αRα(Aα+)(u)]
=_
α[0,1]
[α(Rα(Aα+))(u)] = _
α[0,1]
[α(1 Rα(Aα+)(u))]
and
γR(A)(u) = ^
α[0,1]
[αRα(A)α(u)] = ^
α[0,1]
[αRα(Aα+)(u)]
=^
α[0,1]
[α(Rα(Aα+))(u)] = ^
α[0,1]
[α(1 Rα(Aα+)(u))]
Hence, by the duality of generalized upper and lower IF soft rough approximation operators
(see Theorem 4.7), we can conclude
µR(A)(u) = γR(A)(u) = V
α[0,1]
[α(1 Rα(Aα+)(u))],
γR(A)(u) = µR(A)(u) = W
α[0,1]
[α(1 Rα(Aα+)(u))].
Similar to the above proof, we can obtain
µR(A)(u) = ^
α[0,1]
[α(1 Rα(Aα)(u))] = ^
α[0,1]
[α(1 Rα+(Aα+)(u))]
=^
α[0,1]
[α(1 Rα+(Aα)(u))],
γR(A)(u) = _
α[0,1]
[α(1 Rα(Aα)(u))] = _
α[0,1]
[α(1 Rα+(Aα+)(u))]
=_
α[0,1]
[α(1 Rα+(Aα)(u))].
(3) and (4) It is similar to the proof of Theorem 4.8(3). 2
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5 Application of IF soft rough sets in decision making
In this section, we shall develop an approach to generalized IF soft rough sets based
decision making. In the following, we will define the ring sum operation of IF sets. By the
operation, an approach to generalized intuitionistic fuzzy soft rough sets based decision
making will be presented.
Definition 5.1 Let F, G I F (U). The ring sum operation about IF sets Fand Gcan
be defined as follows:
FG={< u, µF(u) + µG(u)µF(u)·µG(u), γF(u)·γG(u)>|uU}.
Let (U, E, R) be an intuitionistic fuzzy soft approximation space, where Uis the uni-
verse of the discourse, Eis the parameter set, and Ris an intuitionistic fuzzy soft relation
on U×E. Then we can give an algorithm based on generalized IF soft rough sets as
follows:
1. Input the intuitionistic fuzzy soft relation Rfrom Uto E, or the intuitionistic fuzzy
soft set (F, E) over U.
2. Give the optimum normal decision object Awhich is an IF set over E, according
to different needs to decision maker.
3. Compute the generalized IF soft rough approximation operators R(A) and R(A) by
Equations (3) and (4).
4. Compute the choice set
H=R(A)R(A) = {< u,µR(A)(u) + µR(A)(u)µR(A)(u)·µR(A)(u),
γR(A)(u)·γR(A)(u)>|uU}.
5. Choose the top-level threshold value λ= (µ, γ)L,where µ= max
1inµH(ui), γ =
min
1inγH(ui).
6. The decision is u, if IF set H(u)Lλ, that is, µH(u)µand λH(u)γ.
In the last step of the above algorithm, one may go back to the second step and change
decision object so that the final decision is only one, when there exist too many “optimal
choices” to be chosen.
To illustrate the idea of algorithm given above, let us consider the example as follows.
Example 5.2 Reconsider Example 4.6. In Example 4.6, we have computed generalized
IF soft rough approximation operators R(A)and R(A)of the optimum normal decision
object A. Now by using the fourth step of algorithm for generalized IF soft rough sets in
decision making presented in this section, we can obtain
H=R(A)R(A) = {< u1,0.79,0.12 >, < u2,0.65,0.24 >, < u3,0.58,0.30 >,
< u4,0.86,0.06 >, < u5,0.72,0.20 >}.
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Obviously, the optimal decision is u4. Hence, Mr X will buy the house u4.
6 Conclusion
Intuitionistic fuzzy set theory, soft set theory and rough set theory are all mathemat-
ical tools for dealing with uncertainties. This paper is devoted to the discussion of the
combinations of intuitionistic fuzzy set, rough set and soft set. Based on the models pre-
sented in [35], by combining soft set theory with generalized IF rough set theory, a new
soft rough set model called generalized IF soft rough set is proposed and its properties
are derived. Furthermore, the relationships between generalized intuitionistic fuzzy soft
rough sets with the existing generalized soft rough sets are established. Finally, a practical
application based on generalized intuitionistic fuzzy soft rough sets is applied to show the
validity.
Actually, there are at least two aspects in the study of rough set theory: constructive
and axiomatic approaches. In further research, the axiomatization of generalized intu-
itionistic fuzzy soft rough approximation operators is an important and interesting issue
to be addressed.
Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments
and suggestions. This work is supported by the National Natural Science Foundation of
China (No. 11461082) and the Fundamental Research Funds for the Central Universities
of Northwest University for Nationalities (No. 31920130003).
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The basic ideas of rough sets and intuitionistic fuzzy sets (IFSs) are precise statistical instruments that can handle vague knowledge easily. The EDAS (evaluation based on distance from average solution) approach plays an important role in decision-making issues, particularly when multicriteria group decision-making (MCGDM) issues have more competing criteria. The purpose of this paper is to introduce the intuitionistic fuzzy rough Frank EDAS (IFRF-EDAS) methodology based on IF rough averaging and geometric aggregation operators. We proposed various aggregation operators such as IF rough Frank weighted averaging (IFRFWA), IF rough Frank ordered weighted averaging (IFRFOWA), IF rough Frank hybrid averaging (IFRFHA), IF rough Frank weighted geometric (IFRFWG), IF rough Frank ordered weighted geometric (IFRFOWG), and IF rough Frank hybrid geometric (IFRFHG) on the basis of Frank t-norm and Frank t-conorm. Information is given for the basic favorable features of the analyzed operator. For the suggested operators, a new score and precision functions are described. Then, using the suggested method, the IFRF-EDAS method for MCGDM and its stepwise methodology are shown. After this, a numerical example is given for the established model, and a comparative analysis is generally articulated for the investigated models with some previous techniques, showing that the investigated models are much more efficient and useful than the previous techniques.
... Zhang [35] proposed the generalized IFRS based on IF covering. Zhang et al. [36] extended the notion of generalized IF soft rough set based on IF soft relation. Hussain et al. [37]- [39] presented the generalized notions of IFS by using the concept of soft set and rough set under Pythagorean and Orthopair fuzzy environment. ...
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The primitive notions of rough sets and intuitionistic fuzzy set (IFS) are general mathematical tools having the ability to handle the uncertain and imprecise knowledge easily. EDAS (Evaluation based on distance from average solution) method has a significant role in decision making problems, especially when more conflicting criteria exist in multicriteria group decision making (MCGDM) problems. The aim of this manuscript is to present intuitionistic fuzzy rough- EDAS (IFR-EDAS) method based on IF rough averaging and geometric aggregation operators. In addition, we put forward the concept of IF rough weighted averaging (IFRWA), IF rough ordered weighted averaging (IFROWA) and IF rough hybrid averaging (IFRHA) aggregation operators. Furthermore, the concepts of IF rough weighted geometric (IFRWG), IF rough ordered weighted geometric (IFROWG) and IF rough hybrid geometric (IFRHG) aggregation operators are investigated. The basic desirable characteristics of the investigated operator are given in detail. A new score and accuracy functions are defined for the proposed operators. Next, IFR-EDAS model for MCGDM and their stepwise algorithm are demonstrated by utilizing the proposed approach. Finally, a numerical example for the developed model is presented and a comparative study of the investigated models with some existing methods are expressed broadly which show that the investigated models are more effective and useful than the existing approaches.
... Zhang et al. in [29] investigated the notion of IF soft rough set (IFSRS) and proposed its application in DM. The concept of generalized IFSRS was presented by Zhang et al. [30] and studied their applications in DM. Zhan and Sun [31] introduced three classes of coverings based IF rough set model via IFβ-neighbourhoods and IF complementary β-neighbourhood and studied their application in MCDM. ...
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... Zhang et al. in [29] investigated the notion of IF soft rough set (IFSRS) and proposed its application in DM. The concept of generalized IFSRS was presented by Zhang et al. [30] and studied their applications in DM. Zhan and Sun [31] introduced three classes of coverings based IF rough set model via IFβ-neighbourhoods and IF complementary β-neighbourhood and studied their application in MCDM. ...
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Chapter
The Rough set theory is a very successful tool to deal with vague, inconsistent, imprecise and uncertain knowledge. In recent years, rough set theory and its applications have drawn many researchers' interest progressively in one of its hot issues, viz. the field of artificial intelligence. An intuitionistic fuzzy (IF) set, which is a generalization of fuzzy set, has more practical and flexible real‐world proficiency to characterize a complex information and provide a better glimpse to confront uncertainty and ambiguity when compared with those of the fuzzy set. Moreover, rough sets and intuitionistic fuzzy sets deal with the specific aspects of the same problem & imprecision, and their combination IF rough set has been studied by many researchers in the past few years. The present chapter deals with a review of IF rough set theory, their basic concepts, properties, topological structures, logic operators, approximation operators and similarity relations on the basis of axiomatic and constructive approaches. The characterization of IF rough sets based on various operators, similarity relations, distances, IF cut sets, IF coverings and inclusion degrees is also discussed. Moreover, several extensions of IF rough sets and their hybridization with other extended rough set theories are thoroughly surveyed. The applications of IF rough sets in different real‐world problems are also discussed in detail.
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