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Volume 20, Number 4 April 2016

ISSN:1521-1398 PRINT,1572-9206 ONLINE

Journal of

Computational

Analysis and

Applications

EUDOXUS PRESS,LLC

601

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+ν≤1}and denote (µ1, ν1)≤L∗

(µ2, ν2)⇔µ1≤µ2and ν1≥ν2,∀(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 L∗has 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 <L∗is 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)>|x∈U},

where µA:U→[0,1],and γA:U→[0,1],satisfy 0≤µA(x) + γA(x)≤1for all x∈U,

and µA(x)and γA(x)are, respectively, called the degree of membership and the degree of

non-membership of the element x∈Uto 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)>|x∈U}.

Obviously, every fuzzy set A={< x, A(x)>|x∈U}={< x, µA(x)>|x∈U}can be

identiﬁed with the IF set of the form {< x, µA(x),1−µA(x)>|x∈U}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) A⊆Biﬀ µA(x)≤µB(x) and γA(x)≥γB(x) for all x∈U,

(2) A=Biﬀ A⊆Band B⊆A,

(3) A∩B={< x, min{µA(x), µB(x)}, max{γA(x), γB(x)>|x∈U},

(4) A∪B={< x, max{µA(x), µB(x)}, min{γA(x), γB(x)>|x∈U}.

For (α, β )∈L∗,

\

(α, β) denotes a constant IF set:

\

(α, β)(x) = {< x, α, β > |x∈U}

,where α, β ∈[0,1] and α+β≤1; The IF universe set is U= 1U=

[

(1,0) = {< x, 1,0>

|x∈U}and the IF empty set is ∅= 0U=

[

(0,1) = {< x, 0,1>|x∈U}.

Deﬁnition 2.3 ( [33, 35]) Let A={< x, µA(x), γA(x)>|x∈U} ∈ IF (U), and (α, β )∈

L∗. The (α, β )-level cut set of A, denoted by Aβ

α, is deﬁned as follows:

Aβ

α={x∈U|µA(x)≥α, γA(x)≤β}.

Aα={x∈U|µA(x)≥α},and Aα+={x∈U|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={x∈U|γA(x)≤β}and Aβ+={x∈U|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+={x∈U|µA(x)> α, γA(x)≤β},which is called the (α+, β)-level cut set of A;

Aβ+

α={x∈U|µA(x)≥α, γA(x)< β},which is called the (α, β+)-level cut set of A;

Aβ+

α+={x∈U|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+β≤1,

(1) Aβ

α=Aα∩Aβ,

(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:U→P(U)

by Rs(x) = {y∈U|(x, y)∈R}, x ∈U.

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The pair (U, R)is called a crisp approximation space. For any A⊆U, 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) = {x∈U|Rs(x)∩A6=∅}, R(A) = {x∈U|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:E→P(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:E→F(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:U→P(E)by Rs(u) = {x∈E|(u, x)∈R}, u ∈U.

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Ris referred to as serial if for all u∈U, Rs(u)6=∅. The pair (U, E, R)is called a crisp

soft approximation space. For any A⊆E, 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) = {u∈U|Rs(u)∩A6=∅},(1)

R(A) = {u∈U|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(A∩B) = R(A)∩R(B),(CSU2) R(A∪B) = R(A)∪R(B);

(CSL3) A⊆B⇒R(A)⊆R(B),(CSU3) A⊆B⇒R(A)⊆R(B);

(CSL4) R(A∪B)⊇R(A)∪R(B),(CSU4) R(A∩B) = 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),∀A∈E⇔R(∅) = ∅ ⇔ 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:E→IF (U),

where IF (U)is the set of all intuitionistic fuzzy subsets of U. That is, for ∀x∈E, F (x) =

{< u, µF(x)(u), γF(x)(u)>|u∈U} ∈ 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 0≤µR(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 A∈IF (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)>|u∈U},(3)

R(A) = {< u, µR(A)(u), γR(A)(u)>|u∈U}.(4)

where

µR(A)(u) = W

x∈E

[µR(u, x)∧µA(x)], γR(A)(u) = V

x∈E

[γR(u, x)) ∨γA(x)];

µR(A)(u) = V

x∈E

[γR(u, x)∨µA(x)], γR(A)(u) = W

x∈E

[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) = _

x∈E

[µR(u, x)∧µA(x)] + ^

x∈E

[(γR(u, x)) ∨γA(x)]

≤_

x∈E

[(1 −γR(u, x)) ∧(1 −γA(x))] + ^

x∈E

[(γR(u, x)) ∨γA(x)]

= 1 −^

x∈E

[(γR(u, x)) ∨γA(x)] + ^

x∈E

[(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) = 1−µR(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),1−µR(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 A∈IF (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)>|u∈U},

R(A) = {< u, µR(A)(u), γR(A)(u)>|u∈U}.

where

µR(A)(u) = W

x∈E

[µR(u, x)∧µA(x)], γR(A)(u) = V

x∈E

[(1 −µR(u, x)) ∨γA(x)];

µR(A)(u) = V

x∈E

[(1 −µR(u, x)) ∨µA(x)], γR(A)(u) = W

x∈E

[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 A∈F(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)>|u∈U}, R(A) = {< u, µR(A)(u)>|u∈U}.

where µR(A)(u) = W

x∈E

[µR(u, x)∧µA(x)], µR(A)(u) = V

x∈E

[(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(A∩B) = R(A)∩R(B),(GIFSU3) R(A∪B) = R(A)∪R(B);

(GIFSL4) A⊆B⇒R(A)⊆R(B),(GIFSU4) A⊆B⇒R(A)⊆R(B);

(GIFSL5) R(A∪B)⊇R(A)∪R(B),(GIFSU5) R(A∩B) = 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) = {x∈E|µR(u, x)≥α}, α ∈[0,1],

Rα+={(u, x)∈U×E|µR(u, x)> α}, Rα+(u) = {x∈E|µR(u, x)> α}, α ∈[0,1),

Rα={(u, x)∈U×E|γR(u, x)≤α}, Rα(u) = {x∈E|γR(u, x)≤α}, α ∈[0,1],

Rα+={(u, x)∈U×E|γR(u, x)< α}, Rα+(u) = {x∈E|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: ∀u∈U

(1)

µR(A)(u) = _

α∈[0,1]

[α∧Rα(Aα)(u)] = _

α∈[0,1]

[α∧Rα(Aα+)(u)]

=_

α∈[0,1]

[α∧Rα+(Aα)(u)] = _

α∈[0,1]

[α∧Rα+(Aα+)(u)],

(2)

γR(A)(u) = ^

α∈[0,1]

[α∨Rα(Aα)(u)] = ^

α∈[0,1]

[α∨Rα(Aα+)(u)]

=^

α∈[0,1]

[α∨Rα+(Aα)(u)] = ^

α∈[0,1]

[α∨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) For any u∈U, we have that

_

α∈[0,1]

[α∧Rα(Aα)(u)] = sup{α∈[0,1]|u∈Rα(Aα)}

=sup{α∈[0,1]|Rα(u)∩Aα6=∅}

=sup{α∈[0,1]|∃x∈E[x∈Rα(u), x ∈Aα]}

=sup{α∈[0,1]|∃x∈E[µR(u, x)≥α, µA(x)≥α]}

=_

x∈E

[µR(u, x)∧µA(x)] = µR(A)(u)

Similarly, we can prove

µR(A)(u) = W

α∈[0,1]

[α∧Rα(Aα+)(u)] = W

α∈[0,1]

[α∧Rα+(Aα)(u)] = W

α∈[0,1]

[α∧Rα+(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]

[α∨Rα(Aα)(u))] = inf{α∈[0,1]|u∈Rα(Aα)}

=inf{α∈[0,1]|Rα(u)∩Aα6=∅}

=inf{α∈[0,1]|∃x∈E[x∈Rα(u), x ∈Aα]}

=inf{α∈[0,1]|∃x∈E[γR(u, x)≤α, γA(x)≤α]}

=^

x∈E

[γR(u, x)∨γA(x)] = γR(A)(u).

Likewise, we can prove that

γR(A)(u) = ^

α∈[0,1]

[α∨Rα(Aα+)(u)] = ^

α∈[0,1]

[α∨Rα+(Aα)(u)]

=^

α∈[0,1]

[α∨Rα+(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

x∈E

[µR(u, x)∧

µA(x)] > α holds. Then ∃x0∈E, such that µR(u, x0)∧µA(x0)> α, that is, µR(u, x0)> α

and µA(x0)> α. Thus x0∈Rα+(u) and x0∈Aα+. Consequently, Rα+(u)∩Aα+6=∅. By

Deﬁnition 3.5, we have u∈Rα+(Aα+).Hence [R(A)]α+⊆Rα+(Aα+).

On the other hand, for any u∈Rα(Aα),we have Rα(Aα)(u) = 1. Since µR(A)(u) =

W

β∈[0,1]

[β∧Rβ(Aβ)(u)] ≥α∧Rα(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: ∀u∈U

(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 u∈Uwe have

µR(∼A)(u) = _

α∈[0,1]

[α∧Rα(∼A)α(u)] = _

α∈[0,1]

[α∧Rα(∼Aα+)(u)]

=_

α∈[0,1]

[α∧(∼Rα(Aα+))(u)] = _

α∈[0,1]

[α∧(1 −Rα(Aα+)(u))]

and

γR(∼A)(u) = ^

α∈[0,1]

[α∨Rα(∼A)α(u)] = ^

α∈[0,1]

[α∨Rα(∼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:

F⊕G={< u, µF(u) + µG(u)−µF(u)·µG(u), γF(u)·γG(u)>|u∈U}.

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)>|u∈U}.

5. Choose the top-level threshold value λ= (µ, γ)∈L∗,where µ= max

1≤i≤nµH(ui), γ =

min

1≤i≤nγH(ui).

6. The decision is u, if IF set H(u)≥L∗λ, 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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