# Possibility Intuitionistic Fuzzy Soft Set

Authors: Maruah Bashir; Abdul Razak Salleh; Shawkat Alkhazaleh
Journal: Advances in Decision Sciences
 (2012)

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## Abstract

Possibility intuitionistic fuzzy soft set and its operations are introduced, and a few of their properties are studied. An application of possibility intuitionistic fuzzy soft sets in decision making is investigated. A similarity measure of two possibility  intuitionistic fuzzy soft sets has been discussed. An application of this similarity measure in medical diagnosis has been shown.


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## Body

## 1. Introduction

In most real-life problems in social sciences, engineering, medical sciences, and economics the data involved are imprecise in nature. The solutions of such problems involve the use of mathematical principles based on uncertainty and imprecision. A number of theories have been proposed for dealing with uncertainties in an efficient way. Fuzzy set was introduced by Zadeh in [1] as a mathematical way to represent and deal with vagueness in everyday life. Then Atanassov [2] defined the concept of intuitionistic fuzzy set which is more general than fuzzy set. Molodtsov [3] initiated the theory of soft sets as a new mathematical tool for dealing with uncertainties which traditional mathematical tools cannot handle. Maji et al. [4, 5] have further studied the theory of soft sets and used this theory to solve some decision-making problems. Alkhazaleh et al. [6] introduced soft multiset as a generalisation of Molodtsov’s soft set. Alkhazaleh and Salleh [7] defined the concept of soft expert set and they gave an application of this concept to decision making. Also Maji et al. [8] introduced the concept of fuzzy soft set and studied its properties and also Roy and Maji used this theory to solve some decision-making problems [9]. Majumdar and Samanta [10] defined and studied the generalised fuzzy soft sets, where the degree is attached with the parameterizat ion of fuzzy sets while defining a fuzzy soft set. In 2010 Baesho [11] introduced the concept of generalised intuitionistic fuzzy soft sets, where the degree is attached with the parameterization of fuzzy sets while defining an intuitionistic fuzzy soft set (see also Baesho et al. [12]). Dinda et al. [13] introduced the same concept independently, and Agarwal et al. [14] introduced the same concept in 2011. Alkhazaleh et al. defined in [15] the concept of fuzzy parameterized interval-valued fuzzy soft set and gave its applications in decision making and medical diagnosis. Alkhazaleh et al. [16] defined the concept of possibility fuzzy soft set and gave its applications in decision making and medical diagnosis. Maji [17] defined the concept of intuitionistic fuzzy soft set, and Liang and Shi in [18] defined some new operations on intuitionistic fuzzy soft sets and studied some results relating to the properties of these operations. Salleh [19] gave a brief survey from soft set to intuitionistic fuzzy soft set. In this paper, we generalise the concept of possibility fuzzy soft set to the possibility intuitionistic fuzzy soft set. In our generalisation, a possibility of each element in the universe is attached with the parameterization of fuzzy sets while defining an intuitionistic fuzzy soft set. We also give some applications of the possibility intuitionistic fuzzy soft set in decision-making problem and medical diagnosis.


## 2. Preliminaries

In this section we recall some definitions and properties regarding intuitionistic fuzzy soft set and apossibilityfuzzy soft set required in this paper.LetUbe a universe set and E be a set of parameters. Let P(U)denote the power set of Uand A⊆E.Definition 2.1 (see [12]).
ConsiderU and E as a universe set and a set of parameters, respectively. Let P(U)denote the set of all intuitionistic fuzzy sets of U. Let A⊆E. A pair(F,E)is an intuitionistic fuzzy soft set over U, where F is mapping given by F:A→P(U).Definition 2.2 (see [2]).
An intuitionistic fuzzy set (IFS)A in a nonempty set U (a universe of discourse) is an object having the form A={〈x,μA(x),vA(x)〉:x∈U}, where the functions μA(x):U→[0,1], vA(x):U→[0,1], denotes the degree of membership and degree of nonmembership of each element x∈Uto the set A,respectively, and 0≤μA(x)+vA(x)≤1 for all x∈U.The following definitions and propositions are due to Alkhazaleh et al. [16].Definition 2.3.
LetU={x1,x2,…,xn} be the universal set of elements andE={e1,e2,…,em} be the universal set of parameters. The pair (U,E) will be called a soft universe. Let F:E→IU and μ be a fuzzy subset of E, that is, μ:E→IU, where IU is the collection of all fuzzy subsets of U. Let Fμ:E→IU×IU be a function defined as follows: (2.1)Fμ(e)=(F(e)(x),μ(e)(x)),∀x∈U.
Then Fμ is called a possibility fuzzy soft set (PFSS in short) over the soft universe (U,E). For each parameter ei, Fμ(ei)=(F(ei)(x),μ(ei)(x)) indicates not only the degree of belongingness of the elements of U in F(ei),but also the degree of possibility of belongingness of the elements of U in F(ei), which is represented byμ(ei). So one can write Fμ(ei) as follows:
(2.2)Fμ(ei)={(x1F(ei)(x1),μ(ei)(x1)),(x2F(ei)(x2),μ(ei)(x2)),…,(xnF(ei)(xn),μ(ei)(xn))}.
Sometimes one writes Fμ as (Fμ,E). If A⊆E one can also have a PFSS (Fμ,A).Definition 2.4.
LetFμ and Gδ be two PFSSs over (U,E). Fμ is said to be a possibility fuzzy soft subset(PFS subset) of Gδ and one writes Fμ⊆Gδ if(i)
μ(e) is a fuzzy subset of δ(e), for alle∈E,(ii)
F(e) is a fuzzy subset of G(e), for alle∈E.Definition 2.5.
LetFμ and Gδ be two PFSSs over (U,E). Then Fμ and Gδ are said to be equal and we write Fμ=Gδ if Fμ is a PFS subset of Gδ and Gδ is a PFS subset of Fμ. In other words, Fμ=Gδ if the following conditions are satisfied:(i)
μ(e) is equal to δ(e), for alle∈E,(ii)
F(e) is equal to G(e), for alle∈E.Definition 2.6.
A PFSSs is said to be apossibility null fuzzy soft set, denoted byϕ0, if ϕ0:E→IU×IU such that
(2.3)ϕ0(e)=(F(e)(x),μ(e)(x)),∀e∈E,
whereF(e)=0, andμ(e)=0, for alle∈E.Definition 2.7.
A PFSSs is said to be apossibility  absolute  fuzzy soft set, denoted by A1, if A1:E→IU×IU such that
(2.4)A1(e)=(F(e)(x),μ(e)(x)),∀e∈E,
whereF(e)=1 and μ(e)=1,foralle∈E.Definition 2.8.
Union of two PFSSs Fμ and Gδ, denoted by Fμ⋃̃Gδ, is a PFSSs Hν:E→IU×IU defined by
(2.5)Hν(e)=(H(e)(x),ν(e)(x)),∀e∈E,
such that H(e)=s(F(e),G(e)), and ν(e)=s(μ(e),δ(e)) where s is an s-norm.Definition 2.9.
Intersection of two PFSSs Fμ and Gδ, denoted by Fμ⋂̃Gδ, is a PFSSs Hν:E→IU×IU defined by
(2.6)Hν(e)=(H(e)(x),ν(e)(x)),∀e∈E
such thatH(e)=t(F(e),G(e)) andν(e)=t(μ(e),δ(e)) where t is a fuzzy t-norm.Proposition 2.10.
LetFμ be a PFSSs over (U,E). Then the following results hold:(i)
Fμ⋃̃Fμ=Fμ,(ii)
Fμ⋂̃Fμ=Fμ,(iii)
Fμ⋃̃Aμ=Aμ,(iv)
Fμ⋂̃Aμ=Fμ,(v)
Fμ⋃~ϕμ=Fμ,(vi)
Fμ⋂̃ϕμ=ϕμ.Proposition 2.11.
LetFμ, Gδ, and Hν be any three PFSSs over (U,E), then the following results hold:(i)
Fμ⋃̃Gδ=Gδ⋃̃Fμ,(ii)
Fμ⋂̃Gδ=Gδ⋂̃Fμ,(iii)
Fμ⋃̃(Gδ⋃̃Hv)=(Fμ⋃̃Gδ)⋃̃Hv,(iv)
Fμ⋂̃(Gδ⋂̃Hv)=(Fμ⋂̃Gδ)⋂̃Hv.Definition 2.12.
Similaritybetween two GFSSs Fμand Gδ, denoted by S(Fμ,Gδ),is defined as follows:
(2.7)S(Fμ,Gδ)=M(F(e),G(e))⋅m(μ(e),δ(e))
such that, M(F(e),G(e))=maxiMi(F(e),G(e)), where
(2.8)Mi(F(e),G(e))=1-∑j=1n|Fij(e)-Gij(e)|∑j=1n|Fij(e)+Gij(e)|,m(μ(e),δ(e))=1-∑|μ(e)-δ(e)|∑|μ(e)+δ(e)|.Definition 2.13.
Similaritybetween two PFSSs Fμ and Gδ, denoted by S(Fμ,Gδ), is defined as follows:
(2.9)S(Fμ,Gδ)=M(F(e),G(e))⋅M(μ(e),δ(e)),
such that, M(F(e),G(e))=maxiMi(F(e),G(e)) and M(μ(e),δ(e))=maxiMi(μ(e),δ(e)) where
(2.10)Mi(F(e),G(e))=1-∑j=1n|Fij(e)-Gij(e)|∑j=1n|Fij(e)+Gij(e)|,(2.11)Mi(μ(e),δ(e))=1-∑j=1n|μij(e)-δij(e)|∑j=1n|μij(e)+δij(e)|.Definition 2.14.
If(Fμ,A) and (Gδ,B) are two PFSSs then “(Fμ,A) AND(Gδ,B)”, denoted by (Fμ,A)∧(Gδ,B), is defined by
(2.12)(Fμ,A)∧(Gδ,B)=(Hλ,A×B),
where Hλ(α,β)=(H(α,β)(x),ν(α,β)(x)), for all(α,β)∈A×B, such that H(α,β)=t(F(α),G(β)) and ν(α,β)=t(μ(α),δ(β)), for all(α,β)∈A×B, where t is a t-norm.Definition 2.15.
If(Fμ,A) and (Gδ,B) are two PFSSs then “(Fμ,A) OR (Gδ,B)”, denoted by (Fμ,A)∨(Gδ,B), is defined by
(2.13)(Fμ,A)∨(Gδ,B)=(Hλ,A×B),
where Hλ(α,β)=(H(α,β)(x),ν(α,β)(x)),forall(α,β)∈A×B, such that H(α,β)=s(F(α),G(β)) and ν(α,β)=s(μ(α),δ(β)),forall(α,β)∈A×B, where s is an s-norm.Proposition 2.16.
LetFμ and Gδ be any two GFSSs over (U,E). Then the following holds:(i)
S(Fμ,Gδ)=S(Gδ,Fμ),(ii)
0≤S(Fμ,Gδ)≤1,(iii)
Fμ=Gδ⇒S(Fμ,Gδ)=1,(iv)
Fμ⊆Gδ⊆Hλ⇒S(Fμ,Hλ)≤S(Gδ,Hλ),(v)
Fμ⋂̃Gδ=ϕ⇔S(Fμ,Gδ)=0.


## 3. Possibility Intuitionistic Fuzzy Soft Sets

In this section we generalise the concept of possibility fuzzy soft sets as introduced by Alkhazaleh et al. [16]. In our generalisation of a possibility fuzzy soft set, a possibility of each element in the universe is attached with the parameterization of fuzzy sets while defining an intuitionistic fuzzy soft set.Definition 3.1.
LetU={x1,x2,…,xn} be the universal set of elements and E={e1,e2,…,em} be the universal set of parameters. The pair (U,E) will be called a soft universe. Let F:E→(I×I)U×IU where (I×I)U is the collection of all intuitionistic fuzzy subsets of U and IU is the collection of all fuzzy subsets of U. Let p be a fuzzy subset of E, that is, p:E→IU and let Fp:E→(I×I)U×IU be a function defined as follows:
(3.1)Fp(e)=(F(e)(x),p(e)(x)),whereF(e)(x)=(μ(x),ν(x))∀x∈U.
Then Fp is called a possibility intuitionistic  fuzzy soft set (PIFSS in short) over the soft universe (U,E). For each parameter ei,Fp(ei)=(F(ei)(x),p(ei)(x)) indicates not only the degree of belongingness of the elements of U in F(ei), but also the degree of possibility of belongingness of the elements of U in F(ei), which is represented by p(ei). So we can write Fp(ei) as follows:
(3.2)Fp(ei)={(x1F(ei)(x1),p(ei)(x1)),(x2F(ei)(x2),p(ei)(x2)),…,(xnF(ei)(xn),p(ei)(xn))}.
Sometime we write Fp as (Fp,E). If A⊆E we can also have a PIFSS (Fp,A).Example 3.2.
LetU={x1,x2,x3}be a universe set. Let E={e1,e2,e3} be a set of parameters and let p:E→IU. We define a function Fp:E→(I×I)U×IU as follows:
(3.3)Fp(e1)={(x1(0.4,0.3),0.7),(x2(0.7,0.1),0.5),(x3(0.5,0.2),0.6)},Fp(e2)={(x1(0.5,0.1),0.6),(x2(0.6,0),0.5),(x3(0.6,0.3),0.5)},Fp(e3)={(x1(0.7,0),0.5),(x2(0.6,0.2),0.5),(x3(0.5,0.1),0.7)}.
Then Fp is a PIFSS over (U,E). In matrix notation we write
(3.4)Fp=((0.4,0.3),0.7(0.5,0.1),0.6(0.7,0),0.5(0.7,0.1),0.5(0.6,0),0.5(0.6,0.2),0.5(0.5,0.2),0.6(0.6,0.3),0.5(0.5,0.1),0.7).Definition 3.3.
LetFp and Gq be two PIFSSs over (U,E). Fp is said to be a possibility  intuitionistic  fuzzy soft subset (PIFS subset) of Gq and one writes Fp⊆Gq if(i)
p(e) is a fuzzy subset of q(e), for all e∈E,(ii)
F(e) is an intuitionistic fuzzy subset of G(e), for all e∈E.Example 3.4.
LetU={x1,x2,x3}be a universe set. Let E={e1,e2,e3}be a set of parameters and let p:E→IU. We define a function Fp:E→(I×I)U×IU as follows:
(3.5)Fp(e1)={(x1(0.2,0.1),0.4),(x2(0.6,0.3),0.5),(x3(0.5,0.4),0.6)},Fp(e2)={(x1(0.7,0.2),0.5),(x2(0.6,0.4),0.6),(x3(0.8,0.2),0.6)},Fp(e3)={(x1(0,0.7),0.1),(x2(0.5,0.1),0.3),(x3(0.3,0.3),0.1)}.
Let Gq:E→(I×I)U×IUbe another PIFSS over (U,E) defined as follows:
(3.6)Gq(e1)={(x1(0.3,0),0.6),(x2(0.7,0.2),0.6),(x3(0.6,0.3),0.7)},Gq(e2)={(x1(0.8,0.1),0.6),(x2(0.7,0.3),0.7),(x3(0.9,0.1),0.8)},Gq(e3)={(x1(0.1,0.6),0.2),(x2(0.6,0),0.5),(x3(0.5,0.1),0.2)}.
It is clear thatFp is a PIFS subset of Gq.Definition 3.5.
LetFp and Gq be two PIFSSs over (U,E). ThenFp and Gq are said to be equal and one writesFp=Gq if Fp is a PIFS subset of Gqand Gqis a PIFS subset ofFp.
In other words,Fp=Gq if the following conditions are satisfied:
(i)
p(e)is equal to q(e), for all e∈E,(ii)
F(e)is equal to G(e), for alle∈E.Definition 3.6.
A PIFSS is said to be apossibility null intuitionistic fuzzy soft set, denoted by ϕ0, if ϕ0:E→(I×I)U×IU such that
(3.7)ϕ0(e)=(F(e)(x),p(e)(x)),∀e∈E,
whereF(e)=(0,v(e)), andp(e)=0,foralle∈E.Definition 3.7.
A PIFSS is said to be apossibility absolute intuitionistic fuzzy soft set, denoted by A1, if A1:E→(I×I)U×IU such that
(3.8)A1(e)=(F(e)(x),P(e)(x)),∀e∈E,
whereF(e)=(1,0) and P(e)=1,foralle∈E.Example 3.8.
LetU={x1,x2,x3}be a universe set. Let E={e1,e2,e3}be a set of parameters and let p:E→IU. We define a function Fp:E→(I×I)U×IU as follows:
(3.9)Fp(e1)={(x1(0,0.3),0),(x2(0,0.5),0),(x3(0,0.6),0)},Fp(e2)={(x1(0,1),0),(x2(0,0.2),0),(x3(0,0.3),0)},Fp(e3)={(x1(0,0.2),0),(x2(0,0.6),0),(x3(0,0.9),0)}.
Then Fpis a possibility null intuitionistic fuzzy soft set.
Letp:E→IU, and we define the function Fp:E→(I×I)U×IU which is a PIFSS over (U,E)as follows:
(3.10)Fp(e1)={(x1(1,0),1),(x2(1,0),1),(x3(1,0),1)},Fp(e2)={(x1(1,0),1),(x2(1,0),1),(x3(1,0),1)},Fp(e3)={(x1(1,0),1),(x2(1,0),1),(x3(1,0),1)}.
Then Fpis a possibility absolute intuitionistic fuzzy soft set.


## 4. Union and Intersection of PIFSS

In this section we introduce the definitions of union and intersection of PIFSS, derive some properties and give some examples.Definition 4.1.
Union of two PIFSSs Fp and Gq, denoted by Fp⋃̃Gq, is a PIFSSs Hr:E→(I×I)U×IU defined by
(4.1)Hr(e)=(H(e)(x),r(e)(x)),∀e∈E,
such that H(e)=∪Atan(F(e),G(e)) and r(e)=s(p(e),q(e)), where∪Atan is Atanassov union and s is an s-norm.Example 4.2.
LetU={x1,x2,x3}and E={e1,e2,e3}. Let Fμ be a PIFSS defined as follows:
(4.2)Fp(e1)={(x1(0.3,0.4),0.1),(x2(0.7,0.1),0.4),(x3(0.2,0.6),0.6)},Fp(e2)={(x1(0.2,0.6),0.3),(x2(0.2,0.5),0.2),(x3(0.1,0.3),0.4)},Fp(e3)={(x1(0.7,0.1),0.1),(x2(0.2,0.5),0),(x3(0.5,0.3),0.6)}.
Let Gq be another PIFSS over (U,E) defined as follows:
(4.3)Gq(e1)={(x1(0.1,0.4),0.3),(x2(0.3,0.3),0.6),(x3(0,0.5),0.2)},Gq(e2)={(x1(0,0.1),0.1),(x2(0.6,0.1),0.6),(x3(0.7,0),0.6)},Gq(e3)={(x1(0,0),0.3),(x2(0.2,0.1),0.8),(x3(0.3,0.1),0.2)}.
By using the Atanassov union which is the basic intuitionistic fuzzy union we have
(4.4)Hr(e1)={(x1(max⁡(0.3,0.1),min⁡(0.4,0.4)),max⁡(0.1,0.3)),(x1(max⁡(0.7,0.3),min⁡(0.1,0.3)),max⁡(0.4,0.6)),(x1(max⁡(0.2,0),min⁡(0.6,0.5)),max⁡(0.6,0.2))},Hr(e1)={(x1(0.3,0.4),0.3),(x2(0.7,0.1),0.6),(x3(0.2,0.5),0.6)}.
Similarly we get
(4.5)Hr(e2)={(x1(0.2,0.1),0.3),(x2(0.6,0.1),0.6),(x3(0.7,0),0.6)},Hr(e3)={(x1(0.7,0),0.3),(x2(0.2,0.1),0.8),(x3(0.5,0.1),0.6)}.Remark 4.3.
The Atanassov union can be replaced by anyS-norm which is a general intuitionistic fuzzy union (see Fathi [20]).Definition 4.4.
Intersection of two PIFSSs Fp and Gq, denoted by Fp⋂̃Gq, is a PIFSSs Hr:E→(I×I)U×IU defined by
(4.6)Hr(e)=(H(e)(x),r(e)(x)),∀e∈E
such that H(e)=∩Atan(F(e),G(e)) and r(e)=t(p(e),q(e)), where∩Atan  is Atanassov  intersection and t is a t-norm.Example 4.5.
Consider Example4.2 where Fp and Gqare PIFSSs defined as follows:
(4.7)Fp(e1)={(x1(0.3,0.2),0.2),(x2(0.4,0.1),0.3),(x3(0,0.1),0.4)},Fp(e2)={(x1(0,0.1),0.1),(x2(0.2,0.6),0.4),(x3(0.3,0),0.2)},Fp(e3)={(x1(0.7,0.1),0.3),(x2(0.1,0.2),0.1),(x3(0.4,0.1),0.4)},Gq(e1)={(x1(0.7,0.1),0.3),(x2(0.1,0.2),0.1),(x3(0.4,0.1),0.4)},Gq(e2)={(x1(0.1,0.6),0.1),(x2(0.3,0.2),0.3),(x3(0.8,0),0.4)},Gq(e3)={(x1(0,0.5),0.4),(x2(0.1,0.4),0.1),(x3(0.9,0),0.3)}.
By using the Atanassov intersection which is the basic intuitionistic fuzzy union we have
(4.8)Hr(e1)={(x1(min⁡(0.3,0.7),max⁡(0.2,0.1)),min⁡(0.2,0.3)),(x1(min⁡(0.4,0.1),max⁡(0.1,0.2)),min⁡(0.3,0.1)),(x1(min⁡(0,0.4),max⁡(0.1,0.1)),min⁡(0.4,0.4))},={(x1(0.3,0.2),0.2),(x2(0.1,0.2),0.1),(x3(0,0.1),0.4)}.
Similarly we get
(4.9)Hr(e2)={(x1(0,0.6),0.1),(x2(0.2,0.6),0.3),(x3(0.3,0),0.2)},Hr(e3)={(x1(0,0.5),0.3),(x2(0.1,0.4),0.1),(x3(0.4,0.1),0.3)}.Remark 4.6.
The Atanassov intersection can be replaced by anyT-norm which is a general intuitionistic fuzzy intersection (see Fathi [20]).Proposition 4.7.
LetFp, Gq, and Hr be any three PIFSSs over (U,E), then the following results hold:(i)
Fp⋃̃Gq=Gp⋃̃Fq,(ii)
Fp⋂̃Gq=Gp⋂̃Fq,(iii)
Fp⋃̃(Gq⋃̃Hr)=(Fp⋃̃Gq)⋃̃Hr,(iv)
Fp⋂̃(Gq⋂̃Hr)=(Fp⋂̃Gq)⋂̃Hr.Proof.
The proof is straightforward by using the fact that the union and intersection of fuzzy sets and intuitionistic fuzzy sets are commutative and associative.Proposition 4.8.
LetFp be a PFSSs over (U,E). Then the following results hold:(i)
Fp⋃̃Fp=Fp,(ii)
Fp⋂̃Fp=Fp,(iii)
Fp⋃̃Ap=Ap,(iv)
Fp⋂̃Ap=Fp.Proof.
The proof is straightforward by using the definitions of union and intersection.Proposition 4.9.
LetFp, Gq, and Hr be any three PIFSS over (U,E). Then the following results hold:(i)
Fp⋃̃(Gq⋂̃Hr)=(Fp⋃̃Gq)⋂̃(Fp⋃̃Hr),(ii)
Fp⋂̃(Gq⋃̃Hr)=(Fp⋂̃Gq)⋃̃(Fp⋂̃Hq).Proof.
(i) For allx∈E,
(4.10)λF(x)∪̃(G(x)∩̃H(x))(x)=∪{λF(x)(x),λ(G(x)∩̃H(x))(x)}=∪{λF(x)(x),∩(λG(x)(x),λH(x)(x))}={〈x,max⁡(μF(x)(x),min⁡(μG(x)(x),μH(x)(x))),min⁡(νF(x),max⁡(νG(x)(x),νH(x)(x)))〉:x∈U}={〈x,min⁡(max⁡(μF(x)(x),μG(x)(x)),max⁡(μF(x)(x),μH(x)(x))),max⁡(min⁡(νF(x)(x),νG(x)(x)),min⁡(νF(x)(x),νH(x)(x)))〉:x∈U}=∩(∪(λF(x)∪G(x)(x)),λF(x)∪H(x)(x))=λ(F(x)∪G(x))∩(F(x)∪H(x))(x),(4.11)γμ(x)∪̃(δ(x)∩̃ν(x))(x)=max⁡{γp(x)(x),γ(q(x)∩̃r(x))(x)}=max⁡{γp(x)(x),min⁡(γq(x)(x),γr(x)(x))}=min⁡{max⁡(γp(x)(x),γq(x)(x)),max⁡(γp(x)(x),γr(x)(x))}=min⁡{γ(μ(x)∪δ(x))(x),γ(μ(x)∪ν(x))(x)}=γ(μ(x)∪δ(x))∩(μ(x)∪ν(x))(x).
We can use the same method in (i).


## 5. AND and OR Operations on PIFSS with Applications in Decision Making

In this section we introduce the definitions of AND and OR operations on possibility intuitionistic fuzzy soft sets. Applications of possibility fuzzy soft sets in decision-making problem are given.Definition 5.1.
If(Fμ,A)and (Gδ,B),are two PIFSSs then “(Fμ,A) AND(Gδ,B)” denoted by (Fμ,A)∧(Gδ,B)is defined by
(5.1)(Fμ,A)∧(Gδ,B)=(Hλ,A×B),
where Hλ(α,β)=(H(α,β)(x),∧(α,β)(x)),forall(α,β)∈A×B, such that H(α,β)=T(F(α),G(β)) and ∧(α,β)=t(μ(α),δ(β)),for all (α,β)∈A×B, where T is a T-norm and t is a t-norm.Example 5.2.
Suppose the universe consists of three machinesx1,x2, x3 that is, U={x1,x2,x3} and there are three parameters E={e1,e2,e3} which describe their performances according to certain specific task. Suppose a firm wants to buy one such machine depending on any two of the parameters only. Let there be two observations Fμ and Gδ by two experts defined as follows:
(5.2)Fμ(e1)={(x1(0,0.1),0.1),(x2(0.2,0.6),0.4),(x3(0.3,0),0.2)},Fμ(e2)={(x1(0,0.1),0.2),(x2(0.1,0.3),0.6),(x3(0.8,0),0.1)},Fμ(e3)={(x1(0.3,0.1),0.4),(x2(0,0.5),0.1),(x3(0.5,0),0.2)},Gδ(e1)={(x1(0.1,0.6),0.1),(x2(0.3,0.2),0.3),(x3(0.8,0),0.4)},Gδ(e2)={(x1(0,0),0.3),(x2(0.2,0.4),0.2),(x3(0.6,0),0.1)},Gδ(e3)={(x1(0.6,0),0.7),(x2(0.3,0.1),0.4),(x3(0,0),0.4)}.
By using the Atanassov intersection which is the basic intuitionistic fuzzy union we have
(5.3)Hv(e1,e1)={(x1(min⁡(0,0.1),max⁡(0.1,0.6)),min⁡(0.1,0.1)),(x1(min⁡(0.2,0.3),max⁡(0.6,0.2)),min⁡(0.4,0.3)),(x1(min⁡(0.3,0.8),max⁡(0,0)),min⁡(0.2,0.4))}.Hλ(e1,e1)={(x1(0,0.6),0.1),(x2(0.2,0.6),0.3),(x3(0.3,0),0.2)}.
Similarly we get
(5.4)Hλ(e1,e2)={(x1(0,0.1),0.1),(x2(0.2,0.6),0.2),(x3(0.3,0),0.1)},Hλ(e1,e3)={(x1(0,0.1),0.7),(x2(0.2,0.6),0.4),(x3(0,0),0.2)},Hλ(e2,e1)={(x1(0,0.6),0.1),(x2(0.1,0.3),0.3),(x3(0.6,0),0.1)},Hλ(e2,e2)={(x1(0,0.1),0.2),(x2(0.1,0.4),0.2),(x3(0.6,0),0.1)},Hλ(e2,e3)={(x1(0,0.1),0.2),(x2(0.1,0.3),0.4),(x3(0,0),0.1)},Hλ(e3,e1)={(x1(0.1,0.6),0.1),(x2(0,0.5),0.1),(x3(0.5,0),0.2)},Hλ(e3,e2)={(x1(0,0.1),0.3),(x2(0,0.5),0.1),(x3(0.5,0),0.1)},Hλ(e3,e3)={(x1(0.3,0.1),0.4),(x2(0,0.5),0.1),(x3(0,0),0.2)}.
In matrix notation we have
(5.5)(Fμ,A)∧(Gδ,B)=(((0,0.6),0.1)((0,0.1),0.1)((0,0.1),0.7)((0,0.6),0.1)((0.2,0.6),0.3)((0.2,0.6),0.2)((0.2,0.6),0.4)((0.1,0.3),0.3)((0.3,0),0.2)((0.3,0),0.1)((0,0),0.2)((0.6,0),0.1)((0,0.1),0.2)((0.1,0.4),0.2)((0.6,0),0.1)((0,0.1),0.2)((0.1,0.6),0.1)((0,0.1),0.3)((0.3,0.1),0.4)((0.1,0.3),0.4)((0,0.5),0.1)̲((0,0.5),0.1)((0,0.5),0.1)((0,0),0.1)̲((0.5,0),0.2)((0.5,0),0.1)̲((0,0),0.2)).Now to determine the best machine we first calculate the difference between the membership and non-membership values and then we mark the highest numerical grade (indicated in parenthesis) in each row. Now the score of each of such machine is calculated by taking the sum of the products of these different numerical grades with the corresponding value ofλ. The machine with the highest score is the desired machine. We do not consider the different numerical grades of the machine against the pairs (ei,ei),i=1,2,3,as both parameters are the same.Matrix of different numerical grades is shown below:(5.6)((-0.6,0.1)(-0.4,0.3)(0.3,0.2)(-0.1,0.1)(-0.4,0.2)(0.3,0.1)(-0.1,0.7)(-0.4,0.4)(0,0.2)(-0.6,0.1)(-0.2,0.3)(0.6,0.1)(-0.1,0.2)(-0.3,0.2)(0.6,0.1)(-0.1,0.2)(-0.2,0.4)(0,0.1)(-0.5,0.1)(-0.5,0.1)(0.5,0.2)(-0.1,0.3)(-0.5,0.1)(0.5,0.1)(0.2,0.4)(-0.5,0.1)(0,0.2)).Grade Table(5.7)(Hxihighestnumericalgradeλi(e1,e1)x3××(e1,e2)x30.30.1(e1,e3)(e2,e1)(e2,e2)(e2,e3)(e3,e1)(e3,e2)(e3,e3)x3x3x3x3x3x3x100.6×00.50.5×0.20.1×0.10.20.1×).Score(x1)=0,Score(x2)=0,Score(x3)=(0.3×0.1)+(0×0.2)+(0.6×0.1)+(0×0.1)+(0.5×0.2)×(0.5×0.1)=0.24.Then the firm will select the machine with the highest score. Hence, they will buy machinex3.Definition 5.3.
If(Fμ,A) and (Gδ,B)are two PIFSS then “(Fμ,A)OR (Gδ,B)”, denoted by (Fμ,A)∨(Gδ,B),is defined by
(5.8)(Fμ,A)∨(Gδ,B)=(Hλ,A×B),
where Hλ(α,β)=(H(α,β)(x),∨(α,β)(x))forall(α,β)∈A×B, such that H(α,β)=S(F(α),G(β)) and ∨(α,β)=s(μ(α),δ(β)), for all (α,β)∈A×B, where S is an S-norm and s is an s-norm.Example 5.4.
LetU={x1,x2,x3},E={e1,e2,e3}consider Fμ and Gδ as in Example 5.2.  Suppose now the firm wants to buy a machine depending on any one of two parameters. Then we have (Fμ,A)∨(Gδ,B)=(Hλ,A×B) where
(5.9)H(e1,e1)={(x1(max⁡(0,0.1),min⁡(0.1,0.6)),max⁡(0.1,0.1)),(x1(max⁡(0.2,0.3),min⁡(0.6,0.2)),max⁡(0.4,0.3)),(x1(max⁡(0.3,0.8),min⁡(0,0)),max⁡(0.2,0.4))},Hλ(e1,e1)={(x1(0.1,0.1),0.1),(x2(0.3,0.2),0.4),(x3(0.8,0),0.4)}.
Similarly we get
(5.10)Hλ(e1,e2)={(x1(0,0.1),0.3),(x2(0.2,0.4),0.4),(x3(0.6,0),0.2)},Hλ(e1,e3)={(x1(0.6,0),0.7),(x2(0.3,0.1),0.4),(x3(0.3,0),0.4)},Hλ(e2,e1)={(x1(0.1,0.1),0.2),(x2(0.2,0.3),0.6),(x3(0.8,0),0.4)},Hλ(e2,e2)={(x1(0,0.1),0.3),(x2(0.2,0.3),0.6),(x3(0.8,0),0.1)},Hλ(e2,e3)={(x1(0.6,0),0.7),(x2(0.3,0.1),0.6),(x3(0.8,0),0.4)},Hλ(e3,e1)={(x1(0.3,0.1),0.4),(x2(0.3,0.2),0.1),(x3(0.8,0),0.4)},Hλ(e3,e2)={(x1(0.3,0),0.1),(x2(0.2,0.4),0.2),(x3(0.6,0),0.2)},Hλ(e3,e3)={(x1(0.6,0),0.7),(x2(0.3,0.1),0.4),(x3(0.5,0),0.4)}.
In matrix notation we have
(5.11)(Fμ,A)∨(Gδ,B)=(((0.1,0.1),0.1)((0,0.1),0.3)((0.6,0),0.7)((0.1,0.1),0.2)((0.3,0.2),0.4)((0.2,0.4),0.4)((0.3,0.1),0.4)((0.2,0.3),0.6)((0.8,0),0.4)((0.6,0),0.2)((0.3,0),0.4)((0.8,0),0.4)((0,0.1),0.3)((0.2,0.3),0.6)((0.8,0),0.1)((0.6,0),0.7)((0.3,0.1),0.4)((0.3,0),0.1)((0.6,0),0.7)((0.3,0.1),0.6)((0.3,0.2),0.1)̲((0.2,0.4),0.2)((0.3,0.1),0.4)((0.8,0),0.4)̲((0.8,0),0.4)((0.6,0),0.2)̲((0.5,0),0.4)).Now to determine the best machine we first calculate the difference between the membership and non-membership values and then we mark the highest numerical grade (indicated in parenthesis) in each row. Now the score of each of such machine is calculated by taking the sum of the products of these different numerical grades with the corresponding value ofλ. The machine with the highest score is the desired machine. We do not consider the different numerical grades of the machine against the pairs (ei,ei), i=1,2,3,as both parameters are the same.Matrix of different numerical grades is shown below:(5.12)((0,0.1)(0.1,0.4)(0.8,0.4)(-0.1,0.3)(-0.2,0.4)(0.6,0.2)(0.6,0.7)(0.2,0.4)(0.3,0.4)(0,0.2)(-0.1,0.6)(0.8,0.4)(-0.1,0.3)(-0.1,0.6)(0.8,0.1)(0.6,0.7)(0.2,0.6)(0.8,0.4)(0.2,0.4)(0.1,0.1)(0.8,0.4)(0.3,0.1)(-0.2,0.2)(0.6,0.2)(0.6,0.7)(0.2,0.4)(0.5,0.4)).Grade Table(5.13)(Hxihighestnumericalgradeλi(e1,e1)x3××(e1,e2)x30.60.2(e1,e3)(e2,e1)(e2,e2)(e2,e3)(e3,e1)(e3,e2)(e3,e3)x1x3x3x3x3x3x30.60.8×0.80.80.6×0.70.4×0.40.40.2×).Score(x1)=(0.6×0.7)=0.42,Score(x2)=0,Score(x3)=(0.6×0.2)+(0.8×0.4)+(0.8×0.4)+(0.8×0.4)+(0.6×0.2)=1.2.Then the firm will select the machine with the highest score. Hence, they will buy machinex3.


## 6. An Application of PIFSS in Decision Making

In this section we present an application of PIFSS in decision making problem. We shall use the algorithm introduced by Dinda et al. [13]. Suppose that there are three schools in universe U={x1,x2,x3}and the parameter set E={e1,e2,e3,e4,e5,e6}, each, ei, 1≤i≤6 indicates a specific criterion for the schoolse1 stands for “international”.e2 stands for “English”.e3 stands for “high efficiency”.e4 stands for “ modern”.e5 stands for “full day”.e6 stands for “half day”.Suppose Madam X wants to pick a good school for her son on the basis of her wishing parameters among those listed above. Our aim is to find out the most appropriate school for her son.Suppose the wishing parameters of Madam X isA⊆E where A={e1,e3,e6}.Letp:E→IU be a fuzzy subset of E, defined by Madam X.Consider the PIFSS defined as follows:(6.1)Fμ(e1)={(x1(0.3,0.5),0.1),(x2(0.5,0.3),0.4),(x3(0.3,0.4),0.5)},Fμ(e2)={(x1(0,0.8),0.1),(x2(1,0),0.3),(x3(0.9,0.1),0.2)},Fμ(e3)={(x1(0.6,0.3),0.4),(x2(0.5,0.4),0.3),(x3(0.6,0.3),0.1)}.Now we introduce the following operations:(i)
for membership function:α(ei)=μi+γi-μiγi,(ii)
for non-membership functionβ=υiγi, for  i=1,2,3.Actually we have taken these two operations to ascend the membership value and descend the non-membership value ofF(ei)on the basis of the degree of preference of Madam X. Then the PIFSS Fμ(ei) reduced to an intuitionistic fuzzy soft set Ψ(ei)given as follows:(6.2)Ψ(e1)={x1(0.37,0.05),x2(0.78,0.12),x3(0.65,0.2)},Ψ(e2)={x1(0,0.08),x2(1,0),x3(0.92,0.02)},Ψ(e3)={x1(0.76,0.12),x2(0.65,0.12),x3(0.64,0.03)}.Definition 6.1 (see [13]).
A comparison table is a square table in which number of rows and number of columns are equal and both are labeled by object name of the universe such asx1,x2,...,xn and the entries are cij, where cij= the number of parameters for which the value of xi exceeds or equal to the value of xj.Algorithm 6.2.
(i)
Input the setA⊆E of choice of parameters of Madam X.(ii)
Consider the reduced intuitionistic fuzzy soft set.(iii)
Consider the tabular representation of membership function andnon-membership function (see Table1 and Table 4 respectively).(iv)
Compute the comparison table membership function andnon-membership function (see Table2 and Table 5 respectively).(v)
Compute the membership score and non-membership score (see Table3 and Table 6 respectively).(vi)
Compute the final score by subtracting non-membership score frommembership score (see Table7).(vii)
Find the maximum score, if it occurs inith row then Madam X will choose school xi.Table 1
Tabular representation of membership function.
Ue1e3e6x10.3700.76x20.7810.65x30.650.920.64Table 2
Comparison table of the above table.
x1x2x3x1311x2233x3203Table 3
Membership score table.
Row sum (a)Column sum (b)Membership score (a-b)x157−2x2844x35−7−2Table 4
Tabular representation of non-membership function.
e1e3e6x10.050.080.12x20.1200.12x30.20.020.03Table 5
Comparison table of the above table.
x1x2x3x1322x2231x3123Table 6
Non-membership score table.
Row sum (c)Column sum (d)Non-membership score (c-d)x1761x267−1x3660Table 7Membership score (m)Non-membership score (n)Finale score (m-n)x1−213x24−15x3−20−2Clearly the maximum score is 5 scored by the schoolx2Decision:
Madam X will choose the schoolx2. In case, if she does not want to choose x2 due to certain reasons, her second choice will be x1.


## 7. Similarity between Two Possibility Intuitionistic Fuzzy Soft Sets

Similarity measures have extensive application in several areas such as pattern recognition, image processing, region extraction, and coding theory and so forth. We are often interested to know whether two patterns or images are identical or approximately identical or at least to what degree they are identical.Several researchers have studied the problem of similarity measurement between fuzzy sets, intuitionistic fuzzy sets and Liang and Shi [18] have studied the similarity measures on intuitionistic fuzzy sets. Shawkat et al. [16] have studied the similarity between two possibility fuzzy soft sets.In this section we introduce a measure of similarity between two PIFSSs. The set theoretic approach has been taken in this regard because it is easier for calculation and is a very popular method too.Definition 7.1.
Similarity between two PIFSSs Fμ and Gδ, denoted by S(Fμ,Gδ), is defined as follows:(7.1)S(Fμ,Gδ)=M(F(e),G(e))⋅M(μ(e),δ(e)),
such that
(7.2)M(F(e),G(e))=max⁡i⁡Mi(F(e),G(e)),M(μ(e),δ(e))=max⁡i⁡Mi(μ(e),δ(e)),
where
(7.3)Mi(F(e),G(e))=1-1np∑i=1n(ϕF(e)(i)-ϕG(e)(i))pp1≤p≤∞,
such that and
(7.4)ϕF(e)(i)=μF(e)+νF(e)2,ϕG(e)(i)=μG(e)+νG(e)2,Mi(μ(e),δ(e))=1-∑j=1n|μij(e)-δij(e)|∑j=1n|μij(e)+δij(e)|.Definition 7.2.
LetFμand Gδ be two PIFSSs over (U,E). We say that Fμ and Gδare significantly similar if S(Fμ,Gδ)≥1/2.Proposition 7.3.
LetFμ and Gδ be any two PIFSSs over (U,E). Then the following holds:(i)
S(Fμ,Gδ)=S(Gδ,Fμ),(ii)
0≤S(Fμ,Gδ)≤1,(iii)
Fμ=Gδ⇒S(Fμ,Gδ)=1,(iv)
Fμ⊆Gδ⊆Hλ⇒S(Fμ,Hλ)≤S(Gδ,Hλ),(v)
Fμ⋂̃Gδ=ϕ⇔S(Fμ,Gδ)=0.Proof.
The proof is straightforward and follows from Definition6.1.Example 7.4.
Consider Example4.2 where Fμ and Gδare defined as follows:
(7.5)Fμ(e1)={(x1(0.3,0.4),0.1),(x2(0.7,0.1),0.4),(x3(0.2,0.6),0.6)},Fμ(e2)={(x1(0.2,0.6),0.3),(x2(0.2,0.5),0.2),(x3(0.1,0.3),0.4)},Fμ(e3)={(x1(0.7,0.1),0.1),(x2(0.2,0.5),0),(x3(0.5,0.3),0.6)},Gδ(e1)={(x1(0.1,0.4),0.3),(x2(0.3,0.3),0.6),(x3(0,0.5),0.2)},Gδ(e2)={(x1(0,0.1),0.1),(x2(0.6,0.1),0.6),(x3(0.7,0),0.6)},Gδ(e3)={(x1(0,0),0.3),(x2(0.2,0.1),0.8),(x3(0.3,0.1),0.2)}.
Here
(7.6)M1(μ(e),δ(e))=1-∑j=13|μ1j(e)-δ1j(e)|∑j=13|μ1j(e)+δ1j(e)|=1-|(0.1-0.3)|+|(0.4-0.6)|+|(0.6-0.2)||(0.1+0.3)|+|(0.4+0.6)|+|(0.6+0.2)|=0.39.
Similarly we get M2(μ(e),δ(e))=0.38 and M3(μ(e),δ(e))=0.3. Then
(7.7)M(μ(e),δ(e))=max⁡(M1(μ(e),δ(e)),M2(μ(e),δ(e)),M3(μ(e),δ(e)))=0.39,M1(F(e),G(e))=1-1np∑i=1n(ϕF(e)(i)-ϕG(e)(i))pp=1-13((0.35-0.25)+(0.4-0.3)+(0.4-0.25))=0.66.
Similarly we get M2(F(e),G(e))=0.63 and M3(F(e),G(e))=0.48. Then
(7.8)M(F(e),G(e))=max⁡(M1(F(e),G(e)),M2(F(e),G(e)),M3(F(e),G(e)))=0.66.
Hence the similarity between the two PIFSS Fμand Gδ is given by
(7.9)S(Fμ,Gδ)=M(F(e),G(e))⋅M(μ(e),δ(e))=0.66×0.39≅0.26.


## 8. Application of Similarity Measure in Medical Diagnosis

In the following example we will try to estimate the possibility that a sick person having certain visible symptoms is suffering from swamp fever. For this we first construct a model possibility intuitionistic fuzzy soft set for swamp fever and the possibility intuitionistic fuzzy soft set of symptoms for the sick person. Next we find the similarity measure of these two sets. If they are significantly similar then we conclude that the person is possibly suffering from swamp fever.Let our universal set contain only two elements “yes” and “no,” that is,U={y,n}. Here the set of parameters E is the set of certain visible symptoms. Let E=(e1,e2,e3,e4,e5,e6,e7,e8,e9,e10), where e1= trembling, e2= cough with chest congestion, e3= muscles pain, e4= nausea, e5= headache, e6= low heart rate (bradycardia), e7= pain upon moving the eyes, e8=fever, e9= a flushing or pale pink rash comes over the face, e10= vomiting.Our model possibility intuitionistic fuzzy soft set for swamp feverFμ is given in Table 8 and this can be prepared with the help of a physician.Table 8
Model PIFSS for swamp fever.
Fμe1e2e3e4e5e6e7e8e9e10y(1,0)(0,1)(0,1)(1,0)(1,0)(1,0)(1,0)(0,1)(1,0)(1,0)μy1111111111n(0,1)(1,0)(1,0)(0,1)(0,1)(0,1)(0,1)(1,0)(0,1)(0,1)μn1111111111After talking to the sick person we can construct his PIFSSGδ as in Table 9. Now we find the similarity measure of these two sets (as in Example 7.4), here S(Fμ,Gδ)≅0.38<1/2. Hence the two PIFSSs are not significantly similar. Therefore we conclude that the person is not suffering from swamp fever.Table 9
PIFSS for the sick person.
Gαe1e2e3e4e5e6e7e8e9e10y(0.3,0)(0,0.7)(0.5,0.1)(0.3,0)(0.4,0)(0.1,0)(0,0)(0.7,0.1)(0,0)(0.4,0.2)μy0.10.20.100.50.10.20.210.4n(0.6,0.2)(0.1,0)(0.4,0)(0.6,0)(0.4,0.2)(0.6,0.3)(08,0)(0.1,0.3)(0.8,0)(0.5,0.3)μn0.50.30.20.10.30.20.40.10.50


## 9. Conclusion

In this paper we have introduced the concept of possibility intuitionistic fuzzy soft set and studied some of its properties. Applications of this theory have been given to solve a decision-making problem. Similarity measure of two possibility intuitionistic fuzzy soft sets is discussed and an application of this to medical diagnosis has been shown.



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*Source: 404325-2012-03-13.xml*