# Possibility Fuzzy Soft Set

Authors: Shawkat Alkhazaleh; Abdul Razak Salleh; Nasruddin Hassan
Journal: Advances in Decision Sciences
 (2011)

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## Abstract

We introduce the concept of possibility fuzzy soft set and its operation and study some of its properties. We give applications of this theory in solving a decision-making problem. We also introduce a similarity measure of two possibility fuzzy soft sets and discuss their application in a medical diagnosis problem.


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## Body

## 1. Introduction

Fuzzy set was introduced by Zadeh in [1] as a mathematical way to represent and deal with vagueness in everyday life. After that many authors have studied the applications of fuzzy sets in different areas (see Klir and Yuan [2]). Molodtsov [3] initiated the theory of soft sets as a new mathematical tool for dealing with uncertainties which traditional mathematical tools cannot handle. He has shown several applications of this theory in solving many practical problems in economics, engineering, social science, medical science, and so forth. Maji et al. [4, 5] have further studied the theory of soft sets and used this theory to solve some decision-making problems. They have also introduced the concept of fuzzy soft set, a more general concept, which is a combination of fuzzy set and soft set and studied its properties [6], and also Roy and Maji used this theory to solve some decision-making problems [7]. Alkhazaleh et al. [8] introduced soft multiset as a generalization of Molodtsov’s soft set. They also introduced in [9] the concept of fuzzy parameterized interval-valued fuzzy soft set and gave its application in decision making. Zhu and Wen in [10] incorporated Molodtsov’s soft set theory with the probability theory and proposed the notion of probabilistic soft sets. In [11] Chaudhuri et al. defined the concepts of soft relation and fuzzy soft relation and then applied them to solve a number of decision-making problems. Majumdar and Samanta [12] defined and studied the generalised fuzzy soft sets where the degree is attached with the parameterization of fuzzy sets while defining a fuzzy soft set. In this paper, we generalise the concept of fuzzy soft sets as introduced by Maji et al. [6] to the possibility fuzzy soft set. In our generalisation of fuzzy soft set, a possibility of each element in the universe is attached with the parameterization of fuzzy sets while defining a fuzzy soft set. Also we give some applications of the possibility fuzzy soft set in decision-making problem and medical diagnosis.


## 2. Preliminaries

In this section, we recall some definitions and properties regarding fuzzy soft set and generalised fuzzy soft set required in this paper.LetU be a universe set, and let E be a set of parameters. Let P(U) denote the power set of U and A⊆E.Definition 2.1 (see [3]).
A pair (F,E) is called a soft  set over U, where F is a mapping given by F:E→P(U). In other words, a soft set over U is a parameterized family of subsets of the universe U.Definition 2.2 (see [6]).
LetU be an initial universal set, and let E be a set of parameters. Let IU denote the power set of all fuzzy subsets of U. Let A⊆E.A pair (F,E) is called a fuzzy  soft  set over U where F is a mapping given by F:A→IU.The following definitions and propositions are due to Majumdar and Samanta [12].Definition 2.3.
LetU={x1,x2,…,xn} be the universal set of elements, and let E={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→I=[0,1], where IU is the collection of all fuzzy subsets of U. Let Fμ:E→IU×I be a function defined as follows:
(2.1)Fμ(e)=(F(e),μ(e)).
Then Fμ is called a generalized  fuzzy  soft  set (GFSS in short) over the soft universe (U,E). Here for each parameter ei,Fμ(ei)=(F(ei),μ(ei)) indicates not only the degree of belongingness of the elements of U in F(ei) but also the degree of possibility of such belongingness which is represented by μ(ei). So we can write Fμ(ei) as follows:
(2.2)Fμ(ei)=({x1F(ei)(x1),x2F(ei)(x2),…,xnF(ei)(xn)},μ(ei)),
where F(ei)(x1),F(ei)(x2),…,F(ei)(xn) are the degrees of belongingness and μ(ei) is the degree of possibility of such belongingness.Definition 2.4.
LetFμ and Gδ be two GFSSs over (U,E). Fμ is said to be a generalised fuzzy soft subset of Gδ if (i)
μ is a fuzzy subset of δ;(ii)
F(e) is also a fuzzy subset of G(e), forall e∈E.In this case, we write Fμ⊆Gδ.Definition 2.5.
Union of two GFSSs Fμ and Gδ, denoted by Fμ∪̃Gδ, is a GFSS Hν, defined as Hν:E→IU×I such that
(2.3)Hν(e)=(H(e),ν(e)),
where H(e)=s(F(e),G(e)), ν(e)=s(μ(e),δ(e)), and s is any s-norm.Definition 2.6.
Intersection of two GFSSs Fμ and Gδ, denoted by Fμ∩̃Gδ, is a GFSS Hν, defined as Hν:E→IU×I such that
(2.4)Hν(e)=(H(e),ν(e)),
where H(e)=t(F(e),G(e)), ν(e)=t(μ(e),δ(e)), and t is any t-norm.Definition 2.7.
A GFSS is said to be a generalised  null  fuzzy  soft  set, denoted by ϕθ, if ϕθ:E→IU×I such that ϕθ(e)=(F(e),θ(e)), where F(e)=0¯, forall e∈E and θ(e)=0¯ forall e∈E.Definition 2.8.
A GFSS is said to be a generalised  absolute  fuzzy  soft  set, denoted by Ãα, if Ãα:E→IU×I such that Ãα(e)=(A(e),α(e)), where A(e)=1¯, forall e∈E and α(e)=1¯forall e∈E.Proposition 2.9.
LetFμ be a GFSS over (U,E). Then the following holds:(i)
Fμ⊆Fμ∪̃Fμ,(ii)
Fμ∩̃Fμ⊆Fμ,(iii)
Fμ∪̃ϕθ=Fμ,(iv)
Fμ∩̃ϕθ=ϕθ,(v)
Fμ∪̃Ãα=Ãα,(vi)
Fμ∪̃Ãα=Fμ.Proposition 2.10.
LetFμ,Gδ, and Hλ be any three GFSSs over (U,E). Then the following holds:(i)
Fμ∪̃Gδ=Gδ∪̃Fμ,(ii)
Fμ∩̃Gδ=Gδ∩̃Fμ,(iii)
Fμ∪̃(Gδ∪̃Hλ)=(Fμ∩̃Gδ)∪̃Hλ,(iv)
Fμ∩̃(Gδ∩̃Hλ)=(Fμ∩̃Gδ)∩̃Hλ.Definition 2.11.
Similarity between the two GFSSs Fμ and Gδ, denoted by S(Fμ,Gδ), is defined as follows:
(2.5)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
(2.6)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 2.12.
LetFμ and Gδ be two GFSSs over the same universe (U,E). We call the two GFSSs to be significantly  similar if S(Fμ,Gδ)≥1/2.Proposition 2.13.
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 Fuzzy Soft Sets

In this section, we generalise the concept of fuzzy soft sets as introduced by Maji et al. [6]. In our generalisation of fuzzy soft set, a possibility of each element in the universe is attached with the parameterization of fuzzy sets while defining a fuzzy soft set.Definition 3.1.
LetU={x1,x2,…,xn} be the universal set of elements and let E={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:
(3.1)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 by μ(ei). So we can write Fμ(ei) as follows:
(3.2)Fμ(ei)={(x1F(ei)(x1),μ(ei)(x1)),(x2F(ei)(x2),μ(ei)(x2)),…,(xnF(ei)(xn),μ(ei)(xn))}.
Sometime we write Fμ as (Fμ,E). If A⊆E, we can also have a PFSS (Fμ,A).Example 3.2.
LetU={x1,x2,x3} be a set of three blouses. Let E={e1,e2,e3} be a set of qualities where e1 = bright, e2 = cheap, and e3 = colourful, and let μ:E→IU. We define a function Fμ:E→IU×IU as follows:
(3.3)Fμ(e1)={(x10.3,0.7),(x20.7,0.5),(x30.5,0.6)},Fμ(e2)={(x10.5,0.6),(x20.6,0.5),(x30.6,0.5)},Fμ(e3)={(x10.7,0.5),(x20.6,0.5),(x30.5,0.7)}.
Then Fμ is a PFSS over (U,E). In matrix notation, we write
(3.4)Fμ=(0.3,0.70.7,0.50.5,0.60.5,0.60.6,0.50.6,0.50.7,0.50.6,0.50.5,0.7).Definition 3.3.
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), foralle∈E,(ii)
F(e) is a fuzzy subset of G(e),foralle∈E.Example 3.4.
LetU={x1,x2,x3} be a set of three cars, and let E={e1,e2,e3} be a set of parameters where e1 = cheap, e2 = expensive, and e3 = red. Let Fμ be a PFSS over (U,E) defined as follows:
(3.5)Fμ(e1)={(x10.2,0.4),(x20.6,0.5),(x30.5,0.6)},Fμ(e2)={(x10.7,0.5),(x20.6,0.6),(x30.8,0.6)},Fμ(e3)={(x10,0.1),(x20.5,0.3),(x30.3,0.1)}.
Let Gδ:E→IU×IU be another PFSS over (U,E) defined as follows:
(3.6)Gδ(e1)={(x10.3,0.6),(x20.7,0.6),(x30.6,0.7)},Gδ(e2)={(x10.8,0.6),(x20.7,0.7),(x30.9,0.8)},Gδ(e3)={(x10.1,0.2),(x20.6,0.5),(x30.5,0.2)}.
It is clear that Fμ is a PFS subset of Gδ.Definition 3.5.
LetFμ and Gδ be two PFSSs over (U,E). Then Fμ and Gδ are said to be equal, and one writes 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),foralle∈E,(ii)
F(e)is equal to G(e),foralle∈E.Definition 3.6.
A PFSS is said to be a possibility  null  fuzzy  soft  set, denoted by φ0, if φ0:E→IU×IU such that
(3.7)φ0(e)=(F(e)(x),μ(e)(x)),∀e∈E,
where F(e)=0, and μ(e)=0,foralle∈E.Definition 3.7.
A PFSS is said to be a possibility  absolute  fuzzy  soft  set, denoted by A1, if A1:E→IU×IU such that
(3.8)A1(e)=(F(e)(x),μ(e)(x)),∀e∈E,
where F(e)=1 and μ(e)=1,foralle∈E.Example 3.8.
LetU={x1,x2,x3} be a set of three blouses. Let E={e1,e2,e3} be a set of qualities where e1 = bright, e2 = cheap, and e3 = colorful, and let μ:E→IU. We define a function Fμ:E→IU×IU which is a PFSS over (U,E) defined as follows:
(3.9)Fμ(e1)={(x10,0),(x20,0),(x30,0)},Fμ(e2)={(x10,0),(x20,0),(x30,0)},Fμ(e3)={(x10,0),(x20,0),(x30,0)}.
Then Fμ is a possibility null fuzzy soft set.
Letμ:E→IU, and we define the function Fμ:E→IU×IU which is a PFSS over (U,E) as follows:
(3.10)Fμ(e1)={(x11,1),(x21,1),(x31,1)},Fμ(e2)={(x11,1),(x21,1),(x31,1)},Fμ(e3)={(x11,1),(x21,1),(x31,1)}.
Then Fμ is a possibility absolute fuzzy soft set.Definition 3.9.
LetFμ be a PFSS over (U,E). Then the complement of Fμ, denoted by Fμc, is defined by Fμc=Gδ such that δ(e)=c(μ(e)) and G(e)=c(F(e)),foralle∈E, where c is a fuzzy complement.Example 3.10.
Consider the matrix notation in Example3.2:
(3.11)Fμ=(0.3,0.70.7,0.50.5,0.60.5,0.60.6,0.50.6,0.50.7,0.50.6,0.50.5,0.7).
By using the basic fuzzy complement, we have Fμc=Gδ where Gδ(3.12)Gδ=(0.7,0.30.3,0.50.5,0.40.5,0.40.4,0.50.4,0.50.3,0.50.4,0.50.5,0.3).


## 4. Union and Intersection of PFSS

In this section, we introduce the definitions of union and intersection of PFSS, derive some properties, and give some examples.Definition 4.1.
Union of two PFSSs Fμ and Gδ, denoted by Fμ∪̃Gδ, is a PFSS Hν:E→IU×IU defined by
(4.1)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.Example 4.2.
LetU={x1,x2,x3} and E={e1,e2,e3}. Let Fμ be a PFSS defined as follows:
(4.2)Fμ(e1)={(x10.7,0.4),(x20.7,0.6),(x30.6,0.6)},Fμ(e2)={(x10.4,0.6),(x20.8,0.5),(x30.3,0.8)},Fμ(e3)={(x10.2,0.9),(x20.8,0.8),(x30.3,0.6)}.
Let Gδ be another PFSS over (U,E) defined as follows:
(4.3)Gδ(e1)={(x10.6,0.4),(x20.3,0.5),(x30.3,0.5)},Gδ(e2)={(x10.7,0.7),(x20.5,0.6),(x30.4,0.7)},Gδ(e3)={(x10.3,0.9),(x20.4,0.4),(x30.6,0.5)}.
By using the basic fuzzy union, we have Fμ∪̃Gδ=Hv, where
(4.4)Hν(e1)={(x1max⁡⁡(0.7,0.6),max⁡(0.4,0.4)),(x2max⁡⁡(0.7,0.3),max⁡(0.6,0.5)),(x3max⁡(0.6,0.3),max⁡(0.6,0.5))}={(x10.7,0.4),(x20.7,0.6),(x30.6,0.6)}.
Similarly we get
(4.5)Hν(e2)={(x10.7,0.7),(x20.8,0.6),(x30.4,0.8)},Hν(e3)={(x10.3,0.9),(x20.8,0.8),(x30.6,0.6)}.
In matrix notation, we write
(4.6)Hν(e)=(0.7,0.40.7,0.60.6,0.60.7,0.70.8,0.60.4,0.80.3,0.90.8,0.80.6,0.6).Definition 4.3.
Intersection of two PFSSs Fμ and Gδ, denoted by Fμ∩̃Gδ, is a PFSS Hν:E→IU×IU defined by
(4.7)Hν(e)=(H(e)(x),ν(e)(x)),∀e∈E,
such that H(e)=t(F(e),G(e)) and ν(e)=t(μ(e),δ(e)) where t is a fuzzy t-norm.Example 4.4.
Consider the Example4.2 where Fμ and Gδ are PFSSs defined as follows:
(4.8)Fμ(e1)={(x10.7,0.4),(x20.7,0.6),(x30.6,0.6)},Fμ(e2)={(x10.4,0.6),(x20.8,0.5),(x30.3,0.8)},Fμ(e3)={(x10.2,0.9),(x20.8,0.8),(x30.3,0.6)},Gδ(e1)={(x10.6,0.4),(x20.3,0.5),(x30.3,0.5)},Gδ(e2)={(x10.7,0.7),(x20.5,0.6),(x30.4,0.7)},Gδ(e3)={(x10.3,0.9),(x20.4,0.4),(x30.6,0.5)}.
By using the basic fuzzy intersection, we have Fμ∩̃Gδ=Hν, where
(4.9)Hν(e1)={(x1min⁡⁡(0.7,0.6),min⁡⁡(0.4,0.4)),(x2min⁡⁡(0.7,0.3),min⁡⁡(0.6,0.5)),(x3min⁡⁡(0.6,0.3),min⁡⁡(0.6,0.5))}={(x10.6,0.4),(x20.3,0.5),(x30.3,0.5)}.
Similarly we get
(4.10)Hν(e2)={(x10.4,0.6),(x20.5,0.5),(x30.3,0.7)},Hν(e3)={(x10.2,0.9),(x20.4,0.4),(x30.3,0.5)}.
In matrix notation, we write
(4.11)Hν(e)=(0.6,0.40.3,0.50.3,0.50.4,0.60.5,0.50.3,0.70.2,0.90.4,0.40.3,0.5).Proposition 4.5.
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∪̃Hν)=(Fμ∪̃Gδ)∪̃Hν,(iv)
Fμ∩̃(Gδ∩̃Hν)=(Fμ∩̃Gδ)∩̃Hν.Proof.
The proof is straightforward by using the fact that fuzzy sets are commutative and associative.Proposition 4.6.
LetFμ be a PFSS 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.Proof.
The proof is straightforward by using the definitions of union and intersection.Proposition 4.7.
LetFμ,Gδ, and Hν be any three PFSSs over (U,E). Then the following results hold:(i)
Fμ∪̃(Gδ∩̃Hν)=(Fμ∪̃Gδ)∩̃(Fμ∪̃Hν),(ii)
Fμ∩̃(Gδ∪̃Hν)=(Fμ∩̃Gδ)∪̃(Fμ∩̃Hν).Proof.
For allx∈E,
(4.12)λF(x)∪̃(G(x)∩̃H(x))(x)=max⁡{λF(x)(x),λ(G(x)∩̃H(x))(x)}=max⁡{λF(x)(x),min⁡(λG(x)(x),λH(x)(x))}=min⁡{max⁡(λF(x)(x),λG(x)(x)),max⁡(λF(x)(x),λH(x)(x))}=min⁡{λ(F(x)∩̃G(x))(x),λ(F(x)∩̃H(x))(x)}=λ(F(x)∩̃G(x))∪̃(F(x)∩̃H(x))(x),γμ(x)∪̃(δ(x)∩̃ν(x))(x)=max⁡{γμ(x)(x),γ(δ(x)∩̃ν(x))(x)}=max⁡{γμ(x)(x),min⁡(γδ(x)(x),γν(x)(x))}=min⁡{max⁡(γμ(x)(x),γδ(x)(x)),max⁡(γμ(x)(x),γν(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 PFSS with Applications in Decision Making

In this section, we introduce the definitions of AND and OR operations on possibility 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 PFSSs 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),v(α,β)(x)),forall(α,β)∈A×B, such that H(α,β)=t(F(α),G(β)) and v(α,β)=t(μ(α),δ(β)),forall(α,β)∈A×B.Example 5.2.
Suppose the universe consists of three machinesx1,x2,x3,thatis,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)={(x10.6,0.4),(x20.7,0.3),(x30.7,0.5)},Fμ(e2)={(x10.7,0.5),(x20.8,0.6),(x30.4,0.5)},Fμ(e3)={(x10.4,0.4),(x20.6,0.6),(x30.9,0.2)}.Gδ(e1)={(x10.7,0.4),(x21,0.4),(x30.5,1)},Gδ(e2)={(x10.8,0.3),(x20.5,0.7),(x30.9,1)},Gδ(e3)={(x10.4,0.7),(x20,0.4),(x30.6,0.4)}.
Then (Fμ,A)∧(Gδ,B)=(Hλ,A×B) where
(5.3)Hλ(e1,e1)={(x1min⁡(0.6,0.7),min⁡(0.4,0.4)),(x2min⁡(0.7,1),min⁡(0.3,0.4)),(x3min⁡(0.6,0.5),min⁡(0.5,1))}={(x10.6,0.4),(x20.7,0.3),(x30.5,0.5)}.
Similarly we get
(5.4)Hλ(e1,e2)={(x10.6,0.3),(x20.5,0.3),(x30.6,0.5)},Hλ(e1,e3)={(x10.4,0.4),(x20,0.3),(x30.6,0.4)},Hλ(e2,e1)={(x10.7,0.4),(x20.8,0.4),(x30.4,0.5)},Hλ(e2,e2)={(x10.7,0.3),(x20.5,0.6),(x30.4,0.5)},Hλ(e2,e3)={(x10.4,0.5),(x20,0.4),(x30.4,0.4)},Hλ(e3,e1)={(x10.4,0.4),(x20.6,0.4),(x30.5,0.2)},Hλ(e3,e2)={(x10.4,0.3),(x20.5,0.6),(x30.9,0.2)},Hλ(e3,e3)={(x10.4,0.4),(x20,0.4),(x30.6,0.2)}.
In matrix notation, we have
(5.5)(Fμ,A)∧(Gδ,B)=(0.6,0.40.7̲,0.30.5,0.50.6̲,0.30.5,0.30.6̲,0.50.4,0.40,0.30.6̲,0.40.7,0.40.8̲,0.40.4,0.50.7,0.30.5,0.60.9̲,0.20.4̲,0.50,0.40.4̲,0.40.4,0.40.6̲,0.40.5,0.20.4,0.30.5,0.60.9,̲0.20.4,0.40,0.40.6̲,0.2).
Now to determine the best machine, we first mark the highest numerical grade (values with underline mark) in each row. Now the score of each of such machines is calculated by taking the sum of the products of these numerical grades with the corresponding possibility λ. The machine with the highest score is the desired machine. We do not consider the numerical grades of the machine against the pairs (ei,ei),i=1,2,3, as both the parameters are the same. Then the firm will select the machine with the highest score. Hence, they will buy machine x3 (see Table 1).Table 1
Grade table.
HxiHighestnumericalgradeλi(e1,e1)x2××(e1,e2)x1,x30.60.3,0.5(e1,e3)x30.60.4(e2,e1)x20.80.4(e2,e2)x3××(e2,e3)x1,x30.40.5,0.4(e3,e1)x20.60.4(e3,e2)x30.90.2(e3,e3)x3××Score(x1)=(0.6×0.3)+(0.4×0.5)=0.38.Score(x2)=(0.8×0.4)+(0.6×0.4)=0.56.Score(x3)=(0.6×0.5)+(0.6×0.4)+(0.4×0.4)+(0.9×0.2)=0.88.Definition 5.3.
If(Fμ,A) and (Gδ,B) are two PFSSs then “(Fμ,A) OR (Gδ,B)”, denoted by (Fμ,A)∨(Gδ,B), is defined by
(5.6)(Fμ,A)∨(Gδ,B)=(Hλ,A×B),
where Hλ(α,β)=(H(α,β)(x),v(α,β)(x))forall(α,β)∈A×B, such that H(α,β)=s(F(α),G(β)) and v(α,β)=s(μ(α),δ(β)),forall(α,β)∈A×B.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
(5.7)Hλ(e1,e1)={(x1max⁡⁡(0.6,0.7),max⁡⁡(0.4,0.4)),(x2max⁡(0.7,1),max⁡⁡(0.3,0.4)),(x3max⁡⁡(0.6,0.5),max⁡⁡(0.5,1))}={(x10.7,0.4),(x21,0.4),(x30.6,1)}.
Similarly we get
(5.8)Hλ(e1,e2)={(x10.8,0.4),(x20.7,0.7),(x30.9,1)},Hλ(e1,e3)={(x10.6,0.7),(x20.7,0.4),(x30.6,0.5)},Hλ(e2,e1)={(x10.7,0.5),(x21,0.6),(x30.5,1)},Hλ(e2,e2)={(x10.8,0.5),(x20.8,0.7),(x30.9,1)},Hλ(e2,e3)={(x10.7,0.7),(x20.8,0.6),(x30.6,0.5)},Hλ(e3,e1)={(x10.7,0.4),(x21,0.6),(x30.9,1)},Hλ(e3,e2)={(x10.8,0.4),(x20.6,0.7),(x30.9,1)},Hλ(e3,e3)={(x10.4,0.7),(x20.6,0.4),(x30.9,0.4)}.
In matrix notation, we have
(5.9)(Fμ,A)∨(Gδ,B)=(0.7,0.41̲,0.40.6,10.8,0.40.7,0.70.9̲,10.6,0.70.7̲,0.40.6,0.50.7,0.51̲,0.60.5,10.8,0.50.8,0.70.9̲,10.7,0.70.8̲,0.60.6,0.50.7,0.41̲,0.60.9,10.8,0.40.6,0.70.9̲,10.4,0.70.6,0.60.9̲,0.4).
Now to determine the best machine, we first mark the highest numerical grade (value with underline mark) in each row. Now the score of each of such machines is calculated by taking the sum of the products of these numerical grades with the corresponding possibility λ. The machine with the highest score is the desired machine. We do not consider the numerical grades of the machine against the pairs (ei,ei),i=1,2,3, as both the parameters are the same. Then the firm will select the machine with the highest score. Hence, they will buy the machine x2 (see Table 2).Table 2
Grade table.
HxiHighestnumericalgradeλi(e1,e1)x2××(e1,e2)x30.91(e1,e3)x20.70.4(e2,e1)x210.6(e2,e2)x3××(e2,e3)x20.80.6(e3,e1)x210.6(e3,e2)x30.91(e3,e3)x3××Score(x1)=0.Score(x2)=(0.7×0.4)+(1×0.6)+(0.8×0.6)+(1×0.6)=1.96.Score(x3)=(0.9×1)+(0.9×1)=1.8.


## 6. Similarity between Two Possibility Fuzzy Soft Sets

Similarity measures have extensive application in several areas such as pattern recognition, image processing, region extraction, 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, fuzzy numbers, and vague sets. Majumdar and Samanta [12–14] have studied the similarity measure of soft sets, fuzzy soft sets, and generalised fuzzy soft sets.In this section, we introduce a measure of similarity between two PFSSs. The set theoretic approach has been taken in this regard because it is easier for calculation and is a very popular method too.Definition 6.1.
Similarity between two PFSSs Fμ and Gδ, denoted by S(Fμ,Gδ), is defined as follows:
(6.1)S(Fμ,Gδ)=M(F(e),G(e))⋅M(μ(e),δ(e)),
such that
(6.2)M(F(e),G(e))=maxiMi(F(e),G(e)),M(μ(e),δ(e))=maxiMi(μ(e),δ(e)),
where
(6.3)Mi(F(e),G(e))=1-∑j=1n|Fij(e)-Gij(e)|∑j=1n|Fij(e)+Gij(e)|,Mi(μ(e),δ(e))=1-∑j=1n|μij(e)-δij(e)|∑j=1n|μij(e)+δij(e)|.Definition 6.2.
LetFμ and Gδ be two PFSSs over (U,E). We say that Fμ and Gδ are significantly  similar if S(Fμ,Gδ)≥1/2.Proposition 6.3.
LetFμ and Gδ be any two PFSSs 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 6.4.
Consider Example4.2 where Fμ and Gδ are defined as follows:
(6.4)Fμ(e1)={(x10.7,0.4),(x20.7,0.6),(x30.6,0.6)},Fμ(e2)={(x10.4,0.6),(x20.8,0.5),(x30.3,0.8)},Fμ(e3)={(x10.2,0.9),(x20.8,0.8),(x30.3,0.6)},Gδ(e1)={(x10.6,0.4),(x20.3,0.5),(x30.3,0.5)},Gδ(e2)={(x10.7,0.7),(x20.5,0.6),(x30.4,0.7)},Gδ(e3)={(x10.3,0.9),(x20.4,0.4),(x30.6,0.5)}.
Here
(6.5)M1(μ(e),δ(e))=1-∑j=13|μ1j(e)-δ1j(e)|∑j=13|μ1j(e)+δ1j(e)|=1-|(0.4-0.4)|+|(0.6-0.5)|+|(0.6-0.5)||(0.4+0.4)|+|(0.6+0.5)|+|(0.6+0.5)|=0.82.
Similarly we get M2(μ(e),δ(e))=0.77 and M3(μ(e),δ(e))=0.88. Then
(6.6)M(μ(e),δ(e))=max⁡⁡(M1(μ(e),δ(e)),M2(μ(e),δ(e)),M3(μ(e),δ(e)))=0.88,M1(F(e),G(e))=1-∑j=13|F1j(e)-G1j(e)|∑j=13|F1j(e)+G1j(e)|=1-|(0.7-0.6)|+|(0.7-0.3)|+|(0.6-0.3)||(0.7+0.6)|+|(0.7+0.3)|+|(0.6+0.3)|=0.75.
Similarly we get M2(F(e),G(e))=0.77 and M3(F(e),G(e))=0.69. Then
(6.7)M(F(e),G(e))=max⁡⁡(M1(F(e),G(e)),M2(F(e),G(e)),M3(F(e),G(e)))=0.77.
Hence, the similarity between the two PFSSs Fμ and Gδ is given by
(6.8)S(Fμ,Gδ)=M(F(e),G(e))⋅M(μ(e),δ(e))=0.96×0.77≅0.74.


## 7. Application of This 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 dengue fever. For this we first construct a model possibility fuzzy soft set for dengue fever and the possibility 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 dengue 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,e11), where e1 is body temperature, e2is cough with chest congestion, e3is loose motion, e4is chills, e5is headache, e6is low heart rate (bradycardia), e7is pain upon moving the eyes, e8is breathing trouble, e9is a flushing or pale pink rash comes over the face, e10is low blood pressure (hypotension), and e11is Loss of appetite.Our model possibility fuzzy soft set for dengue feverMμ is given in Table 3, and this can be prepared with the help of a physician.Table 3
Model PFSS for dengue fever.
Mμe1e2e3e4e5e6e7e8e9e10e11y10011110110μy11111111111n01100001001μn11111111111After talking to the sick person, we can construct his PFSSGδ as in Table 4. Now we find the similarity measure of these two sets (as in Example 6.4), here S(Mμ,Gδ)≅0.43<1/2. Hence the two PFSSs are not significantly similar. Therefore, we conclude that the person is not suffering from dengue fever.Table 4
PFSS for the sick person.
Fαe1e2e3e4e5e6e7e8e9e10e11y0.30.70.50.30.40.100.700.40.2μy0.60.20.10.50.50.80.70.210.40.5n0.60.10.40.50.40.60.70.10.80.50.4μn0.50.60.40.50.30.60.40.10.50.60.7


## 8. Conclusion

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



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*Source: 479756-2011-09-08.xml*