# Generalised Interval-Valued Fuzzy Soft Set

Authors: Shawkat Alkhazaleh; Abdul Razak Salleh
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

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


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## Body

## 1. Introduction

Molodtsov [1] initiated the theory of soft set 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. Presently, work on the soft set theory is progressing rapidly. Maji et al. [2, 3], Roy and Maji [4] have further studied the theory of soft set and used this theory to solve some decision making problems. Maji et al. [5] 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.  Zou and Xiao  [6] introduced soft set and fuzzy soft set into the incomplete environment, respectively. Alkhazaleh et al. [7] introduced the concept of soft multiset as a generalisation of soft set. They also defined the concepts of fuzzy parameterized interval-valued fuzzy soft set [8] and possibility fuzzy soft set [9] and gave their applications in decision making and medical diagnosis. Alkhazaleh and Salleh [10] introduced the concept of a soft expert set, where the user can know the opinion of all experts in one model without any operations. Even after any operation, the user can know the opinion of all experts. In 2011, Salleh [11] gave a brief survey from soft set to intuitionistic fuzzy soft set. Majumdar and Samanta [12] introduced and studied generalised fuzzy soft set where the degree is attached with the parameterization of fuzzy sets while defining a fuzzy soft set. Yang et al. [13] presented the concept of interval-valued fuzzy soft set by combining the interval-valued fuzzy set [14, 15] and soft set models. In this paper, we generalise the concept of fuzzy soft set as introduced by Maji et al. [5] to generalised interval-valued fuzzy soft set. In our generalisation of fuzzy soft set, a degree is attached with the parameterization of fuzzy sets while defining an interval-valued fuzzy soft set. Also, we give some applications of generalised interval-valued fuzzy soft set in decision making problem and medical diagnosis.


## 2. Preliminary

In this section, we recall some definitions and properties regarding fuzzy soft set and generalised fuzzy soft set required in this paper.Definition 2.1 (see [15]).
An interval-valued fuzzy setX̃ on a universe U is a mapping such that
(2.1)X̃:U⟶Int([0,1]),
where Int([0,1]) stands for the set of all closed subintervals of [0,1], the set of all interval-valued fuzzy sets on U is denoted by P̃(U).Suppose thatX̃∈P̃(U), for all x∈U, μx(x)=[μx-(x),μx+(x)] is called the degree of membership of an element x to X. μx-(x) and μx+(x) are referred to as the lower and upper degrees of membership of x to X where 0⩽μx-(x)⩽μx+(x)⩽1.Definition 2.2 (see [14]).
The subset, complement, intersection, and union of the interval-valued fuzzy sets are defined as follows. LetX̃,Ỹ∈P̃(U), then(a)
the complement ofX̃ is denoted by X̃c where
(2.2)μX̃c(x)=1-μX̃(x)=[1-μX̃+(x),1-μX̃-(x)],(b)
the intersection ofX̃ and Ỹ is denoted by X̃∩Ỹ where
(2.3)μX̃∩Ỹ(x)=inf⁡[μX̃(x),μỸ(x)]=[inf⁡(μX̃-(x),μỸ-(x)),inf⁡(μX̃+(x),μỸ+(x))],(c)
the union ofX̃ and Ỹ is denoted by X̃∪Ỹ where
(2.4)μX̃∪Ỹ(x)=sup⁡[μX̃(x),μỸ(x)]=[sup⁡(μX̃-(x),μỸ-(x)),sup⁡(μX̃+(x),μỸ+(x))];(d)
X is a subset of Ydenoted by X⊆Y if μX-(x)≤μY-(x) and μX+(x)≤μY+(x).Definition 2.3 (see [14]).
The compatibility measureφ̅(A,B) of an interval-valued fuzzy set A with an interval-valued fuzzy set B (A is a reference set) is given by
(2.5)φ̅(A,B)=[φ-(A,B),φ+(A,B)],
such that
(2.6)φ-(A,B)=min⁡[φ1(A,B),φ2(A,B)],φ+(A,B)=max⁡[φ1(A,B),φ2(A,B)],
where
(2.7)φ1(A,B)=maxx∈X{min⁡[μA-(x),μB-(x)]}maxx∈X[μA-(x)],φ2(A,B)=maxx∈X{min⁡[μA+(x),μB+(x)]}maxx∈X[μA+(x)].Theorem 2.4 (see [14]).
Consider arbitrary, nonempty interval-valued fuzzy setsA, B, and C from the family of ivf(X) and a compatibility measure in the sense of Definition 2.3. Then,(a)
Aφ̅(A,A)=[1,1]={1},(b)
φ̅(A,B)=[0,0]={0}⇔A∩B=∅,(c)
in generalφ̅(A,B)≠φ̅(B,A).LetU be a universal set and E a set of parameters. Let P(U) denote the power set of U and A⊆E. Molodtsov [1] defined soft set as follows.Definition 2.5.
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 2.6 (see [5]).
LetU be a 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
(2.8)F:A⟶IU.Definition 2.7 (see [12]).
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→IU, where IU is the collection of all fuzzy subsets of U, and let μ be a fuzzy subset of E. Let Fμ:E→IU×I be a function defined as follows:
(2.9)Fμ(e)=(F(e),μ(e)).
Then, Fμ is called a generalised fuzzy soft set (GFSS in short) over the soft set (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:
(2.10)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 degree of belongingness and μ(ei) is the degree of possibility of such belongingness.Definition 2.8 (see [13]).
LetU be an initial universe and E a set of parameters. P̃(U) denotes the set of all interval-valued fuzzy sets of U. Let A⊆E. A pair (F̃,A) is an interval-valued fuzzy soft set over U, where F̃ is a mapping given by F̃:A→P̃(U).


## 3. Generalised Interval-Valued Fuzzy Soft Set

In this section, we generalise the concept of interval-valued fuzzy soft sets as introduced in [13]. In our generalisation of interval-valued fuzzy soft set, a degree is attached with the parameterization of fuzzy sets while defining an interval-valued fuzzy soft set.Definition 3.1.
LetU={x1,x2,…,xn} be the universal set of elements and E={e1,e2,…,em} the universal set of parameters. The pair (U,E) will be called a soft universe. Let F̃:E→P̃(U) and μ be a fuzzy set of E, that is,μ:E→I=[0,1], where P̃(U) is the set of all interval-valued fuzzy subsets on U. Let F̃μ:E→P̃(U)×I be a function defined as follows:
(3.1)F̃μ(e)=(F̃(e),μ(e)).
Then, F̃μ is called a generalised interval-valued fuzzy soft set (GIVFSS in short) over the soft universe (U,E). For each parameter ei, F̃μ(ei)=(F̃(ei)(x),μ(ei)) indicates not only the degree of belongingness of the elements of U in F̃(ei) but also the degree of possibility of such belongingness which is represented by μ(ei). So we can write F̃μ(ei) as follows:
(3.2)F̃μ(ei)=({x1F̃(ei)(x1),x2F̃(ei)(x2),…,xnF̃(ei)(xn)},μ(ei)).Example 3.2.
LetU={x1,x2,x3} be a set of universe, E={e1,e2,e3} a set of parameters, and let μ:E→I. Define a function F̃μ:E→P̃(U)×I as follows:
(3.3)F̃μ(e1)=({x1[0.3,0.6],x2[0.7,0.8],x3[0.5,0.8]},0.6),F̃μ(e2)=({x1[0.1,0.4],x2[0,0.3],x3[0.1,0.5]},0.5),F̃μ(e3)=({x1[0.7,0.8],x2[0.1,0.2],x3[0,0.4]},0.3).
Then, F̃μ is a GIVFSS over (U,E).
In matrix notation, we write(3.4)F̃μ=([0.3,0.6][0.7,0.8][0.5,0.8]0.6[0.1,0.4][0,0.3][0.1,0.5]0.5[0.7,0.8][0.1,0.2][0,0.4]0.3).Definition 3.3.
LetF̃μ and G̃δ be two GIVFSSs over (U,E). F̃μ is called a generalised interval-valued fuzzy soft subset of G̃δ, and we write F̃μ⊆G̃δ if(a)
μ(e) is a fuzzy subset of δ(e) for all e∈E,(b)
F̃(e) is an interval-valued fuzzy subset of G̃(e) for all e∈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, e3 = red. Let F̃μ be a GIVFSS over (U,E) defined as follows:
(3.5)F̃μ(e1)=({x1[0.1,0.3],x2[0.5,0.7],x3[0.3,0.5]},0.4),F̃μ(e2)=({x1[0,0.3],x2[0,0.2],x3[0.1,0.3]},0.4),F̃μ(e3)=({x1[0.5,0.6],x2[0.1,0.1],x3[0.1,0.3]},0.1).
Let G̃δ be another GIVFSS over (U,E) defined as follows:
(3.6)G̃δ(e1)=({x1[0.3,0.6],x2[0.7,0.8],x3[0.5,0.8]},0.6),G̃δ(e2)=({x1[0.2,0.4],x2[0.2,0.3],x3[0.3,0.5]},0.5),G̃δ(e3)=({x1[0.7,0.8],x2[0.2,0.4],x3[0.2,0.5]},0.3).
It is clear that F̃μ is a GIVFS subset of G̃δ.Definition 3.5.
Two GIVFSSsF̃μ and G̃δ over (U,E) are said to be equal, and we write F̃μ=G̃δ if F̃μ is a GIVFS subset of G̃δ and G̃δ is a GIVFS subset of F̃μ. In other words, F̃μ=G̃δ if the following conditions are satisfied: (a)
μ(e) is equal to δ(e) for all e∈E,(b)
F̃(e) is equal to G̃(e) for all e∈E.Definition 3.6.
A GIVFSS is called a generalised null interval-valued fuzzy soft set, denoted by∅̃μ if ∅̃μ:E→P̃(U)×I such that
(3.7)∅̃μ(e)=(F̃(e)(x),μ(e)),
where F̃(e)=[0,0]=[0] and μ(e)=0 for all e∈E.Definition 3.7.
A GIVFSS is called a generalised absolute interval-valued fuzzy soft set, denoted byÃμ if Ãμ:E→P̃(U)×I such that
(3.8)Ãμ(e)=(F̃(e)(x),μ(e)),
where F̃(e)=[1,1]=[1], and μ(e)=1 for all e∈E.


## 4. Basic Operations on GIVFSS

In this section, we introduce some basic operations on GIVFSS, namely, complement, union and intersection and we give some properties related to these operations.Definition 4.1.
LetF̃μ be a GIVFSS over (U,E). Then, the complement of F̃μ, denoted by F̃μc and is defined by F̃μc=G̃δ, such that δ(e)=c(μ(e)) and G̃(e)=c̃(F̃(e)) for all e∈E, where c is a fuzzy complement and c̃ is an interval-valued fuzzy complement.Example 4.2.
Consider a GIVFSSF̃μ over (U,E) as in Example 3.2:
(4.1)F̃μ=([0.3,0.6][0.7,0.8][0.5,0.8]0.6[0.1,0.4][0,0.3][0.1,0.5]0.5[0.7,0.8][0.1,0.2][0,0.4]0.3).
By using the basic fuzzy complement for μ(e) and interval-valued fuzzy complement for F̃(e), we have F̃μc=Gδ where
(4.2)Gδ=([0.4,0.7][0.2,0.3][0.2,0.5]0.4[0.6,0.9][0.7,1][0.5,0.9]0.5[0.2,0.3][0.8,0.9][0.6,1]0.7).Proposition 4.3.
LetF̃μ be a GIVFSS over (U,E). Then, the following holds:
(4.3)(F̃μc)c=F̃μ.Proof.
SinceF̃μc=G̃δ, then
(4.4)(F̃μc)c=G̃δc
but, from Definition 4.1,  G̃δ=(c̃(F̃(e)),c(μ(e))), then
(4.5)G̃δc=(c̃(c̃(F̃(e))),c(c(μ(e))))=(F̃(e),μ(e))=F̃μ.Definition 4.4.
Union of two GIVFSSs(F̃μ,A) and (G̃δ,B), denoted by F̃μ∪G̃δ, is a GIVFSS (H̃ν,C) where C=A∪B and H̃ν:E→P̃(U)×I is defined by
(4.6)H̃ν(e)=(H̃(e),ν(e))
such that H̃(e)=F̃(e)∪G̃(e) and ν(e)=s(μ(e),δ(e)), where s is an s-norm and H̃(e)=[sup(μF̃(e)-,μG̃(e)-),sup(μF̃(e)+,μG̃(e)+)].Example 4.5.
Consider GIVFSSF̃μ and G̃δ as in Example 3.4. By using the interval-valued fuzzy union and basic fuzzy union, we have F̃μ∪G̃δ=H̃v, where
(4.7)H̃v(e1)=({x1[sup⁡(0.1,0.3),sup⁡(0.3,0.6)],x2[sup⁡(0.5,0.7),sup⁡(0.7,0.8)],x3[sup⁡(0.3,0.5),sup⁡(0.5,0.8)]},max⁡(0.4,0.6)x1[sup⁡(0.1,0.3),sup⁡(0.3,0.6)])=({x1[0.3,0.6],x2[0.7,0.8],x3[0.5,0.8]},0.6).
Similarly, we get
(4.8)H̃v(e2)=({x1[0.2,0.4],x2[0.2,0.3],x3[0.3,0.5]},0.5),H̃v(e3)=({x1[0.7,0.8],x2[0.2,0.4],x3[0.2,0.5]},0.3).
In matrix notation, we write
(4.9)H̃v(e)=([0.3,0.6][0.7,0.8][0.5,0.8]0.6[0.2,0.4][0.2,0.3][0.3,0.5]0.5[0.7,0.8][0.2,0.4][0.2,0.5]0.3).Proposition 4.6.
LetF̃μ, G̃δ, and H̃v be any three GIVFSSs. Then, the following results hold.(a)
F̃μ∪G̃δ=G̃δ∪F̃μ.(b)
F̃μ∪(G̃δ∪H̃v)=(F̃μ∪G̃δ)∪H̃v.(c)
F̃μ∪F̃μ⊆F̃μ.(d)
F̃μ∪Ãμ=Ãμ.(e)
F̃μ∪∅̃μ=F̃μ.Proof.
(a)F̃μ∪G̃δ=H̃ν.
From Definition4.4, we have H̃ν(e)=(H̃(e)(x),ν(e)) such that H̃(e)=F̃(e)∪G̃(e) and ν(e)=s(μ(e),δ(e)).
ButH̃(e)=F̃(e)∪G̃(e)=G̃(e)∪F̃(e) (since union of interval-valued fuzzy sets is commutative) and ν(e)=s(μ(e),δ(e))=s(δ(e),μ(e)) (since s-norm is commutative), then G̃δ∪F̃μ=H̃ν.
(b) The proof is straightforward from Definition4.4.
(c) The proof is straightforward from Definition4.4.
(d) The proof is straightforward from Definition4.4.
(e) The proof is straightforward from Definition4.4.Definition 4.7.
Intersection of two GIVFSSs(F̃μ,A) and (G̃δ,B), denoted by F̃μ∩G̃δ, is a GIVFSS (H̃ν,C) where C=A∪B and H̃ν:E→P̃(U)×I is defined by
(4.10)H̃ν(e)=(H̃(e),ν(e))
such that H̃(e)=F̃(e)∩G̃(e) and ν(e)=t(μ(e),δ(e)), where t is a t-norm and H̃(e)=[inf(μF̃(e)-,μG̃(e)-),inf(μF̃(e)+,μG̃(e)+)].Example 4.8.
Consider GIVFSSF̃μ and G̃δ as in Example 4.5. By using the interval-valued fuzzy intersection and basic fuzzy intersection, we have F̃μ∩G̃δ=H̃v, where
(4.11)H̃v(e1)=({x1[inf⁡(0.1,0.3),inf⁡(0.3,0.6)],x2[inf⁡(0.5,0.7),inf⁡(0.7,0.8)],x3[inf⁡(0.3,0.5),inf⁡(0.5,0.8)]}min⁡(0.4,0.6)x1[inf⁡(0.1,0.3),inf⁡(0.3,0.6)])=({x1[0.1,0.3],x2[0.5,0.7],x3[0.3,0.5]},0.4).
Similarly, we get
(4.12)H̃v(e2)=({x1[0,0.3],x2[0,0.2],x3[0.1,0.3]},0.4),H̃v(e3)=({x1[0.5,0.6],x2[0.1,0.1],x3[0.1,0.3]},0.1).
In matrix notation, we write
(4.13)H̃v(e)=([0.1,0.3][0.5,0.7][0.3,0.5]0.4[0,0.3][0,0.2][0.1,0.3]0.4[0.5,0.6][0.1,0.1][0.1,0.3]0.1).Proposition 4.9.
LetF̃μ, G̃δ, and H̃v be any three GIVFSSs. Then, the following results hold.(a)
F̃μ∩G̃δ=G̃δ∩F̃μ.(b)
F̃μ∩(G̃δ∩H̃v)=(F̃μ∩G̃δ)∩H̃v.(c)
F̃μ∩F̃μ⊆F̃μ.(d)
F̃μ∩Ãμ=F̃μ.(e)
F̃μ∩∅̃μ=∅̃μ.Proof.
(a)F̃μ∩G̃δ=H̃ν.
From Definition4.7, we have H̃ν(e)=(H̃(e)(x),ν(e)) such that H̃(e)=F̃(e)∩G̃(e) and ν(e)=t(μ(e),δ(e)).
ButH̃(e)=F̃(e)∩G̃(e)=G̃(e)∩F̃(e) (since intersection of interval-valued fuzzy sets is commutative) and ν(e)=t(μ(e),δ(e))=t(δ(e),μ(e)) (since t-norm is commutative), then G̃δ∩F̃μ=H̃ν.
(b) The proof is straightforward from Definition4.7.
(c) The proof is straightforward from Definition4.7.
(d) The proof is straightforward from Definition4.7.
(e) The proof is straightforward from Definition4.7.Proposition 4.10.
LetF̃μ and G̃δ be any two GIVFSSs. Then the DeMorgan’s Laws hold:(a)
(F̃μ∪G̃δ)c=G̃δc∩F̃μc.(b)
(F̃μ∩G̃δ)c=G̃δc∪F̃μc.Proof.
(a) Consider(4.14)F̃μc∩G̃δc=((c̃(F(e)),c(μ(e)))∩(c̃(G(e)),c(δ(e))))=((c̃(F(e))∩c̃(G(e))),(c(μ(e))∩c(δ(e))))=(((F(e))∪(G(e)))c̃,(μ(e)∪δ(e))c)=(F̃μ∪G̃δ)c.
(b) The proof is similar to the above.Proposition 4.11.
LetF̃μ, G̃δ, and H̃v be any three GIVFSSs. Then, the following results hold. (a)
F̃μ∪(G̃δ∩H̃v)=(F̃μ∪G̃δ)∩(F̃μ∪H̃v).(b)
F̃μ∩(G̃δ∪H̃v)=(F̃μ∩G̃δ)∪(F̃μ∩H̃v).Proof.
(a) For allx∈E,
(4.15)λF̃(x)∪(G̃(x)∩H̃(x))(x)=[sup⁡(λF̃(x)-(x),λG̃(x)∩H̃(x)-(x)),sup⁡(λF̃(x)+(x),λG̃(x)∩H̃(x)+(x))]=[sup⁡(λF̃(x)-(x),inf⁡(λG̃(x)-(x),λH̃(x)-(x))),sup⁡(λF̃(x)+(x),inf⁡(λG̃(x)+(x),λH̃(x)+(x)))]=[inf⁡(sup⁡(λF̃(x)-(x),λG̃(x)-(x)),sup⁡(λF̃(x)-(x),λH̃(x)-(x))),inf⁡(sup⁡(λF̃(x)+(x),λG̃(x)+(x)),sup⁡(λF̃(x)+(x),λH̃(x)+(x)))]=λ(F̃(x)∪G̃(x))∩(F̃(x)∪H̃(x))(x),γμ(x)∪(δ(x)∩ν(x))(x)=max⁡{γμ(x)(x),γδ(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⁡{γμ(x)∪δ(x)(x),γμ(x)∪ν(x)(x)}=γ(μ(x)∪δ(x))∩̃(μ(x)∪ν(x))(x).
(b) Similar to the proof of (a).


## 5. AND and OR Operations on GIVFSS with Application

In this section, we give the definitions of AND and OR operations on GIVFSS. An application of this operations in decision making problem has been shown.Definition 5.1.
If(F̃μ,A) and (G̃δ,B) are two GIVFSSs, then “(F̃μ,A) AND (G̃δ,B)” denoted by (F̃μ,A)∧(G̃δ,B) is defined by
(5.1)(F̃μ,A)∧(G̃δ,B)=(H̃λ,A×B),
where H̃λ(α,β)=(H(α,β),λ(α,β)) for all (α,β)∈A×B, such that H̃(α,β)=F̃(α)∩G̃(β) and λ(α,β)=t(μ(α),δ(β)), for all (α,β)∈A×B, where t is a t-norm.Example 5.2.
Suppose the universe consists of three machinesx1,x2,x3, that is,U={x1,x2,x3}, and consider the set of 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 F̃μ and G̃δ by two experts A and B, respectively, defined as follows:
(5.2)F̃μ(e1)=({x1[0.1,0.3],x2[0.5,0.7],x3[0.3,0.5]},0.4),F̃μ(e2)=({x1[0,0.3],x2[0,0.2],x3[0.1,0.3]},0.4),F̃μ(e3)=({x1[0.5,0.6],x2[0.1,0.1],x3[0.1,0.3]},0.1),G̃μ(e1)=({x1[0.3,0.5],x2[0.2,0.6],x3[0.4,0.5]},0.3),G̃μ(e2)=({x1[0.3,0.5],x2[0.4,0.6],x3[0,0.3]},0.1),G̃μ(e3)=({x1[0.1,0.6],x2[0.4,0.7],x3[0.2,0.3]},0.2).
To find the AND between the two GIVFSSs, we have (F̃μ,A) AND (G̃δ,B)=(H̃λ,A×B), where
(5.3)H̃λ(e1,e1)=({x1[0.1,0.3],x2[0.2,0.6],x3[0.3,0.5]},0.3),H̃λ(e1,e2)=({x1[0.1,0.3],x2[0.4,0.6],x3[0,0.3]},0.1),H̃λ(e1,e3)=({x1[0.1,0.3],x2[0.4,0.7],x3[0.2,0.3]},0.2),H̃λ(e2,e1)=({x1[0,0.3],x2[0,0.2],x3[0.1,0.3]},0.3),H̃λ(e2,e2)=({x1[0,0.3],x2[0,0.2],x3[0,0.3]},0.1),H̃λ(e2,e3)=({x1[0,0.3],x2[0,0.2],x3[0.1,0.3]},0.2),H̃λ(e3,e1)=({x1[0.3,0.5],x2[0.1,0.1],x3[0.1,0.3]},0.1),H̃λ(e3,e2)=({x1[0.3,0.5],x2[0.1,0.1],x3[0,0.3]},0.1),H̃λ(e3,e3)=({x1[0.1,0.6],x2[0.1,0.1],x3[0.1,0.3]},0.1).
Now, to determine the best machine, we first compute the numerical grade rp∈P(xi) for each p∈P such that
(5.4)rp∈P(xi)=∑x∈U((ci--μH(pi)-(x))+(ci+-μH(pi)+(x))).
The result is shown in Tables 1 and 2. Let P={p1=(e1,e1),p2=(e1,e2),…,p9=(e3,e3)}.  Now, we mark the highest numerical grade (indicated in parenthesis) in each row excluding the last row which is the grade of such belongingness of a machine against each pair of parameters (see Table 3). Now, the score of each such machine is calculated by taking the sum of the products of these numerical grades with the corresponding value of μ. 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:Score(x1)
=(1*0.1)+(1.1*0.1)=0.21,Score(x2)
=(1.3*0.1)+(1.3*0.2)=0.39,Score(x3)
=(0.3*0.3)+(0.3*0.2)=0.15.The firm will select the machine with the highest score. Hence, they will buy machine x2.Table 1
(H̃λ,A×B).
x1x2x3μ(e1,e1)[0.1,0.3][0.2,0.6][0.3,0.5]0.3(e1,e2)[0.1,0.3][0.4,0.6][0,0.3]0.1(e1,e3)[0.1,0.3][0.4,0.7][0.2,0.3]0.2(e2,e1)[0,0.3][0,0.2][0.1,0.3]0.3(e2,e2)[0,0.3][0,0.2][0,0.3]0.1(e2,e3)[0,0.3][0,0.2][0.1,0.3]0.2(e3,e1)[0.3,0.5][0.1,0.1][0.1,0.3]0.1(e3,e2)[0.3,0.5][0.1,0.1][0,0.3]0.1(e3,e3)[0.1,0.6][0.1,0.1][0.1,0.3]0.1Table 2
Numerical graderp∈P(xi).
p1p2p3p4p5p6p7p8p9x1−0.8−0.5−0.8000(1)(1.1)(0.8)x2(0.4)(1.3)(1.3)−0.3−0.2−0.3−0.8−0.7−0.7x3(0.4)−0.8−0.5(0.3)(0.1)(0.3)−0.2−0.4−0.1μ0.30.10.20.30.10.20.10.10.1Table 3
Grade table.
p1p2p3p4p5p6p7p8p9xix2,x3x2x2x3x3x3x1x1x1Highest grade—1.31.30.3—0.311.1—μ0.30.10.20.30.10.20.10.10.1Definition 5.3.
If(F̃μ,A) and (G̃δ,B) are two GIVFSSs, then “(F̃μ,A) OR (G̃δ,B)” denoted by (F̃μ,A)∨(G̃δ,B) is defined by
(5.5)(F̃μ,A)∨(G̃δ,B)=(H̃λ,A×B),
where H̃λ(α,β)=(H(α,β),λ(α,β)) for all (α,β)∈A×B, such that H̃(α,β)=F̃(α)∩G̃(β) and λ(α,β)=s(μ(α),δ(β)), for all (α,β)∈A×B, where s is an s-norm.Example 5.4.
Consider Example5.2. To find the OR between the two GIVFSSs, we have (F̃μ,A) OR (G̃δ,B)=(H̃λ,A×B), where
(5.6)H̃λ(e1,e1)=({x1[0.3,0.5],x2[0.5,0.7],x3[0.4,0.5]},0.4),H̃λ(e1,e2)=({x1[0.3,0.5],x2[0.5,0.7],x3[0.4,0.5]},0.4),H̃λ(e1,e3)=({x1[0.1,0.6],x2[0.5,0.7],x3[0.3,0.5]},0.4),H̃λ(e2,e1)=({x1[0.3,0.5],x2[0.,0.6],x3[0.4,0.5]},0.4),H̃λ(e2,e2)=({x1[0.3,0.5],x2[0.4,0.6],x3[0.1,0.3]},0.4),H̃λ(e2,e3)=({x1[0.1,0.6],x2[0.4,0.7],x3[0.2,0.3]},0.4),H̃λ(e3,e1)=({x1[0.5,0.6],x2[0.2,0.6],x3[0.4,0.5]},0.3),H̃λ(e3,e2)=({x1[0.5,0.6],x2[0.4,0.6],x3[0.1,0.3]},0.1),H̃λ(e3,e3)=({x1[0.5,0.6],x2[0.4,0.7],x3[0.2,0.3]},0.2).Remark 5.5.
We use the same method in Example5.2 for the OR operation if the firm wants to buy one such machine depending on any one of the parameters only.Proposition 5.6.
Let(F̃μ,A) and (G̃δ,B) be any two GIVFSSs. Then, the following results hold:(a)
((F̃μ,A)∧(G̃δ,B))c=(F̃μ,A)c∨(G̃δ,B)c,(b)
((F̃μ,A)∨(G̃δ,B))c=(F̃μ,A)c∧(G̃δ,B)c.Proof.
Straightforward from Definitions4.1, 5.1, and 5.3.Proposition 5.7.
Let(F̃μ,A), (G̃δ,B), and (H̃λ,C) be any three GIVFSSs. Then, the following results hold:(a)
(F̃μ,A)∧((G̃δ,B)∧(H̃λ,C))=((F̃μ,A)∧(G̃δ,B))∧(H̃λ,C),(b)
(F̃μ,A)∨((G̃δ,B)∨(H̃λ,C))=((F̃μ,A)∨(G̃δ,B))∨(H̃λ,C),(c)
(F̃μ,A)∨((G̃δ,B)∧(H̃λ,C))=((F̃μ,A)∨(G̃δ,B))∧((F̃μ,A)∨(H̃λ,C)),(d)
(F̃μ,A)∧((G̃δ,B)∨(H̃λ,C))=((F̃μ,A)∧(G̃δ,B))∨((F̃μ,A)∧(H̃λ,C)).Proof.
Straightforward from Definitions5.1 and 5.3.Remark 5.8.
The commutativity property does not hold for AND and OR operations sinceA×B≠B×A.


## 6. Similarity between Two GIVFSS

In this section, we give a measure of similarity between two GIVFSSs. We are taking the set theoretic approach because it is easier to calculate on and is a very popular method too.Definition 6.1.
Similarity between two GIVFSSsF̃μ and G̃δ, denoted by S(F̃μ,G̃δ), is defined by
(6.1)S(F̃μ,G̃δ)=[φ-(F̃,G̃)⋅m(μ,δ),φ+(F̃,G̃)⋅m(μ,δ)],suchthatφ-(F̃,G̃)=min⁡(φ1(F̃,G̃),φ2(F̃,G̃)),φ+(F̃,G̃)=max⁡(φ1(F̃,G̃),φ2(F̃,G̃)),
where
(6.2)φ1(F̃,G̃)={0,ifμF̃i-(x)=0,∀i,∑inmaxx∈X{min⁡(μF̃i-(x),μG̃i-(x))}∑inmaxx∈X[μF̃i-(x)],otherwise,φ2(F̃,G̃)=∑inmaxx∈X{min⁡(μF̃i+(x),μG̃i+(x))}∑inmaxx∈X(μF̃i+(x)),m(μ(e),δ(e))=1-∑|μ(e)-δ(e)|∑|μ(e)+δ(e)|.Definition 6.2.
LetF̃μ and G̃δ be two GIVFSSs over the same universe (U,E). We say that two GIVFSS are significantly similar if φ-(F̃,G̃)·m(μ,δ)⩾1/2.Theorem 6.3.
LetF̃μ, G̃δ, and H̃λ be any three GIVFSSs over (U,E). Then, the following hold: (a)
in generalS(F̃μ,G̃δ)≠S(G̃δ,F̃μ),(b)
φ-(F̃,G̃)≥0 and φ+(F̃,G̃)≤1,(c)
F̃μ=G̃δ⇒S(F̃μ,G̃δ)=1,(d)
F̃μ⊆G̃δ⊆H̃λ⇒S(F̃μ,H̃λ)⩽S(G̃δ,H̃λ),(e)
F̃μ∩̃G̃δ=∅⇔S(F̃μ,G̃δ)=0.Proof.
(a) The proof is straightforward and follows from Definition6.1.
(b) From Definition6.1, we have(6.3)φ1(F̃,G̃)={0,ifμF̃i-(x)=0,∀i,∑inmaxx∈X{min⁡(μF̃i-(x),μG̃i-(x))}∑inmaxx∈X(μF̃i-(x)),otherwise.
IfμF̃i-(x)=0, for all i, then φ-(F̃,G̃)=0, and, if μF̃i-(x)≠0, for some i, then it is clear that φ-(F̃,G̃)≥0.
Also sinceφ+(F̃,G̃)=max(φ1(F̃,G̃),φ2(F̃,G̃)), suppose that φ1(F̃,G̃)=1 and φ2(F̃,G̃)=1, then φ+(F̃,G̃)=1, that means, if φ1(F̃,G̃)<1 and φ2(F̃,G̃)<1, then φ+(F̃,G̃)≤1.
(c) The proof is straightforward and follows from Definition6.1.
(d) The proof is straightforward and follows from Definition6.1.
(e) The proof is straightforward and follows from Definition6.1.Example 6.4.
LetF̃μ be GIVFSS over (U,E) defined as follows:
(6.4)F̃μ(e1)=({x1[0.3,0.7],x2[0.4,0.8],x3[0.1,0.3]},0.4),F̃μ(e2)=({x1[0.5,0.6],x2[0.1,0.3],x3[0,0.4]},0.6),F̃μ(e3)=({x1[0.7,0.9],x2[0.1,0.5],x3[0.8,1]},0.8).
Let G̃δ be another GIVFSS over (U,E) defined as follows:
(6.5)G̃δ(e1)=({x1[0.1,0.4],x2[0.5,0.7],x3[0.2,0.3]},0.3),G̃δ(e2)=({x1[0.6,0.8],x2[0.5,0.6],x3[0.4,0.8]},0.7),G̃δ(e3)=({x1[0.4,0.7],x2[0.3,0.5],x3[0.5,0.7]},0.6).
Here,
(6.6)m(μ(e),δ(e))=1-∑|μ(e)-δ(e)|∑|μ(e)+δ(e)|=1-|(0.4-0.3)|+|(0.6-0.7)|+|(0.8-0.6)||(0.4+0.3)|+|(0.6+0.7)|+|(0.80+0.6)|≅0.8824,φ1(F̃,G̃)=(max⁡{min⁡(0.3,0.1),min⁡(0.4,0.5),min⁡(0.1,0.2)}max⁡(0.3,0.4,0.1)+max⁡(0.5,0.1,0)+max⁡(0.7,0.1,0.8)+max⁡{min⁡(0.5,0.6),min⁡(0.1,0.5),min⁡(0,0.4)}max⁡(0.3,0.4,0.1)+max⁡(0.5,0.1,0)+max⁡(0.7,0.1,0.8)+max⁡{min⁡(0.7,0.4),min⁡(0.1,0.3),min⁡(0.8,0.5)})max⁡(0.3,0.4,0.1)+max⁡(0.5,0.1,0)+max⁡(0.7,0.1,0.8)=max⁡{0.1,0.4,0.1}+max⁡{0.5,0.1,0}+max⁡{0.4,0.1,0.5}max⁡(0.3,0.4,0.1)+max⁡(0.5,0.1,0)+max⁡(0.7,0.1,0.8)=0.4+0.5+0.50.4+0.5+0.8=0.824,φ2(F̃,G̃)=(max⁡{min⁡(0.7,0.4),min⁡(0.8,0.7),min⁡(0.3,0.3)}max⁡(0.7,0.8,0.3)+max⁡(0.6,0.3,0.4)+max⁡(0.9,0.5,1)+max⁡{min⁡(0.6,0.8),min⁡(0.3,0.6),min⁡(0.4,0.8)}max⁡(0.7,0.8,0.3)+max⁡(0.6,0.3,0.4)+max⁡(0.9,0.5,1)+max⁡{min⁡(0.9,0.7),min⁡(0.5,0.5),min⁡(1,0.7)})max⁡(0.7,0.8,0.3)+max⁡(0.6,0.3,0.4)+max⁡(0.9,0.5,1)=max⁡{0.4,0.7,0.3}+max⁡{0.6,0.3,0.4}+max⁡{0.7,0.5,0.7}max⁡(0.7,0.8,0.3)+max⁡(0.6,0.3,0.4)+max⁡(0.9,0.5,1)=0.7+0.6+0.70.8+0.6+1=0.833.
Then,
(6.7)φ-(F̃,G̃)=min⁡(φ1(F̃,G̃),φ2(F̃,G̃))=min⁡(0.824,0.833)=0.824,φ+(F̃,G̃)=max⁡(φ1(F̃,G̃),φ2(F̃,G̃))=max⁡(0.824,0.833)=0.833.
Hence,  the similarity between the two GIVFSSs F̃μ and G̃δ will be
(6.8)S(F̃μ,G̃δ)=[φ-(F̃,G̃)⋅m(μ,δ),φ+(F̃,G̃)⋅m(μ,δ)]=[(0.824)⋅(0.8824),(0.833)⋅(0.8824)]=[0.727,0.735].
Therefore, F̃μ and G̃δ are significantly similar.


## 7. Application of Similarity Measure in Medical Diagnosis

In this section, 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 GIVFSS model for dengue fever and the GIVFSS 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}, where e1 = body temperature, e2 = cough with chest congestion, e3 = loose motion, e4 = chills, e5 = headache, e6 = low heart rate (bradycardia), and e7 = pain upon moving the eyes. Our model GIVFSS for dengue fever Mμ is given in Table 4, and this can be prepared with the help of a physician.Table 4
Model GIVFSS for dengue fever.
Mμe1e2e3e4e5e6e7y1001111n0110000μ1111111Now, after talking to the sick person, we can construct his GIVFSSGδ as in Table 5.Table 5
GIVFSS for the sick person.
Fαe1e2e3e4e5e6e7y[0.3,0.4][0.2,0.5][0,0.2]1[0.4,0.6]0[0.3,0.4]n[0.6,0.9][0.5,0.7][0.6,0.8][0.3,0.5]1[0.4,0.6][0.3,0.5]δ0.30.50.40.60.10.50.2Now, we find the similarity measure of these two sets by using the same method as in Example6.4, where, after the calculation, we get φ-(M̃,G̃)m(μ,δ)≅0.22<1/2. Hence the two GIVFSSs are not significantly similar. Therefore, we conclude that the person is not suffering from dengue fever.


## 8. Conclusion

In this paper, we have introduced the concept of generalised interval-valued fuzzy soft set and studied some of its properties. The complement, union, intersection, “AND,” and “OR” operations have been defined on the interval-valued fuzzy soft sets. An application of this theory is given in solving a decision making problem. Similarity measure of two generalised interval-valued fuzzy soft sets is discussed, and its application to medical diagnosis has been shown.



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*Source: 870504-2012-05-27.xml*