# On Generalised Interval-Valued Fuzzy Soft Sets

Authors: Xiaoqiang Zhou; Qingguo Li; Lankun Guo
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Soft set theory, initiated by Molodtsov, can be used as a new mathematical tool for
dealing with imprecise, vague, and uncertain problems. In this paper, the concepts of two types of
generalised interval-valued fuzzy soft set are proposed and their basic properties are studied. The
lattice structures of generalised interval-valued fuzzy soft set are also discussed. Furthermore, an
application of the new approach in decision making based on generalised interval-valued fuzzy
soft set is developed.


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## Body

## 1. Introduction

Most of our real-life problems in social science, economics, medical science, engineering, environmental science, and many other fields have various uncertainties. To deal with these uncertainties, many kinds of theories have been proposed such as theory of probability [1], fuzzy set theory [2], rough set theory [3], intuitionistic fuzzy set theory [4], and interval mathematics [5–7]. Unfortunately, each of these theories has its inherent difficulties, which was pointed out by Molodtsov in [8]. To overcome these difficulties, Molodtsov [8] proposed the soft set theory, which has become a new completely generic mathematical tool for modeling uncertainties.Recently, the soft set theory has been widely focused in theory and application after Molodtsov’s work. Maji and Biswas [9] first introduced the concepts of soft subset, soft superset, soft equality, null soft set, and absolute soft set. They also gave some operations on soft set and verified De Morgan’s laws. Ali et al. [10] corrected some errors of former studies and defined some new operations on soft sets. Afterwards, Ali et al. [11] further studied some important properties associated with the new operations and investigated some algebraic structures of soft sets. Sezgin and Atagün [12] extended the theoretical aspect of operations on soft sets. Soft mappings, soft equality, kernels and closures of soft set relations, and soft set relation mappings were presented in [13–15]. On the other hand, soft set theory has a rich potential for application in many fields. Especially, it has been successfully applied to soft decision making [16–18] and some algebra structures such as groups [19, 20], ordered semigroups [21], rings [22], semirings [23], BCK/BCI-algebras [24–26], d-algebras [27], and BL-algebras [28].Clearly, all of these works mentioned above are based on the classical soft set theory. To improve the capability of soft set theory in dealing with more complex real-life problems, some fuzzy extensions of soft set theory have been studied by many scholars [29–36]. Particularly, Maji et al. [29] firstly proposed the concept of the fuzzy soft set. Roy and Maji [30] presented an application of fuzzy soft set in decision making. Yang et al. [31] defined the interval-valued fuzzy soft set which is based on a combination of the interval-valued fuzzy set and soft set. Majumdar and Samanta [32] generalized the concept of fuzzy soft sets; that is, a degree of which is attached with the parameterization of fuzzy sets while defining a fuzzy soft set.However, in many practical applications, specially in fuzzy decision-making problems, the membership functions of objects and parameters are very individual, which are dependent on evaluation of experts in general and thus cannot be lightly confirmed. For example, concerning the fuzzy concept “capability”, there are three experts who give their evaluations to that of someone as 0.6, 0.76, and 0.8, respectively. Clearly, it is more practical and reasonable to evaluate someone’s capability by an interval-valued data [0.6, 0.8] than a certain single value. In this case, therefore, we can make use of interval-valued fuzzy sets which assign to each object or parameter an interval that approximates the “real’’ (but unknown) membership degree. This paper aims to further generalize the concept of generalised fuzzy soft sets by combining the generalised fuzzy soft sets [32] and interval-valued fuzzy sets [7] and obtain a new soft set model named generalised interval-valued fuzzy soft set. It can be viewed as an interval-valued fuzzy extension of the generalised fuzzy soft set theory [32] or a generalization of the interval-valued fuzzy soft set theory [31].The rest of this paper is organized as follows. In Section2, the notions of soft set, fuzzy soft set, generalised fuzzy soft set, and interval-valued fuzzy soft set are recalled. In Section 3, the concept and operations of generalised interval-valued fuzzy soft sets are proposed and some of their properties are investigated. Section 4 studies the lattice structures of generalised interval-valued fuzzy soft set. Section 5 introduces the concept of generalised comparison table, which is applied to decision making based on generalised interval-valued fuzzy soft set. Some illustrative examples are also employed to show that the method presented here is not only reasonable but also more efficient in practical applications. Finally, Section 6 presents the conclusion.


## 2. Preliminary

In this section, we briefly review the concepts of soft sets, fuzzy soft sets, generalised fuzzy soft sets, interval-valued fuzzy soft set, and so on. Further details could be found in [7, 8, 29, 31, 32, 37]. Throughout this paper, unless otherwise stated, U refers to an initial universe, E is a set of parameters, P(U) is the power set of U, and α,β,γ are fuzzy subset of A,B,C⊆E, respectively.Definition 2.1 (see [8]).
A pair(F,A) is called a soft set over U where F is a mapping given by F:A→P(U).In other words, a soft set overU is a parameterized family of subsets of the universe U. For ε∈A, F(ε) may be considered as the set of ε-elements of the soft set (F,A) or as the set of ε-approximate elements of the soft set.Definition 2.2 (see [29]).
Let𝒫(U) denote the set of all fuzzy subsets of U.Then a pair (F̃,A) is called a fuzzy soft set over U,where F̃ is a mapping from A to 𝒫(U).From the definition, it is clear thatF̃(e) is a fuzzy set on U for any e∈A. The modified definition of fuzzy soft set by Majumdar and Samanta is as follows.Definition 2.3 (see [32]).
LetU be an initial universal set, E a set of parameters, and the pair (U,E) a soft universe. Let F:E→𝒫(U) and μ be a fuzzy subset of E; that is, μ:E→[0,1].Let Fμ:E→𝒫(U)×[0,1] be a function defined as follows: Fμ(e)=(F(e),μ(e)),where F(e)∈𝒫(U).Then Fμ is called a generalised fuzzy soft set over (U,E).Definition 2.4 (see [7]).
An interval-valued fuzzy setX on a universe U is a mapping X: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 onU is denoted by ℱ(U). Suppose that X∈ℱ(U), for all h∈U,μX(h)=[μX-(h),μX+(h)] is called the degree of membership of an element h to X. And μX-(h) and μX+(h) are referred to as the lower and upper degrees of membership of h to X, where 0≤μX-(h)≤μX+(h)≤1.Definition 2.5 (see [7]).
LetX and Y be two interval-valued fuzzy sets on universe U.Then the union, intersection, and complement of vague sets are defined as follows:
(2.1)X∪Y={h[μX-(h)∨μY-(h),μX+(h)∧μY+(h)]∣h∈U},X∩Y={h[μX-(h)∧μY-(h),μX+(h)∨μY+(h)]∣h∈U},Xc={h[1-μX+(h),1-μX-(h)]∣h∈U}.Definition 2.6 (see [31]).
LetU be an initial universe, let E be a set of parameters, and let A⊆E.ℱ(U) denotes the set of all interval-valued fuzzy sets of U.A pair (F,A) is an interval-valued fuzzy soft set over U,where F is a mapping given by F:A→ℱ(U).An interval-valued fuzzy soft set is a parameterized family of interval-valued fuzzy subsets ofU. For each parameter e∈A, F(e) is actually an interval-valued fuzzy set of U, and it can be written as F(e)={(h/μF(e)(h)):h∈U}, where μF(e)(h) is the interval-valued fuzzy membership degree that object h holds on parameter e.Definition 2.7 (see [37]).
At-norm is an increasing, associative, and commutative mapping T:[0,1]×[0,1]→[0,1] that satisfies the boundary condition: T(a,1)=a for all a∈[0,1].The commonly used continuoust-norms are T(a,b)=min(a,b), T(a,b)=max{0,a+b-1}, and T(a,b)=a·b.Definition 2.8 (see [37]).
At-conorm is an increasing, associative, and commutative mapping S:[0,1]×[0,1]→[0,1] that satisfies the boundary condition: S(a,0)=a for all a∈[0,1].The commonly used continuoust-conorms are S(a,b)=max(a,b), S(a,b)=a+b-a·b, and S(a,b)=min{1,a+b}.


## 3. Generalised Interval-Valued Fuzzy Soft Set

Obviously, by combining generalised soft set and the interval-valued fuzzy set, it is natural to define the generalised interval-valued fuzzy soft set model. We first define two types of generalised interval-valued fuzzy soft set as follows.Definition 3.1.
LetU be an initial universe and E a set of parameters, A⊆E,F̃:A→ℱ(U),and let α be a fuzzy sets of A, that is, α:A→[0,1].Define a function F̃α:A→ℱ(U)×[0,1] as F̃α(e)=(F̃(e)={h/μF̃(e)(h)},α(e)), where μF̃(e)(h)=[μF̃(e)-(h),μF̃(e)+(h)] is an interval value is called the degree of membership an element htoF̃(e), and α(e) is called the degree of possibility of such belongness. Then F̃α is called type 1 generalised interval-valued fuzzy soft set over the soft universe (U,E).Here for each parametere, F̃α(e) indicates not only the degree of belongingness of elements of U in F̃(e) but also the degree of preference of such belongingness which is represented by α(e).Definition 3.2.
LetU be an initial universe and E a set of parameters, A⊆E,F̃:A→ℱ(U),and let α be an interval-valued fuzzy sets of A; that is, α:A→Int([0,1]), where Int([0,1]) stands for the set of all closed subintervals of [0,1].Define a function F̃α:A→ℱ(U)×Int([0,1]) as F̃α(e)=(F̃(e)={h/μF̃(e)(h)},α(e)), where μF̃(e)(h)=[μF̃(e)-(h),μF̃(e)+(h)] and α(e)=[α-(e),α+(e)] are interval values. Then F̃α is called type 2 generalised interval-valued fuzzy soft set over the soft universe (U,E).It is clear that ifα-(e)=α+(e) holds for each a∈A, then the type 2 generalised interval-valued fuzzy soft set will degenerate to the type 1 generalised interval-valued fuzzy soft set. And if μF̃(e)-(h)=μF̃(e)+(h) also holds for each a∈A, then type 1 generalised interval-valued fuzzy soft set will degenerate to generalised fuzzy soft set [32].In this paper, the type 2 generalised interval-valued fuzzy soft set is denoted by GIVFS set in short. To illustrate this idea, let us consider the following example.Example 3.3.
LetU={h1,h2,h3} be a set of mobile telephones and A={e1,e2,e3}∈E a set of parameters. The ei(i=1,2,3) stand for the parameters “expensive”, “beautiful”, and “multifunctional”, respectively. Let F̃α:A→𝒫(U)×Int([0,1]) be a function given as follows:
(3.1)F̃α(e1)=({h1[0.8,0.9],h2[0.6,0.7],h3[0.5,0.6]},[0.7,0.8]),F̃α(e2)=({h1[0.7,0.8],h2[0.3,0.4],h3[0.5,0.7]},[0.6,0.7]),F̃α(e3)=({h1[0.5,0.6],h2[0.5,0.7],h3[0.7,0.8]},[0.8,0.9]).
Then F̃α is a GIVFS set.Definition 3.4.
LetF̃α and G̃β be GIVFS sets over (U,E). Then F̃α is called a GIVFS subset of G̃β if(1)
A⊆B;(2)
F̃(e) is an interval-valued fuzzy subset of G̃(e) for any e∈A; that is, μF̃(e)-(h)≤μG̃(e)-(h) and μF̃(e)+(h)≤μG̃(e)+(h) for any h∈U and e∈A;(3)
α is an interval-valued fuzzy subset of β; that is, α-(e)≤β-(e) and α+(e)≤β+(e) for any e∈A.
In this case, the above relationship is denoted byF̃α⋐G̃β. And G̃β is said to be a GIVFS superset of F̃α.Definition 3.5.
LetF̃α and G̃β be GIVFS sets over (U,E).Then F̃α and G̃β are said to be GIVFS equal if and only if F̃α⋐G̃β and G̃β⋐F̃α.Definition 3.6.
The relative complement of a GIVFS setF̃α is denoted by F̃αr and is defined by F̃αr=(F̃r,αr), where F̃r:A→ℱ(U) is a mapping given by F̃r(e)={h/μF̃r(e)(h)} and αr:A→Int([0,1])isa mapping given by αr(e) for all h∈U,e∈A, where μF̃r(e)(h)=[μF̃r(e)-(h),μF̃r(e)+(h)]=[1-μF̃(e)+(h),1-μF̃(e)-(h)], αr(e)=[αr-(e),αr+(e)]=[1-α+(e),1-α-(e)].Example 3.7.
We consider theGIVFS set F̃α given in Example 3.3 and define a GIVFS set G̃β as follows:
(3.2)G̃β(e1)=({h1[0.7,0.8],h2[0.4,0.5],h3[0.4,0.6]},[0.5,0.6]),G̃β(e2)=({h1[0.5,0.6],h2[0.2,0.4],h3[0.5,0.6]},[0.3,0.4]).
Then G̃β is a GIVFS subset of F̃α, and the relative complement of a GIVFS set G̃β is
(3.3)G̃βr(e1)=({h1[0.2,0.3],h2[0.5,0.6],h3[0.4,0.6]},[0.4,0.5]),G̃βr(e2)=({h1[0.4,0.5],h2[0.6,0.8],h3[0.4,0.5]},[0.6,0.7]).Definition 3.8.
Let1¯=[1,1].A GIVFS set F̃α over (U,E) is said to be relative absolute GIVFS set denoted by Ω̃A, if μF̃(e)(h)=1¯ and α(e)=1¯ for all h∈U and e∈A.Definition 3.9.
Let0¯=[0,0].A GIVFS set F̃α over (U,E) is said to be relative null GIVFS set, denoted by Φ̃A, if μF̃(e)(h)=0¯ and α(e)=0¯ for all h∈U and e∈A.Definition 3.10.
The union of two GIVFS setsF̃α and G̃β over (U,E) denoted by F̃α∪̃G̃β is a GIVFS set H̃γ and defined as H̃γ:A∪B→ℱ(U)×Int([0,1]) such that, for all h∈U and e∈A∪B,
(3.4)H̃γ(e)={({hμF̃(e)(h)},α(e)),ife∈A-B,({hμG̃(e)(h)},β(e)),ife∈B-A,({hμH̃(e)(h)},γ(e)),ife∈A∩B,
where μH̃(e)(h)=S(μF̃(e)(h),μG̃(e)(h))=[S(μF̃(e)-(h),μG̃(e)-(h)),S(μF̃(e)+(h),μG̃(e)+(h))] and γ(e)=S(α(e),β(e))=[S(α-(e),β-(e)),S(α+(e),β+(e))].Definition 3.11.
The intersection of twoGIVFS sets F̃α and G̃β over (U,E) denoted by F̃α⋒G̃β is a GIVFS set H̃γ and defined as H̃γ:A∩B→ℱ(U)×Int([0,1]) such that, for all h∈U and e∈A∩B,H̃γ(e)=({h/μH̃(e)(h)},γ(e)), where μH̃(e)(h)=T(μF̃(e)(h),μG̃(e)(h))=[T(μF̃(e)-(h),μG̃(e)-(h)),T(μF̃(e)+(h),μG̃(e)+(h))] and γ(e)=T(α(e),β(e))=[T(α-(e),β-(e)),T(α+(e),β+(e))].Example 3.12.
We consider the GIVFS setsF̃α and G̃β given in Examples 3.3 and 3.7, respectively, and consider S(x,y)=max{x,y} and T(x,y)=min{x,y}. Then
(3.5)(F̃α∪̃G̃β)(e1)=({h1[0.8,0.9],h2[0.6,0.7],h3[0.5,0.6]},[0.7,0.8]),(F̃α∪̃G̃β)(e2)=({h1[0.7,0.8],h2[0.3,0.4],h3[0.5,0.7]},[0.6,0.7]),(F̃α∪̃G̃β)(e3)=({h1[0.5,0.6],h2[0.5,0.7],h3[0.7,0.8]},[0.8,0.9]),(F̃α⋒G̃β)(e1)=({h1[0.7,0.8],h2[0.4,0.5],h3[0.4,0.6]},[0.5,0.6]),(F̃α⋒G̃β)(e2)=({h1[0.5,0.6],h2[0.2,0.4],h3[0.5,0.6]},[0.3,0.4]).Proposition 3.13.
LetF̃α be a GIVFS set over (U,E). Then the following holds(1)
F̃α⋒Ω̃A=F̃α,(2)
F̃α∪̃Ω̃A=Ω̃A,(3)
F̃α⋒Φ̃A=Φ̃A,(4)
F̃α∪̃Φ̃A=F̃α.Proof.
It is easily obtained from Definitions3.8–3.11.Theorem 3.14.
LetF̃α,  G̃β,  and H̃γ be GIVFS sets over (U,E). Then the following holds(1)
F̃α⋒G̃β=G̃β⋒F̃α,(2)
F̃α⋒(G̃β⋒H̃γ)=(F̃α⋒G̃β)⋒H̃γ,(3)
F̃α∪̃G̃β=G̃β∪̃F̃α,(4)
F̃α∪̃(G̃β∪̃H̃γ)=(F̃α∪̃G̃β)∪̃H̃γ.Proof.
It is easily obtained from Definitions3.10 and 3.11.Definition 3.15.
The restricted union of two GIVFS setsF̃α and G̃β over (U,E) denoted by F̃α⋓G̃β is a GIVFS set H̃γ and defined as H̃γ:A∩B→ℱ(U)×Int([0,1]) such that, for all h∈U and e∈A∩B,H̃γ(e)=({h/μH̃(e)(h)},γ(e)), where μH̃(e)(h)=S(μF̃(e)(h),μG̃(e)(h))=[S(μF̃(e)-(h),μG̃(e)-(h)),S(μF̃(e)+(h),μG̃(e)+(h))] and γ(e)=S(α(e),β(e))=[S(α-(e),β-(e)),S(α+(e),β+(e))].Definition 3.16.
The extended intersection of two GVS setsF̃α and G̃β over (U,E), denoted by F̃α∩̃G̃β, is a GVS set H̃γ:A∪B→ℱ(U)×Int([0,1]) which is defined as, for all h∈U,e∈A∪B,(3.6)H̃γ(e)={({hμF̃(e)(h)},α(e)),ife∈A-B,({hμG̃(e)(h)},β(e)),ife∈B-A,({hμH̃(e)(h)},γ(e)),ife∈A∩B,
where μH̃(e)(h)=T(μF̃(e)(h),μG̃(e)(h))=[T(μF̃(e)-(h),μG̃(e)-(h)), T(μF̃(e)+(h),μG̃(e)+(h))] and γ(e)=T(α(e),β(e))=[T(α-(e),β-(e)),T(α+(e),β+(e))].Theorem 3.17.
LetF̃α,  G̃β,  and H̃γ be three GIVFS sets over (U,E). Then the following holds:(1)
F̃α⋓G̃β=G̃β⋓F̃α,(2)
F̃α⋓(G̃β⋓H̃γ)=(F̃α⋓G̃β)⋓H̃γ,(3)
F̃α∩̃G̃β=G̃β∩̃F̃α,(4)
F̃α∩̃(G̃β∩̃H̃γ)=(F̃α∩̃G̃β)∩̃H̃γ.Proof.
It is easily obtained from Definitions3.15 and 3.16.Theorem 3.18.
LetF̃α and  G̃β be two  GIVFS sets over  (U,E). Then the following holds: (1)
(F̃α⋒G̃β)r=(F̃α)r⋓(G̃β)r,(2)
(F̃α⋓G̃β)r=(F̃α)r⋒(G̃β)r.Proof.
(1) Suppose thatF̃α⋒G̃β=H̃γ, then C=A∩B, and, foralle∈C,h∈U,
(3.7)μH̃(e)(h)=T(μF̃(e)(h),μG̃(e)(h))=[T(μF̃(e)-(h),μG̃(e)-(h)),T(μF̃(e)+(h),μG̃(e)+(h))],γ(e)=T(α(e),β(e))=[T(α-(e),β-(e)),T(α+(e),β+(e))].
Moreover, we have (F̃α⋒G̃β)r=H̃γr, C=A∩B, and foralle∈C,h∈U,
(3.8)μH̃r(e)(h)=[1-T(μF̃(e)+(h),μG̃(e)+(h)),1-T(μF̃(e)-(h),μG̃(e)-(h))],γr(e)=[1-T(α+(e),β+(e)),1-T(α-(e),β-(e))].
Assume that the parameters set of a GIVFS set J̃δ is denoted D, and F̃αr⋓G̃βr=J̃δ. Then D=A∩B. Since
(3.9)μF̃r(e)(h)=[1-μF̃(e)+(h),1-μF̃(e)-(h)],αr(e)=[1-α+(e),1-α-(e)],μG̃r(e)(h)=[1-μG̃(e)+(h),1-μG̃(e)-(h)],βr(e)=[1-β+(e),1-β-(e)].
Then, for each e∈D,h∈U,
(3.10)μJ̃(e)(h)=S(μF̃r(e)(h),μG̃r(e)(h))=[S(1-μF̃(e)+(h),1-μG̃(e)+(h)),S(1-μF̃(e)-(h),1-μG̃(e)-(h))]=[1-T(μF̃(e)+(h),μG̃(e)+(h)),1-T(μF̃(e)-(h),μG̃(e)-(h))]=μH̃r(e)(h),δ(e)=S(αr(e),βr(e))=[S(1-α+(e),1-β+(e)),S(1-α-(e),1-β-(e))]=[1-T(α+(e),β+(e)),1-T(α-(e),β-(e))]=γr(e).
Therefore, H̃γr and J̃δ are the same GIVFS sets. Thus, (F̃α⋒G̃β)r=(F̃α)r⋓(G̃β)r.
(2) The proof is similar to that of (1).Definition 3.19.
The “AND” of two GIVFS setsF̃α and G̃β over (U,E), denoted by F̃α⌅G̃β, is defined as H̃γ:A×B→ℱ(U)×Int([0,1]) such that for all h∈U and (a,b)∈A×B,H̃γ(a,b)=({h/μH̃(a,b)(h)},γ(a,b)), where μH̃(a,b)(h)=T(μF̃(a)(h),μG̃(b)(h))=[T(μF̃(a)-(h),μG̃(b)-(h)),T(μF̃(a)+(h),μG̃(b)+(h))] and γ(a,b)=T(α(a),β(b))=[T(α-(a),β-(b)),T(α+(a),β+(b))].Definition 3.20.
The “OR” of two GIVFS setsF̃α and G̃β over (U,E), denoted by F̃α⊻G̃β, is defined as H̃γ:A×B→ℱ(U)×Int([0,1]) such that for all h∈U and (a,b)∈A×B,H̃γ(a,b)=({h/μH̃(a,b)(h)},γ(a,b)),where μH̃(a,b)(h)=S(μF̃(a)(h),μG̃(b)(h))=[S(μF̃(a)-(h),μG̃(b)-(h)),S(μF̃(a)+(h),μG̃(b)+(h))] and γ(a,b)=S(α(a),β(b))=[S(α-(a),β-(b)),S(α+(a),β+(b))].Theorem 3.21.
LetF̃α,  G̃β, and  H̃γ be three GIVFS sets over  (U,E). Then the following holds (1)
F̃α⊻(G̃β⊻H̃γ)=(F̃α⊻G̃β)⊻H̃γ,(2)
F̃α⌅(G̃β⌅H̃γ)=(F̃α⌅G̃β)⌅H̃γ.Proof.
It is easily obtained from Definitions3.19 and 3.20.Theorem 3.22.
LetF̃α and  G̃β be two GIVFS sets over  (U,E). Then the following holds (1)
(F̃α⊻G̃β)r=(F̃α)r⌅(G̃β)r,(2)
(F̃α⌅G̃β)r=(F̃α)r⊻(G̃β)r.Proof.
(1) Suppose thatF̃α⊻G̃β=H̃γ, then C=A×B, and, forall(a,b)∈C,h∈U,
(3.11)μH̃(a,b)(h)=S(μF̃(a)(h),μG̃(b)(h))=[S(μF̃(a)-(h),μG̃(b)-(h)),S(μF̃(a)+(h),μG̃(b)+(h))],γ(a,b)=S(α(a),β(b))=[S(α-(a),β-(b)),S(α+(a),β+(b))].
Moreover, we have (F̃α⊻G̃β)r=H̃γr, C=A×B, and forall(a,b)∈C,h∈U,
(3.12)μH̃r(a,b)(h)=[1-S(μF̃(a)+(h),μG̃(b)+(h)),1-S(μF̃(a)-(h),μG̃(b)-(h))],γr(a,b)=[1-S(α+(a),β+(b)),1-S(α-(a),β-(b))].
Assume that the parameters set of a GIVFS set J̃δ is denoted D, and F̃αr⌅G̃βr=J̃δ. Then D=A×B. Since foralla∈A,b∈B,h∈U,
(3.13)μF̃r(a)(h)=[1-μF̃(a)+(h),1-μF̃(a)+(h)],αr(a)=[1-α+(a),1-α-(a)],μG̃r(b)(h)=[1-μG̃(b)+(h),1-μG̃(b)-(h)],βr(b)=[1-β+(b),1-β-(b)],
then, for each (a,b)∈D,h∈U,
(3.14)μJ̃(a,b)(h)=T(μF̃r(a)(h),μG̃r(b)(h))=[T(1-μF̃(a)+(h),1-μG̃(b)+(h)),T(1-μF̃(a)-(h),1-μG̃(b)-(h))]=[1-S(μF̃(a)+(h),μG̃(b)+(h)),1-S(μF̃(a)-(h),μG̃(b)-(h))]=μH̃r(a,b)(h),δ(a,b)=T(αr(a),βr(b))=[T(1-α+(a),1-β+(b)),T(1-α-(a),1-β-(b))]=[1-S(α+(a),β+(b)),1-S(α-(a),β-(b))]=γr(a,b).
Therefore, H̃γr and J̃δ are the same GIVFS sets. Thus, (F̃α⊻G̃β)r=(F̃α)r⌅(G̃β)r.
(2) The proof is similar to that of (1).


## 4. The Lattice Structures of GIVFS Sets

The lattice structures of soft sets have been studied by Qin and Hong in [14]. In this section, we will discuss the lattice structures of GIVFS sets. The following proposition shows the idempotent law with respect to operations ∪̃ and ⋒ does not hold in general.Proposition 4.1.
LetF̃α be a GIVFS sets over (U,E). Then the following holds(1)
F̃α⋐(F̃α∪̃F̃α),(2)
(F̃α⋒F̃α)⋐F̃α.To illuminate the above proposition, we give an example as follows.Example 4.2.
We consider the GIVFS setF̃α given in Example 3.3. We have that the following(1)
IfS(a,b)=a+b-a·b, then (F̃α∪̃F̃α)(e1)=({h1/[0.96,0.99],h2/[0.84,0.91], h3/[0.75,0.84]},[0.91,0.96])⊇F̃α(e1), (F̃α∪̃F̃α)(e2)⊇F̃α(e2), and (F̃α∪̃F̃α)(e3)⊇F̃α(e3); that is, (F̃α∪̃F̃α)⊇F̃α.(2)
IfS(a,b)=min(1,a+b), then (F̃α∪̃F̃α)(e1)=({h1/[1.0,1.0],h2/[1.0,1.0], h3/[1.0,1.0]},[1.0,1.0])⊇F̃α(e1), (F̃α∪̃F̃α)(e2)⊇F̃α(e2), and (F̃α∪̃F̃α)(e3)⊇F̃α(e3); that is, (F̃α∪̃F̃α)⊇F̃α.(3)
ifT(a,b)=a·b, then (F̃α⋒F̃α)(e1)=({h1/[0.64,0.81],h2/[0.36,0.49], h3/[0.25,0.36]},[0.49,0.64])⊆F̃α(e1), (F̃α⋒F̃α)(e2)⊆F̃α(e2) and (F̃α⋒F̃α)(e3)⊆F̃α(e3), that is, (F̃α⋒F̃α)⋐F̃α;(4)
IfT(a,b)=max(0,a+b-1), then (F̃α⋒F̃α)(e1)=({h1/[0.6,0.8],h2/[0.2,0.4], h3/[0.0,0.2]},[0.4,0.6])⊆F̃α(e1), (F̃α⋒F̃α)(e2)⊆F̃α(e2) and (F̃α⋒F̃α)(e3)⊆F̃α(e3); that is, (F̃α⋒F̃α)⋐F̃α.
For convenience, let𝔖̃(U,E) denote the set of all GIVFS sets over (U,E); that is, 𝔖̃(U,E)={F̃α∣A⊆E,F:A→ℱ(U),α:A→Int([0,1])}.From Proposition4.1, we can see that (𝔖̃(U,E),⋒,∪̃) is not a lattice in general. However, if T(a,b)=min(a,b) and S(a,b)=max(a,b), then the idempotent law and absorption law with respect to operations ∪̃ and ⋒ hold. In the remainder of this section, we always consider T(a,b)=min(a,b) and S(a,b)=max(a,b).Theorem 4.3.
LetA,B⊆E,  F̃α, and  G̃β be GIVFS sets over  (U,E). Then the following hold: (1)
(F̃α⋒F̃α)=F̃α,(2)
(F̃α∪̃F̃α)=F̃α,(3)
(F̃α∪̃G̃β)⋒F̃α=F̃α,(4)
(F̃α⋒̃G̃β)∪̃F̃α=F̃α.Proof.
(1) and (2) are trivial to prove. We prove only (3) since (4) can be proved similarly.
Suppose that the parameter sets of two GIVFS setsJ̃δ and K̃η are denoted by M and N, respectively. Let F̃α∪̃G̃β=J̃δ and (F̃α∪̃G̃β)⋒F̃α=K̃η. Then M=A∪B, N=(A∪B)∩A=A. And, for each e∈A and h∈U,(i)
ife∉B, then μK̃(e)(h)=T(μJ̃(e)(h),μF̃(e)(h))=min(μF̃(e)(h),μF̃(e)(h))=μF̃(e)(h), and η(e)=T(α(e),α(e))=min(α(e),α(e))=α(e),(ii)
ife∈B, then μK̃(e)(h)=min(μJ̃(e)(h),μF̃(e)(h))=min(max(μF̃(e)(h), μG̃(e)(h)),μF̃(e)(h))=μF̃(e)(h), and η(e)=T(S(α(e),β(e)),α(e))=min(max(α(e),β(e)),α(e))=α(e).Thus K̃η=F̃α; that is, (F̃α∪̃G̃β)⋒F̃α=F̃α.Theorem 4.4.
LetA,B,C⊆E,  F̃α,  G̃β, and  H̃γ be GIVFS sets over  (U,E). Then the following hold: (1)
F̃α⋒(G̃β∪̃H̃γ)=(F̃α⋒G̃β)∪̃(F̃α⋒H̃γ),(2)
F̃α∪̃(G̃β⋒H̃γ)=(F̃α∪̃G̃β)⋒(F̃α∪̃H̃γ).Proof.
(1) Suppose that the parameter sets of two GIVFS setsJ̃δ and K̃η are denoted by M and N, respectively. Let F̃α⋒(G̃β∪̃H̃γ)=J̃δ and (F̃α⋒G̃β)∪̃(F̃α⋒H̃γ)=K̃η. Then M=A∩(B∪C)=(A∩B)∪(A∩C)=N. And, for each e∈M,h∈U, it follows that e∈A and e∈B∪C,(i)
ife∈A,e∉B,e∈C, then μJ̃(e)(h)=T(μF̃(e)(h),μH̃(e)(h))=min(μF̃(e)(h),  μH̃(e)(h))=μK̃(e)(h),  and δ(e)=T(α(e),γ(e))=min(α(e),γ(e))=η(e),(ii)
ife∈A,e∈B,e∉C, then μJ̃(e)(h)=T(μF̃(e)(h),μG̃(e)(h))=min(μF̃(e)(h),   μG̃(e)(h))=μK̃(e)(h),  and δ(e)=T(α(e),β(e))=min(α(e),β(e))=η(e),(iii)
ife∈A,e∈B,e∈C, then μJ̃(e)(h)=min(μF̃(e)(h), max(μG̃(e)(h),μH̃(e)(h)))=max(min(μF̃(e)(h),μG̃(e)(h)),    min((μF̃(e)(h),μH̃(e)(h)))=μK̃(e)(h), and δ(e)=T(α(e),S(β(e),γ(e)))=min(α(e),max(β(e),γ(e)))=max(min(α(e),β(e)), min(α(e),γ(e)))=S(T(α(e),β(e)),T(α(e),γ(e)))=η(e).Thus J̃δ=K̃η; that is, F̃α⋒(G̃β∪̃H̃γ)=(F̃α⋒G̃β)∪̃(F̃α⋒H̃γ).
(2) The proof is similar to that of (1).Theorem 4.5.
(1)(𝔖̃(U,E),⋒,∪̃) is a distributive lattice.
(2) Let≤1 be the order relation in 𝔖̃(U,E) and F̃α,G̃β∈𝔖̃(U,E). One has F̃α≤1G̃β if and only if A⊆B,μF̃(e)(h)≤μG̃(e)(h) and α(e)≤β(e) for all e∈A and h∈U.Proof.
(1) The proof is straightforward from Theorems3.14, 4.3, and 4.4.
(2) Suppose thatF̃α≤1G̃β. Then F̃α∪̃G̃β=G̃β. So by Definition 3.10, we have A∪B=B, max(μF̃(e)(h),μG̃(e)(h))=μG̃(e)(h), and max(α(e),β(e))=β(e) for all e∈A and h∈U. It follows that A⊆B, μF̃(e)(h)≤μG̃(e)(h) and α(e)≤β(e) for all e∈A and h∈U. Conversely, suppose that A⊆B, μF̃(e)(h)≤μG̃(e)(h) and α(e)≤β(e) for all e∈A and h∈U. We can easily verify that F̃α∪̃G̃β=G̃β. Thus F̃α≤1G̃β.For operators⋓ and ∩̃, we can obtain similar results as follows.Theorem 4.6.
LetF̃α and  G̃β be GIVFS sets over  (U,E). Then the following hold: (1)
(F̃α⋓F̃α)=F̃α,(2)
(F̃α∩̃F̃α)=F̃α,(3)
(F̃α⋓G̃β)∩̃F̃α=F̃α,(4)
(F̃α∩̃G̃β)⋓F̃α=F̃α.Theorem 4.7.
LetF̃α,  G̃β and  H̃γ be  GIVFS sets over (U,E). Then the following hold: (1)
F̃α∩̃(G̃β⋓H̃γ)=(F̃α∩̃G̃β)⋓(F̃α∩̃H̃γ),(2)
F̃α⋓(G̃β∩̃H̃γ)=(F̃α⋓G̃β)∩̃(F̃α⋓H̃γ).Theorem 4.8.
(1)(𝔖̃(U,E),⋓,∩̃) is a distributive lattice.
(2) Let≤2 be the order relation in 𝔖̃(U,E) and F̃α,G̃β∈𝔖̃(U,E). F̃α≤2G̃β if and only if B⊆A,μF̃(e)(h)≤μG̃(e)(h) and α(e)≤β(e) for all e∈B.It is worth noting that(𝔖̃(U,E),⋒,⋓) and (𝔖̃(U,E),∩̃,∪̃) are not lattices, as the absorption laws of them do not hold necessarily. To illustrate this, we give an example as follows.Example 4.9.
LetU={h1,h2,h3}be the universe,E={e1,e2,e3}the set of parameters,A={e1,e2},B={e2,e3}. The GIVFS setsF̃αandG̃βover(U,E)are given as(4.1)F̃α(e1)=({h1[0.5,0.7],h2[0.3,0.4],h3[0.6,0.7]},[0.8,0.9]),F̃α(e2)=({h1[0.6,0.8],h2[0.2,0.3],h3[0.7,0.9]},[0.4,0.5]),G̃β(e2)=({h1[0.1,0.3],h2[0.4,0.5],h3[0.5,0.6]},[0.6,0.8]),G̃β(e3)=({h1[0.3,0.4],h2[0.5,0.8],h3[0.4,0.6]},[0.5,0.7]).
Suppose that(F̃α⋓G̃β)⋒F̃α=H̃γ. Then C=A∩B={e2}≠A. So H̃γ≠F̃α, that is, (F̃α⋓G̃β)⋒F̃α≠F̃α.
Again, suppose that the parameters set of a GIVFS setJ̃δ is denoted by D, and (F̃α∩̃G̃β)∪̃F̃α=J̃δ. Then D=A∪B={e1,e2,e3}≠A, Therefore, J̃δ≠F̃α, that is, (F̃α∩̃G̃β)∪̃F̃α≠F̃α.


## 5. An Application of GIVFS Sets

In this section we present a simple application of GIVFS set in an interval-valued fuzzy decision making problem. We first give the following definition.Definition 5.1.
LetF̃α be a GIVFS set, hi,hj∈U,ek∈A. One says membership value of hj lowerly exceeds or equals to the membership value of hi with respect to the parameter ek if μF̃(ek)-(hi)≤μF̃(ek)-(hj). The corresponding characteristic function is defined as follows:
(5.1)fek-(hi,hj)={1,ifμF̃(ek)-(hi)≤μF̃(ek)-(hj),0,otherwise.Definition 5.2.
LetF̃α be a GIVFS set, hi,hj∈U,ek∈A. One says membership value of hj upperly exceeds or equals to the membership value of hi with respect to the parameter ek if μF̃(ek)+(hi)≤μF̃(ek)+(hj). The corresponding characteristic function is defined as follows:
(5.2)fek+(hi,hj)={1,ifμF̃(ek)+(hi)≤μF̃(ek)+(hj);0,otherwise.Remark 5.3.
LetF̃α be a GIVFS set, hi,hj∈U,andek∈A. For convenience, we denote the vectors (fek-(hi,hj),fek+(hi,hj)) and (α-(ek),α+(ek)) as fek(hi,hj)⃗ and α(ek)⃗, respectively.Now we can define the generalised comparison table about GIVFS setF̃α.Definition 5.4.
LetF̃α be a GIVFS set. The generalised comparison table about F̃α is a square table in which the number of rows and number of columns are equal. Both rows and columns are labeled by the object names of the universe such as h1,h2,…,hn, and the entries are Cij, given as follows:
(5.3)Cij=∑k=1m(fek(hi,hj)⃗⋅α(ek)⃗),i,j=1,2,…,n.Clearly, fori,j=1,…,n, k=1,…,m, 0≤Cij≤2m, and Cii=∑k=1m(α-(ek)+α+(ek)), where n and m are the numbers of objects and parameters present in a GIVFS set, respectively.Remark 5.5.
The generalised comparison table is different from the comparison table in [30]. First, the comparison in the generalised comparison table is between two interval values, instead of two single values. Second, the entries Cij of the generalised comparison table are numbers of real interval [0,1] in general, instead of single values 0 and 1. Hence, the generalised comparison table is an extension of the comparison table in [30]. If each interval degenerates to a point and α(e)=1 for each e∈A, then the generalised comparison table will be degenerate to the comparison table in [30].In the generalised comparison table, the row sum and the column sum of an objecthi are denoted by pi and qi, respectively, and the score of an object hi is denoted as Si which can be given by Si=pi-qi. Now we present an algorithm as follows.Algorithm 5.6.
(1) Input the objects setU and the parameter set A⊆E.
(2) Consider the GIVFS setF̃α in tabular form.
(3) By calculating the entriesCij, construct generalised comparison table.
(4) Compute the score of eachhi using row sum and the column sum.
(5) The optimal decision is to selecthk if the score of hk is maximum.
(6) Ifk has more than one value then any one of hk may be chosen.To illustrate the basic idea of the above algorithm, let us consider the following example.Example 5.7.
Let us consider a GIVFS set which describes the capability of the candidates who are wanted to fill a position for a company. Suppose that there are six candidates in the universeU={h1,h2,h3,h4,h5,h6} under consideration, and E={e1,e2,e3,e4,e5,e6} is the set of decision parameters, where ei(i=1,2,3,4,5,6) stands for the parameters “experience”, “computer knowledge”, “young age”, “higher education”, “good health”, and “over-married”, respectively.Here, the degree of possibility of belongingness of the parameterei can be interpreted as the degree of importance of the parameter to the position. Our purpose is to find out the best candidate for the company based on her expected parameters. Suppose that the company do not consider the parameter “over-married”; that is, the degree of importance of parameter e6 is regarded as 0¯. In this case, let A={e1,e2,e3,e4,e5}⊂E, and let α:A→Int([0,1]) be an interval-valued fuzzy subset of A, which is given by the company as follows: α(e1)=[0.7,0.8], α(e2)=[0.5,0.6],α(e3)=[0.8,0.9],α(e4)=[0.6,0.7], α(e5)=[0.4,0.5]. And consider the GIVFS set F̃α as follows:
(5.4)F̃α(e1)=({h1[0.70,0.85],h2[0.85,0.90],h3[0.65,0.75],h4[0.80,0.90],h5[0.60,0.70],h6[0.65,0.80]},[0.7,0.8]),F̃α(e2)=({h1[0.75,0.80],h2[0.60,0.70],h3[0.60,0.70],h4[0.70,0.75],h5[0.80,0.90],h6[0.70,0.80]},[0.5,0.6]),F̃α(e3)=({h1[0.80,0.90],h2[0.55,0.66],h3[0.65,0.80],h4[0.68,0.75],h5[0.70,0.80],h6[0.75,0.85]},[0.8,0.9]),F̃α(e4)=({h1[0.70,0.80],h2[0.65,0.75],h3[0.70,0.78],h4[0.62,0.70],h5[0.72,0.82],h6[0.80,0.90]},[0.6,0.7]),F̃α(e5)=({h1[0.65,0.75],h2[0.60,0.70],h3[0.80,0.90],h4[0.60,0.76],h5[0.75,0.85],h6[0.70,0.75]},[0.4,0.5]).The tabular representation of the GIVFS setF̃α is given in Table 1.Table 1
Tabular representation of the GIVFS setF̃α.
e1e2e3e4e5h1[0.70,0.85][0.75,0.80][0.80,0.90][0.70,0.80][0.65,0.75]h2[0.85,0.90][0.60,0.70][0.55,0.66][0.65,0.75][0.60,0.70]h3[0.65,0.75][0.60,0.70][0.65,0.80][0.70,0.78][0.80,0.90]h4[0.80,0.90][0.70,0.75][0.68,0.75][0.62,0.70][0.60,0.76]h5[0.60,0.70][0.80,0.90][0.70,0.80][0.72,0.82][0.75,0.85]h6[0.65,0.80][0.70,0.80][0.75,0.85][0.80,0.90][0.70,0.75]α[0.7,0.8][0.5,0.6][0.8,0.9][0.6,0.7][0.4,0.5]It is easy to calculate the entriesCij by the formula 5.3. For example, let us calculate C21. Firstly, we compute fek(h2,h1)⃗ for each ek∈A, where fe1(h2,h1)⃗=(0,0), fek(h2,h1)⃗=(1,1),k=2,3,4,5. Secondly, we can obtain C21=5.0 by computing ∑k=1m(fek(h2,h1)⃗·α(ek)⃗), where α(e1)⃗=(0.7,0.8), α(e2)⃗=(0.5,0.6), α(e3)⃗=(0.8,0.9), α(e4)⃗=(0.6,0.7), α(e5)⃗=(0.4,0.5). And the generalised comparison table about the GIVFS set F̃α is given in Table 2.Table 2
The generalised comparison table aboutF̃α.
h1h2h3h4h5h6h16.505.005.604.503.204.80h21.506.502.603.201.501.50h31.505.006.503.103.301.60h42.004.503.406.501.502.50h53.305.004.105.006.502.00h62.805.005.604.504.506.50From Table2, we can obtain the row sum and column sum and compute the score of each hi, which are presented in Table 3.Table 3
The score ofhi about F̃α.
Row sum (pi)Column sum (qi)The score (Si)h129.6017.6012.00h216.8031.00−14.20h321.0027.80−6.80h420.4026.80−6.40h525.9020.505.40h628.9018.9010.00From Table3, it is clear that the maximum score is S1=12.00. So h1 could be selected as the optimal alternative.It is worth noting that, unlike [30], the decision result depends not only on F̃(e) but also on α(e). For example, consider the GIVFS set G̃β with data as in Table 4, where B=A and G̃(e)=F̃(e), but β(e)≠α(e) for each e∈B.Table 4
Tabular representation of the GIVFS setG̃β.
e1e2e3e4e5h1[0.70,0.85][0.75,0.80][0.80,0.90][0.70,0.80][0.65,0.75]h2[0.85,0.90][0.60,0.70][0.55,0.66][0.65,0.75][0.60,0.70]h3[0.65,0.75][0.60,0.70][0.65,0.80][0.70,0.78][0.80,0.90]h4[0.80,0.90][0.70,0.75][0.68,0.75][0.62,0.70][0.60,0.76]h5[0.60,0.70][0.80,0.90][0.70,0.80][0.72,0.82][0.75,0.85]h6[0.65,0.80][0.70,0.80][0.75,0.85][0.80,0.90][0.70,0.75]β[0.7,0.8][0.5,0.6][0.4,0.5][0.8,0.9][0.6,0.7]The generalised comparison table and the score ofhi about the GIVFS set G̃β can be seen in Tables 5 and 6, respectively.Table 5
The generalised comparison table aboutG̃β.
h1h2h3h4h5h6h16.505.005.204.302.404.20h21.506.502.603.801.501.50h32.105.006.503.503.302.00h42.204.103.006.501.502.70h54.105.003.705.006.502.40h63.605.005.204.304.106.50Table 6
The score ofhi about G̃β.
Row sum (pi)Column sum (qi)The score (Si)h127.6020.007.60h217.4030.60−13.20h322.4026.20−3.80h420.0027.40−7.40h526.7019.307.40h628.7019.309.40From Table6, it is clear that the maximum score is S6=9.40. Hence, the optimal alternative is h6, but not h1.


## 6. Conclusion

This paper can be viewed as a continuation of the study of Majumdar and Samanta [32], Yang et al. [31], and Roy and Maji [30]. We extended the generalised fuzzy soft set and defined two types of generalised interval-valued fuzzy soft set and studied some of their properties. We also gave the application of GIVFS sets in dealing with some decision-making problems by defining generalised comparison table.



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