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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 479783, 18 pages
http://dx.doi.org/10.1155/2012/479783
Research Article

On Generalised Interval-Valued Fuzzy Soft Sets

1College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
2College of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China
3College of Computer and Communication, Hunan University, Changsha 410082, China

Received 10 August 2011; Revised 18 November 2011; Accepted 22 November 2011

Academic Editor: Jong Hae Kim

Copyright © 2012 Xiaoqiang Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [57]. 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 [1315]. 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 [1618] and some algebra structures such as groups [19, 20], ordered semigroups [21], rings [22], semirings [23], BCK/BCI-algebras [2426], 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 [2936]. 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 Section 2, 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,𝐶𝐸, respectively.

Definition 2.1 (see [8]). A pair (𝐹,𝐴) is called a soft set over 𝑈 where 𝐹 is a mapping given by 𝐹𝐴𝑃(𝑈).

In other words, a soft set over 𝑈 is a parameterized family of subsets of the universe 𝑈. For 𝜀𝐴, 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]). Let 𝒫(𝑈) denote the set of all fuzzy subsets of 𝑈. Then a pair (𝐹,𝐴) is called a fuzzy soft set over 𝑈, where 𝐹 is a mapping from 𝐴 to 𝒫(𝑈).

From the definition, it is clear that 𝐹(𝑒) is a fuzzy set on 𝑈 for any 𝑒𝐴. The modified definition of fuzzy soft set by Majumdar and Samanta is as follows.

Definition 2.3 (see [32]). Let 𝑈 be an initial universal set, 𝐸 a set of parameters, and the pair (𝑈,𝐸) a soft universe. Let 𝐹𝐸𝒫(𝑈) and 𝜇 be a fuzzy subset of 𝐸; that is, 𝜇𝐸[0,1]. Let 𝐹𝜇𝐸𝒫(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 set 𝑋 on a universe 𝑈 is a mapping 𝑋𝑈Int([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 (𝑈). Suppose that 𝑋(𝑈), for all 𝑈,𝜇𝑋()=[𝜇𝑋(),𝜇+𝑋()] is called the degree of membership of an element to 𝑋. And 𝜇𝑋() and 𝜇+𝑋() are referred to as the lower and upper degrees of membership of to 𝑋, where 0𝜇𝑋()𝜇+𝑋()1.

Definition 2.5 (see [7]). Let 𝑋 and 𝑌 be two interval-valued fuzzy sets on universe 𝑈. Then the union, intersection, and complement of vague sets are defined as follows: 𝑋𝑌=𝜇𝑋()𝜇𝑌(),𝜇+𝑋()𝜇+𝑌,()𝑈𝑋𝑌=𝜇𝑋()𝜇𝑌(),𝜇+𝑋()𝜇+𝑌(,𝑋)𝑈𝑐=1𝜇+𝑋(),1𝜇𝑋.()𝑈(2.1)

Definition 2.6 (see [31]). Let 𝑈 be an initial universe, let 𝐸 be a set of parameters, and let 𝐴𝐸. (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 𝐹𝐴(𝑈).

An interval-valued fuzzy soft set is a parameterized family of interval-valued fuzzy subsets of 𝑈. For each parameter 𝑒𝐴, 𝐹(𝑒) is actually an interval-valued fuzzy set of 𝑈, and it can be written as 𝐹(𝑒)={(/𝜇𝐹(𝑒)())𝑈}, where 𝜇𝐹(𝑒)() is the interval-valued fuzzy membership degree that object holds on parameter 𝑒.

Definition 2.7 (see [37]). A 𝑡-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 continuous 𝑡-norms are 𝑇(𝑎,𝑏)=min(𝑎,𝑏), 𝑇(𝑎,𝑏)=max{0,𝑎+𝑏1}, and 𝑇(𝑎,𝑏)=𝑎𝑏.

Definition 2.8 (see [37]). A 𝑡-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 continuous 𝑡-conorms are 𝑆(𝑎,𝑏)=max(𝑎,𝑏), 𝑆(𝑎,𝑏)=𝑎+𝑏𝑎𝑏, 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. Let 𝑈 be an initial universe and 𝐸 a set of parameters, 𝐴𝐸, 𝐹𝐴(𝑈), and let 𝛼 be a fuzzy sets of 𝐴, that is, 𝛼𝐴[0,1]. Define a function 𝐹𝛼𝐴(𝑈)×[0,1] as 𝐹𝛼(𝑒)=(𝐹(𝑒)={/𝜇𝐹(𝑒)()},𝛼(𝑒)), where 𝜇𝐹(𝑒)()=[𝜇𝐹(𝑒)(),𝜇+𝐹(𝑒)()] is an interval value is called the degree of membership an element to 𝐹(𝑒), and 𝛼(𝑒) is called the degree of possibility of such belongness. Then 𝐹𝛼 is called type 1 generalised interval-valued fuzzy soft set over the soft universe (𝑈,𝐸).

Here for each parameter 𝑒, 𝐹𝛼(𝑒) indicates not only the degree of belongingness of elements of 𝑈 in 𝐹(𝑒) but also the degree of preference of such belongingness which is represented by 𝛼(𝑒).

Definition 3.2. Let 𝑈 be an initial universe and 𝐸 a set of parameters, 𝐴𝐸, 𝐹𝐴(𝑈), and let 𝛼 be an interval-valued fuzzy sets of 𝐴; that is, 𝛼𝐴Int([0,1]), where Int([0,1]) stands for the set of all closed subintervals of [0,1]. Define a function 𝐹𝛼𝐴(𝑈)×Int([0,1]) as 𝐹𝛼(𝑒)=(𝐹(𝑒)={/𝜇𝐹(𝑒)()},𝛼(𝑒)), where 𝜇𝐹(𝑒)()=[𝜇𝐹(𝑒)(),𝜇+𝐹(𝑒)()] and 𝛼(𝑒)=[𝛼(𝑒),𝛼+(𝑒)] are interval values. Then 𝐹𝛼 is called type 2 generalised interval-valued fuzzy soft set over the soft universe (𝑈,𝐸).

It is clear that if 𝛼(𝑒)=𝛼+(𝑒) holds for each 𝑎𝐴, then the type 2 generalised interval-valued fuzzy soft set will degenerate to the type 1 generalised interval-valued fuzzy soft set. And if 𝜇𝐹(𝑒)()=𝜇+𝐹(𝑒)() also holds for each 𝑎𝐴, 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. Let 𝑈={1,2,3} 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 𝐹𝛼𝐴𝒫(𝑈)×Int([0,1]) be a function given as follows: 𝐹𝛼𝑒1=1[],0.8,0.92[],0.6,0.73[],[],𝐹0.5,0.60.7,0.8𝛼𝑒2=1[],0.7,0.82[],0.3,0.43[],[],𝐹0.5,0.70.6,0.7𝛼𝑒3=1[],0.5,0.62[],0.5,0.73[],[].0.7,0.80.8,0.9(3.1) Then 𝐹𝛼 is a GIVFS set.

Definition 3.4. Let 𝐹𝛼 and 𝐺𝛽 be GIVFS sets over (𝑈,𝐸). Then 𝐹𝛼 is called a GIVFS subset of 𝐺𝛽 if(1)𝐴𝐵;(2)𝐹(𝑒) is an interval-valued fuzzy subset of 𝐺(𝑒) for any 𝑒𝐴; that is, 𝜇𝐹(𝑒)()𝜇𝐺(𝑒)() and 𝜇+𝐹(𝑒)()𝜇+𝐺(𝑒)() for any 𝑈 and 𝑒𝐴;(3)𝛼 is an interval-valued fuzzy subset of 𝛽; that is, 𝛼(𝑒)𝛽(𝑒) and 𝛼+(𝑒)𝛽+(𝑒) for any 𝑒𝐴.
In this case, the above relationship is denoted by 𝐹𝛼𝐺𝛽. And 𝐺𝛽 is said to be a GIVFS superset of 𝐹𝛼.

Definition 3.5. Let 𝐹𝛼 and 𝐺𝛽 be GIVFS sets over (𝑈,𝐸). Then 𝐹𝛼 and 𝐺𝛽 are said to be GIVFS equal if and only if 𝐹𝛼𝐺𝛽 and 𝐺𝛽𝐹𝛼.

Definition 3.6. The relative complement of a GIVFS set 𝐹𝛼 is denoted by 𝐹𝑟𝛼 and is defined by 𝐹𝑟𝛼𝐹=(𝑟,𝛼𝑟), where 𝐹𝑟𝐴(𝑈) is a mapping given by 𝐹𝑟𝐹(𝑒)={/𝜇𝑟(𝑒)()} and 𝛼𝑟𝐴Int([0,1]) is a mapping given by 𝛼𝑟(𝑒) for all 𝑈,𝑒𝐴, where 𝜇𝐹𝑟(𝑒)()=[𝜇𝐹𝑟(𝑒)(),𝜇+𝐹𝑟(𝑒)()]=[1𝜇+𝐹(𝑒)(),1𝜇𝐹(𝑒)()], 𝛼𝑟(𝑒)=[𝛼𝑟(𝑒),𝛼𝑟+(𝑒)]=[1𝛼+(𝑒),1𝛼(𝑒)].

Example 3.7. We consider the GIVFS set 𝐹𝛼 given in Example 3.3 and define a GIVFS set 𝐺𝛽 as follows: 𝐺𝛽𝑒1=1[],0.7,0.82[],0.4,0.53[],[],𝐺0.4,0.60.5,0.6𝛽𝑒2=1[],0.5,0.62[],0.2,0.43[],[].0.5,0.60.3,0.4(3.2) Then 𝐺𝛽 is a GIVFS subset of 𝐹𝛼, and the relative complement of a GIVFS set 𝐺𝛽 is 𝐺𝑟𝛽𝑒1=1[],0.2,0.32[],0.5,0.63[],[],𝐺0.4,0.60.4,0.5𝑟𝛽𝑒2=1[],0.4,0.52[],0.6,0.83[],[].0.4,0.50.6,0.7(3.3)

Definition 3.8. Let 1=[1,1]. A GIVFS set 𝐹𝛼 over (𝑈,𝐸) is said to be relative absolute GIVFS set denoted by Ω𝐴, if 𝜇𝐹(𝑒)()=1 and 𝛼(𝑒)=1 for all 𝑈 and 𝑒𝐴.

Definition 3.9. Let 0=[0,0]. A GIVFS set 𝐹𝛼 over (𝑈,𝐸) is said to be relative null GIVFS set, denoted by Φ𝐴, if 𝜇𝐹(𝑒)()=0 and 𝛼(𝑒)=0 for all 𝑈 and 𝑒𝐴.

Definition 3.10. The union of two GIVFS sets 𝐹𝛼 and 𝐺𝛽 over (𝑈,𝐸) denoted by 𝐹𝛼𝐺𝛽 is a GIVFS set 𝐻𝛾 and defined as 𝐻𝛾𝐴𝐵(𝑈)×Int([0,1]) such that, for all 𝑈 and 𝑒𝐴𝐵, 𝐻𝛾(𝑒)=𝜇𝐹(𝑒)(),𝛼(𝑒),if𝑒𝐴𝐵,𝜇𝐺(𝑒)(),𝛽(𝑒),if𝑒𝐵𝐴,𝜇𝐻(𝑒)(),𝛾(𝑒),if𝑒𝐴𝐵,(3.4) where 𝜇𝐻(𝑒)()=𝑆(𝜇𝐹(𝑒)(),𝜇𝐺(𝑒)())=[𝑆(𝜇𝐹(𝑒)(),𝜇𝐺(𝑒)()),𝑆(𝜇+𝐹(𝑒)(),𝜇+𝐺(𝑒)())] and 𝛾(𝑒)=𝑆(𝛼(𝑒),𝛽(𝑒))=[𝑆(𝛼(𝑒),𝛽(𝑒)),𝑆(𝛼+(𝑒),𝛽+(𝑒))].

Definition 3.11. The intersection of two GIVFS sets 𝐹𝛼 and 𝐺𝛽 over (𝑈,𝐸) denoted by 𝐹𝛼𝐺𝛽 is a GIVFS set 𝐻𝛾 and defined as 𝐻𝛾𝐴𝐵(𝑈)×Int([0,1]) such that, for all 𝑈 and 𝑒𝐴𝐵, 𝐻𝛾(𝑒)=({/𝜇𝐻(𝑒)()},𝛾(𝑒)), where 𝜇𝐻(𝑒)()=𝑇(𝜇𝐹(𝑒)(),𝜇𝐺(𝑒)())=[𝑇(𝜇𝐹(𝑒)(),𝜇𝐺(𝑒)()), 𝑇(𝜇+𝐹(𝑒)(),𝜇+𝐺(𝑒)())] 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 sets 𝐹𝛼 and 𝐺𝛽 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 𝐹𝛼𝐺𝛽𝑒1=1[],0.8,0.92[],0.6,0.73[],[],𝐹0.5,0.60.7,0.8𝛼𝐺𝛽𝑒2=1[],0.7,0.82[],0.3,0.43[],[],𝐹0.5,0.70.6,0.7𝛼𝐺𝛽𝑒3=1[],0.5,0.62[],0.5,0.73[],[],𝐹0.7,0.80.8,0.9𝛼𝐺𝛽𝑒1=1[],0.7,0.82[],0.4,0.53[],[],𝐹0.4,0.60.5,0.6𝛼𝐺𝛽𝑒2=1[],0.5,0.62[],0.2,0.43[],[].0.5,0.60.3,0.4(3.5)

Proposition 3.13. Let 𝐹𝛼 be a GIVFS set over (𝑈,𝐸). Then the following holds(1)𝐹𝛼Ω𝐴=𝐹𝛼, (2)𝐹𝛼Ω𝐴=Ω𝐴, (3)𝐹𝛼Φ𝐴=Φ𝐴, (4)𝐹𝛼Φ𝐴=𝐹𝛼.

Proof. It is easily obtained from Definitions 3.83.11.

Theorem 3.14. Let 𝐹𝛼, 𝐺𝛽, and 𝐻𝛾 be GIVFS sets over (𝑈,𝐸). Then the following holds(1)𝐹𝛼𝐺𝛽=𝐺𝛽𝐹𝛼, (2)𝐹𝛼𝐺(𝛽𝐻𝛾𝐹)=(𝛼𝐺𝛽𝐻)𝛾, (3)𝐹𝛼𝐺𝛽=𝐺𝛽𝐹𝛼, (4)𝐹𝛼𝐺(𝛽𝐻𝛾𝐹)=(𝛼𝐺𝛽)𝐻𝛾.

Proof. It is easily obtained from Definitions 3.10 and 3.11.

Definition 3.15. The restricted union of two GIVFS sets 𝐹𝛼 and 𝐺𝛽 over (𝑈,𝐸) denoted by 𝐹𝛼𝐺𝛽 is a GIVFS set 𝐻𝛾 and defined as 𝐻𝛾𝐴𝐵(𝑈)×Int([0,1]) such that, for all 𝑈 and 𝑒𝐴𝐵, 𝐻𝛾(𝑒)=({/𝜇𝐻(𝑒)()},𝛾(𝑒)), where 𝜇𝐻(𝑒)()=𝑆(𝜇𝐹(𝑒)(),𝜇𝐺(𝑒)())=[𝑆(𝜇𝐹(𝑒)(),𝜇𝐺(𝑒)()), 𝑆(𝜇+𝐹(𝑒)(),𝜇+𝐺(𝑒)())] 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 sets 𝐹𝛼 and 𝐺𝛽 over (𝑈,𝐸), denoted by 𝐹𝛼𝐺𝛽, is a GVS set 𝐻𝛾𝐴𝐵(𝑈)×Int([0,1]) which is defined as, for all 𝑈,𝑒𝐴𝐵,𝐻𝛾(𝑒)=𝜇𝐹(𝑒)(),𝛼(𝑒),if𝑒𝐴𝐵,𝜇𝐺(𝑒)(),𝛽(𝑒),if𝑒𝐵𝐴,𝜇𝐻(𝑒)(),𝛾(𝑒),if𝑒𝐴𝐵,(3.6) where 𝜇𝐻(𝑒)()=𝑇(𝜇𝐹(𝑒)(),𝜇𝐺(𝑒)())=[𝑇(𝜇𝐹(𝑒)(),𝜇𝐺(𝑒)()), 𝑇(𝜇+𝐹(𝑒)(),𝜇+𝐺(𝑒)())] and 𝛾(𝑒)=𝑇(𝛼(𝑒),𝛽(𝑒))=[𝑇(𝛼(𝑒),𝛽(𝑒)),𝑇(𝛼+(𝑒),𝛽+(𝑒))].

Theorem 3.17. Let 𝐹𝛼, 𝐺𝛽, and 𝐻𝛾 be three GIVFS sets over (𝑈,𝐸). Then the following holds:(1)𝐹𝛼𝐺𝛽=𝐺𝛽𝐹𝛼, (2)𝐹𝛼𝐺(𝛽𝐻𝛾𝐹)=(𝛼𝐺𝛽𝐻)𝛾, (3)𝐹𝛼𝐺𝛽=𝐺𝛽𝐹𝛼, (4)𝐹𝛼𝐺(𝛽𝐻𝛾𝐹)=(𝛼𝐺𝛽)𝐻𝛾.

Proof. It is easily obtained from Definitions 3.15 and 3.16.

Theorem 3.18. Let 𝐹𝛼 and 𝐺𝛽 be two GIVFS sets over (𝑈,𝐸). Then the following holds: (1)(𝐹𝛼𝐺𝛽)𝑟𝐹=(𝛼)𝑟𝐺(𝛽)𝑟,(2)(𝐹𝛼𝐺𝛽)𝑟𝐹=(𝛼)𝑟𝐺(𝛽)𝑟.

Proof. (1) Suppose that 𝐹𝛼𝐺𝛽=𝐻𝛾, then 𝐶=𝐴𝐵, and, forall𝑒𝐶,𝑈, 𝜇𝐻(𝑒)𝜇()=𝑇𝐹(𝑒)(),𝜇𝐺(𝑒)=𝑇𝜇()𝐹(𝑒)(),𝜇𝐺(𝑒)𝜇(),𝑇+𝐹(𝑒)(),𝜇+𝐺(𝑒),=()𝛾(𝑒)=𝑇(𝛼(𝑒),𝛽(𝑒))𝑇(𝛼(𝑒),𝛽(𝛼𝑒)),𝑇+(𝑒),𝛽+(.𝑒)(3.7) Moreover, we have (𝐹𝛼𝐺𝛽)𝑟=𝐻𝑟𝛾, 𝐶=𝐴𝐵, and forall𝑒𝐶,𝑈, 𝜇𝐻𝑟(𝑒)𝜇()=1𝑇+𝐹(𝑒)(),𝜇+𝐺(𝑒)𝜇(),1𝑇𝐹(𝑒)(),𝜇𝐺(𝑒),𝛾()𝑟𝛼(𝑒)=1𝑇+(𝑒),𝛽+(𝑒),1𝑇(𝛼(𝑒),𝛽.(𝑒))(3.8) Assume that the parameters set of a GIVFS set 𝐽𝛿 is denoted 𝐷, and 𝐹𝑟𝛼𝐺𝑟𝛽=𝐽𝛿. Then 𝐷=𝐴𝐵. Since 𝜇𝐹𝑟(𝑒)()=1𝜇+𝐹(𝑒)(),1𝜇𝐹(𝑒)(),𝛼𝑟(𝑒)=1𝛼+(𝑒),1𝛼,𝜇𝐺(𝑒)𝑟(𝑒)()=1𝜇+𝐺(𝑒)(),1𝜇𝐺(𝑒)(),𝛽𝑟(𝑒)=1𝛽+(𝑒),1𝛽.(𝑒)(3.9) Then, for each 𝑒𝐷,𝑈, 𝜇𝐽(𝑒)𝜇𝐹()=𝑆𝑟(𝑒)𝐺(),𝜇𝑟(𝑒)=𝑆()1𝜇+𝐹(𝑒)(),1𝜇+𝐺(𝑒)(),𝑆1𝜇𝐹(𝑒)(),1𝜇𝐺(𝑒)=𝜇()1𝑇+𝐹(𝑒)(),𝜇+𝐺(𝑒)(𝜇),1𝑇𝐹(𝑒)(),𝜇𝐺(𝑒)(𝐻)=𝜇𝑟(𝑒)(),𝛿(𝑒)=𝑆(𝛼𝑟(𝑒),𝛽𝑟=𝑆(𝑒))1𝛼+(𝑒),1𝛽+(𝑒),𝑆(1𝛼(𝑒),1𝛽=𝛼(𝑒))1𝑇+(𝑒),𝛽+(𝑒),1𝑇(𝛼(𝑒),𝛽(𝑒))=𝛾𝑟(𝑒).(3.10) Therefore, 𝐻𝑟𝛾 and 𝐽𝛿 are the same GIVFS sets. Thus, (𝐹𝛼𝐺𝛽)𝑟𝐹=(𝛼)𝑟𝐺(𝛽)𝑟.
(2) The proof is similar to that of (1).

Definition 3.19. The “AND” of two GIVFS sets 𝐹𝛼 and 𝐺𝛽 over (𝑈,𝐸), denoted by 𝐹𝛼𝐺𝛽, is defined as 𝐻𝛾𝐴×𝐵(𝑈)×Int([0,1]) such that for all 𝑈 and (𝑎,𝑏)𝐴×𝐵, 𝐻𝛾(𝑎,𝑏)=({/𝜇𝐻(𝑎,𝑏)()},𝛾(𝑎,𝑏)), where 𝜇𝐻(𝑎,𝑏)()=𝑇(𝜇𝐹(𝑎)(),𝜇𝐺(𝑏)())=[𝑇(𝜇𝐹(𝑎)(),𝜇𝐺(𝑏)()), 𝑇(𝜇+𝐹(𝑎)(),𝜇+𝐺(𝑏)())] 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 sets 𝐹𝛼 and 𝐺𝛽 over (𝑈,𝐸), denoted by 𝐹𝛼𝐺𝛽, is defined as 𝐻𝛾𝐴×𝐵(𝑈)×Int([0,1]) such that for all 𝑈 and (𝑎,𝑏)𝐴×𝐵, 𝐻𝛾(𝑎,𝑏)=({/𝜇𝐻(𝑎,𝑏)()},𝛾(𝑎,𝑏)), where 𝜇𝐻(𝑎,𝑏)()=𝑆(𝜇𝐹(𝑎)(),𝜇𝐺(𝑏)())=[𝑆(𝜇𝐹(𝑎)(),𝜇𝐺(𝑏)()), 𝑆(𝜇+𝐹(𝑎)(),𝜇+𝐺(𝑏)())] and 𝛾(𝑎,𝑏)=𝑆(𝛼(𝑎),𝛽(𝑏))=[𝑆(𝛼(𝑎),𝛽(𝑏)),𝑆(𝛼+(𝑎),𝛽+(𝑏))].

Theorem 3.21. Let 𝐹𝛼, 𝐺𝛽, and 𝐻𝛾 be three GIVFS sets over (𝑈,𝐸). Then the following holds (1)𝐹𝛼𝐺(𝛽𝐻𝛾𝐹)=(𝛼𝐺𝛽𝐻)𝛾, (2)𝐹𝛼𝐺(𝛽𝐻𝛾𝐹)=(𝛼𝐺𝛽𝐻)𝛾.

Proof. It is easily obtained from Definitions 3.19 and 3.20.

Theorem 3.22. Let 𝐹𝛼 and 𝐺𝛽 be two GIVFS sets over (𝑈,𝐸). Then the following holds (1)(𝐹𝛼𝐺𝛽)𝑟𝐹=(𝛼)𝑟𝐺(𝛽)𝑟, (2)(𝐹𝛼𝐺𝛽)𝑟𝐹=(𝛼)𝑟𝐺(𝛽)𝑟.

Proof. (1) Suppose that 𝐹𝛼𝐺𝛽=𝐻𝛾, then 𝐶=𝐴×𝐵, and, forall(𝑎,𝑏)𝐶,𝑈, 𝜇𝐻(𝑎,𝑏)𝜇()=𝑆𝐹(𝑎)(),𝜇𝐺(𝑏)=𝑆𝜇()𝐹(𝑎)(),𝜇𝐺(𝑏)𝜇(),𝑆+𝐹(𝑎)(),𝜇+𝐺(𝑏),=()𝛾(𝑎,𝑏)=𝑆(𝛼(𝑎),𝛽(𝑏))𝑆(𝛼(𝑎),𝛽(𝛼𝑏)),𝑆+(𝑎),𝛽+(.𝑏)(3.11) Moreover, we have (𝐹𝛼𝐺𝛽)𝑟=𝐻𝑟𝛾, 𝐶=𝐴×𝐵, and forall(𝑎,𝑏)𝐶,𝑈, 𝜇𝐻𝑟(𝑎,𝑏)𝜇()=1𝑆+𝐹(𝑎)(),𝜇+𝐺(𝑏)𝜇(),1𝑆𝐹(𝑎)(),𝜇𝐺(𝑏),𝛾()𝑟𝛼(𝑎,𝑏)=1𝑆+(𝑎),𝛽+(𝑏),1𝑆(𝛼(𝑎),𝛽.(𝑏))(3.12) Assume that the parameters set of a GIVFS set 𝐽𝛿 is denoted 𝐷, and 𝐹𝑟𝛼𝐺𝑟𝛽=𝐽𝛿. Then 𝐷=𝐴×𝐵. Since forall𝑎𝐴,𝑏𝐵,𝑈, 𝜇𝐹𝑟(𝑎)()=1𝜇+𝐹(𝑎)(),1𝜇+𝐹(𝑎)(),𝛼𝑟(𝑎)=1𝛼+(𝑎),1𝛼,𝜇𝐺(𝑎)𝑟(𝑏)()=1𝜇+𝐺(𝑏)(),1𝜇𝐺(𝑏)(),𝛽𝑟(𝑏)=1𝛽+(𝑏),1𝛽,(𝑏)(3.13) then, for each (𝑎,𝑏)𝐷,𝑈, 𝜇𝐽(𝑎,𝑏)𝜇𝐹()=𝑇𝑟(𝑎)𝐺(),𝜇𝑟(𝑏)=𝑇()1𝜇+𝐹(𝑎)(),1𝜇+𝐺(𝑏)(),𝑇1𝜇𝐹(𝑎)(),1𝜇𝐺(𝑏)=𝜇()1𝑆+𝐹(𝑎)(),𝜇+𝐺(𝑏)(𝜇),1𝑆𝐹(𝑎)(),𝜇𝐺(𝑏)(𝐻)=𝜇𝑟(𝑎,𝑏)(),𝛿(𝑎,𝑏)=𝑇(𝛼𝑟(𝑎),𝛽𝑟=𝑇(𝑏))1𝛼+(𝑎),1𝛽+(𝑏),𝑇(1𝛼(𝑎),1𝛽=𝛼(𝑏))1𝑆+(𝑎),𝛽+(𝑏),1𝑆(𝛼(𝑎),𝛽(𝑏))=𝛾𝑟(𝑎,𝑏).(3.14) Therefore, 𝐻𝑟𝛾 and 𝐽𝛿 are the same GIVFS sets. Thus, (𝐹𝛼𝐺𝛽)𝑟𝐹=(𝛼)𝑟𝐺(𝛽)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. Let 𝐹𝛼 be a GIVFS sets over (𝑈,𝐸). Then the following holds(1)𝐹𝛼𝐹(𝛼𝐹𝛼), (2)(𝐹𝛼𝐹𝛼𝐹)𝛼.

To illuminate the above proposition, we give an example as follows.

Example 4.2. We consider the GIVFS set 𝐹𝛼 given in Example 3.3. We have that the following(1)If 𝑆(𝑎,𝑏)=𝑎+𝑏𝑎𝑏, then (𝐹𝛼𝐹𝛼)(𝑒1)=({1/[0.96,0.99],2/[0.84,0.91], 3𝐹/[0.75,0.84]},[0.91,0.96])𝛼(𝑒1), (𝐹𝛼𝐹𝛼)(𝑒2𝐹)𝛼(𝑒2), and (𝐹𝛼𝐹𝛼)(𝑒3𝐹)𝛼(𝑒3); that is, (𝐹𝛼𝐹𝛼𝐹)𝛼.(2)If 𝑆(𝑎,𝑏)=min(1,𝑎+𝑏), then (𝐹𝛼𝐹𝛼)(𝑒1)=({1/[1.0,1.0],2/[1.0,1.0], 3𝐹/[1.0,1.0]},[1.0,1.0])𝛼(𝑒1), (𝐹𝛼𝐹𝛼)(𝑒2𝐹)𝛼(𝑒2), and (𝐹𝛼𝐹𝛼)(𝑒3𝐹)𝛼(𝑒3); that is, (𝐹𝛼𝐹𝛼𝐹)𝛼.(3)if 𝑇(𝑎,𝑏)=𝑎𝑏, then (𝐹𝛼𝐹𝛼)(𝑒1)=({1/[0.64,0.81],2/[0.36,0.49], 3𝐹/[0.25,0.36]},[0.49,0.64])𝛼(𝑒1), (𝐹𝛼𝐹𝛼)(𝑒2𝐹)𝛼(𝑒2) and (𝐹𝛼𝐹𝛼)(𝑒3𝐹)𝛼(𝑒3), that is, (𝐹𝛼𝐹𝛼𝐹)𝛼;(4)If 𝑇(𝑎,𝑏)=max(0,𝑎+𝑏1), then (𝐹𝛼𝐹𝛼)(𝑒1)=({1/[0.6,0.8],2/[0.2,0.4], 3𝐹/[0.0,0.2]},[0.4,0.6])𝛼(𝑒1), (𝐹𝛼𝐹𝛼)(𝑒2𝐹)𝛼(𝑒2) and (𝐹𝛼𝐹𝛼)(𝑒3𝐹)𝛼(𝑒3); that is, (𝐹𝛼𝐹𝛼𝐹)𝛼.
For convenience, let 𝔖(𝑈,𝐸) denote the set of all GIVFS sets over (𝑈,𝐸); that is, 𝐹𝔖(𝑈,𝐸)={𝛼𝐴𝐸,𝐹𝐴(𝑈),𝛼𝐴Int([0,1])}.

From Proposition 4.1, we can see that (𝔖(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. Let 𝐴,𝐵𝐸, 𝐹𝛼, and 𝐺𝛽 be GIVFS sets over (𝑈,𝐸). Then the following hold: (1)(𝐹𝛼𝐹𝛼𝐹)=𝛼, (2)(𝐹𝛼𝐹𝛼𝐹)=𝛼, (3)(𝐹𝛼𝐺𝛽𝐹)𝛼=𝐹𝛼,(4)(𝐹𝛼𝐺𝛽)𝐹𝛼=𝐹𝛼.

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 sets 𝐽𝛿 and 𝐾𝜂 are denoted by 𝑀 and 𝑁, respectively. Let 𝐹𝛼𝐺𝛽=𝐽𝛿 and (𝐹𝛼𝐺𝛽𝐹)𝛼=𝐾𝜂. Then 𝑀=𝐴𝐵, 𝑁=(𝐴𝐵)𝐴=𝐴. And, for each 𝑒𝐴 and 𝑈,(i)if 𝑒𝐵, then 𝜇𝐾(𝑒)()=𝑇(𝜇𝐽(𝑒)(),𝜇𝐹(𝑒)())=min(𝜇𝐹(𝑒)(),𝜇𝐹(𝑒)())=𝜇𝐹(𝑒)(), and 𝜂(𝑒)=𝑇(𝛼(𝑒),𝛼(𝑒))=min(𝛼(𝑒),𝛼(𝑒))=𝛼(𝑒),(ii)if 𝑒𝐵, then 𝜇𝐾(𝑒)()=min(𝜇𝐽(𝑒)(),𝜇𝐹(𝑒)())=min(max(𝜇𝐹(𝑒)(), 𝜇𝐺(𝑒)()),𝜇𝐹(𝑒)())=𝜇𝐹(𝑒)(), and 𝜂(𝑒)=𝑇(𝑆(𝛼(𝑒),𝛽(𝑒)),𝛼(𝑒))=min(max(𝛼(𝑒),𝛽(𝑒)),𝛼(𝑒))=𝛼(𝑒).Thus 𝐾𝜂=𝐹𝛼; that is, (𝐹𝛼𝐺𝛽𝐹)𝛼=𝐹𝛼.

Theorem 4.4. Let 𝐴,𝐵,𝐶𝐸, 𝐹𝛼, 𝐺𝛽, and 𝐻𝛾 be GIVFS sets over (𝑈,𝐸). Then the following hold: (1)𝐹𝛼𝐺(𝛽𝐻𝛾𝐹)=(𝛼𝐺𝛽)𝐹(𝛼𝐻𝛾),(2)𝐹𝛼𝐺(𝛽𝐻𝛾𝐹)=(𝛼𝐺𝛽𝐹)(𝛼𝐻𝛾).

Proof. (1) Suppose that the parameter sets of two GIVFS sets 𝐽𝛿 and 𝐾𝜂 are denoted by 𝑀 and 𝑁, respectively. Let 𝐹𝛼𝐺(𝛽𝐻𝛾𝐽)=𝛿 and (𝐹𝛼𝐺𝛽)𝐹(𝛼𝐻𝛾𝐾)=𝜂. Then 𝑀=𝐴(𝐵𝐶)=(𝐴𝐵)(𝐴𝐶)=𝑁. And, for each 𝑒𝑀,𝑈, it follows that 𝑒𝐴 and 𝑒𝐵𝐶,(i)if 𝑒𝐴,𝑒𝐵,𝑒𝐶, then 𝜇𝐽(𝑒)()=𝑇(𝜇𝐹(𝑒)(),𝜇𝐻(𝑒)())=min(𝜇𝐹(𝑒)(),  𝜇𝐻(𝑒)())=𝜇𝐾(𝑒)(), and 𝛿(𝑒)=𝑇(𝛼(𝑒),𝛾(𝑒))=min(𝛼(𝑒),𝛾(𝑒))=𝜂(𝑒),(ii)if 𝑒𝐴,𝑒𝐵,𝑒𝐶, then 𝜇𝐽(𝑒)()=𝑇(𝜇𝐹(𝑒)(),𝜇𝐺(𝑒)())=min(𝜇𝐹(𝑒)(),   𝜇𝐺(𝑒)())=𝜇𝐾(𝑒)(), and 𝛿(𝑒)=𝑇(𝛼(𝑒),𝛽(𝑒))=min(𝛼(𝑒),𝛽(𝑒))=𝜂(𝑒),(iii) if 𝑒𝐴,𝑒𝐵,𝑒𝐶, then 𝜇𝐽(𝑒)()=min(𝜇𝐹(𝑒)(), max(𝜇𝐺(𝑒)(),𝜇𝐻(𝑒)()))=max(min(𝜇𝐹(𝑒)(),𝜇𝐺(𝑒)()),   min((𝜇𝐹(𝑒)(),𝜇𝐻(𝑒)()))=𝜇𝐾(𝑒)(), 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 𝐽𝛿=𝐾𝜂; that is, 𝐹𝛼𝐺(𝛽𝐻𝛾𝐹)=(𝛼𝐺𝛽)𝐹(𝛼𝐻𝛾).
(2) The proof is similar to that of (1).

Theorem 4.5. (1)  (𝔖(𝑈,𝐸),,) is a distributive lattice.
(2) Let 1 be the order relation in 𝔖(𝑈,𝐸) and 𝐹𝛼,𝐺𝛽𝔖(𝑈,𝐸). One has 𝐹𝛼1𝐺𝛽 if and only if 𝐴𝐵,𝜇𝐹(𝑒)()𝜇𝐺(𝑒)() and 𝛼(𝑒)𝛽(𝑒) for all 𝑒𝐴 and 𝑈.

Proof. (1) The proof is straightforward from Theorems 3.14, 4.3, and 4.4.
(2) Suppose that 𝐹𝛼1𝐺𝛽. Then 𝐹𝛼𝐺𝛽=𝐺𝛽. So by Definition 3.10, we have 𝐴𝐵=𝐵, max(𝜇𝐹(𝑒)(),𝜇𝐺(𝑒)())=𝜇𝐺(𝑒)(), and max(𝛼(𝑒),𝛽(𝑒))=𝛽(𝑒) for all 𝑒𝐴 and 𝑈. It follows that 𝐴𝐵, 𝜇𝐹(𝑒)()𝜇𝐺(𝑒)() and 𝛼(𝑒)𝛽(𝑒) for all 𝑒𝐴 and 𝑈. Conversely, suppose that 𝐴𝐵, 𝜇𝐹(𝑒)()𝜇𝐺(𝑒)() and 𝛼(𝑒)𝛽(𝑒) for all 𝑒𝐴 and 𝑈. We can easily verify that 𝐹𝛼𝐺𝛽=𝐺𝛽. Thus 𝐹𝛼1𝐺𝛽.

For operators and , we can obtain similar results as follows.

Theorem 4.6. Let 𝐹𝛼 and 𝐺𝛽 be GIVFS sets over (𝑈,𝐸). Then the following hold: (1)(𝐹𝛼𝐹𝛼𝐹)=𝛼, (2)(𝐹𝛼𝐹𝛼𝐹)=𝛼, (3)(𝐹𝛼𝐺𝛽)𝐹𝛼=𝐹𝛼,(4)(𝐹𝛼𝐺𝛽𝐹)𝛼=𝐹𝛼.

Theorem 4.7. Let 𝐹𝛼, 𝐺𝛽 and 𝐻𝛾 be 𝐺𝐼𝑉𝐹𝑆 sets over (U,E). Then the following hold: (1)𝐹𝛼𝐺(𝛽𝐻𝛾𝐹)=(𝛼𝐺𝛽𝐹)(𝛼𝐻𝛾), (2)𝐹𝛼𝐺(𝛽𝐻𝛾𝐹)=(𝛼𝐺𝛽)𝐹(𝛼𝐻𝛾).

Theorem 4.8. (1)(𝔖(𝑈,𝐸),,) is a distributive lattice.
(2) Let 2 be the order relation in 𝔖(𝑈,𝐸) and 𝐹𝛼,𝐺𝛽𝔖(𝑈,𝐸). 𝐹𝛼2𝐺𝛽 if and only if 𝐵𝐴,𝜇𝐹(𝑒)()𝜇𝐺(𝑒)() and 𝛼(𝑒)𝛽(𝑒) for all 𝑒𝐵.

It is worth noting that (𝔖(𝑈,𝐸),,) and (𝔖(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. Let 𝑈={1,2,3} be the universe, 𝐸={𝑒1,𝑒2,𝑒3} the set of parameters, 𝐴={𝑒1,𝑒2}, 𝐵={𝑒2,𝑒3}. The GIVFS sets 𝐹𝛼 and 𝐺𝛽 over (𝑈,𝐸) are given as𝐹𝛼𝑒1=1[],0.5,0.72[],0.3,0.43[],[],𝐹0.6,0.70.8,0.9𝛼𝑒2=1[],0.6,0.82[],0.2,0.33[],[],𝐺0.7,0.90.4,0.5𝛽𝑒2=1[],0.1,0.32[],0.4,0.53[],[],𝐺0.5,0.60.6,0.8𝛽𝑒3=1[],0.3,0.42[],0.5,0.83[],[].0.4,0.60.5,0.7(4.1)
Suppose that (𝐹𝛼𝐺𝛽𝐹)𝛼=𝐻𝛾. Then 𝐶=𝐴𝐵={𝑒2}𝐴. So 𝐻𝛾𝐹𝛼, that is, (𝐹𝛼𝐺𝛽𝐹)𝛼𝐹𝛼.
Again, suppose that the parameters set of a GIVFS set 𝐽𝛿 is denoted by 𝐷, and (𝐹𝛼𝐺𝛽)𝐹𝛼=𝐽𝛿. Then 𝐷=𝐴𝐵={𝑒1,𝑒2,𝑒3}𝐴, Therefore, 𝐽𝛿𝐹𝛼, that is, (𝐹𝛼𝐺𝛽)𝐹𝛼𝐹𝛼.

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. Let 𝐹𝛼 be a GIVFS set, 𝑖,𝑗𝑈,𝑒𝑘𝐴. One says membership value of 𝑗 lowerly exceeds or equals to the membership value of 𝑖 with respect to the parameter 𝑒𝑘 if 𝜇𝐹(𝑒𝑘)(𝑖)𝜇𝐹(𝑒𝑘)(𝑗). The corresponding characteristic function is defined as follows: 𝑓𝑒𝑘𝑖,𝑗=1,if𝜇𝐹𝑒𝑘𝑖𝜇𝐹𝑒𝑘𝑗,0,otherwise.(5.1)

Definition 5.2. Let 𝐹𝛼 be a GIVFS set, 𝑖,𝑗𝑈,𝑒𝑘𝐴. One says membership value of 𝑗 upperly exceeds or equals to the membership value of 𝑖 with respect to the parameter 𝑒𝑘 if 𝜇+𝐹(𝑒𝑘)(𝑖)𝜇+𝐹(𝑒𝑘)(𝑗). The corresponding characteristic function is defined as follows: 𝑓+𝑒𝑘𝑖,𝑗=1,if𝜇+𝐹𝑒𝑘𝑖𝜇+𝐹𝑒𝑘𝑗;0,otherwise.(5.2)

Remark 5.3. Let 𝐹𝛼 be a GIVFS set, 𝑖,𝑗𝑈,and𝑒𝑘𝐴. For convenience, we denote the vectors (𝑓𝑒𝑘(𝑖,𝑗),𝑓+𝑒𝑘(𝑖,𝑗)) and (𝛼(𝑒𝑘),𝛼+(𝑒𝑘)) as 𝑓𝑒𝑘(𝑖,𝑗) and 𝛼(𝑒𝑘), respectively.

Now we can define the generalised comparison table about GIVFS set 𝐹𝛼.

Definition 5.4. Let 𝐹𝛼 be a GIVFS set. The generalised comparison table about 𝐹𝛼 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 1,2,,𝑛, and the entries are 𝐶𝑖𝑗, given as follows: 𝐶𝑖𝑗=𝑚𝑘=1𝑓𝑒𝑘𝑖,𝑗𝛼𝑒𝑘,𝑖,𝑗=1,2,,𝑛.(5.3)

Clearly, for 𝑖,𝑗=1,,𝑛, 𝑘=1,,𝑚, 0𝐶𝑖𝑗2𝑚, and 𝐶𝑖𝑖=𝑚𝑘=1(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 𝑒𝐴, 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 object 𝑖 are denoted by 𝑝𝑖 and 𝑞𝑖, respectively, and the score of an object 𝑖 is denoted as 𝑆𝑖 which can be given by 𝑆𝑖=𝑝𝑖𝑞𝑖. Now we present an algorithm as follows.

Algorithm 5.6. (1) Input the objects set 𝑈 and the parameter set 𝐴𝐸.
(2) Consider the GIVFS set 𝐹𝛼 in tabular form.
(3) By calculating the entries 𝐶𝑖𝑗, construct generalised comparison table.
(4) Compute the score of each 𝑖 using row sum and the column sum.
(5) The optimal decision is to select 𝑘 if the score of 𝑘 is maximum.
(6) If 𝑘 has more than one value then any one of 𝑘 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 universe 𝑈={1,2,3,4,5,6} 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 parameter 𝑒𝑖 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 𝛼𝐴Int([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 𝐹𝛼 as follows: 𝐹𝛼𝑒1=1[],0.70,0.852[],0.85,0.903[],0.65,0.754[],0.80,0.905[],0.60,0.706[],[],𝐹0.65,0.800.7,0.8𝛼𝑒2=1[],0.75,0.802[],0.60,0.703[],0.60,0.704[],0.70,0.755[],0.80,0.906[],[],𝐹0.70,0.800.5,0.6𝛼𝑒3=1[],0.80,0.902[],0.55,0.663[],0.65,0.804[],0.68,0.755[],0.70,0.806[],[],𝐹0.75,0.850.8,0.9𝛼𝑒4=1[],0.70,0.802[],0.65,0.753[],0.70,0.784[],0.62,0.705[],0.72,0.826[],[],𝐹0.80,0.900.6,0.7𝛼𝑒5=1[],0.65,0.752[],0.60,0.703[],0.80,0.904[],0.60,0.765[],0.75,0.856[],[].0.70,0.750.4,0.5(5.4)

The tabular representation of the GIVFS set 𝐹𝛼 is given in Table 1.

tab1
Table 1: Tabular representation of the GIVFS set 𝐹𝛼.

It is easy to calculate the entries 𝐶𝑖𝑗 by the formula 5.3. For example, let us calculate 𝐶21. Firstly, we compute 𝑓𝑒𝑘(2,1) for each 𝑒𝑘𝐴, where 𝑓𝑒1(2,1)=(0,0), 𝑓𝑒𝑘(2,1)=(1,1),𝑘=2,3,4,5. Secondly, we can obtain 𝐶21=5.0 by computing 𝑚𝑘=1(𝑓𝑒𝑘(2,1)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 𝐹𝛼 is given in Table 2.

tab2
Table 2: The generalised comparison table about 𝐹𝛼.

From Table 2, we can obtain the row sum and column sum and compute the score of each 𝑖, which are presented in Table 3.

tab3
Table 3: The score of 𝑖 about 𝐹𝛼.

From Table 3, it is clear that the maximum score is 𝑆1=12.00. So 1 could be selected as the optimal alternative.

It is worth noting that, unlike [30], the decision result depends not only on 𝐹(𝑒) but also on 𝛼(𝑒). For example, consider the GIVFS set 𝐺𝛽 with data as in Table 4, where 𝐵=𝐴 and 𝐺(𝑒)=𝐹(𝑒), but 𝛽(𝑒)𝛼(𝑒) for each 𝑒𝐵.

tab4
Table 4: Tabular representation of the GIVFS set 𝐺𝛽.

The generalised comparison table and the score of 𝑖 about the GIVFS set 𝐺𝛽 can be seen in Tables 5 and 6, respectively.

tab5
Table 5: The generalised comparison table about 𝐺𝛽.
tab6
Table 6: The score of 𝑖 about 𝐺𝛽.

From Table 6, it is clear that the maximum score is 𝑆6=9.40. Hence, the optimal alternative is 6, but not 1.

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.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (no. 11071061) and the National Basic Research Program of China (no. 2011CB311808).

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