#### 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 [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 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, refers to an initial universe, is a set of parameters, is the power set of , and are fuzzy subset of , respectively.

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

In other words, a soft set over is a parameterized family of subsets of the universe . For , may be considered as the set of -elements of the soft set or as the set of -approximate elements of the soft set.

*Definition 2.2 (see [29]). *Let denote the set of all fuzzy subsets of *. *Then a pair is called a fuzzy soft set over *, *where is a mapping from to .

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

*Definition 2.3 (see [32]). *Let be an initial universal set, a set of parameters, and the pair a soft universe. Let and be a fuzzy subset of ; that is, *. *Let be a function defined as follows: *, *where *. *Then is called a generalised fuzzy soft set over *. *

*Definition 2.4 (see [7]). *An interval-valued fuzzy set on a universe is a mapping , where stands for the set of all closed subintervals of *. *

The set of all interval-valued fuzzy sets on is denoted by . Suppose that , for all is called the degree of membership of an element to . And and are referred to as the lower and upper degrees of membership of to , where .

*Definition 2.5 (see [7]). *Let and be two interval-valued fuzzy sets on universe *. *Then the union, intersection, and complement of vague sets are defined as follows:

*Definition 2.6 (see [31]). *Let be an initial universe, let be a set of parameters, and let *. * denotes the set of all interval-valued fuzzy sets of *. *A pair is an interval-valued fuzzy soft set over *, *where is a mapping given by *. *

An interval-valued fuzzy soft set is a parameterized family of interval-valued fuzzy subsets of . For each parameter , is actually an interval-valued fuzzy set of , and it can be written as , where is the interval-valued fuzzy membership degree that object holds on parameter .

*Definition 2.7 (see [37]). *A *-*norm is an increasing, associative, and commutative mapping that satisfies the boundary condition: for all *. *

The commonly used continuous -norms are , , and .

*Definition 2.8 (see [37]). *A -conorm is an increasing, associative, and commutative mapping that satisfies the boundary condition: for all *. *

The commonly used continuous -conorms are , , and .

#### 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 be an initial universe and a set of parameters, *, **, *and let be a fuzzy sets of , that is, *. *Define a function as , where is an interval value is called the degree of membership an element * to *, and is called the degree of possibility of such belongness. Then is called type 1 generalised interval-valued fuzzy soft set over the soft universe *. *

Here for each parameter , indicates not only the degree of belongingness of elements of in but also the degree of preference of such belongingness which is represented by .

*Definition 3.2. *Let be an initial universe and a set of parameters, *, **, *and let be an interval-valued fuzzy sets of ; that is, , where stands for the set of all closed subintervals of *. *Define a function as , where and are interval values. Then is called type 2 generalised interval-valued fuzzy soft set over the soft universe *. *

It is clear that if holds for each , then the type 2 generalised interval-valued fuzzy soft set will degenerate to the type 1 generalised interval-valued fuzzy soft set. And if also holds for each , 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 be a set of mobile telephones and a set of parameters. The stand for the parameters “expensive”, “beautiful”, and “multifunctional”, respectively. Let be a function given as follows:
Then is a GIVFS set.

*Definition 3.4. *Let and be GIVFS sets over *.* Then is called a GIVFS subset of if(1);(2) is an interval-valued fuzzy subset of for any ; that is, and for any and ;(3) is an interval-valued fuzzy subset of ; that is, and for any .

In this case, the above relationship is denoted by . And is said to be a GIVFS superset of .

*Definition 3.5. *Let and be GIVFS sets over *. *Then and are said to be GIVFS equal if and only if and *. *

*Definition 3.6. *The relative complement of a GIVFS set is denoted by and is defined by , where is a mapping given by and * is *a mapping given by for all , where , *. *

*Example 3.7. *We consider the set given in Example 3.3 and define a set as follows:
Then is a GIVFS subset of , and the relative complement of a GIVFS set is

*Definition 3.8. *Let *. *A GIVFS set over is said to be relative absolute GIVFS set denoted by , if and for all and *. *

*Definition 3.9. *Let *. *A GIVFS set over is said to be relative null GIVFS set, denoted by , if and for all and *. *

*Definition 3.10. *The union of two GIVFS sets and over denoted by is a GIVFS set and defined as such that, for all and ,
where and .

*Definition 3.11. *The intersection of two sets and over denoted by is a GIVFS set and defined as such that, for all and *, *, where *, * and .

*Example 3.12. *We consider the GIVFS sets and given in Examples 3.3 and 3.7, respectively, and consider and . Then

Proposition 3.13. *Let be a GIVFS set over . Then the following holds*(1)*,
*(2)*,
*(3)*,
*(4)*. *

*Proof. *It is easily obtained from Definitions 3.8–3.11.

Theorem 3.14. *Let , , and be GIVFS sets over . Then the following holds*(1)*,
*(2)*,
*(3)*,
*(4)*. *

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

*Definition 3.15. *The restricted union of two GIVFS sets and over denoted by is a GIVFS set and defined as such that, for all and *, *, where *, * and .

*Definition 3.16. *The extended intersection of two GVS sets and over , denoted by , is a GVS set which is defined as, for all *,*
where , and .

Theorem 3.17. *Let , , and be three GIVFS sets over . Then the following holds:*(1)*,
*(2)*,
*(3)*,
*(4)*. *

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

Theorem 3.18. *Let and be two GIVFS sets over . Then the following holds: *(1)*,*(2)*. *

*Proof. *(1) Suppose that , then , and, ,
Moreover, we have , , and ,
Assume that the parameters set of a GIVFS set is denoted , and . Then . Since
Then, for each ,
Therefore, and are the same GIVFS sets. Thus, .

(2) The proof is similar to that of (1).

*Definition 3.19. *The “AND” of two GIVFS sets and over , denoted by , is defined as such that for all and *, *, where *, * and .

*Definition 3.20. *The “OR” of two GIVFS sets and over , denoted by , is defined as such that for all and *, **, *where *, * and .

Theorem 3.21. *Let , , and be three GIVFS sets over . Then the following holds *(1)*,
*(2)*. *

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

Theorem 3.22. *Let and be two GIVFS sets over . Then the following holds *(1)*,
*(2)*. *

*Proof. *(1) Suppose that , then , and, ,
Moreover, we have , , and ,
Assume that the parameters set of a GIVFS set is denoted , and . Then . Since ,
then, for each ,
Therefore, and are the same GIVFS sets. Thus, .

(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 be a sets over . Then the following holds*(1)*,
*(2)*. *

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

*Example 4.2. *We consider the GIVFS set given in Example 3.3. We have that the following(1)If , then , , , and ; that is, .(2)If , then , , , and ; that is, .(3)if , then , , and , that is, ;(4)If , then , , and ; that is, .

For convenience, let denote the set of all GIVFS sets over ; that is, .

From Proposition 4.1, we can see that is not a lattice in general. However, if and , then the idempotent law and absorption law with respect to operations and hold. In the remainder of this section, we always consider and .

Theorem 4.3. *Let , , and be GIVFS sets over . Then the following hold: *(1)*,
*(2)*,
*(3)*,*(4)*.*

*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 and are denoted by and , respectively. Let and . Then , . And, for each and ,(i)if , then , and ,(ii)if , then , , and .Thus ; that is, .

Theorem 4.4. *Let , , , and be GIVFS sets over . Then the following hold: *(1)*,*(2)*.*

*Proof. *(1) Suppose that the parameter sets of two GIVFS sets and are denoted by and , respectively. Let and . Then . And, for each , it follows that and ,(i)if , then , , and ,(ii)if , then , , and ,(iii) if , then , , , and , .Thus ; that is, .

(2) The proof is similar to that of (1).

Theorem 4.5. *
(1) is a distributive lattice.**
(2) Let be the order relation in and . One has if and only if and for all and .*

*Proof. *(1) The proof is straightforward from Theorems 3.14, 4.3, and 4.4.

(2) Suppose that . Then . So by Definition 3.10, we have , , and for all and . It follows that , and for all and . Conversely, suppose that , and for all and . We can easily verify that . Thus .

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

Theorem 4.6. *Let and be GIVFS sets over . Then the following hold: *(1)*,
*(2)*,
*(3)*,*(4)*.*

Theorem 4.7. *Let , and be sets over (U,E). Then the following hold: *(1)*,
*(2)*. *

Theorem 4.8. *
(1) is a distributive lattice.**
(2) Let be the order relation in and . if and only if and for all .*

It is worth noting that and 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 ** be the universe, ** the set of parameters, **, **. The GIVFS sets ** and ** over ** are given as*

Suppose that . Then . So , that is, .

Again, suppose that the parameters set of a GIVFS set is denoted by , and . Then , Therefore, , that is, .

#### 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 be a GIVFS set, . One says membership value of lowerly exceeds or equals to the membership value of with respect to the parameter if . The corresponding characteristic function is defined as follows:

*Definition 5.2. *Let be a GIVFS set, . One says membership value of upperly exceeds or equals to the membership value of with respect to the parameter if . The corresponding characteristic function is defined as follows:

*Remark 5.3. *Let be a GIVFS set, . For convenience, we denote the vectors and as and , respectively.

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

*Definition 5.4. *Let be a GIVFS set. The generalised comparison table about 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 , and the entries are , given as follows:

Clearly, for , , , and , where and 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 of the generalised comparison table are numbers of real interval 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 for each , 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 are denoted by and , respectively, and the score of an object is denoted as which can be given by . Now we present an algorithm as follows.

*Algorithm 5.6. *(1) Input the objects set and the parameter set .

(2) Consider the GIVFS set in tabular form.

(3) By calculating the entries , construct generalised comparison table.

(4) Compute the score of each using row sum and the column sum.

(5) The optimal decision is to select if the score of is maximum.

(6) If has more than one value then any one of 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 under consideration, and is the set of decision parameters, where 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 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 is regarded as . In this case, let , and let be an interval-valued fuzzy subset of , which is given by the company as follows: , ,,, . And consider the GIVFS set as follows:

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

It is easy to calculate the entries by the formula 5.3. For example, let us calculate . Firstly, we compute for each , where , . Secondly, we can obtain by computing , where , , , , . And the generalised comparison table about the GIVFS set is given in Table 2.

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

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

It is worth noting that, unlike [30], the decision result depends not only on but also on . For example, consider the GIVFS set with data as in Table 4, where and , but for each .

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