Generalized interval-valued hesitant intuitionistic fuzzy soft sets

Abstract

This paper aims to extend the Interval-valued Intuitionistic Hesitant Fuzzy Set to a Generalized Interval-valued Hesitant Intuitionistic Fuzzy Soft Set (GIVHIFSS). Definition of a GIVHIFSS and some of their operations are defined, and some of their properties are studied. In these GIVHIFSSs, the authors have defined complement, null, and absolute. Soft binary operations like operations union, intersection, a subset are also defined. Here is also verified De Morgan’s laws and the algebraic structure of GIVHIFSSs. Finally, by using the comparison table, a different approach to GIVHIFSS based decision-making is presented.

Keywords

Soft sets intuitionistic fuzzy soft sets hesitant fuzzy soft sets interval-valued fuzzy soft sets

1 Introduction

Since various types of problems are uncertain, frequently in real life, such as in agriculture and health, sometimes researchers take in data that are not emphatic, proper, and deterministic. Such uncertainty problems can be handled with theories, such as probability, Fuzzy Sets (FSs), Intuitionistic Fuzzy Set, etc. Related to FS, many theoretical mathematical problems can be solved using this concept, such as the research by Nazra [15], which is one of the studies in Fuzzy algebra. In developing the FS theory, it can not be released from the structure of algebra.

In FS theory, we study the membership degree (shortly, MD) of an element in an FS, which is expressed by a single value in the interval [0,1]. With the development of the theories and applications of FSs in various fields, in reality, it may not always appropriate that the non-membership degree (shortly, NMD) is NMD:=1 - MD. So, as a development of the FS, it was introduced by Atanassov [1] a concept called an Intuitionistic Fuzzy Set (IFS).

Since the theories above still have limitations, Molodtsov [14] has initiated a new concept called Soft Set (SS), a new approach to dealing with matters that contain uncertainty or obscurity. Maji et al. [11] are those who first introduced SS theory in the decision-making problems. Then Maji et al. [9] introduced the concept of Fuzzy Soft Sets (FSSs), which is a combination of SS and FS.

In real life, we find that the MD of an element is not always single, but it is a set of several different values in [0,1]. To overcome this problem, Torra and Narukawa [23] used a concept, namely Hesitant Fuzzy Sets (HFSs), as a development of the FS theory. The study on HFS continues to grow, along with the development of the study on SS. Wang et al. [24, 25] have conducted a study by combining HFS and FSS called Hesitant Fuzzy Soft Sets (HFSSs).

The study on Intuitionistic Fuzzy Soft Sets (IFSSs) has been done by Maji et al. [10, 12], which is an amalgamation of the concepts of SS and IFS. Then the IFSS concept was generalized by Babitha and John [3] and Dinda et al. [4] who study Generalized Intuitionistic Fuzzy Soft Sets (GIFSSs) and their applications. Related to GIFSSs, Nazra et al. [18] constructed a kind of fuzzy sets, called Generalized intuitionistic fuzzy soft matrices. Then related to the study on HFSs, Qian et al. [20] have reviewed the generalization of it, and then Nazra et al. [16] have conducted preliminary research on Hesitant Intuitionistic Fuzzy Soft Sets (HIFSSs), which combines the concept of HFS and IFSS. As a generalization of the concept of HIFSSs introduced by Nazra et al. [16] and inspired by the results of research by Babitha and John [3], Dinda et al. [4] and Qian et al. [20] then Nazra et al. [17] have researched with the theme Generalized Hesitant Intuitionistic Fuzzy Soft Sets (GHIFSSs).

In 1987, Gorzalczany [7] developed a concept called Interval-valued Hesitant Fuzzy Sets (IVHFSs). This concept was born because decision-makers often have insufficient knowledge or information to determine the MD and NMD of an element in an FS. So, the researchers define the MD as an interval in the interval [0,1]. This study is called Interval-valued Fuzzy Sets (IVFSs).

The study on IVFS continues to grow, such as Zhang et al. [28], who initiated the concept of Interval-valued Hesitant Fuzzy Soft Sets (IVHFSSs) as a combination of IVFS, HFS, and SS. Then Peng and Yang [19] and Sooraj et al. [22] have developed IVHFSSs to be Interval-valued Intuitionistic Hesitant Fuzzy Soft Sets (IVIHFSSs). Jiang et al. [8] have reviewed the Interval-valued Intuitionistic Fuzzy Soft Sets (IVIFSS) and proved the algebraic properties. Dugenci [6] studied the distance measure of Interval-valued Intuitionistic Fuzzy Sets (IVIFSs) and their applications while Ding and Wu [5] studied Interval-valued Hesitant Fuzzy Sets (IVHFSs) and their applications in multi-criteria decision-making problems.

In this article is introduced a concept called Generalized Interval-valued Hesitant Intuitionistic Fuzzy Soft Sets (GIVHIFSSs). This concept is what distinguishes this study from Dugenci [6] and Ding and Wu [5]. This concept is also a development of the article Nazra et al. [16, 17], Peng and Yang [19] and Sooraj et al. [22]. Here we construct a new definition of an Interval-valued Hesitant Intuitionistic Fuzzy Sets (IVHIFS) and its operations and prove some properties that are different from those in Zhang [27]. Zhang [27] extends the HFS to interval-valued intuitionistic fuzzy environments and allows the membership of an element to be a set of several possible Interval-valued Intuitionistic Fuzzy numbers. This case happens if there are more than one decision-makers who give the Interval-valued Intuitionistic Fuzzy number. However, due to lack of knowledge or data, sometimes a decision-maker cannot provide an evaluation with a single subinterval of [0,1] either for MD or NMD in an Interval-valued Intuitionistic Fuzzy Number. This motivation explains why we introduce a concept called Interval-valued Hesitant Intuitionistic Fuzzy Sets (IVHIFSs), where a decision-maker can provide the membership and non-membership values of an element to be a set of several subintervals of [0,1]. Moreover, by combining IVHIFSs, SSs, and FSs, we define a GIVHIFSS and its operations. The given definition of the GIVHIFSS is more realistic as it contains a degree of preference corresponding to each parameter. Finally, we obtain some properties related to such operations. In the last section, we present the application of GIVHIFSSs in decision-making problems.

2 Preliminaries

In this section, we recall the definition of a Fuzzy Set, a Fuzzy Soft Set, an Intuitionistic Fuzzy Set and the related operations and properties. Zadeh, in [26] (see also [13]), introduced the concept of Fuzzy Sets. A Fuzzy Set (FS) over a set of objects O is a set Fs = {(w, α(w)) |w ∈ O} where α: O → [0, 1]. Here α and α(w) are called the membership function of Fs and the membership value of w in Fs, respectively. From now on we write α in place of Fs.

Suppose that α and β are FSs over O. In [26] and [13], it was presented several operations in FSs. Some of them are as follows.

The complement of the fuzzy set α over O is α^c : = {(w, α(w)) |w ∈ O} ^c = {(w, α^c(w)) |w ∈ O}, with α^c(w) : =1 - α(w).

The union of two FSs α and β over O is α ∪ β : = {(w, α(w) ∨ β(w)) |w ∈ O}, with α(w) ∨ β(w) : = max {α(w) , β(w)}.

The intersection of two FSs α and β over O is α ∩ β : = {(w, α(w) ∧ β(w)) |w ∈ O}, with α(w) ∧ β(w) : = min {α(w) , β(w)}.

The empty FS over O is ∅ : = {(w, 0) |w ∈ O}.

The universal FS over O is 1 : = {(w, 1) |w ∈ O}.

The algebraic sum of two FSs α and β over O is α ⊕ β = α + β - αβ : = {(w, α(w) + β(w) - α(w) β(w)) |w ∈ O}.

The algebraic product of two FSs α and β over O is α ⊗ β = αβ : = {(w, α(w) β(w)) |w ∈ O}.

Different from the above; we define two other operations of FSs, namely:

The t-product of α is tα : = {(w, 1 -(1 - α(w)) ^t) |w ∈ O}, t > 0.

The t-power of α is α^t : = {(w, (α(w)) ^t) |w ∈ O}, t > 0.

The following algebraic properties of FSs, related to some of the above operations, have been explained clearly in [26] and [13].

Proposition 2.1. [26] Let α, β and γ be three FSs over O. Then the following statements are valid.

(α^c) ^c = α.

α ∪ β = β ∪ α.

α ∩ β = β ∩ α.

(α ∪ β) ^c = β^c ∩ α^c.

(α ∩ β) ^c = β^c ∪ α^c.

α ∩(β ∩ γ) =(α ∩ β) ∩ γ.

α ∪(β ∪ γ) =(α ∪ β) ∪ γ.

(α ⊕ β) ^c = α^c ⊗ β^c.

(α ⊗ β) ^c = α^c ⊕ β^c.

Related to our definition ((8) and (9) above), we obtain the following properties.

Proposition 2.2. Let α, and β be two FSs over O and t > 0. Then

(α^c) ^t =(tα) ^c,

(α ⊗ β) ^t = α^t ⊗ β^t.

Proof. By definition, for any w ∈ O, (α^c(w)) ^t =(1 - α(w)) ^t = 1 -(1 -(1 - α(w)) ^t) =(1 -(1 - α(w)) ^t) ^c. Hence i) is proved. For ii), since (α(w) β(w)) ^t =(α(w)) ^t(β(w)) ^t for any w ∈ O then the theorem is proved. □

Now, we recall the definition of a Fuzzy Soft Set (FSS) as a hybrid model involving FS and SS.

Definition 2.3. [9] Let O be a set of objects, T is a parameter set, P ⊆ T. A Fuzzy Soft Set over O is a pair (f_P, P) where f_P: P → I^O with I^O is the collection of all Fuzzy Sets over O.

As a generalization of FS, we recall the definition of an Intuitionistic Fuzzy Set (IFS).

Definition 2.4. [1] Let O be a set. An Intuitionistic Fuzzy Set over O is the set $If = {< w, α_{If} (w), γ_{If} (w) > | w \in O}$ where α_If: O → [0, 1] and γ_If: O → [0, 1], such that 0 ≤ α_If(w) + γ_If(w) ≤1 for all w ∈ O. α_If(w) and γ_If(w) are called the membership and the non-membership values (degrees) of w, respectively.

3 Interval-valued hesitant intuitionistic fuzzy sets

The study on Generalized Interval-valued Hesitant Intuitionistic Fuzzy Soft Sets (GIVHIFSS) is related to a concept called Interval-valued Hesitant Intuitionistic Fuzzy Sets (IVHIFSs). So, in this section, we construct the definition of IVHIFSs and their operations. Then, we derive some of the related properties. However, Zhang [27] introduced the definition of Interval-valued Intuitionistic Hesitant Fuzzy Sets, which is different from our definition.

Now we recollect the definition of Interval-valued Intuitionistic Fuzzy Sets (IVIFSs) [2].

Definition 3.1. [2] Let O be a universe of objects and Int[0, 1] denotes all closed subintervals of the interval [0, 1]. An Interval-valued Intuitionistic Fuzzy Set over O is a set having the form: $I_{v} = {< w, α (w), α^{'} (w) > | w \in O}$ where $α : O \to Int [0, 1]$ $α^{'} : O \to Int [0, 1]$ with the condition $0 \leq \sup (α (w)) + \sup (α^{'} (w)) \leq 1 .$ The intervals α(w) and α′(w) are the membership and non-membership degrees of the element w ∈ O to the set I_v, respectively.

By combining the definition of IVIFSs and HFSs, we construct our new definition, IVHIFSs.

Definition 3.2. Let O be a set of objects. An Interval-valued Hesitant Intuitionistic Fuzzy Set (IVHIFS) over O is the set $\tilde{W} = {< x, I_{\tilde{w}} (x) > | x \in O} .$ where $I_{\tilde{w}} (x) = (α_{I_{\tilde{w}} (x)}, β_{I_{\tilde{w}} (x)})$ . Here $α_{I_{\tilde{W}} (x)} = {a = [α_{a}^{-}, α_{a}^{+}] | a \in Int [0, 1]}$ and $β_{I_{\tilde{W}} (x)} = {b = [β_{b}^{-}, β_{b}^{+}] | b \in Int [0, 1]}$ are sets of feasible membership and non-membership degrees of x ∈ O to the set $\tilde{W}$ , respectively, and for any x, $\max {α_{a}^{+}} + \max {β_{b}^{+}} \leq 1$ . For simplification, we denote $\tilde{I} = I_{\tilde{w}} (x)$ which is called an Interval-valued Hesitant Intuitionistic Fuzzy Element (IVHIFE). The collection of all Interval-valued Hesitant Intuitionistic Fuzzy Sets over O is denoted by $\tilde{IV} (O)$ .

Example 3.3. Let O = {x₁, x₂} be a set of objects. Then the Interval-valued Hesitant Intuitionistic Fuzzy Set over O is given as $\tilde{W} = {< x_{i}, I_{\tilde{w}} (x_{i}) = (α_{I_{\tilde{w}} (x_{i})}, β_{I_{\tilde{w}} (x_{i})}) > | x_{i} \in O} .$ where $α_{I_{\tilde{w}} (x_{1})} : = {[0.2, 0.25], [0.25, 0.4], [0.1, 0.26]},$ $β_{I_{\tilde{w}} (x_{1})} : = {[0.3, 0.55], [0.4, 0.5], [0.5, 0.6]},$ $α_{I_{\tilde{w}} (x_{2})} : = {[0.1, 0.2], [0.25, 0.3]},$ $β_{I_{\tilde{w}} (x_{2})} : = {[0.5, 0.68], [0.4, 0.65]} .$ We see that for x₁, $\max {α_{a}^{+}} = 0.4$ and $\max {β_{b}^{+}} = 0.6$ , so that $\max {α_{a}^{+}} + \max {β_{b}^{+}} \leq 1$ . Such condition hold also for x₂. In this IVHIFS there are two IVHIFEs, namely $\begin{matrix} {\tilde{I}}_{1} & = I_{\tilde{w}} (x_{1}) = (α_{I_{\tilde{w}} (x_{1})}, β_{I_{\tilde{w}} (x_{1})}) \\ = ({[0.2, 0.25], [0.25, 0.4], [0.1, 0.26]}, \\ {[0.3, 0.55], [0.40, 0.5], [0.5, 0.60]}) \\ and \\ {\tilde{I}}_{2} & = I_{\tilde{w}} (x_{2}) = (α_{I_{\tilde{w}} (x_{2})}, β_{I_{\tilde{w}} (x_{2})}) \\ = ({[0.1, 0.20], [0.25, 0.30]}, \\ {[0.5, 0.68], [0.40, 0.65]}) . \end{matrix}$

Now, using the definition of IVHIFEs above, we derive some operations in IVHIFEs.

Definition 3.4. Suppose that $\tilde{I} = (α_{\tilde{I}}, β_{\tilde{I}}), {\tilde{I}}_{1} = (α_{{\tilde{I}}_{1}}, β_{{\tilde{I}}_{1}})$ and ${\tilde{I}}_{2} = (α_{{\tilde{I}}_{2}}, β_{{\tilde{I}}_{2}})$ are three IVHIFEs. Then, it is defined some operations as follows.

The complement of $\tilde{I}$ is ${\tilde{I}}^{c} = (β_{\tilde{I}}, α_{\tilde{I}})$

The union of two IVHIFEs ${\tilde{I}}_{1}$ and ${\tilde{I}}_{2}$ is ${\tilde{I}}_{1} \cup {\tilde{I}}_{2} = ({⋃_{a_{1} \in α_{{\tilde{I}}_{1}} a_{2} \in α_{{\tilde{I}}_{2}}} {[α_{a_{1}}^{-} \lor α_{a_{2}}^{-}, α_{a_{1}}^{+} \lor α_{a_{2}}^{+}]},}, {⋃_{b_{1} \in β_{{\tilde{I}}_{1}} b_{2} \in β_{{\tilde{I}}_{2}}} {[β_{b_{1}}^{-} \land β_{b_{2}}^{-}, β_{b_{1}}^{+} \land β_{b_{2}}^{+}]}})$ .

The intersection of two IVHIFEs ${\tilde{I}}_{1}$ and ${\tilde{I}}_{2}$ is ${\tilde{I}}_{1} \cap {\tilde{I}}_{2} = ({⋃_{a_{1} \in α_{{\tilde{I}}_{1}} a_{2} \in α_{{\tilde{I}}_{2}}} {[α_{a_{1}}^{-} \land α_{a_{2}}^{-}, α_{a_{1}}^{+} \land α_{a_{2}}^{+}]}}, {⋃_{b_{1} \in β_{{\tilde{I}}_{1}} b_{2} \in β_{{\tilde{I}}_{2}}} {[β_{b_{1}}^{-} \lor β_{b_{2}}^{-}, β_{b_{1}}^{+} \lor β_{b_{2}}^{+}]}})$ .

The algebraic sum of two IVHIFEs ${\tilde{I}}_{1}$ and ${\tilde{I}}_{2}$ is ${\tilde{I}}_{1} \oplus {\tilde{I}}_{2} = ({⋃_{a_{1} \in α_{{\tilde{I}}_{1}} a_{2} \in α_{{\tilde{I}}_{2}}} {[α_{a_{1}}^{-} + α_{a_{2}}^{-} - α_{a_{1}}^{-} α_{a_{2}}^{-}, α_{a_{1}}^{+} + α_{a_{2}}^{+} - α_{a_{1}}^{+} α_{a_{2}}^{+}]}}, {⋃_{b_{1} \in β_{{\tilde{I}}_{1}} b_{2} \in β_{{\tilde{I}}_{2}}} {[β_{b_{1}}^{-} β_{b_{2}}^{-}, β_{b_{1}}^{+} β_{b_{2}}^{+}]}})$ .

The algebraic product of two IVHIFEs ${\tilde{I}}_{1}$ and ${\tilde{I}}_{2}$ is ${\tilde{I}}_{1} \otimes {\tilde{I}}_{2} = ({⋃_{a_{1} \in α_{{\tilde{I}}_{1}} a_{2} \in α_{{\tilde{I}}_{2}}} {[α_{a_{1}}^{-} α_{a_{2}}^{-}, α_{a_{1}}^{+} α_{a_{2}}^{+}]}}, {⋃_{b_{1} \in β_{{\tilde{I}}_{1}} b_{2} \in β_{{\tilde{I}}_{2}}} {[β_{b_{1}}^{-} + β_{b_{2}}^{-} - β_{b_{1}}^{-} β_{b_{2}}^{-}, β_{b_{1}}^{+} + β_{b_{2}}^{+} - β_{b_{1}}^{+} β_{b_{2}}^{+}]}})$ .

The t-product of $\tilde{I}$ is $t \tilde{I} = ({⋃_{a \in α_{\tilde{I}}} {[1 - (1 - α_{a}^{-})^{t}, 1 - (1 - α_{a}^{+})^{t}]}}, {⋃_{b \in β_{\tilde{I}}} {[(β_{b}^{-})^{t}, (β_{b}^{+})^{t}]}})$ , t > 0.

The t-power of $\tilde{I}$ is ${\tilde{I}}^{t} = ({⋃_{a \in α_{\tilde{I}}} {[(α_{a}^{-})^{t}, (α_{a}^{+})^{t}]}}, {⋃_{b \in β_{\tilde{I}}} {[1 - (1 - β_{b}^{-})^{t}, 1 - (1 - β_{b}^{+})^{t}]} .})$ , t > 0.

Example 3.5. From Example 3.3 we apply Definition 3.4 (2). We have $\begin{matrix} {\tilde{I}}_{1} \cup {\tilde{I}}_{2} = & ({[\max {0.2, 0.1}, \max {0.25, 0.20}], \\ [\max {0.2, 0.25}, \max {0.25, 0.30}], \\ [\max {0.25, 0.1}, \max {0.40, 0.20}], \\ [\max {0.25, 0.25}, \max {0.40, 0.30}], \\ [\max {0.1, 0.1}, \max {0.26, 0.20}], \\ [\max {0.1, 0.25}, \max {0.26, 0.30}]}, \\ {[\min {0.3, 0.5}, \min {0.55, 0.68}], \\ [\min {0.3, 0.4}, \min {0.55, 0.65}], \\ [\min {0.4, 0.5}, \min {0.50, 0.68}], \\ [\min {0.4, 0.4}, \min {0.50, 0.65}], \\ [\min {0.5, 0.5}, \min {0.6, 0.68}], \\ [\min {0.5, 0.4}, \min {0.6, 0.65}]}) . \end{matrix}$

By applying Definition 3.4 above, we derive the following operations in IVHIFSs. The following are some properties of IVHIFSs, where the features are related to those in Definition 3.4 above.

Definition 3.6. Given two IVHIFSs $\tilde{W} = {< w, I_{\tilde{w}} (w) > | w \in O}$ , $\tilde{Z} = {< w, I_{\tilde{z}} (w)) > | w \in O}$ on O. The following are some operations in IVHIFSs.

The complement of $\tilde{W}$ is ${\tilde{W}}^{c} = {< w, I_{\tilde{w}}^{c} (w) > | w \in O}$ .

The union of two IVHIFSs $\tilde{W}$ and $\tilde{Z}$ is $\tilde{W} \cup \tilde{Z} = {< w, I_{\tilde{w}} (w) \cup I_{\tilde{z}} (w) > | w \in O}$ .

The intersection of two IVHIFSs $\tilde{W}$ and $\tilde{Z}$ is $\tilde{W} \cap \tilde{Z} = {< w, I_{\tilde{w}} (w) \cap I_{\tilde{z}} (w) > | w \in O}$ .

The algebraic sum of two IVHIFSs $\tilde{W}$ and $\tilde{Z}$ is $\tilde{W} \oplus \tilde{Z} = {< w, I_{\tilde{w}} (w) \oplus I_{\tilde{z}} (w) > | w \in O}$ .

The algebraic product of two IVHIFSs $\tilde{W}$ and $\tilde{Z}$ is $\tilde{W} \otimes \tilde{Z} = {< w, I_{\tilde{w}} (w) \otimes I_{\tilde{z}} (w) > | w \in O}$ .

The t-product of $\tilde{W}$ is $t \tilde{W} = {< w, t I_{\tilde{w}} (w) > | w \in O}$ , t > 0.

The t-power of $\tilde{W}$ is ${\tilde{W}}^{t} = {< w, (I_{\tilde{w}} (w))^{t} > | w \in O}$ , t > 0.

Example 3.7. Let O = {x₁, x₂} be a set of objects. Given two IVHIFSs $\tilde{W}$ and $\tilde{Z}$ over O as follows. $\begin{matrix} \tilde{W} & = {< x_{i}, I_{\tilde{w}} (x_{i}) > | x_{i} \in O} where \\ I_{\tilde{w}} (x_{1}) & = (α_{I_{\tilde{w}} (x_{1})}, β_{I_{\tilde{w}} (x_{1})}) \\ = ({[0.2, 0.25], [0.25, 0.4]}, \\ {[0.3, 0.55], [0.40, 0.5]}), \\ I_{\tilde{w}} (x_{2}) & = (α_{I_{\tilde{w}} (x_{2})}, β_{I_{\tilde{w}} (x_{2})}) \\ = ({[0.1, 0.20], [0.25, 0.30]}, \\ {[0.5, 0.68], [0.40, 0.65]}) . \end{matrix}$ $\begin{matrix} \tilde{Z} & = {< x_{i}, I_{\tilde{z}} (x_{i}) > | x_{i} \in O} where \\ I_{\tilde{z}} (x_{1}) & = (α_{I_{\tilde{z}} (x_{1})}, β_{I_{\tilde{z}} (x_{1})}) \\ = ({[0.25, 0.20], [0.20, 0.45]}, \\ {[0.35, 0.50], [0.45, 0.50]}), \\ I_{\tilde{z}} (x_{2}) & = (α_{I_{\tilde{z}} (x_{2})}, β_{I_{\tilde{z}} (x_{2})}) \\ = ({[0.15, 0.20], [0.20, 0.35]}, \\ {[0.55, 0.60], [0.45, 0.65]}) . \end{matrix}$ We have $\tilde{W} \cup \tilde{Z} = {< x_{i}, I_{\tilde{w}} (x_{i}) \cup I_{\tilde{z}} (x_{i}) > | x_{i} \in O}$ where, by applying Definition 3.4 (2), $\begin{matrix} I_{\tilde{w}} (x_{1}) \cup I_{\tilde{z}} (x_{1}) = & ({[0.25, 0.25], [0.2, 0.45], \\ [0.25, 0.4], [0.25, 0.45]}, \\ {[0.3, 0.5], [0.3, 0.5], \\ [0.35, 0.5], [0.4, 0.5]}) and \end{matrix}$ $\begin{matrix} I_{\tilde{w}} (x_{2}) \cup I_{\tilde{z}} (x_{2}) = & ({[0.15, 0.2], [0.2, 0.35], \\ [0.25, 0.3], [0.25, 0.35]}, \\ {[0.5, 0.6], [0.45, 0.65], \\ [0.4, 0.6], [0.4, 0.65]}) . \end{matrix}$

Definition 3.8. Let $\tilde{W} = {< x, (α_{I_{\tilde{w}} (x)}, β_{I_{\tilde{w}} (x)}) > | x \in O}$ and $\tilde{Z} = {< x, (α_{I_{\tilde{z}} (x)}, β_{I_{\tilde{z}} (x)}) > | x \in O}$ be two IVHIFSs over O, where $\begin{matrix} α_{I_{\tilde{w}} (x)} & = {a = [α_{a}^{-}, α_{a}^{+}] | a \in Int [0, 1]}, \\ β_{I_{\tilde{w}} (x)} & = {b = [β_{b}^{-}, β_{b}^{+}] | b \in Int [0, 1]}, \\ α_{I_{\tilde{z}} (x)} & = {c = [α_{c}^{-}, α_{c}^{+}] | c \in Int [0, 1]}, \\ β_{I_{\tilde{z}} (x)} & = {d = [β_{d}^{-}, β_{d}^{+}] | d \in Int [0, 1]} . \end{matrix}$ Now, $\tilde{W}$ is called an Interval-valued Hesitant Intuitionistic Fuzzy subset of $\tilde{Z}$ , denoted by $\tilde{W} \tilde{\subset} \tilde{Z}$ , if ∀x ∈ O, $\max {α_{a}^{+}} \leq \min {α_{c}^{-}}$ and $\min {β_{b}^{-}} \geq \max {β_{d}^{+}}$ .

Example 3.9. Let O = {x₁, x₂} be a set of objects. Given two IVHIFSs $\tilde{W}$ and $\tilde{Z}$ over O as follows. $\begin{matrix} \tilde{W} & = {< x_{i}, I_{\tilde{w}} (x_{i}) > | x_{i} \in O} where \\ I_{\tilde{w}} (x_{1}) & = (α_{I_{\tilde{w}} (x_{1})}, β_{I_{\tilde{w}} (x_{1})}) \\ = ({[0.2, 0.25], [0.25, 0.4]}, \\ {[0.3, 0.55], [0.40, 0.5]}), \\ I_{\tilde{w}} (x_{2}) & = (α_{I_{\tilde{w}} (x_{2})}, β_{I_{\tilde{w}} (x_{2})}) \\ = ({[0.1, 0.20], [0.25, 0.30]}, \\ {[0.5, 0.68], [0.40, 0.65]}) . \end{matrix}$ $\begin{matrix} \tilde{Z} & = {< x_{i}, I_{\tilde{z}} (x_{i}) > | x_{i} \in O} where \\ I_{\tilde{z}} (x_{1}) & = (α_{I_{\tilde{z}} (x_{1})}, β_{I_{\tilde{z}} (x_{1})}) \\ = ({[0.45, 0.50], [0.5, 0.65]}, \\ {[0.15, 0.20], [0.25, 0.3]}), \\ I_{\tilde{z}} (x_{2}) & = (α_{I_{\tilde{z}} (x_{2})}, β_{I_{\tilde{z}} (x_{2})}) \\ = ({[0.35, 0.4], [0.3, 0.35]}, \\ {[0.25, 0.30], [0.35, 0.4]}) . \end{matrix}$ It is clear that $\tilde{W} \tilde{\subset} \tilde{Z}$ .

Related to Definition 3.8, we obtain the following proposition.

Proposition 3.10. Let $\tilde{W}$ and $\tilde{Z}$ be two IVHIFSs over O. Suppose that $\tilde{W} \tilde{\subset} \tilde{Z}$ . Then

${\tilde{W}}^{c} \cap {\tilde{Z}}^{c}$ $\tilde{\subset}$ $(\tilde{W} \cap \tilde{Z})^{c}$ ,

$(\tilde{W} \cup \tilde{Z})^{c}$ $\tilde{\subset}$ ${\tilde{W}}^{c} \cup {\tilde{Z}}^{c}$ .

Proof. i) Let $\tilde{W} = {< x, (α_{I_{\tilde{w}} (x)}, β_{I_{\tilde{w}} (x)}) > | x \in O}$ and $\tilde{Z} = {< x, (α_{I_{\tilde{z}} (x)}, β_{I_{\tilde{z}} (x)}) > | x \in O}$ be two IVHIFSs over O, where $\begin{matrix} α_{I_{\tilde{w}} (x)} & = {a = [α_{a}^{-}, α_{a}^{+}] | a \in Int [0, 1]}, \\ β_{I_{\tilde{w}} (x)} & = {b = [β_{b}^{-}, β_{b}^{+}] | b \in Int [0, 1]}, \\ α_{I_{\tilde{z}} (x)} & = {c = [α_{c}^{-}, α_{c}^{+}] | c \in Int [0, 1]}, \\ β_{I_{\tilde{z}} (x)} & = {d = [β_{d}^{-}, β_{d}^{+}] | d \in Int [0, 1]} . \end{matrix}$ By definition, $\tilde{Y} = \tilde{W} \cap \tilde{Z} = {< x, (α_{I_{\tilde{y}} (x)}, β_{I_{\tilde{y}} (x)}) > | x \in O}$ with $\begin{matrix} α_{I_{\tilde{y}} (x)} = & {[\min {α_{a}^{-}, α_{c}^{-}}, \min {α_{a}^{+}, α_{c}^{+}}] | \\ a \in α_{I_{\tilde{w}} (x)}, c \in α_{I_{\tilde{z}} (x)}}, \\ β_{I_{\tilde{y}} (x)} = & {[\max {β_{b}^{-}, β_{d}^{-}}, \max {β_{b}^{+}, β_{d}^{+}}] | \\ b \in β_{I_{\tilde{w}} (x)}, d \in β_{I_{\tilde{z}} (x)}} . \end{matrix}$ Then, ${\tilde{Y}}^{c} = (\tilde{W} \cap \tilde{Z})^{c} = {< x, (β_{I_{\tilde{y}} (x)}, α_{I_{\tilde{y}} (x)}) > | x \in O}$ .

On the other hand,

$\tilde{H} = {\tilde{W}}^{c} \cap {\tilde{Z}}^{c} = {< x, (α_{I_{\tilde{h}} (x)}, β_{I_{\tilde{h}} (x)}) > | x \in O}$ with $\begin{matrix} α_{I_{\tilde{h}} (x)} = & {[\min {β_{b}^{-}, β_{d}^{-}}, \min {β_{b}^{+}, β_{d}^{+}}] | \\ b \in β_{I_{\tilde{w}} (x)}, d \in β_{I_{\tilde{z}} (x)}}, \\ β_{I_{\tilde{h}} (x)} = & {[\max {α_{a}^{-}, α_{c}^{-}}, \max {α_{a}^{+}, α_{c}^{+}}] | \\ a \in α_{I_{\tilde{w}} (x)}, c \in α_{I_{\tilde{z}} (x)}} . \end{matrix}$ Since $\tilde{W} \tilde{\subset} \tilde{Z}$ , by Definition 3.8, $\begin{matrix} \max {α_{a}^{+}} \leq \min {α_{c}^{-}}, \\ \min {β_{b}^{-}} \geq \max {β_{d}^{+}} . \end{matrix}$ For any b and d, $β_{d}^{-} \leq \max {β_{d}^{+}} \leq \min {β_{b}^{-}} \leq β_{b}^{+}$ implies $\min {β_{b}^{+}, β_{d}^{+}} = β_{d}^{+}$ and $\max {β_{b}^{-}, β_{d}^{-}} = β_{b}^{-}$ . Therefore, $\begin{matrix} \max {\min {β_{b}^{+}, β_{d}^{+}}} = \max {β_{d}^{+}} \leq \\ \min {β_{b}^{-}} = \min {\max {β_{b}^{-}, β_{d}^{-}}} . \end{matrix}$ (1)

On the other hand, for any a and c, $α_{a}^{-} \leq \max {α_{a}^{+}} \leq \min {α_{c}^{-}} \leq α_{c}^{+}$ implies $\max {α_{a}^{-}, α_{c}^{-}} = α_{c}^{-}$ and $\min {α_{a}^{+}, α_{c}^{+}} = α_{a}^{+}$ . Therefore, $\begin{matrix} \min {\max {α_{a}^{-}, α_{c}^{-}}} = \min {α_{c}^{-}} \geq \\ \max {α_{a}^{+}} = \max {\min {α_{a}^{+}, α_{c}^{+}}} . \end{matrix}$ (2) Because of (1) and (2), then Proposition 3.10 i) is proved.

The proof of Proposition 3.10 ii) is similar. □

Refer to Definition 3.4; we obtain the following properties in IVHIFEs.

Proposition 3.11. Let $\tilde{I}, {\tilde{I}}_{1}, {\tilde{I}}_{2}, {\tilde{I}}_{3}$ be IVHIFEs on w ∈ O. Then, we obtain the following properties.

$((\tilde{I})^{c})^{c} = \tilde{I}$ .

${\tilde{I}}_{1} \cup {\tilde{I}}_{2} = {\tilde{I}}_{2} \cup {\tilde{I}}_{1}$ .

${\tilde{I}}_{1} \cap {\tilde{I}}_{2} = {\tilde{I}}_{2} \cap {\tilde{I}}_{1}$ .

$({\tilde{I}}_{1} \cup {\tilde{I}}_{2})^{c} = {\tilde{I}}_{1}^{c} \cap {\tilde{I}}_{2}^{c}$ .

$({\tilde{I}}_{1} \cap {\tilde{I}}_{2})^{c} = {\tilde{I}}_{1}^{c} \cup {\tilde{I}}_{2}^{c}$ .

${\tilde{I}}_{1} \cap ({\tilde{I}}_{2} \cap {\tilde{I}}_{3}) = ({\tilde{I}}_{1} \cap {\tilde{I}}_{2}) \cap {\tilde{I}}_{3}$ .

${\tilde{I}}_{1} \cup ({\tilde{I}}_{2} \cup {\tilde{I}}_{3}) = ({\tilde{I}}_{1} \cup {\tilde{I}}_{2}) \cup {\tilde{I}}_{3}$ .

${\tilde{I}}_{1} \cup ({\tilde{I}}_{2} \cap {\tilde{I}}_{3}) = ({\tilde{I}}_{1} \cup {\tilde{I}}_{2}) \cap ({\tilde{I}}_{1} \cup {\tilde{I}}_{3})$ .

${\tilde{I}}_{1} \cap ({\tilde{I}}_{2} \cup {\tilde{I}}_{3}) = ({\tilde{I}}_{1} \cap {\tilde{I}}_{2}) \cup ({\tilde{I}}_{1} \cap {\tilde{I}}_{3})$ .

$({\tilde{I}}_{1} \oplus {\tilde{I}}_{2})^{c} = {\tilde{I}}_{1}^{c} \otimes {\tilde{I}}_{2}^{c}$ .

$({\tilde{I}}_{1} \otimes {\tilde{I}}_{2})^{c} = {\tilde{I}}_{1}^{c} \oplus {\tilde{I}}_{2}^{c}$ .

$({\tilde{I}}^{c})^{t} = (t \tilde{I})^{c}$ , t > 0.

$t ({\tilde{I}}^{c}) = ({\tilde{I}}^{t})^{c}$ , t > 0.

Proof. By using Definition 3.4 (1), then i) is proved. ii), iii), vi) and vii) hold, because operations ∨ and ∧ satisfy commutative and associative laws. Using Definition 3.4 (1), (2) and (3), then iv) and v) are proved. Since max {a, min {b, c}} = min {max {a, b} , max {a, c}} and min {a, max {b, c}} = max {min {a, b} , min {a, c}}, then viii) and ix) hold. Using Definition 3.4 (1), (4) and (5), then x) and xi) are proved. xii) and xiii) hold because of Definition 3.4 (1), (6) and (7). □

As a consequence of Proposition 3.11 and by using Definition 3.6, we obtain the following corollary.

Corollary 3.12. Given IVHIFSs $\tilde{W} = {< w, I_{\tilde{w}} (w) > | w \in O}$ , ${\tilde{W}}_{i} = {< w, I_{{\tilde{w}}_{i}} (w) > | w \in O}$ i = 1, 2, 3 on O. Then, the following properties hold.

$((\tilde{W})^{c})^{c} = \tilde{W}$ .

${\tilde{W}}_{1} \cup {\tilde{W}}_{2} = {\tilde{W}}_{2} \cup {\tilde{W}}_{1}$ .

${\tilde{W}}_{1} \cap {\tilde{W}}_{2} = {\tilde{W}}_{2} \cap {\tilde{W}}_{1}$ .

$({\tilde{W}}_{1} \cup {\tilde{W}}_{2})^{c} = {\tilde{W}}_{1}^{c} \cap {\tilde{W}}_{2}^{c}$ .

$({\tilde{W}}_{1} \cap {\tilde{W}}_{2})^{c} = {\tilde{W}}_{1}^{c} \cup {\tilde{W}}_{2}^{c}$ .

${\tilde{W}}_{1} \cap ({\tilde{W}}_{2} \cap {\tilde{W}}_{3}) = ({\tilde{W}}_{1} \cap {\tilde{W}}_{2}) \cap {\tilde{W}}_{3}$ .

${\tilde{W}}_{1} \cup ({\tilde{W}}_{2} \cup {\tilde{W}}_{3}) = ({\tilde{W}}_{1} \cup {\tilde{W}}_{2}) \cup {\tilde{W}}_{3}$ .

${\tilde{W}}_{1} \cup ({\tilde{W}}_{2} \cap {\tilde{W}}_{3}) = ({\tilde{W}}_{1} \cup {\tilde{W}}_{2}) \cap ({\tilde{W}}_{1} \cup {\tilde{W}}_{3})$ .

${\tilde{W}}_{1} \cap ({\tilde{W}}_{2} \cup {\tilde{W}}_{3}) = ({\tilde{W}}_{1} \cap {\tilde{W}}_{2}) \cup ({\tilde{W}}_{1} \cap {\tilde{W}}_{3})$ .

$({\tilde{W}}_{1} \oplus {\tilde{W}}_{2})^{c} = {\tilde{W}}_{1}^{c} \otimes {\tilde{W}}_{2}^{c}$ .

$({\tilde{W}}_{1} \otimes {\tilde{W}}_{2})^{c} = {\tilde{W}}_{1}^{c} \oplus {\tilde{W}}_{2}^{c}$ .

$({\tilde{W}}^{c})^{t} = (t \tilde{W})^{c}$ , t > 0.

$t ({\tilde{W}}^{c}) = ({\tilde{W}}^{t})^{c}$ , t > 0.

4 Generalized interval-valued hesitant intuitionistic fuzzy soft sets

The study on Generalized Interval-valued Hesitant Intuitionistic Fuzzy Soft Sets (GIVHIFSSs) is significantly related to the concept of IVHIFSs because a GIVHIFSS is a combination of IVHIFS, FSS, and FS. In this section, definitions and operations related to GIVHIFSSs are defined, and proofs of the related result are given.

Definition 4.1. Suppose that $\tilde{IV} (O)$ is the collection of all IVHIFSs on a set of objects O, E is a set of parameters, ψ is an FS over a parameter set P ⊂ E and $\tilde{W} : P \to \tilde{IV} (O)$ is a map. A pair $({\hat{W}}_{ψ}, P)$ is called a Generalized Interval-valued Hesitant Intuitionistic Fuzzy Soft Set (GIVHIFSS) over O, where ${\hat{W}}_{ψ} : P \to \tilde{IV} (O) \times [0, 1]$ , defined by ${\hat{W}}_{ψ} (p) = (\tilde{W} (p), ψ (p)) = ({< w, I_{\tilde{W} (p)} (w) > | w \in O}, ψ (p)) = ({< w, (α_{I_{\tilde{W} (p)} (w)}, β_{I_{\tilde{W} (p)} (w)}) > | w \in O}, ψ (p))$ $\in \tilde{IV} (O) \times [0, 1]$ .

Here, we write $({\hat{W}}_{ψ}, P) = {< p, \tilde{W} (p), ψ (p) > | p \in P}$ with $\tilde{W} (p)$ is an IVHIFS corresponding to a parameter p. The complement of a GIVHIFSS is defined by $({\hat{W}}_{ψ}, P)^{c} = {< p, {\tilde{W}}^{c} (p), ψ^{c} (p) > | p \in P}$ .

A GIVHIFSS $({\hat{W}}_{ψ}, P)$ can be expressed in the Tabular Representation, Table 1.

Table 1
Tabular Representation of a GIVHIFSS

ψ(p₁) … ψ(p_j) … ψ(p_m)

w ₁ ( $α_{I_{\tilde{W} (p_{1})} (w_{1})}, β_{I_{\tilde{W} (p_{1})} (w_{1})}$ ) … ( $α_{I_{\tilde{W} (p_{j})} (w_{1})}, β_{I_{\tilde{W} (p_{j})} (w_{1})}$ ) ( $α_{I_{\tilde{W} (p_{m})} (w_{1})}, β_{I_{\tilde{W} (p_{m})} (w_{1})}$ )

… … … … … …

w _i ( $α_{I_{\tilde{W} (p_{1})} (w_{i})}, β_{I_{\tilde{W} (p_{1})} (w_{i})}$ ) … ( $α_{I_{\tilde{W} (p_{j})} (w_{i})}, β_{I_{\tilde{W} (p_{j})} (w_{i})}$ ) ( $α_{I_{\tilde{W} (p_{m})} (w_{i})}, β_{I_{\tilde{W} (p_{m})} (w_{i})}$ )

… … … … … …

w _n ( $α_{I_{\tilde{W} (p_{1})} (w_{n})}, β_{I_{\tilde{W} (p_{1})} (w_{n})}$ ) … ( $α_{I_{\tilde{W} (p_{j})} (w_{n})}, β_{I_{\tilde{W} (p_{j})} (w_{n})}$ ) ( $α_{I_{\tilde{W} (p_{m})} (w_{n})}, β_{I_{\tilde{W} (p_{m})} (w_{n})}$ )

Example 4.2. Let O = {x₁, x₂} be a set of objects and E = {e₁, e₂, e₃} is a set of parameters. Then, it is defined a Fuzzy Set over P = {e₁, e₂}, Fs = {(e₁, ψ(e₁)) , (e₂, ψ(e₂))} with ψ(e₁) =0.3 and ψ(e₂) =0.7.

It is given two IVHIFSs $\tilde{W}$ and $\tilde{Z}$ over O as in Example 3.9. $\begin{matrix} \tilde{W} & = {< x_{i}, I_{\tilde{w}} (x_{i}) > | x_{i} \in O} where \\ I_{\tilde{w}} (x_{1}) & = (α_{I_{\tilde{w}} (x_{1})}, β_{I_{\tilde{w}} (x_{1})}) \\ = ({, [0.25, 0.4]}, \\ {[0.3, 0.55], [0.40, 0.5]}), \\ I_{\tilde{w}} (x_{2}) & = (α_{I_{\tilde{w}} (x_{2})}, β_{I_{\tilde{w}} (x_{2})}) \\ = ({[0.1, 0.20], [0.25, 0.30]}, \\ {[0.5, 0.68], [0.40, 0.65]}) . \end{matrix}$ $\begin{matrix} \tilde{Z} & = {< x_{i}, I_{\tilde{z}} (x_{i}) > | x_{i} \in O} where \\ I_{\tilde{z}} (x_{1}) & = (α_{I_{\tilde{z}} (x_{1})}, β_{I_{\tilde{z}} (x_{1})}) \\ = ({[0.25, 0.20], [0.20, 0.45]}, \\ {[0.35, 0.50], [0.45, 0.50]}), \\ I_{\tilde{z}} (x_{2}) & = (α_{I_{\tilde{z}} (x_{2})}, β_{I_{\tilde{z}} (x_{2})}) \\ = ({[0.15, 0.20], [0.20, 0.35]}, \\ {[0.55, 0.60], [0.45, 0.65]}) . \end{matrix}$

We can define a GIVHIFSS $({\hat{W}}_{ψ}, P)$ over O, where ${\hat{W}}_{ψ} (e_{1}) = (\tilde{W}, 0.3)$ and ${\hat{W}}_{ψ} (e_{2}) = (\tilde{Z}, 0.7)$ .

Now, we introduce some operations such as union, intersection, particular elements of GIVHIFSSs, and a subset of a GIVHIFSS.

Definition 4.3. Let $({\hat{W}}_{ψ}, P)$ and $({\hat{Z}}_{ξ}, Q)$ be two GIVHIFSSs over O. The union of such two sets is a GIVHIFSS, denoted by $({\hat{W}}_{ψ}, P) \hat{\cup} ({\hat{Z}}_{ξ}, Q) = : ({\hat{T}}_{τ}, R)$ , where R = P ∪ Q, and ∀ r ∈ R, $\begin{matrix} \tilde{T} (r) = {\begin{matrix} \tilde{W} (r) & if r \in P - Q, \\ \tilde{Z} (r) & if r \in Q - P, \\ \tilde{W} (r) \cup \tilde{Z} (r) & if r \in P \cap Q, \end{matrix} \\ τ (r) = {\begin{matrix} ψ (r) & if r \in P - Q, \\ ξ (r) & if r \in Q - P, \\ ψ (r) \lor ξ (r) & if r \in P \cap Q . \end{matrix} \end{matrix}$

Definition 4.4. Let $({\hat{W}}_{ψ}, P)$ and $({\hat{Z}}_{ξ}, Q)$ be two GIVHIFSSs over O. The intersection of such two sets is a GIVHIFSS, denoted by $({\hat{W}}_{ψ}, P) \hat{\cap} ({\hat{Z}}_{ξ}, Q) = : ({\hat{T}}_{τ}, R)$ , where R = P∩ Q ≠ ∅, and ∀ r ∈ R, $\tilde{T} (r) = \tilde{W} (r) \cap \tilde{Z} (r)$ and τ(r) = ψ(r) ∧ ξ(r) .

Given a GIVHIFSS $({\hat{W}}_{ψ}, P) = {< p, \tilde{W} (p), ψ (p) > | p \in P}$ . The null GIVHIFSS is denoted by $({\hat{\emptyset}}_{\emptyset}, A)$ , if $\tilde{W} (p) : = \tilde{\emptyset}$ , ψ(p) : =0 and the universal GIVHIFSS is denoted by $({\hat{1}}_{1}, P)$ , if $\tilde{W} (p) : = \tilde{1}$ , ψ(p) : =1.

Here, two IVHIFSs $\tilde{\emptyset} : = {< x, ({0}, {1}) > | x \in O}$ and $\tilde{1} : = {< x, ({1}, {0}) > | x \in O}$ are called the null IVHIFS and the universal IVHIFS respectively. Note that {0} = {[0, 0]} and {1} = {[1, 1]}.

Definition 4.5. Let $({\hat{W}}_{ψ}, P)$ and $({\hat{Z}}_{ξ}, Q)$ be two GIVHIFSSs over O, with ${\hat{W}}_{ψ} (p) = (\tilde{W} (p), ψ (p))$ and ${\hat{Z}}_{ξ} (q) = (\tilde{Z} (q), ξ (q))$ . $({\hat{W}}_{ψ}, P)$ is called a Generalized Interval-valued Hesitant Intuitionistic Fuzzy subset of $({\hat{Z}}_{ξ}, Q)$ , denoted by $({\hat{W}}_{ψ}, P) \hat{\subset} ({\hat{Z}}_{ξ}, Q)$ , if:

P ⊆ Q,

ψ(p) ≤ ξ(p) ∀ p ∈ P,

$\tilde{W} \tilde{\subset} \tilde{Z}$ .

As a consequence of Definitions 4.3, 4.4, 4.5, and Proposition 3.10, we obtain the following corollary.

Corollary 4.6. Let $({\hat{W}}_{ψ}, P)$ and $({\hat{Z}}_{ξ}, Q)$ be two GIVHIFSSs over O. Suppose that $({\hat{W}}_{ψ}, P) \hat{\subset} ({\hat{Z}}_{ξ}, Q)$ . Then

$({\hat{W}}_{ψ}, P)^{c} \hat{\cap} ({\hat{Z}}_{ξ}, Q)^{c}$ $\hat{\subset}$ $(({\hat{W}}_{ψ}, P) \hat{\cap} ({\hat{Z}}_{ξ}, Q))^{c}$ ,

$(({\hat{W}}_{ψ}, P) \hat{\cup} ({\hat{Z}}_{ξ}, Q))^{c}$ $\hat{\subset}$ $({\hat{W}}_{ψ}, P)^{c} \hat{\cup} ({\hat{Z}}_{ξ}, Q)^{c}$ .

Theorem 4.7. Let $({\hat{W}}_{ψ}, P)$ and $({\hat{Z}}_{ξ}, P)$ be two GIVHIFSSs over O. Then the following De Morgan’s laws hold.

$({\hat{W}}_{ψ}, P) \hat{\cup} ({\hat{Z}}_{ξ}, P))^{c} = ({\hat{W}}_{ψ}, P)^{c} \hat{\cap} ({\hat{Z}}_{ξ}, P)^{c}$ .

$(({\hat{W}}_{ψ}, P) \hat{\cap} ({\hat{Z}}_{ξ}, P))^{c} = ({\hat{W}}_{ψ}, P)^{c} \hat{\cup} ({\hat{Z}}_{ξ}, P)^{c}$ .

Proof. Here, we give the proof of ii). For i) is similar.

Let $(({\hat{W}}_{ψ}, P) \hat{\cap} ({\hat{Z}}_{ξ}, P))^{c}$ = $({\hat{T}}_{γ}, P)^{c}$ = ${< p, {\tilde{T}}^{c} (p), γ^{c} (p) > | p \in P}$ where ∀p ∈ P, $\tilde{T} (p) = \tilde{W} (p) \cap \tilde{Z} (p)$ and γ(p) = ψ(p) ∧ ξ(p). By using Proposition 3.11 v) and the definition of the complement of a FS, ${\tilde{T}}^{c} (p) = {\tilde{W}}^{c} (p) \cup {\tilde{Z}}^{c} (p)$ and γ^c(p) =1 - γ(p) = ψ^c(p) ∨ β^c(p). Remember that 1 - min {ψ(p) , ξ(p)}=max {1 - ψ(p) , 1 - ξ(p)}. Hence Theorem 4.7 ii) is proved. □

Theorem 4.8. Let $({\hat{W}}_{ψ}, P)$ , $({\hat{Z}}_{ξ}, Q)$ and $({\hat{G}}_{γ}, R)$ be three GIVHIFSSs over O. We obtain the following associative laws.

$(({\hat{W}}_{ψ}, P) \hat{\cup} ({\hat{Z}}_{ξ}, Q)) \hat{\cup} ({\hat{G}}_{γ}, R)$ = $({\hat{W}}_{ψ}, P) \hat{\cup} (({\hat{Z}}_{ξ}, Q) \hat{\cup} ({\hat{G}}_{γ}, R))$ .

$(({\hat{W}}_{ψ}, P) \hat{\cap} ({\hat{Z}}_{ξ}, Q)) \hat{\cap} ({\hat{G}}_{γ}, R)$ = $({\hat{W}}_{ψ}, P) \hat{\cap} (({\hat{Z}}_{ξ}, Q) \hat{\cap} ({\hat{G}}_{γ}, R))$ .

Proof. By Definition 4.3 and 4.4, Proposition 2.1 f) and g), and Proposition 3.11 vi) and vii), the theorem is proved. □

Let us consider the other operations on GIVHIFSSs denoted by "∨" and "∧."

Definition 4.9. Let $({\hat{W}}_{ψ}, P)$ and $({\hat{Z}}_{ξ}, Q)$ be two GIVHIFSSs over O. The operations "∧" and "∨" on such GIVHIFSSs are defined as follows.

$({\hat{W}}_{ψ}, P) \land ({\hat{Z}}_{ξ}, Q) : = ({\hat{T}}_{γ}, P \times Q)$ where ∀(p, q) ∈ P × Q, ${\hat{T}}_{γ} : P \times Q \to \tilde{IV} (O) \times [0, 1]$ and γ: P × Q → [0, 1] which are defined by $\tilde{T} (p, q) : = \tilde{W} (p) \cap \tilde{Z} (q)$ , γ(p, q) = ψ(p) ∧ ξ(q) .

$({\hat{W}}_{ψ}, P) \lor ({\hat{Z}}_{ξ}, Q) : = ({\hat{N}}_{β}, P \times Q)$ , ∀(p, q) ∈ P × Q, ${\hat{N}}_{β} : P \times Q \to \tilde{IV} (O) \times [0, 1]$ and β: P × Q → [0, 1] which are defined by $\tilde{N} (p, q) : = \tilde{W} (p) \cup \tilde{Z} (q)$ , β(p, q) = ψ(p) ∨ ξ(q) .

As a consequence of the above definition, we obtain some theorems as follows.

Theorem 4.10. Let $({\hat{W}}_{ψ}, P)$ and $({\hat{Z}}_{ξ}, Q)$ be two GIVHIFSSs over O. The following De Morgan laws hold.

$(({\hat{W}}_{ψ}, P) \land ({\hat{Z}}_{ξ}, Q))^{c} = ({\hat{W}}_{ψ}, P)^{c} \lor ({\hat{Z}}_{ξ}, Q)^{c}$ .

$(({\hat{W}}_{ψ}, P) \lor ({\hat{Z}}_{ξ}, Q))^{c} = ({\hat{W}}_{ψ}, P)^{c} \land ({\hat{Z}}_{ξ}, Q)^{c}$ .

Proof. Let us prove ii). For i) is similar. Consider $(({\hat{W}}_{ψ}, P) \lor ({\hat{Z}}_{ξ}, Q))^{c} = ({\hat{T}}_{γ}, P \times Q)^{c}$ = ${< (p, q), {\tilde{T}}^{c} (p, q), γ^{c} (p, q) >} | (p, q) \in P \times Q}$ , with ${\tilde{T}}^{c} (p, q) = (\tilde{W} (p) \cup \tilde{Z} (q))^{c} = {\tilde{W}}^{c} (p) \cap {\tilde{Z}}^{c} (q)$ , and γ^c(p, q) =(ψ(p) ∨ ξ(q)) ^c = ψ^c(p) ∧ ξ^c(q) . By definition, Theorem 4.7 ii) is proved. □

Theorem 4.11. Let $({\hat{W}}_{ψ}, P)$ , $({\hat{Z}}_{ξ}, Q)$ and $({\hat{G}}_{γ}, R)$ be three GIVHIFSSs over O. Then we obtain the following associative laws.

$(({\hat{W}}_{ψ}, P) \land ({\hat{Z}}_{ξ}, Q)) \land ({\hat{G}}_{γ}, R)$ = $({\hat{W}}_{ψ}, P) \land (({\hat{Z}}_{ξ}, Q) \land ({\hat{G}}_{γ}, R))$ .

$(({\hat{W}}_{ψ}, P) \lor ({\hat{Z}}_{ξ}, Q)) \lor ({\hat{G}}_{γ}, R)$ = $({\hat{W}}_{ψ}, P) \lor (({\hat{Z}}_{ξ}, Q) \lor ({\hat{G}}_{γ}, R))$ .

Proof. We prove ii). To prove i) is similar with ii).

By Definition 4.9 and Proposition 3.11 vii), $({\hat{W}}_{ψ}, P) \lor (({\hat{Z}}_{ξ}, Q) \lor ({\hat{G}}_{γ}, R))$ = $({\hat{W}}_{ψ}, P) \lor ({\hat{T}}_{α}, Q \times R)$ = $({\hat{N}}_{β}, P \times Q \times R)$ where ∀(q, r) ∈ Q × R, $\tilde{T} (q, r) = \tilde{Z} (q) \cup \tilde{G} (r)$ , α(q, r) = ξ(q) ∨ γ(r) and ∀(p, q, r) ∈ P × Q × R, $\tilde{N} (p, q, r) = \tilde{W} (p) \cup \tilde{T} (q, r)$ = $\tilde{W} (p) \cup (\tilde{Z} (q) \cup \tilde{G} (r))$ = $(\tilde{W} (p) \cup \tilde{Z} (q)) \cup \tilde{G} (r)$ : = $\tilde{M} (p, q) \cup \tilde{G} (r)$ with $\tilde{M} (p, q) = \tilde{W} (p) \cup \tilde{Z} (q)$ , β(p, q, r) = ψ(p) ∨ α(q, r)=ψ(p) ∨(ξ(q) ∨ γ(r))= (ψ(p) ∨ ξ(q)) ∨ γ(r): =τ(p, q) ∨ γ(r) with τ(p, q) = ψ(p) ∨ ξ(q). Now consider that $({\hat{M}}_{τ}, P \times Q) \lor ({\hat{G}}_{γ}, R)$ = $(({\hat{W}}_{ψ}, P) \lor ({\hat{Z}}_{ξ}, Q)) \lor ({\hat{G}}_{γ}, R)$ . Hence the theorem is proved. □

Finally, when we consider the collection of all GIVHIFSSs over O, we get the following corollary as an algebraic structure of such collection.

Corollary 4.12. Let $\tilde{GI}$ be the collection of all GIVHIFSSs $({\hat{W}}_{ψ}, P)$ over O for any ψ. Then the following hold.

$\tilde{GI}$ is closed under operations $\hat{\cap}$ and $\hat{\cup}$ .

$\tilde{GI}$ satisfies the associative law under operations $\hat{\cap}$ and $\hat{\cup}$ .

There exist a GIVHIFSS $({\hat{1}}_{1}, P)$ such that $({\hat{W}}_{ψ}, P) \hat{\cap} ({\hat{1}}_{1}, P)$ = $({\hat{W}}_{ψ}, P)$ $\forall ({\hat{W}}_{α}, P) \in \tilde{GI}$ .

There exist a GIVHIFSS $({\hat{\emptyset}}_{\emptyset}, P)$ such that $({\hat{W}}_{ψ}, P) \hat{\cup} ({\hat{\emptyset}}_{\emptyset}, P) = ({\hat{W}}_{ψ}, P)$ , $\forall ({\hat{W}}_{α}, P) \in \tilde{GI}$ .

Hence,

\tilde{GI}

is a monoid (semigroup with the identity element) over binary operations

\hat{\cap}

and

\hat{\cup}

5 Application to decision-making with generalized interval-valued hesitant intuitionistic fuzzy soft information

In this section, we present an application of GIVHIFSSs in the decision-making problem to find out the best superior variety of rice seeds for a farmer based on his/her evaluation. For this, we will use a method introduced by Roy and Maji [21], namely, using a comparison table.

First of all, we introduce some definitions which are helpful in this section.

Definition 5.1. Given a GIVHIFSS $({\hat{W}}_{ψ}, P)$ over O where for p ∈ P, ${\hat{W}}_{ψ} (p) = (\tilde{W} (p), ψ (p))$ , $\tilde{W} (p) = {< w, I_{\tilde{W} (p)} (w) > | w \in O} =$ ${< w, (α_{I_{\tilde{W} (p)} (w)}, β_{I_{\tilde{W} (p)} (w)}) > | w \in O}$ , $α_{I_{\tilde{W} (p)} (w)} = {a = [α_{a}^{-}, α_{a}^{+}] | a \in Int [0, 1]}$ and $β_{I_{\tilde{W} (p)} (w)} = {b = [β_{b}^{-}, β_{b}^{+}] | b \in Int [0, 1]}$ . We define a function $AV (I_{\tilde{W} (p)} (w)) = ([\frac{\sum_{a \in α_{I_{\tilde{W} (p)} (w)}} α_{a}^{-}}{| α_{I_{\tilde{W} (p)} (w)} |}, \frac{\sum_{a \in α_{I_{\tilde{W} (p)} (w)}} α_{a}^{+}}{| α_{I_{\tilde{W} (p)} (w)} |}],$ $[\frac{\sum_{b \in β_{I_{\tilde{W} (p)} (w)}} β_{b}^{-}}{| β_{I_{\tilde{W} (p)} (w)} |}, \frac{\sum_{b \in β_{I_{\tilde{W} (p)} (w)}} β_{b}^{+}}{| β_{I_{\tilde{W} (p)} (w)} |}])$ called the average intuitionistic function of $I_{\tilde{W} (p)} (w)$ .

While we apply Definition 5.1 to a GIVHIFSS $({\hat{W}}_{ψ}, P)$ over O above, then the corresponding $\tilde{W} (p) = {< w, I_{\tilde{W} (p)} (w) > | w \in O}$ reduced to an IVIFS over O as in Definition 3.1.

Definition 5.2. Given a GIVHIFSS $({\hat{W}}_{ψ}, P)$ over O and function AV as in Definition 5.1. Defined a pair $({\hat{\hat{W}}}_{ψ}, P)$ over O called a Reduced Generalized Interval-valued Intuitionistic Fuzzy Soft set (RGIVIFSS), where for p ∈ P, ${\hat{\hat{W}}}_{ψ} (p) = (\tilde{\tilde{W}} (p), ψ (p))$ , $\tilde{\tilde{W}} (p) = {< w, AV (I_{\tilde{W} (p)} (w)) > | w \in O}$ .

Definition 5.3. Given two $AV (I_{\tilde{W} (p)} (w_{1})) = ([\frac{\sum_{a \in α_{I_{\tilde{W} (p)} (w_{1})}} α_{a}^{-}}{| α_{I_{\tilde{W} (p)} (w_{1})} |}, \frac{\sum_{a \in α_{I_{\tilde{W} (p)} (w_{1})}} α_{a}^{+}}{| α_{I_{\tilde{W} (p)} (w_{1})} |}],$ $[\frac{\sum_{b \in β_{I_{\tilde{W} (p)} (w_{1})}} β_{b}^{-}}{| β_{I_{\tilde{W} (p)} (w_{1})} |}, \frac{\sum_{b \in β_{I_{\tilde{W} (p)} (w_{1})}} β_{b}^{+}}{| β_{I_{\tilde{W} (p)} (w_{1})} |}])$ and $AV (I_{\tilde{W} (p)} (w_{2})) = ([\frac{\sum_{a \in α_{I_{\tilde{W} (p)} (w_{2})}} α_{a}^{-}}{| α_{I_{\tilde{W} (p)} (w_{2})} |}, \frac{\sum_{a \in α_{I_{\tilde{W} (p)} (w_{2})}} α_{a}^{+}}{| α_{I_{\tilde{W} (p)} (w_{2})} |}],$ $[\frac{\sum_{b \in β_{I_{\tilde{W} (p)} (w_{2})}} β_{b}^{-}}{| β_{I_{\tilde{W} (p)} (w_{2})} |}, \frac{\sum_{b \in β_{I_{\tilde{W} (p)} (w_{2})}} β_{b}^{+}}{| β_{I_{\tilde{W} (p)} (w_{2})} |}])$ .

$AV (I_{\tilde{W} (p)} (w_{1}))$ is greater than or equal to $AV (I_{\tilde{W} (p)} (w_{2}))$ , denoted by $AV (I_{\tilde{W} (p)} (w_{1}))$ ⋟ $AV (I_{\tilde{W} (p)} (w_{2}))$ , if $\frac{\sum_{a \in α_{I_{\tilde{W} (p)} (w_{1})}} α_{a}^{-}}{| α_{I_{\tilde{W} (p)} (w_{1})} |}$ ≥ $\frac{\sum_{a \in α_{I_{\tilde{W} (p)} (w_{2})}} α_{a}^{+}}{| α_{I_{\tilde{W} (p)} (w_{2})} |}$ and $\frac{\sum_{b \in β_{I_{\tilde{W} (p)} (w_{1})}} β_{b}^{+}}{| β_{I_{\tilde{W} (p)} (w_{1})} |}$ ≤ $\frac{\sum_{b \in β_{I_{\tilde{W} (p)} (w_{2})}} β_{b}^{-}}{| β_{I_{\tilde{W} (p)} (w_{2})} |} .$

Revert to the definition of comparison table in [21]; we define a similar table according to our an RGIVIFSS, called R-comparison table.

Definition 5.4. (R-comparison table) An R-comparison table is a square table with the same number of rows and columns, and both labeled by the objects w_i ∈ O. An R-Comparison table is obtained from an RGIVIFSS as in Definition 5.2 in which the entries are x_ij where x_ij = ∑_p∈Pψ(p) * c with $c = {\begin{matrix} 1 if AV (I_{\tilde{W} (p)} (w_{i})) ⋟ AV (I_{\tilde{W} (p)} (w_{j})), \\ 0 else . \end{matrix}$ Here, x_ij is related to objects w_i and w_j.

To apply Generalized Interval-valued Hesitant Intuitionistic Fuzzy Soft information in the decision-making problem, we may apply the following algorithm.

Algorithm:

Input a GIVHIFSS in a Tabular Representation.

Consider the RGIVIFSS obtained from the GIVHIFSS in the Tabular Representation.

Compute the R-comparison table of the RGIVIFSS.

Compute the score of the i-th object w_i, namely, sc(w_i) = ∑_jx_ij - ∑_kx_ki.

If the maximum score occurs in the t-th object, such an object is the best choice.

If the maximum score has more than one value, then one of the corresponding objects may be chosen.

Example 5.5. Suppose that a farmer wishes to find out the most appropriate superior variety of rice seed. Let O = {w₁ = Umbul - umbul, w₂ = IR64, w₃ = Mentik Wangi, w₄ = Sandarum, w₅ = Ulo} be a set of superior varieties of rice seeds. The farmer wishes to select based on some criteria as in the set of parameters P = {p₁ = low seed prices, p₂ = high production yield, p₃ =high selling price, p₄ =convenience in selling agricultural products}. Let the preference of the criteria for the farmer be described by the fuzzy set ψ: P → [01] over P as follows, ψ(p₁) =0.5, ψ(p₂) =0.7, ψ(p₃) =0.7 and ψ(p₄) =0.6.

After analyzing all parameters for each rice seed, the farmer presents his/her evaluation in the GIVHIFSS as in Table 2.

Table 2
Tabular Representation of the GIVHIFSS

ψ(p₁) ψ(p₂) ψ(p₃) ψ(p₄)

w ₁ ({, [0.5,0.6]}, ({ [0.65,0.7], [0.55,0.6]} ({ [0.5,0.6], [0.45,0.55]} ({ [0.45,0.6], [0.4,0.55]}

{[0.2,0.3], [0.25,0.3]}) {[0.2,0.25], [0.2,0.3]}) {[0.3,0.4], [0.1,0.25]}) {[0.35,0.4], [0.1,0.2]})

w ₂ ({ [0.5,0.55], [0.4,0.5]}, ({ [0.3,0.45], [0.35,0.5]} ({ [0.35,0.4], [0.25,0.35]} ({ [0.5,0.6], [0.6,0.7]}

{[0.4,0.5], [0.35,0.4]}) {[0.45,0.5], [0.35,0.45]}) {[0.5,0.6], [0.4,0.5]}) {[0.1,0.15], [0.15,0.2]})

w ₃ ({ [0.7,0.8], [0.6,0.7]}, ({ [0.7,0.75], [0.65,0.7]} ({ [0.55,0.65], [0.65,0.75]} ({ [0.5,0.6], [0.65,0.7]}

{[0.1,0.2], [0.15,0.2]}) {[0.1,0.25], [0.15,0.25]}) {[0.2,0.25], [0.1,0.25]}) {[0.25,0.3], [0.15,0.25]})

w ₄ ({ [0.4,0.5], [0.5,0.6]}, ({ [0.45,0.5], [0.55,0.6]} ({ [0.5,0.6], [0.6,0.7]} ({ [0.6,0.65], [0.55,0.6]}

{[0.3,0.35], [0.25,0.4]}) {[0.2,0.35], [0.25,0.3]}) {[0.1,0.25], [0.25,0.3]}) {[0.1,0.2], [0.2,0.25]})

w ₅ ({ [0.5,0.6], [0.4,0.55]}, ({ [0.55,0.6], [0.45,0.55]} ({ [0.4,0.6], [0.4,0.55]} ({ [0.6,0.7], [0.5,0.55]}

{[0.1,0.3], [0.2,0.35]}) {[0.2,0.3], [0.25,0.35]}) {[0.3,0.35], [0.35,0.4]}) {[0.25,0.35], [0.35,0.45]})

	ψ(p₁)	ψ(p₂)	ψ(p₃)	ψ(p₄)
w ₁	({, [0.5,0.6]},	({ [0.65,0.7], [0.55,0.6]}	({ [0.5,0.6], [0.45,0.55]}	({ [0.45,0.6], [0.4,0.55]}
	{[0.2,0.3], [0.25,0.3]})	{[0.2,0.25], [0.2,0.3]})	{[0.3,0.4], [0.1,0.25]})	{[0.35,0.4], [0.1,0.2]})
w ₂	({ [0.5,0.55], [0.4,0.5]},	({ [0.3,0.45], [0.35,0.5]}	({ [0.35,0.4], [0.25,0.35]}	({ [0.5,0.6], [0.6,0.7]}
	{[0.4,0.5], [0.35,0.4]})	{[0.45,0.5], [0.35,0.45]})	{[0.5,0.6], [0.4,0.5]})	{[0.1,0.15], [0.15,0.2]})
w ₃	({ [0.7,0.8], [0.6,0.7]},	({ [0.7,0.75], [0.65,0.7]}	({ [0.55,0.65], [0.65,0.75]}	({ [0.5,0.6], [0.65,0.7]}
	{[0.1,0.2], [0.15,0.2]})	{[0.1,0.25], [0.15,0.25]})	{[0.2,0.25], [0.1,0.25]})	{[0.25,0.3], [0.15,0.25]})
w ₄	({ [0.4,0.5], [0.5,0.6]},	({ [0.45,0.5], [0.55,0.6]}	({ [0.5,0.6], [0.6,0.7]}	({ [0.6,0.65], [0.55,0.6]}
	{[0.3,0.35], [0.25,0.4]})	{[0.2,0.35], [0.25,0.3]})	{[0.1,0.25], [0.25,0.3]})	{[0.1,0.2], [0.2,0.25]})
w ₅	({ [0.5,0.6], [0.4,0.55]},	({ [0.55,0.6], [0.45,0.55]}	({ [0.4,0.6], [0.4,0.55]}	({ [0.6,0.7], [0.5,0.55]}
	{[0.1,0.3], [0.2,0.35]})	{[0.2,0.3], [0.25,0.35]})	{[0.3,0.35], [0.35,0.4]})	{[0.25,0.35], [0.35,0.45]})

Then, by using Definition 5.2, we obtain the corresponding RGIVIFSS as in Table 3.

Table 3

Tabular Representation of the RGIVIFSS

	ψ(p₁)	ψ(p₂)	ψ(p₃)	ψ(p₄)
w ₁	([0.55,0.65], [0.23,0.3])	([0.6,0.65], [0.2,0.28])	([0.48,0.58], [0.2,0.33])	([0.43,0.58], [0.23,0.3])
w ₂	([0.45,0.53], [0.38,0.45])	([0.33,0.48], [0.4,0.48])	([0.3,0.38], [0.45,0.55])	([0.55,0.65], [0.13,0.18])
w ₃	([0.65,0.75], [0.13,0.2])	([0.68,0.73], [0.13,0.25])	([0.6,0.7], [0.15,0.25])	([0.58,0.65], [0.2,0.28])
w ₄	([0.45,0.55], [0.28,0.38])	([0.5,0.55], [0.23,0.33])	([0.55,0.65], [0.18,0.28])	([0.58,0.63], [0.15,0.23])
w ₅	([0.45,0.58], [0.15,0.33])	([0.5,0.58], [0.23,0.33])	([0.4,0.58], [0.33,0.38])	([0.55,0.63], [0.3,0.4])

In step (iii), the R-comparison table, Table 4, is obtained.

Table 4

R-comparison table

	w ₁	w ₂	w ₃	w ₄	w ₅
w ₁	5	1.9	0	0	0
w ₂	0	5	0	0	0
w ₃	0.5	1.9	5	0.5	0.7
w ₄	0.6	1.4	0	5	0
w ₅	0	1.4	0	0	5

Then we compute the score of the object w_i as in Table 5.

Table 5

Tabular score

	Row sum of w_i	Column sum of w_i	sc(w_i)
w ₁	6.9	6.1	0.8
w ₂	5	11.6	-6.6
w ₃	8.6	5	3.6
w ₄	7	5.5	1.5
w ₅	6.4	5.7	0.7

Finally, since the maximum score is sc(w₃) =3.6, then the most appropriate superior variety of rice seed is w₃.

6 Conclusion

This article has introduced the notion of Generalized Interval-valued Hesitant Intuitionistic Fuzzy Soft sets (GIVHIFSSs) with their operations. Based on these operations, we obtained some properties, such as commutative, associative, a subset, De Morgan’s laws, and an algebraic structure of GIVHIFSSs. By using the comparison table and a different approach, the application of this concept to solve an agriculture problem has been investigated. It is expected that with our new notion, the potential theoretical development will be open in future, such as distance and similar measures in GIVHIFSSs. Therefore, several realistic uncertain problems can be solved with more perfect results.

Footnotes

Acknowledgements

A research fund from Universitas Andalas supports this research under the contract of Professor’s acceleration research cluster scheme (Batch II), No. 14/UN.16.17/PP.PGB/LPPM/2018.

References

Atanassov

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96.

Atanassov

and Gargov

, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31 (1989), 343–349.

Babitha

K.V.

and John

S.J.

, Generalized intuitionistic fuzzy soft sets and its applications, Gen Math Notes 7(2) (2011), 1–14.

Dinda

, Bera

and Samanta

T.K.

, Generalised intuitionistic fuzzy soft sets and an adjustable approach to decision making, Annals of Fuzzy Mathematics and Informatics 4(2) (2012), 207–215.

Ding

and Wu

, An Improved Interval-Valued Hesitant Fuzzy Multi-Criteria Group Decision-Making Method and Applications, Mathematical and Computational Applications 21 (2016), 12.

Dugenci

, A new distance measure for interval valued intuitionistic fuzzy sets and its application to group decision making problems with incomplete weights information, Applied Soft Computing 41 (2016), 120–134.

Gorzalczany

M.B.

, A Method Of Inference In Approximate Reasoning Based On Interval-Valued Fuzzy Sets, Fuzzy Sets and Systems 21 (1987), 1–17.

Jiang

, Tang

, Chen

, Liu

and Tang

, Interval-valued intuitionistic fuzzy soft sets and their properties, Computers and Mathematics with Application 60 (2010), 906–918.

Maji

P.K.

, Biswas

and Roy

A.R.

, Fuzzy Soft Sets, Journal of Fuzzy Mathematics 9 (2001), 589–602.

10.

Maji

P.K.

, Biswas

and Roy

A.R.

, Intuitionistic fuzzy soft sets, The Journal of Fuzzy Mathematics 9 (2001), 677–693.

11.

Maji

P.K.

, Roy

A.R.

and Biswan

, An Aplication of Soft Sets in a decision making problems, Computer and Mathematics with Applications 44 (2002), 1070–1083.

12.

Maji

P.K.

, Biswas

and Roy

A.R.

, On intuitionistic fuzzy soft sets, The Journal of Fuzzy Mathematics 12 (2004), 669–683.

13.

Mizumoto

, Fuzzy Sets and Their Operations, Information and Control 48 (1981), 30–48.

14.

Molodtsov

, Soft set theory-first result, Computers and Mathematics with Applications 37 (1999), 19–31.

15.

Nazra

, Ideal and its Fuzzification in implicative semigroups, International Journal of Pure and Applied Mathematics 104(4) (2015), 543–549.

16.

Nazra

, Syafruddin

R.L.

and Wicaksono

G.C.

, Hesitant Intuitionistic Fuzzy Soft Sets, Journal of Physics: Conference Series 890 (2017), 012118.

17.

Nazra

, Syafruddin

G.C.W.

and Syafwan

, Generalized Hesitant Intuitionistic Fuzzy Soft Sets, Journal of Physics: Conference Series 983 (2018), 012127.

18.

Nazra

, Asdi

and Zulvera , Generalized intuitionistic fuzzy soft matrices and their application, Journal of Physics: Conference Series 1321(2) (2019), 022086.

19.

Peng

X.D.

and Yang

, Approaches to interval-valued intuitionistic hesitant fuzzy soft sets based decision making, Annals of Fuzzy Mathematics and Informatics 10(4) (2015), 657–680.

20.

Qian

, Wang

and Feng

X.Q.

, Generalized hesitant fuzzy sets and their application in decision support system, Knowledge-Based Systems 37 (2013), 357–365.

21.

Roy

A.R.

and Maji

P.K.

, A fuzzy soft set theoretic approach to decision-making problems, Journal of Computational and Applied Mathematics 203 (2007), 412–418.

22.

Sooraj

T.R.

, Mohanty

R.K.

and Tripathy

B.K.

, A New Approach to Interval-Valued Intuitionistic Hesitant Fuzzy Soft Sets and Their Application in Decision Making, Smart Computing and Informatics (2018), 243–253.

23.

Torra

and Narukawa

, On hesitant fuzzy sets and decision, In: Proc. of the 2009 IEEE Int. Conf. on Fuzzy Systems, Jeju Island, Korea (2009), 1378–1382.

24.

Wang

, Li

and Chen

, Hesitant Fuzzy Soft Set and Its Applications in Multicriteria Decision Making, Journal of Applied Mathematics (2014), Article ID 643785.

25.

Wang

, Li

and Chen

, Hesitant Fuzzy Soft Sets with Application in Multicriteria Group Decision Making Problems, The Scientific World Journal (2015), Article ID 806983.

26.

Zadeh

L.A.

, Fuzzy set, Information and Control 8 (1965), 338–353.

27.

Zhang

, Interval-Valued Intuitionistic Hesitant Fuzzy Aggregation Operators and Their Application in Group Decision- Making, Journal of Applied Mathematics (2013), Article ID 670285.

28.

Zhang

, Xiong

and Ma

, On Interval-Valued Hesitant Fuzzy Soft Sets, Mathematical Problems in Engineering (2015), 17.

	ψ(p₁)	…	ψ(p_j)	…	ψ(p_m)
w ₁	( $α_{I_{\tilde{W} (p_{1})} (w_{1})}, β_{I_{\tilde{W} (p_{1})} (w_{1})}$ )	…	( $α_{I_{\tilde{W} (p_{j})} (w_{1})}, β_{I_{\tilde{W} (p_{j})} (w_{1})}$ )		( $α_{I_{\tilde{W} (p_{m})} (w_{1})}, β_{I_{\tilde{W} (p_{m})} (w_{1})}$ )
…	…	…	…	…	…
w _i	( $α_{I_{\tilde{W} (p_{1})} (w_{i})}, β_{I_{\tilde{W} (p_{1})} (w_{i})}$ )	…	( $α_{I_{\tilde{W} (p_{j})} (w_{i})}, β_{I_{\tilde{W} (p_{j})} (w_{i})}$ )		( $α_{I_{\tilde{W} (p_{m})} (w_{i})}, β_{I_{\tilde{W} (p_{m})} (w_{i})}$ )
…	…	…	…	…	…
w _n	( $α_{I_{\tilde{W} (p_{1})} (w_{n})}, β_{I_{\tilde{W} (p_{1})} (w_{n})}$ )	…	( $α_{I_{\tilde{W} (p_{j})} (w_{n})}, β_{I_{\tilde{W} (p_{j})} (w_{n})}$ )		( $α_{I_{\tilde{W} (p_{m})} (w_{n})}, β_{I_{\tilde{W} (p_{m})} (w_{n})}$ )