Interval-Valued Intuitionistic Hesitant Fuzzy Aggregation Operators and Their Application in Group Decision-Making

Abstract

Hesitant fuzzy sets, permitting the membership of an element to be a set of several possible values, can be used as an efficient mathematical tool for modelling people’s hesitancy in daily life. In this paper, we extend the hesitant fuzzy set to interval-valued intuitionistic fuzzy environments and propose the concept of interval-valued intuitionistic hesitant fuzzy set, which allows the membership of an element to be a set of several possible interval-valued intuitionistic fuzzy numbers. The aim of this paper is to develop a series of aggregation operators for interval-valued intuitionistic hesitant fuzzy information. Then, some desired properties of the developed operators are studied, and the relationships among these operators are discussed. Furthermore, we apply these aggregation operators to develop an approach to multiple attribute group decision-making with interval-valued intuitionistic hesitant fuzzy information. Finally, a numerical example is provided to illustrate the application of the developed approach.

1. Introduction

In many practical problems, when defining the membership degree of an element, the difficulty of establishing the membership degree is not because we have a margin of error (as in intuitionistic fuzzy sets [1] and interval-valued fuzzy sets [2]) or some possibility distribution (as in type 2 fuzzy sets [3]) on the possibility values, but because we have several possible numerical values. To deal with such cases, Torra [4] introduced the concept of hesitant fuzzy set to permit the membership of an element to be a set of several possible values between 0 and 1, which can depict the human’s hesitance more objectively and precisely.

It should be noted that hesitant fuzzy sets permit the membership of an element to be a set of several possible values. All these possible values are crisp real numbers that belong to [0,1]. However, in the process of some practical decision-makings, sometimes, due to the time pressure and lack of knowledge or data or the decision makers’ (DMs) limited attention and information processing capacities, the DMs cannot provide their evaluations with a single numerical value, a margin of error, some possibility distribution on the possible values, several possible numerical values, several possible interval numbers, or several possible intuitionistic fuzzy numbers but several possible interval-valued intuitionistic fuzzy numbers. For example, to get a reasonable decision result, a decision organization, which contains a lot of decision makers, is required to estimate the degree that an alternative satisfies an attribute. Suppose there are three cases: some decision makers provide ([0.5,0.7],[0.2,0.3]), some assign ([0.2,0.3],[0.5,0.6]), and the others provide ([0.4,0.6],[0.1,0.3]), and these three parts cannot persuade each other to change their opinions. We can easily see that such cases cannot be dealt with by fuzzy sets [5], hesitant fuzzy sets, and their extensions, such as interval-valued fuzzy sets, intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets, type 2 fuzzy sets, interval-valued hesitant fuzzy sets [6], and generalized hesitant fuzzy sets [7]. Thus, it is very necessary to introduce a new extension of hesitant fuzzy sets to address this issue. The aim of this paper is to present the notion of interval-valued intuitionistic hesitant fuzzy set, which extends the hesitant fuzzy set to interval-valued intuitionistic fuzzy environments and permits the membership of an element to be a set of several possible interval-valued intuitionistic fuzzy numbers. Thus, interval-valued intuitionistic hesitant fuzzy set is a very useful tool to deal with the situations in which the experts hesitate between several possible interval-valued intuitionistic fuzzy numbers to assess the degree to which an alternative satisfies an attribute. In the previous example, the degree to which the alternative satisfies the attribute can be represented by an interval-valued intuitionistic hesitant fuzzy set {([0.5,0.7],[0.2,0.3]),([0.2,0.3],[0.5,0.6]),([0.4,0.6],[0.1,0.3])}. Moreover, in many multiple attribute group decision-making (MAGDM) problems, considering that the estimations of the attribute values are interval-valued intuitionistic hesitant fuzzy sets, it therefore is very necessary to give some aggregation techniques to aggregate the interval-valued intuitionistic hesitant fuzzy information. However, we are aware that the existing aggregation techniques have difficulty in coping with group decision-making problems with interval-valued intuitionistic hesitant fuzzy information. Therefore, we in the current paper propose a series of aggregation operators for aggregating the interval-valued intuitionistic hesitant fuzzy information and investigate some properties of these operators. Then, based on these aggregation operators, we develop an approach to MAGDM with interval-valued intuitionistic hesitant fuzzy information. Moreover, we use a numerical example to show the application of the developed approach.

In order to do this, this paper is organized as follows. Section 2 introduces some concepts and properties of interval-valued intuitionistic hesitant fuzzy sets. In Section 3, we present a series of aggregation operators for interval-valued intuitionistic hesitant fuzzy information and examine the relationships among these aggregation operators. Section 4 develops an approach to group decision-makings with interval-valued intuitionistic hesitant fuzzy information. In the sequel, the application of the developed approach in group decision-making problems is shown by an illustrative example in Section 5. The final section offers some concluding remarks.

2. Preliminaries

In this section, we will briefly introduce the basic notions of hesitant fuzzy sets [4], interval-valued intuitionistic fuzzy sets [8], and interval-valued intuitionistic hesitant fuzzy sets.

2.1. Hesitant Fuzzy Sets and Hesitant Fuzzy Elements

Definition 1 (see [4]). Let 𝑋 be a fixed set; a hesitant fuzzy set (HFS) on 𝑋 is given in terms of a function that when applied to 𝑋 returns a subset of [0,1].

To be easily understood, we express the HFS by a mathematical symbol 𝐸=𝑥,ℎ_𝐸(𝑥)∣𝑥∈𝑋, (1)

where ℎ_𝐸(𝑥) is a set of some values in [0,1], denoting the possible membership degrees of the element 𝑥∈𝑋 to the set 𝐸. For convenience, Xia and Xu [9] called ℎ=ℎ_𝐸(𝑥) a hesitant fuzzy element (HFE) and 𝐻 the set of all HFEs.

Given three HFEs represented by ℎ, ℎ₁, and ℎ₂, Torra [4] defined some operations on them, which can be described as ℎ^𝑐=,ℎ1−𝛾∣𝛾∈ℎ₁∪ℎ₂=𝛾₁∨𝛾₂∣𝛾₁∈ℎ₁,𝛾₂∈ℎ₂,ℎ₁∩ℎ₂=𝛾₁∧𝛾₂∣𝛾₁∈ℎ₁,𝛾₂∈ℎ₂. (2)

Furthermore, in order to aggregate hesitant fuzzy information, Xia and Xu [9] defined some new operations on the HFEs ℎ, ℎ₁, and ℎ₂: ℎ₁⊕ℎ₂=𝛾₁+𝛾₂−𝛾₁𝛾₂∣𝛾₁∈ℎ₁,𝛾₂∈ℎ₂,ℎ₁⊗ℎ₂=𝛾₁𝛾₂∣𝛾₁∈ℎ₁,𝛾₂∈ℎ₂,ℎ^𝜆=𝛾^𝜆∣𝛾∈ℎ,𝜆ℎ=1−1−𝛾^𝜆∣𝛾∈ℎ. (3)

To compare the HFEs, Xia and Xu [9] defined the following comparison laws.

Definition 2 (see [9]). For an HFE ℎ,  𝑠(ℎ)=∑_𝛾∈ℎ𝛾/#ℎ is called the score function of ℎ, where #ℎ is the number of the elements in ℎ. For two HFEs ℎ₁ and ℎ₂, if 𝑠(ℎ₁)>𝑠(ℎ₂), then ℎ₁>ℎ₂; if 𝑠(ℎ₁)=𝑠(ℎ₂), then ℎ₁=ℎ₂.

2.2. Interval-Valued Intuitionistic Fuzzy Sets and Interval-Valued Intuitionistic Fuzzy Numbers

Definition 3 (see [8]). Let 𝑋 be an ordinary nonempty set. An interval-valued intuitionistic fuzzy set 𝐴 in 𝑋 is an object that has the form 𝐴=𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)∣𝑥∈𝑋, (4)
where 𝜇_𝐴(𝑥)=[𝜇⁻_𝐴(𝑥),𝜇⁺_𝐴(𝑥)]⊂[0,1] and 𝜈_𝐴(𝑥)=[𝜈⁻_𝐴(𝑥),𝜈⁺_𝐴(𝑥)]⊂[0,1] satisfy 0≤𝜇⁺_𝐴(𝑥)+𝜈⁺_𝐴(𝑥)≤1 for all 𝑥∈𝑋 and are, respectively, called the membership degree and the nonmembership degree of the element 𝑥∈𝑈 to 𝐴.

Xu [10] called each pair (𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)) in 𝐴 an interval-valued intuitionistic fuzzy number (IVIFN), and, for convenience, each IVIFN can be simply denoted by 𝛼=(𝜇_𝛼,𝜈_𝛼), where 𝜇_𝛼=[𝜇⁻_𝛼,𝜇⁺_𝛼]⊂[0,1], 𝜈_𝛼=[𝜈⁻_𝛼,𝜈⁺_𝛼]⊂[0,1], and 𝜇⁺_𝛼+𝜈⁺_𝛼≤1. Let Ω be the set of all IVIFNs.

Xu [10] introduced the following operational laws for IVIFNs.

Definition 4 (see [10]). Let 𝛼=(𝜇_𝛼,𝜈_𝛼)=([𝜇⁻_𝛼,𝜇⁺_𝛼],[𝜈⁻_𝛼,𝜈⁺_𝛼]), 𝛼₁=(𝜇_𝛼1,𝜈_𝛼1)=([𝜇⁻_𝛼1,𝜇⁺_𝛼1],[𝜈⁻_𝛼1,𝜈⁺_𝛼1]), and 𝛼₂=(𝜇_𝛼2,𝜈_𝛼2)=([𝜇⁻_𝛼2,𝜇⁺_𝛼2],[𝜈⁻_𝛼2,𝜈⁺_𝛼2]) be any three IVIFNs. Then, 𝛼^𝑐=𝜈⁻_𝛼,𝜈⁺_𝛼,𝜇⁻_𝛼,𝜇⁺_𝛼,𝛼₁∨𝛼₂=𝜇⁻_𝛼1∨𝜇⁻_𝛼2,𝜇⁺_𝛼1∨𝜇⁺_𝛼2,𝜈⁻_𝛼1∧𝜈⁻_𝛼2,𝜈⁺_𝛼1∧𝜈⁺_𝛼2,𝛼₁∧𝛼₂=𝜇⁻_𝛼1∧𝜇⁻_𝛼2,𝜇⁺_𝛼1∧𝜇⁺_𝛼2,𝜈⁻_𝛼1∨𝜈⁻_𝛼2,𝜈⁺_𝛼1∨𝜈⁺_𝛼2,𝛼₁⊕𝛼₂=𝜇⁻_𝛼1+𝜇⁻_𝛼2−𝜇⁻_𝛼1𝜇⁻_𝛼2,𝜇⁺_𝛼1+𝜇⁺_𝛼2−𝜇⁺_𝛼1𝜇⁺_𝛼2,𝜈⁻_𝛼1𝜈⁻_𝛼2,𝜈⁺_𝛼1𝜈⁺_𝛼2,𝛼₁⊗𝛼₂=𝜇⁻_𝛼1𝜇⁻_𝛼2,𝜇⁺_𝛼1𝜇⁺_𝛼2,𝜈⁻_𝛼1+𝜈⁻_𝛼2−𝜈⁻_𝛼1𝜈⁻_𝛼2,𝜈⁺_𝛼1+𝜈⁺_𝛼2−𝜈⁺_𝛼1𝜈⁺_𝛼2,𝜆𝛼=1−1−𝜇⁻_𝛼^𝜆,1−1−𝜇⁺_𝛼^𝜆,𝜈⁻_𝛼^𝜆,𝜈⁺_𝛼^𝜆,𝛼𝜆>0,_𝜆=𝜇⁻_𝛼^𝜆,𝜇⁺_𝛼^𝜆,1−1−𝜈⁻_𝛼^𝜆,1−1−𝜈⁺_𝛼^𝜆,𝜆>0. (5)

Theorem 5 (see [10]). Let 𝛼=(𝜇_𝛼,𝜈_𝛼),  𝛼₁=(𝜇_𝛼1,𝜈_𝛼1), and 𝛼₂=(𝜇_𝛼2,𝜈_𝛼2) be any three IVIFNs. Then, 𝛼₁⊕𝛼₂,  𝛼₁⊗𝛼₂,  𝜆𝛼, and 𝛼^𝜆 are also IVIFNs.

Xu [10] introduced the score function 𝑠(𝛼)=(1/2)(𝜇⁻_𝛼−𝜈⁻_𝛼+𝜇⁺_𝛼−𝜈⁺_𝛼) to get the score of 𝛼 and defined an accuracy function ℎ(𝛼)=(1/2)(𝜇⁻_𝛼+𝜈⁻_𝛼+𝜇⁺_𝛼+𝜈⁺_𝛼) to evaluate the accuracy degree of 𝛼. Xu [10] gave an order relation between two IVIFNs 𝛼₁ and 𝛼₂.(1)If 𝑠(𝛼₁)>𝑠(𝛼₂), then 𝛼₁>𝛼₂.(2)If 𝑠(𝛼₁)=𝑠(𝛼₂), then the following hold.(a)If ℎ(𝛼₁)>ℎ(𝛼₂), then 𝛼₁>𝛼₂.(b)If ℎ(𝛼₁)=ℎ(𝛼₂), then 𝛼₁=𝛼₂.(c)If ℎ(𝛼₂)>ℎ(𝛼₁), then 𝛼₂>𝛼₁.

2.3. Interval-Valued Intuitionistic Hesitant Fuzzy Sets and Interval-Valued Intuitionistic Hesitant Fuzzy Elements

In the following, we propose the concept of interval-valued intuitionistic hesitant fuzzy sets, which permit the membership of an element to be a set of several possible interval-valued intuitionistic fuzzy numbers. The motivation is that when defining the membership degree of an element, the difficulty of establishing the membership degree is not because we have a margin of error (as in intuitionistic fuzzy sets) or some possibility distribution (as in type 2 fuzzy sets) on the possible values, but because we have several possible interval-valued intuitionistic fuzzy numbers.

Definition 6. Let 𝑋 be a fixed set; an interval-valued intuitionistic hesitant fuzzy set (IVIHFS) on 𝑋 is given in terms of a function that when applied to 𝑋 returns a subset of Ω.

To be easily understood, we express the IVIHFS by a mathematical symbol 𝐸=𝑥,ℎ𝐸(𝑥)∣𝑥∈𝑋, (6)

where ℎ𝐸(𝑥) is a set of some IVIFNs in Ω, denoting the possible membership degree intervals and nonmembership degree intervals of the element 𝑥∈𝑋 to the set 𝐸. For convenience, we call ℎ=ℎ𝐸(𝑥) an interval-valued intuitionistic hesitant fuzzy element (IVIHFE) and 𝐻 the set of all IVIHFEs. If 𝛼∈ℎ, then 𝛼 is an IVIFN, and it can be denoted by 𝛼=(𝜇_𝛼,𝜈_𝛼)=([𝜇⁻_𝛼,𝜇⁺_𝛼],[𝜈⁻_𝛼,𝜈⁺_𝛼]).

For any 𝛼∈ℎ, if 𝛼 is a real number in [0,1], then ℎ reduces to a hesitant fuzzy element (HFE) [9]; if 𝛼 is a closed subinterval of the unit interval, then ℎ reduces to an interval-valued hesitant fuzzy element (IVHFE) [6]; if 𝛼 is an intuitionistic fuzzy number (IFN) [11], then ℎ reduces to an intuitionistic hesitant fuzzy element (IHFE). Therefore, HFEs, IVHFEs, and IHFEs are special cases of IVIHFEs.

Definition 7. Given three IVIHFEs represented by ℎ, ℎ₁, and ℎ₂, one defines some operations on them, which can be described as ℎ^𝑐=𝛼^𝑐∣𝛼∈ℎ=𝜈⁻_𝛼,𝜈⁺_𝛼,𝜇⁻_𝛼,𝜇⁺_𝛼ℎ,ℎ∣𝛼∈₁∪ℎ₂=𝛼₁∨𝛼₂∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=𝜇⁻_𝛼1∨𝜇⁻_𝛼2,𝜇⁺_𝛼1∨𝜇⁺_𝛼2,𝜈⁻_𝛼1∧𝜈⁻_𝛼2,𝜈⁺_𝛼1∧𝜈⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,ℎ₁∩ℎ₂=𝛼₁∧𝛼₂∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=𝜇⁻_𝛼1∧𝜇⁻_𝛼2,𝜇⁺_𝛼1∧𝜇⁺_𝛼2,𝜈⁻_𝛼1∨𝜈⁻_𝛼2,𝜈⁺_𝛼1∨𝜈⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,ℎ₁⊗ℎ₂=𝛼₁⊗𝛼₂∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=𝜇⁻_𝛼1𝜇⁻_𝛼2,𝜇⁺_𝛼1𝜇⁺_𝛼2,𝜈⁻_𝛼1+𝜈⁻_𝛼2−𝜈⁻_𝛼1𝜈⁻_𝛼2,𝜈⁺_𝛼1+𝜈⁺_𝛼2−𝜈⁺_𝛼1𝜈⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,ℎ₁⊕ℎ₂=𝛼₁⊕𝛼₂∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=𝜇⁻_𝛼1+𝜇⁻_𝛼2−𝜇⁻_𝛼1𝜇⁻_𝛼2,𝜇⁺_𝛼1+𝜇⁺_𝛼2−𝜇⁺_𝛼1𝜇⁺_𝛼2,𝜈⁻_𝛼1𝜈⁻_𝛼2,𝜈⁺_𝛼1𝜈⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,𝜆ℎ=ℎ=𝜆𝛼∣𝛼∈1−1−𝜇⁻_𝛼^𝜆,1−1−𝜇⁺_𝛼^𝜆,𝜈⁻_𝛼^𝜆,𝜈⁺_𝛼^𝜆ℎℎ∣𝛼∈,𝜆>0,_𝜆=𝛼^𝜆ℎ=𝜇∣𝛼∈_−𝛼𝜆,𝜇⁺_𝛼^𝜆,1−1−𝜈⁻_𝛼^𝜆,1−1−𝜈⁺_𝛼^𝜆∣𝛼∈ℎ,𝜆>0. (7)

Theorem 8. Let ℎ, ℎ₁, and ℎ₂ be three IVIHFEs, and 𝜆≥0. Then, ℎ₁⊕ℎ₂, ℎ₁⊗ℎ₂, 𝜆ℎ, and ℎ^𝜆 are also IVIHFEs.

Proof. Since ℎ={([𝜇⁻_𝛼,𝜇⁺_𝛼],[𝜈⁻_𝛼,𝜈⁺_𝛼])∣𝛼∈ℎ}, ℎ₁={([𝜇⁻_𝛼1,𝜇⁺_𝛼1],[𝜈⁻_𝛼1,𝜈⁺_𝛼1])∣𝛼₁∈ℎ₁}, and ℎ₂={([𝜇⁻_𝛼2,𝜇⁺_𝛼2],[𝜈⁻_𝛼2,𝜈⁺_𝛼2])∣𝛼₂∈ℎ₂} are three IVIHFEs, we have 0≤𝜇⁻_𝛼,𝜇⁺_𝛼,𝜈⁻_𝛼,𝜈⁺_𝛼,𝜇⁻_𝛼1,𝜇⁺_𝛼1,𝜈⁻_𝛼1,𝜈⁺_𝛼1,𝜇⁻_𝛼2,𝜇⁺_𝛼2,𝜈⁻_𝛼2,𝜈⁺_𝛼2≤1, 𝜇⁺_𝛼+𝜈⁺_𝛼≤1,  𝜇⁺_𝛼1+𝜈⁺_𝛼1≤1, and 𝜇⁺_𝛼2+𝜈⁺_𝛼2≤1. Then, we can obtain 0≤𝜇⁻_𝛼1+𝜇⁻_𝛼2−𝜇⁻_𝛼1𝜇⁻_𝛼2=1−1−𝜇⁻_𝛼11−𝜇⁻_𝛼2≤1−1−𝜇⁺_𝛼11−𝜇⁺_𝛼2=𝜇⁺_𝛼1+𝜇⁺_𝛼2−𝜇⁺_𝛼1𝜇⁺_𝛼2≤1,0≤𝜈⁻_𝛼1𝜈⁻_𝛼2≤𝜈⁺_𝛼1𝜈⁺_𝛼2𝜇≤1,_+𝛼1+𝜇⁺_𝛼2−𝜇⁺_𝛼1𝜇⁺_𝛼2+𝜈⁺_𝛼1𝜈⁺_𝛼2=1−1−𝜇⁺_𝛼11−𝜇⁺_𝛼2+𝜈⁺_𝛼1𝜈⁺_𝛼2≤1−1−𝜇⁺_𝛼11−𝜇⁺_𝛼2+1−𝜇⁺_𝛼11−𝜇⁺_𝛼2=1,0≤𝜇⁻_𝛼1𝜇⁻_𝛼2≤𝜇⁺_𝛼1𝜇⁺_𝛼2≤1,0≤𝜈⁻_𝛼1+𝜈⁻_𝛼2−𝜈⁻_𝛼1𝜈⁻_𝛼2=1−1−𝜈⁻_𝛼11−𝜈⁻_𝛼2≤1−1−𝜈⁺_𝛼11−𝜈⁺_𝛼2=𝜈⁺_𝛼1+𝜈⁺_𝛼2−𝜈⁺_𝛼1𝜈⁺_𝛼2𝜇≤1,_+𝛼1𝜇⁺_𝛼2+𝜈⁺_𝛼1+𝜈⁺_𝛼2−𝜈⁺_𝛼1𝜈⁺_𝛼2=𝜇⁺_𝛼1𝜇⁺_𝛼2+1−1−𝜈⁺_𝛼11−𝜈⁺_𝛼2≤1−𝜈⁺_𝛼11−𝜈⁺_𝛼2+1−1−𝜈⁺_𝛼11−𝜈⁺_𝛼2=1,0≤1−1−𝜇⁻_𝛼^𝜆≤1−1−𝜇⁺_𝛼^𝜆𝜈≤1,0≤⁻_𝛼^𝜆≤𝜈⁺_𝛼^𝜆≤1,1−1−𝜇⁺_𝛼^𝜆+𝜈⁺_𝛼^𝜆≤1−1−𝜇⁺_𝛼^𝜆+1−𝜇⁺_𝛼^𝜆𝜇=1,0≤⁻_𝛼^𝜆≤𝜇⁺_𝛼^𝜆≤1,0≤1−1−𝜈⁻_𝛼^𝜆≤1−1−𝜈⁺_𝛼^𝜆𝜇≤1,_+𝛼𝜆+1−1−𝜈⁺_𝛼^𝜆≤1−𝜈⁺_𝛼^𝜆+1−1−𝜈⁺_𝛼^𝜆=1. (8)
Thus, ℎ₁⊕ℎ₂,  ℎ₁⊗ℎ₂,  𝜆ℎ, and ℎ^𝜆 are IVIHFEs.

Theorem 9. Let ℎ, ℎ₁, and ℎ₂ be three IVIHFEs, and 𝜆, 𝜆₁, and 𝜆₂>0. Then, one has ℎ^𝑐₁∪ℎ^𝑐₂=ℎ₁∩ℎ₂^𝑐,ℎ^𝑐₁∩ℎ^𝑐₂=ℎ₁∪ℎ₂^𝑐,ℎ^𝑐𝜆=𝜆ℎ^𝑐,𝜆ℎ^𝑐=ℎ^𝜆𝑐,ℎ^𝑐₁⊕ℎ^𝑐₂=ℎ₁⊗ℎ₂^𝑐,ℎ^𝑐₁⊗ℎ^𝑐₂=ℎ₁⊕ℎ₂^𝑐,ℎ₁⊕ℎ₂=ℎ₂⊕ℎ₁,𝜆ℎ₁⊕ℎ₂=𝜆ℎ₁⊕𝜆ℎ₂,𝜆₁𝜆₂ℎ=𝜆₁𝜆₂ℎ,ℎ₁⊗ℎ₂=ℎ₂⊗ℎ₁,ℎ^𝜆₁⊗ℎ^𝜆₂=ℎ₁⊗ℎ₂^𝜆,ℎ^𝜆1𝜆2=ℎ^𝜆1𝜆2. (9)

Proof. Consider ℎ^𝑐₁∪ℎ^𝑐₂=𝜈⁻_𝛼1,𝜈⁺_𝛼1,𝜇⁻_𝛼1,𝜇⁺_𝛼1∣𝛼₁∈ℎ₁∪𝜈⁻_𝛼2,𝜈⁺_𝛼2,𝜇⁻_𝛼2,𝜇⁺_𝛼2∣𝛼₂∈ℎ₂=𝜈⁻_𝛼1∨𝜈⁻_𝛼2,𝜈⁺_𝛼1∨𝜈⁺_𝛼2,𝜇⁻_𝛼1∧𝜇⁻_𝛼2,𝜇⁺_𝛼1∧𝜇⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=𝜇⁻_𝛼1∧𝜇⁻_𝛼2,𝜇⁺_𝛼1∧𝜇⁺_𝛼2,𝜈⁻_𝛼1∨𝜈⁻_𝛼2,𝜈⁺_𝛼1∨𝜈⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂^𝑐=ℎ₁∩ℎ₂^𝑐,ℎ^𝑐₁∩ℎ^𝑐₂=𝜈⁻_𝛼1,𝜈⁺_𝛼1,𝜇⁻_𝛼1,𝜇⁺_𝛼1∣𝛼₁∈ℎ₁∩𝜈⁻_𝛼2,𝜈⁺_𝛼2,𝜇⁻_𝛼2,𝜇⁺_𝛼2∣𝛼₂∈ℎ₂=𝜈⁻_𝛼1∧𝜈⁻_𝛼2,𝜈⁺_𝛼1∧𝜈⁺_𝛼2,𝜇⁻_𝛼1∨𝜇⁻_𝛼2,𝜇⁺_𝛼1∨𝜇⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=𝜇⁻_𝛼1∨𝜇⁻_𝛼2,𝜇⁺_𝛼1∨𝜇⁺_𝛼2,𝜈⁻_𝛼1∧𝜈⁻_𝛼2,𝜈⁺_𝛼1∧𝜈⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂^𝑐=ℎ₁∪ℎ₂^𝑐,ℎ^𝑐𝜆=𝜈⁻_𝛼,𝜈⁺_𝛼,𝜇⁻_𝛼,𝜇⁺_𝛼∣𝛼∈ℎ^𝜆=𝜈⁻_𝛼^𝜆,𝜈⁺_𝛼^𝜆,1−1−𝜇⁻_𝛼^𝜆,1−1−𝜇⁺_𝛼^𝜆∣𝛼∈ℎ=𝜆ℎ^𝑐,𝜆ℎ^𝑐=𝜆𝜈⁻_𝛼,𝜈⁺_𝛼,𝜇⁻_𝛼,𝜇⁺_𝛼ℎ=∣𝛼∈1−1−𝜈⁻_𝛼^𝜆,1−1−𝜈⁺_𝛼^𝜆,𝜇⁻_𝛼^𝜆,𝜇⁺_𝛼^𝜆∣𝛼∈ℎ=ℎ^𝜆𝑐,ℎ^𝑐₁⊕ℎ^𝑐₂=𝜈⁻_𝛼1,𝜈⁺_𝛼1,𝜇⁻_𝛼1,𝜇⁺_𝛼1∣𝛼₁∈ℎ₁⊕𝜈⁻_𝛼2,𝜈⁺_𝛼2,𝜇⁻_𝛼2,𝜇⁺_𝛼2∣𝛼₂∈ℎ₂=𝜈⁻_𝛼1+𝜈⁻_𝛼2−𝜈⁻_𝛼1𝜈⁻_𝛼2,𝜈⁺_𝛼1+𝜈⁺_𝛼2−𝜈⁺_𝛼1𝜈⁺_𝛼2,𝜇⁻_𝛼1𝜇⁻_𝛼2,𝜇⁺_𝛼1𝜇⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=𝜇⁻_𝛼1𝜇⁻_𝛼2,𝜇⁺_𝛼1𝜇⁺_𝛼2,𝜈⁻_𝛼1+𝜈⁻_𝛼2−𝜈⁻_𝛼1𝜈⁻_𝛼2,𝜈⁺_𝛼1+𝜈⁺_𝛼2−𝜈⁺_𝛼1𝜈⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂^𝑐=ℎ₁⊗ℎ₂^𝑐,ℎ^𝑐₁⊕ℎ^𝑐₂=𝜈⁻_𝛼1,𝜈⁺_𝛼1,𝜇⁻_𝛼1,𝜇⁺_𝛼1∣𝛼₁∈ℎ₁⊗𝜈⁻_𝛼2,𝜈⁺_𝛼2,𝜇⁻_𝛼2,𝜇⁺_𝛼2∣𝛼₂∈ℎ₂=𝜈⁻_𝛼1𝜈⁻_𝛼2,𝜈⁺_𝛼1𝜈⁺_𝛼2,𝜇⁻_𝛼1+𝜇⁻_𝛼2−𝜇⁻_𝛼1𝜇⁻_𝛼2,𝜇⁺_𝛼1+𝜇⁺_𝛼2−𝜇⁺_𝛼1𝜇⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=𝜇⁻_𝛼1+𝜇⁻_𝛼2−𝜇⁻_𝛼1𝜇⁻_𝛼2,𝜇⁺_𝛼1+𝜇⁺_𝛼2−𝜇⁺_𝛼1𝜇⁺_𝛼2,𝜈⁻_𝛼1𝜈⁻_𝛼2,𝜈⁺_𝛼1𝜈⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂^𝑐=ℎ₁⊕ℎ₂^𝑐,ℎ₁⊕ℎ₂=𝜇⁻_𝛼1+𝜇⁻_𝛼2−𝜇⁻_𝛼1𝜇⁻_𝛼2,𝜇⁺_𝛼1+𝜇⁺_𝛼2−𝜇⁺_𝛼1𝜇⁺_𝛼2,𝜈⁻_𝛼1𝜈⁻_𝛼2,𝜈⁺_𝛼1𝜈⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=𝜇⁻_𝛼2+𝜇⁻_𝛼1−𝜇⁻_𝛼2𝜇⁻_𝛼1,𝜇⁺_𝛼2+𝜇⁺_𝛼1−𝜇⁺_𝛼2𝜇⁺_𝛼1,𝜈⁻_𝛼2𝜈⁻_𝛼1,𝜈⁺_𝛼2𝜈⁺_𝛼1∣𝛼₂∈ℎ₂,𝛼₁∈ℎ₁=ℎ₂⊕ℎ₁,𝜆ℎ₁⊕ℎ₂=𝜆𝜇⁻_𝛼1+𝜇⁻_𝛼2−𝜇⁻_𝛼1𝜇⁻_𝛼2,𝜇⁺_𝛼1+𝜇⁺_𝛼2−𝜇⁺_𝛼1𝜇⁺_𝛼2,𝜈⁻_𝛼1𝜈⁻_𝛼2,𝜈⁺_𝛼1𝜈⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=1−1−𝜇⁻_𝛼1−𝜇⁻_𝛼2+𝜇⁻_𝛼1𝜇⁻_𝛼2^𝜆,1−1−𝜇⁺_𝛼1−𝜇⁺_𝛼2+𝜇⁺_𝛼1𝜇⁺_𝛼2^𝜆,𝜈⁻_𝛼1𝜈⁻_𝛼2^𝜆,𝜈⁺_𝛼1𝜈⁺_𝛼2^𝜆∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=1−1−𝜇⁻_𝛼1^𝜆1−𝜇⁻_𝛼2^𝜆,1−1−𝜇⁺_𝛼1^𝜆1−𝜇⁺_𝛼2^𝜆,𝜈⁻_𝛼1^𝜆𝜈⁻_𝛼2^𝜆,𝜈⁺_𝛼1^𝜆𝜈⁺_𝛼2^𝜆∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,𝜆ℎ₁⊕𝜆ℎ₂=1−1−𝜇⁻_𝛼1^𝜆,1−1−𝜇⁺_𝛼1^𝜆,𝜈⁻_𝛼1^𝜆,𝜈⁺_𝛼1^𝜆∣𝛼₁∈ℎ₁⊕1−1−𝜇⁻_𝛼2^𝜆,1−1−𝜇⁺_𝛼2^𝜆,𝜈⁻_𝛼2^𝜆,𝜈⁺_𝛼2^𝜆∣𝛼₂∈ℎ₂=1−1−𝜇⁻_𝛼1^𝜆+1−1−𝜇⁻_𝛼2^𝜆−1−1−𝜇⁻_𝛼1^𝜆1−1−𝜇⁻_𝛼2^𝜆,1−1−𝜇⁺_𝛼1^𝜆+1−1−𝜇⁺_𝛼2^𝜆−1−1−𝜇⁺_𝛼1^𝜆1−1−𝜇⁺_𝛼2^𝜆,𝜈⁻_𝛼1^𝜆𝜈⁻_𝛼2^𝜆,𝜈⁺_𝛼1^𝜆𝜈⁺_𝛼2^𝜆∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=1−1−𝜇⁻_𝛼1^𝜆1−𝜇⁻_𝛼2^𝜆,1−1−𝜇⁺_𝛼1^𝜆1−𝜇⁺_𝛼2^𝜆,𝜈⁻_𝛼1^𝜆𝜈⁻_𝛼2^𝜆,𝜈⁺_𝛼1^𝜆𝜈⁺_𝛼2^𝜆∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂. (10)
Thus, we have 𝜆(ℎ₁⊕ℎ₂)=𝜆ℎ₁⊕𝜆ℎ₂.
Moreover, 𝜆₁𝜆₂ℎ=1−1−𝜇⁻_𝛼^𝜆1𝜆2,1−1−𝜇⁺_𝛼^𝜆1𝜆2,𝜈⁻_𝛼^𝜆1𝜆2,𝜈⁺_𝛼^𝜆1𝜆2ℎ,𝜆∣𝛼∈₁𝜆₂ℎ=𝜆₁1−1−𝜇⁻_𝛼^𝜆2,1−1−𝜇⁺_𝛼^𝜆2,𝜈⁻_𝛼^𝜆2,𝜈⁺_𝛼^𝜆2ℎ=∣𝛼∈1−1−1+1−𝜇⁻_𝛼^𝜆2𝜆1,1−1−1+1−𝜇⁺_𝛼^𝜆2𝜆1,𝜈⁻_𝛼^𝜆2𝜆1,𝜈⁺_𝛼^𝜆2𝜆1ℎ=∣𝛼∈1−1−𝜇⁻_𝛼^𝜆1𝜆2,1−1−𝜇⁺_𝛼^𝜆1𝜆2,𝜈⁻_𝛼^𝜆1𝜆2,𝜈⁺_𝛼^𝜆1𝜆2∣𝛼∈ℎ. (11)
Thus, we have that (𝜆₁𝜆₂)ℎ=𝜆₁(𝜆₂ℎ).
Take ℎ₁⊗ℎ₂=𝜇⁻_𝛼1𝜇⁻_𝛼2,𝜇⁺_𝛼1𝜇⁺_𝛼2,𝜈⁻_𝛼1+𝜈⁻_𝛼2−𝜈⁻_𝛼1𝜈⁻_𝛼2,𝜈⁺_𝛼1+𝜈⁺_𝛼2−𝜈⁺_𝛼1𝜈⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=𝜇⁻_𝛼2𝜇⁻_𝛼1,𝜇⁺_𝛼2𝜇⁺_𝛼1,𝜈⁻_𝛼2+𝜈⁻_𝛼1−𝜈⁻_𝛼2𝜈⁻_𝛼1,𝜈⁺_𝛼2+𝜈⁺_𝛼1−𝜈⁺_𝛼2𝜈⁺_𝛼1∣𝛼₂∈ℎ₂,𝛼₁∈ℎ₁=ℎ₂⊗ℎ₁,ℎ^𝜆₁⊗ℎ^𝜆₂=𝜇⁻_𝛼1^𝜆,𝜇⁺_𝛼1^𝜆,1−1−𝜈⁻_𝛼1^𝜆,1−1−𝜈⁺_𝛼1^𝜆∣𝛼₁∈ℎ₁⊗𝜇⁻_𝛼2^𝜆,𝜇⁺_𝛼2^𝜆,1−1−𝜈⁻_𝛼2^𝜆,1−1−𝜈⁺_𝛼2^𝜆∣𝛼₂∈ℎ₂=𝜇⁻_𝛼1^𝜆𝜇⁻_𝛼2^𝜆,𝜇⁺_𝛼1^𝜆𝜇⁺_𝛼2^𝜆,1−1−𝜈⁻_𝛼1^𝜆+1−1−𝜈⁻_𝛼2^𝜆−1−1−𝜈⁻_𝛼1^𝜆1−1−𝜈⁻_𝛼2^𝜆,1−1−𝜈⁺_𝛼1^𝜆+1−1−𝜈⁺_𝛼2^𝜆−1−1−𝜈⁺_𝛼1^𝜆×1−1−𝜈⁺_𝛼2^𝜆∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=𝜇⁻_𝛼1^𝜆𝜇⁻_𝛼2^𝜆,𝜇⁺_𝛼1^𝜆𝜇⁺_𝛼2^𝜆,1−1−𝜈⁻_𝛼1^𝜆1−𝜈⁻_𝛼2^𝜆,1−1−𝜈⁺_𝛼1^𝜆1−𝜈⁺_𝛼2^𝜆∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,ℎ₁⊗ℎ₂^𝜆=𝜇⁻_𝛼1𝜇⁻_𝛼2,𝜇⁺_𝛼1𝜇⁺_𝛼2,𝜈⁻_𝛼1+𝜈⁻_𝛼2−𝜈⁻_𝛼1𝜈⁻_𝛼2,𝜈⁺_𝛼1+𝜈⁺_𝛼2−𝜈⁺_𝛼1𝜈⁺_𝛼2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂^𝜆=𝜇⁻_𝛼1𝜇⁻_𝛼2^𝜆,𝜇⁺_𝛼1𝜇⁺_𝛼2^𝜆,1−1−𝜈⁻_𝛼1−𝜈⁻_𝛼2+𝜈⁻_𝛼1𝜈⁻_𝛼2^𝜆,1−1−𝜈⁺_𝛼1−𝜈⁺_𝛼2+𝜈⁺_𝛼1𝜈⁺_𝛼2^𝜆∣ℎ=𝜇𝛼∈_−𝛼1𝜇⁻_𝛼2^𝜆,𝜇⁺_𝛼1𝜇⁺_𝛼2^𝜆,1−1−𝜈⁻_𝛼1^𝜆1−𝜈⁻_𝛼2^𝜆,1−1−𝜈⁺_𝛼1^𝜆1−𝜈⁺_𝛼2^𝜆∣𝛼∈ℎ. (12)
Thus, we have that ℎ^𝜆₁⊗ℎ^𝜆₂=(ℎ₁⊗ℎ₂)^𝜆.
Consider ℎ^𝜆1𝜆2=𝜇⁻_𝛼^𝜆1𝜆2,𝜇⁺_𝛼^𝜆1𝜆2,1−1−𝜈⁻_𝛼^𝜆1𝜆2,1−1−𝜈⁺_𝛼^𝜆1𝜆2∣ℎ,ℎ𝛼∈_𝜆1𝜆2=𝜇⁻_𝛼^𝜆1,𝜇⁺_𝛼^𝜆1,1−1−𝜈⁻_𝛼^𝜆1,1−1−𝜈⁺_𝛼^𝜆1∣𝛼∈ℎ^𝜆2=𝜇⁻_𝛼^𝜆1𝜆2,𝜇⁺_𝛼^𝜆1𝜆2,1−1−1+1−𝜈⁻_𝛼^𝜆1𝜆2,1−1−1+1−𝜈⁺_𝛼^𝜆1𝜆2∣ℎ=𝜇𝛼∈_{−𝛼𝜆1𝜆2},𝜇⁺_𝛼^𝜆1𝜆2,1−1−𝜈⁻_𝛼^𝜆1𝜆2,1−1−𝜈⁺_𝛼^𝜆1𝜆2∣𝛼∈ℎ. (13)
Thus, we have that ℎ^𝜆1𝜆2=(ℎ^𝜆1)^𝜆2.

To compare the IVIHFEs, we define the following comparison laws.

Definition 10. For an IVIHFE ℎ, 𝑠(ℎ)=∑_𝛼∈ℎ𝑠(𝛼)/#ℎ is called the score function of ℎ, where #ℎ is the number of the elements in ℎ. ℎ(ℎ)=∑_𝛼∈ℎℎ(𝛼)/#ℎ is called the accuracy function of ℎ. For any two IVIHFEs ℎ₁ and ℎ₂,(1)if 𝑠(ℎ₁)>𝑠(ℎ₂), then ℎ₁>ℎ₂;(2)if 𝑠(ℎ₁)=𝑠(ℎ₂), then the following hold.(a)If ℎ(ℎ₁)>ℎ(ℎ₂), then ℎ₁>ℎ₂.(b)If ℎ(ℎ₁)=ℎ(ℎ₂), then ℎ₁=ℎ₂.(c)If ℎ(ℎ₂)>ℎ(ℎ₁), then ℎ₂>ℎ₁.

3. Aggregation Operators for Interval-Valued Intuitionistic Hesitant Fuzzy Information

In the current section, we propose a series of operators for aggregating the interval-valued intuitionistic hesitant fuzzy information and investigate some desired properties of these operators.

3.1. The IVIHFWA, IVIHFWG, GIVIHFWA, and GIVIHFWG Operators

Definition 11. Let ℎ_𝑖 (𝑖=1,2,…,𝑛) be a collection of IVIHFEs, and let 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 be the weight vector of ℎ_𝑖 (𝑖=1,2,…,𝑛) with 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1. An interval-valued intuitionistic hesitant fuzzy weighted averaging (IVIHFWA) operator is a mapping 𝐻^𝑛→𝐻 such that IVIHFWAℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝑤_𝑖ℎ_𝑖. (14)
If 𝑤=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇 especially, then the IVIHFWA operator reduces to the interval-valued intuitionistic hesitant fuzzy averaging (IVIHFA) operator: IVIHFAℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=11𝑛ℎ_𝑖. (15)

Theorem 12. Let ℎ_𝑖  (𝑖=1,2,…,𝑛) be a collection of IVIHFEs. Then, their aggregated value calculated using the IVIHFWA operator is an IVIHFE and 𝐼𝑉𝐼𝐻𝐹𝑊𝐴ℎ₁,ℎ₂,…,ℎ_𝑛=1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^𝑤𝑖,1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^𝑤𝑖,_𝑛𝑖=1𝜈⁻_𝛼𝑖^𝑤𝑖,^𝑛_𝑖=1𝜈⁺_𝛼𝑖^𝑤𝑖∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛. (16)

Proof. The first result follows quickly from Definition 11 and Theorem 8. In the following, we prove the second result by using mathematical induction on 𝑛. First, we show that (16) holds for 𝑛=2.
Because 𝑤₁ℎ₁=1−1−𝜇⁻_𝛼1^𝑤1,1−1−𝜇⁺_𝛼1^𝑤1,𝜈⁻_𝛼1^𝑤1,𝜈⁺_𝛼1^𝑤1∣𝛼₁∈ℎ₁,𝑤₂ℎ₂=1−1−𝜇⁻_𝛼2^𝑤2,1−1−𝜇⁺_𝛼2^𝑤2,𝜈⁻_𝛼2^𝑤2,𝜈⁺_𝛼2^𝑤2∣𝛼₂∈ℎ₂, (17) we have 𝑤₁ℎ₁⊕𝑤₂ℎ₂=1−1−𝜇⁻_𝛼1^𝑤1,1−1−𝜇⁺_𝛼1^𝑤1,𝜈⁻_𝛼1^𝑤1,𝜈⁺_𝛼1^𝑤1∣𝛼₁∈ℎ₁⊕1−1−𝜇⁻_𝛼2^𝑤2,1−1−𝜇⁺_𝛼2^𝑤2,𝜈⁻_𝛼2^𝑤2,𝜈⁺_𝛼2^𝑤2∣𝛼₂∈ℎ₂=1−1−𝜇⁻_𝛼1^𝑤1+1−1−𝜇⁻_𝛼2^𝑤2−1−1−𝜇⁻_𝛼1^𝑤11−1−𝜇⁻_𝛼2^𝑤2,1−1−𝜇⁺_𝛼1^𝑤1+1−1−𝜇⁺_𝛼2^𝑤2−1−1−𝜇⁺_𝛼1^𝑤11−1−𝜇⁺_𝛼2^𝑤2,𝜈⁻_𝛼1^𝑤1𝜈⁻_𝛼2^𝑤2,𝜈⁺_𝛼1^𝑤1𝜈⁺_𝛼2^𝑤2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂=1−1−𝜇⁻_𝛼1^𝑤11−𝜇⁻_𝛼2^𝑤2,1−1−𝜇⁺_𝛼1^𝑤11−𝜇⁺_𝛼2^𝑤2,𝜈⁻_𝛼1^𝑤1𝜈⁻_𝛼2^𝑤2,𝜈⁺_𝛼1^𝑤1𝜈⁺_𝛼2^𝑤2∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂. (18)
If (16) holds for 𝑛=𝑘, in other words, IVIHFWAℎ₁,ℎ₂,…,ℎ_𝑘=1−^𝑘_𝑖=11−𝜇⁻_𝛼𝑖^𝑤𝑖,1−^𝑘_𝑖=11−𝜇⁺_𝛼𝑖^𝑤𝑖,_𝑘𝑖=1𝜈⁻_𝛼𝑖^𝑤𝑖,^𝑘_𝑖=1𝜈⁺_𝛼𝑖^𝑤𝑖∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑘∈ℎ_𝑘, (19) then, when 𝑛=𝑘+1, IVIHFE operations yield IVIHFWAℎ₁,ℎ₂,…,ℎ_𝑘,ℎ_𝑘+1=^𝑘+1_𝑖=1𝑤_𝑖ℎ_𝑖=^𝑘_𝑖=1𝑤_𝑖ℎ_𝑖⊕𝑤_𝑘+1ℎ_𝑘+1=1−^𝑘_𝑖=11−𝜇⁻_𝛼𝑖^𝑤𝑖,1−^𝑘_𝑖=11−𝜇⁺_𝛼𝑖^𝑤𝑖,_𝑘𝑖=1𝜈⁻_𝛼𝑖^𝑤𝑖,^𝑘_𝑖=1𝜈⁺_𝛼𝑖^𝑤𝑖∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑘∈ℎ_𝑘⊕1−1−𝜇⁻_𝛼𝑘+1^𝑤𝑘+1,1−1−𝜇⁺_𝛼𝑘+1^𝑤𝑘+1,𝜈⁻_𝛼𝑘+1^𝑤𝑘+1,𝜈⁺_𝛼𝑘+1^𝑤𝑘+1∣𝛼_𝑘+1∈ℎ_𝑘+1=1−^𝑘_𝑖=11−𝜇⁻_𝛼𝑖^𝑤𝑖+1−1−𝜇⁻_𝛼𝑘+1^𝑤𝑘+1−1−^𝑘_𝑖=11−𝜇⁻_𝛼𝑖^𝑤𝑖1−1−𝜇⁻_𝛼𝑘+1^𝑤𝑘+1,1−^𝑘_𝑖=11−𝜇⁺_𝛼𝑖^𝑤𝑖+1−1−𝜇⁺_𝛼𝑘+1^𝑤𝑘+1−1−^𝑘_𝑖=11−𝜇⁺_𝛼𝑖^𝑤𝑖1−1−𝜇⁺_𝛼𝑘+1^𝑤𝑘+1,_𝑘𝑖=1𝜈⁻_𝛼𝑖^𝑤𝑖𝜈⁻_𝛼𝑘+1^𝑤𝑘+1,_𝑘𝑖=1𝜈⁺_𝛼𝑖^𝑤𝑖𝜈⁺_𝛼𝑘+1^𝑤𝑘+1∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑘∈ℎ_𝑘,𝛼_𝑘+1∈ℎ_𝑘+1=1−^𝑘+1_𝑖=11−𝜇⁻_𝛼𝑖^𝑤𝑖,1−^𝑘+1_𝑖=11−𝜇⁺_𝛼𝑖^𝑤𝑖,_𝑘+1𝑖=1𝜈⁻_𝛼𝑖^𝑤𝑖,^𝑘+1_𝑖=1𝜈⁺_𝛼𝑖^𝑤𝑖∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑘∈ℎ_𝑘,𝛼_𝑘+1∈ℎ_𝑘+1. (20)
In other words, (16) holds for 𝑛=𝑘+1. Equation (16) therefore holds for all 𝑛.
This completes the proof of Theorem 12.

Definition 13. Let ℎ_𝑖  (𝑖=1,2,…,𝑛) be a collection of IVIHFEs, and let 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 be the weight vector of ℎ_𝑖  (𝑖=1,2,…,𝑛) with 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1. An interval-valued intuitionistic hesitant fuzzy weighted geometric (IVIHFWG) operator is a mapping 𝐻^𝑛→𝐻 such that IVIHFWGℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1ℎ^𝑤𝑖_𝑖. (21)
Especially, if 𝑤=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇, then the IVIHFWG operator reduces to the interval-valued intuitionistic hesitant fuzzy averaging (IVIHFA) operator: IVIHFGℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1ℎ_𝑖^1/𝑛. (22)

Theorem 14. Let ℎ_𝑖  (𝑖=1,2,…,𝑛) be a collection of IVIHFEs. Then, their aggregated value calculated using the IVIHFWG operator is an IVIHFE, and 𝐼𝑉𝐼𝐻𝐹𝑊𝐺ℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝜇⁻_𝛼𝑖^𝑤𝑖,^𝑛_𝑖=1𝜇⁺_𝛼𝑖^𝑤𝑖,1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^𝑤𝑖,1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^𝑤𝑖∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛. (23)

In the following, by combining the IVIHFWA and IVIHFWG operators with the generalized mean [12], we develop the generalized interval-valued intuitionistic hesitant fuzzy weighted averaging (GIVIHFWA) operator and the generalized interval-valued intuitionistic hesitant fuzzy weighted geometric (GIVIHFWG) operator, respectively. The main characteristic of the GIVIHFWA and GIVIHFWG operators is that they have an additional parameter 𝜆 controlling the power to which the argument values are raised. Different from the IVIHFWA and IVIHFWG operators, the GIVIHFWA and GIVIHFWG operators extend them with addition of a parameter controlling the power to which the argument values are raised. When we use different choices of the parameters 𝜆, we will get some special cases.

Definition 15. Let ℎ_𝑖 (𝑖=1,2,…,𝑛) be a collection of IVIHFEs, and let 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 be the weight vector of ℎ_𝑖 (𝑖=1,2,…,𝑛) with 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1.(1) A generalized interval-valued intuitionistic hesitant fuzzy weighted averaging (GIVIHFWA) operator is a mapping 𝐻^𝑛→𝐻, where GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝑤_𝑖ℎ^𝜆_𝑖^1/𝜆 (24) with 𝜆>0.(2) A generalized interval-valued intuitionistic hesitant fuzzy weighted geometric (GIVIHFWG) operator is a mapping 𝐻^𝑛→𝐻, where GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=1𝜆^𝑛_𝑖=1𝜆ℎ_𝑖^𝑤𝑖 (25) with 𝜆>0.

If 𝜆=1, then the GIVIHFWA operator reduces to the IVIHFWA operator and the GIVIHFWG operator reduces to the IVIHFWG operator.

Using IVIHFE operations and mathematical induction on 𝑛, (24) and (25) can be transformed into the following forms: GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛,GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛. (26)(27)

Example 16. Suppose that ℎ₁={([0.2,0.3],[0.5,0.6]),([0.5,08],[0.1,0.2]),([0.1,0.3],[0.4,0.5])}, ℎ₂={([0.4,0.6],[0.3,0.4]),([0.3,0.5],[0.1,0.2])}, and ℎ₃={([0.5,0.5],[0.2,0.3]),([0.2,0.4],[0.3,0.6]),([0.8,0.9],[0.1,0.1])} are three IVIHFEs, and 𝑤=(0.6,0.3,0.1)^𝑇 is their weight vector. Then, by Definition 15, we can obtain GIVIHFWA₁ℎ₁,ℎ₂,ℎ₃=IVIHFWAℎ₁,ℎ₂,ℎ₃=],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,([0.2998,0.42780.3914,0.4957])([0.2661,0.41720.4076,0.5313])([0.3611,0.51280.3652,0.4441])([0.2667,0.38810.2815,0.4026])([0.2314,0.37690.2932,0.4315])([0.3309,0.47910.2627,0.3607])([0.4719,0.73010.1490,0.2564])([0.4465,0.72520.1552,0.2748])([0.5181,0.77030.1390,0.2297])([0.4469,0.71150.1072,0.2083])([0.4203,0.70620.1116,0.2232])([0.4953,0.75440.1000,0.1866])([0.2486,0.42780.3424,0.4443])([0.2124,0.41720.3565,0.4762])([0.3144,0.51280.3194,0.3981])([0.2130,0.38810.2462,0.3609])([0.1751,0.37690.2564,0.3868])([0.2819,0.47910.2297,0.3234])GIVIHFWA₅ℎ₁,ℎ₂,ℎ₃=],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,([0.3645,0.49090.3596,0.4559])([0.3191,0.48350.3843,0.5008])([0.5306,0.64150.3150,0.3536])([0.3329,0.42620.2255,0.3312])([0.2488,0.41250.2344,0.3471])([0.5248,0.62500.2070,0.2803])([0.4782,0.74750.1410,0.2453])([0.4658,0.74640.1452,0.2536])([0.5696,0.78420.1317,0.2158])([0.4690,0.74000.1064,0.2072])([0.4555,0.73890.1094,0.2134])([0.5653,0.77860.1000,0.1847])([0.3624,0.49090.3314,0.4306])([0.3154,0.48350.3512,0.4661])([0.5302,0.64150.2939,0.3411])([0.3298,0.42620.2142,0.3204])([0.2382,0.41250.2223,0.3351])([0.5243,0.62500.1972,0.2728])GIVIHFWG₁ℎ₁,ℎ₂,ℎ₃=IVIHFWGℎ₁,ℎ₂,ℎ₃=],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,([0.2699,0.38870.4203,0.5223])([0.2462,0.38010.4280,0.5483])([0.2828,0.41220.4134,0.5101])([0.2475,0.36800.3749,0.4792])([0.2259,0.35990.3832,0.5075])([0.2595,0.39030.3675,0.4659])([0.4676,0.70020.1751,0.2759])([0.4267,0.68470.1861,0.3153])([0.4901,0.74250.1654,0.2575])([0.4290,0.66290.1105,0.2106])([0.3914,0.64830.1223,0.2536])([0.4496,0.70300.1000,0.1905])([0.1780,0.38870.3533,0.4538])([0.1625,0.38010.3618,0.4835])([0.1866,0.41220.3456,0.4399])([0.1633,0.36800.3026,0.4046])([0.1490,0.35990.3119,0.4370])([0.1712,0.39030.2944,0.3894])GIVIHFWG₅ℎ₁,ℎ₂,ℎ₃=],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[.([0.2525,0.35680.4555,0.5505])([0.2357,0.35170.4565,0.5658])([0.2541,0.36010.4554,0.5500])([0.2382,0.35020.4522,0.5442])([0.2229,0.34540.4532,0.5602])([0.2395,0.35340.4520,0.5437])([0.4611,0.63550.2382,0.3230])([0.3971,0.59200.2501,0.4080])([0.4684,0.68150.2363,0.3185])([0.4051,0.58170.1326,0.2213])([0.3596,0.55400.1907,0.3840])([0.4098,0.60460.1000,0.1960])([0.1551,0.35680.3698,0.4667])([0.1472,0.35170.3720,0.4952])([0.1558,0.36010.3694,0.4657])([0.1484,0.35020.3617,0.4536])([0.1410,0.34540.3641,0.4851])([0.1491,0.35340.3613,0.4525]) (28)

Theorem 17. Let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of IVIHFEs, 𝜆>0, and 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 is the weight vector of ℎ_𝑖(𝑖=1,2,…,𝑛) with 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1. Then, the GIVIHFWA operator 𝐺𝐼𝑉𝐼𝐻𝐹𝑊𝐴_𝜆(ℎ₁,ℎ₂,…,ℎ_𝑛) is monotonically increasing with respect to the parameter 𝜆.

Proof. According to the proof of Theorem 3.8 in [13], we can obtain that (1−∏^𝑛_𝑖=1(1−(𝜇⁻_𝛼𝑖)^𝜆)^𝑤𝑖)^1/𝜆 and (1−∏^𝑛_𝑖=1(1−(𝜇⁺_𝛼𝑖)^𝜆)^𝑤𝑖)^1/𝜆 are monotonically increasing with respect to the parameter 𝜆. Furthermore, 1−(1−∏^𝑛_𝑖=1(1−(1−𝜈⁻_𝛼𝑖)^𝜆)^𝑤𝑖)^1/𝜆 and 1−(1−∏^𝑛_𝑖=1(1−(1−𝜈⁺_𝛼𝑖)^𝜆)^𝑤𝑖)^1/𝜆 are monotonically decreasing with respect to the parameter 𝜆. Therefore, by Definition 10, 𝑠GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1+1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1+1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}×^𝑛_𝑖=1#ℎ_𝑖⁻¹ (29)
is monotonically increasing with respect to the parameter 𝜆, which implies that GIVIHFWA_𝜆(ℎ₁,ℎ₂,…,ℎ_𝑛) is monotonically increasing with respect to the parameter 𝜆.

Theorem 18. Let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of IVIHFEs, 𝜆>0, and 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 is the weight vector of ℎ_𝑖(𝑖=1,2,…,𝑛) with 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1. Then, the GIVIHFWG operator 𝐺𝐼𝑉𝐼𝐻𝐹𝑊𝐺_𝜆(ℎ₁,ℎ₂,…,ℎ_𝑛) is monotonically decreasing with respect to the parameter 𝜆.

Proof. According to the proof of Theorem 17, we can obtain that 1−(1−∏^𝑛_𝑖=1(1−(1−𝜇⁻_𝛼𝑖)^𝜆)^𝑤𝑖)^1/𝜆 and 1−(1−∏^𝑛_𝑖=1(1−(1−𝜇⁺_𝛼𝑖)^𝜆)^𝑤𝑖)^1/𝜆 are monotonically decreasing with respect to the parameter 𝜆. Furthermore, (1−∏^𝑛_𝑖=1(1−(𝜈⁻_𝛼𝑖)^𝜆)^𝑤𝑖)^1/𝜆 and (1−∏^𝑛_𝑖=1(1−(𝜈⁺_𝛼𝑖)^𝜆)^𝑤𝑖)^1/𝜆 are monotonically increasing with respect to the parameter 𝜆. Therefore, by Definition 10, 𝑠GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}×^𝑛_𝑖=1#ℎ_𝑖⁻¹ (30)
is monotonically decreasing with respect to the parameter 𝜆, which implies that GIVIHFWG_𝜆(ℎ₁,ℎ₂,…,ℎ_𝑛) is monotonically decreasing with respect to the parameter 𝜆.

Lemma 19 (see [14, 15]). Let 𝑥_𝑖>0, 𝜆_𝑖>0, 𝑖=1,2,…,𝑛, and ∑^𝑛_𝑖=1𝜆_𝑖=1, then ^𝑛_𝑖=1𝑥_𝑖^𝜆𝑖≤^𝑛_𝑖=1𝜆_𝑖𝑥_𝑖 (31) with equality if and only if 𝑥₁=𝑥₂=⋯=𝑥_𝑛.

Theorem 20. Let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of IVIHFEs having the weight vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 such that 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1, 𝜆>0; then 𝐼𝑉𝐼𝐻𝐹𝑊𝐺ℎ₁,ℎ₂,…,ℎ_𝑛≤𝐺𝐼𝑉𝐼𝐻𝐹𝑊𝐴_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛. (32)

Proof. For any 𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛, from Lemma 19, we have ^𝑛_𝑖=1𝜇⁻_𝛼𝑖^𝑤𝑖=^𝑛_𝑖=1𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≤^𝑛_𝑖=1𝑤_𝑖𝜇⁻_𝛼𝑖^𝜆1/𝜆=1−^𝑛_𝑖=1𝑤_𝑖1−𝜇⁻_𝛼𝑖^𝜆1/𝜆≤1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},_𝑛𝑖=1𝜇⁺_𝛼𝑖^𝑤𝑖=^𝑛_𝑖=1𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≤^𝑛_𝑖=1𝑤_𝑖𝜇⁺_𝛼𝑖^𝜆1/𝜆=1−^𝑛_𝑖=1𝑤_𝑖1−𝜇⁺_𝛼𝑖^𝜆1/𝜆≤1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^𝑤𝑖≥1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≥1−^𝑛_𝑖=1𝑤_𝑖1−𝜈⁻_𝛼𝑖^𝜆1/𝜆≥1−1−^𝑛_𝑖=1𝑤_𝑖1−1−𝜈⁻_𝛼𝑖^𝜆1/𝜆≥1−1−^𝑛_𝑖=1𝑤_𝑖1−1−𝜈⁻_𝛼𝑖^𝜆1/𝜆≥1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^𝑤𝑖≥1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≥1−^𝑛_𝑖=1𝑤_𝑖1−𝜈⁺_𝛼𝑖^𝜆1/𝜆≥1−1−^𝑛_𝑖=1𝑤_𝑖1−1−𝜈⁺_𝛼𝑖^𝜆1/𝜆≥1−1−^𝑛_𝑖=1𝑤_𝑖1−1−𝜈⁺_𝛼𝑖^𝜆1/𝜆≥1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}. (33)(34)
By Definition 10, we have 𝑠IVIHFWGℎ₁,ℎ₂,…,ℎ_𝑛=12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛^𝑛𝑖=1𝜇⁻_𝛼𝑖^𝑤𝑖−1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^𝑤𝑖+^𝑛_𝑖=1𝜇⁺_𝛼𝑖^𝑤𝑖−1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^𝑤𝑖×#IVIHFWGℎ₁,ℎ₂,…,ℎ_𝑛⁻¹≤12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}×#GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛⁻¹=𝑠GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛. (35)
If 𝑠(IVIHFWG(ℎ₁,ℎ₂,…,ℎ_𝑛))<𝑠(GIVIHFWA_𝜆(ℎ₁,ℎ₂,…,ℎ_𝑛)), then by Definition 10, we have IVIHFWGℎ₁,ℎ₂,…,ℎ_𝑛<𝑠GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛. (36)
If 𝑠(IVIHFWG(ℎ₁,ℎ₂,…,ℎ_𝑛))=𝑠(GIVIHFWA_𝜆(ℎ₁,ℎ₂,…,ℎ_𝑛)), that is, 12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛^𝑛𝑖=1𝜇⁻_𝛼𝑖^𝑤𝑖−1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^𝑤𝑖+^𝑛_𝑖=1𝜇⁺_𝛼𝑖^𝑤𝑖−1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^𝑤𝑖×#IVIHFWGℎ₁,ℎ₂,…,ℎ_𝑛⁻¹=12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}×#GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛⁻¹, (37)
then, by the conditions that ^𝑛_𝑖=1𝜇⁻_𝛼𝑖^𝑤𝑖−1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^𝑤𝑖≤1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},_𝑛𝑖=1𝜇⁺_𝛼𝑖^𝑤𝑖−1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^𝑤𝑖≤1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},∀𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛, (38) we have ^𝑛_𝑖=1𝜇⁻_𝛼𝑖^𝑤𝑖−1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^𝑤𝑖=1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},_𝑛𝑖=1𝜇⁺_𝛼𝑖^𝑤𝑖−1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^𝑤𝑖=1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}. (39)
Furthermore, by the conditions that ^𝑛_𝑖=1𝜇⁻_𝛼𝑖^𝑤𝑖≤1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^𝑤𝑖≥1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},_𝑛𝑖=1𝜇⁺_𝛼𝑖^𝑤𝑖≤1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^𝑤𝑖≥1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},∀𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛, (40) we have ^𝑛_𝑖=1𝜇⁻_𝛼𝑖^𝑤𝑖=1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^𝑤𝑖=1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},_𝑛𝑖=1𝜇⁺_𝛼𝑖^𝑤𝑖=1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^𝑤𝑖=1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}. (41)
Thus, ℎIVIHFWGℎ₁,ℎ₂,…,ℎ_𝑛=12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛^𝑛𝑖=1𝜇⁻_𝛼𝑖^𝑤𝑖+1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^𝑤𝑖+^𝑛_𝑖=1𝜇⁺_𝛼𝑖^𝑤𝑖+1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^𝑤𝑖×#IVIHFWGℎ₁,ℎ₂,…,ℎ_𝑛⁻¹=12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}×#GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛⁻¹=ℎGIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛, (42)
which implies that IVIHFWGℎ₁,ℎ₂,…,ℎ_𝑛=GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛. (43)
Based on the previous analysis, we can conclude that (32) always holds.

According to Theorem 20, we can conclude that the values obtained by the IVIHFWG operator are not bigger than the ones obtained by the GIVIHFWA operator for any 𝜆>0.

If we let 𝜆=1 in Theorem 20, then we can obtain the following result.

Corollary 21. Suppose that ℎ_𝑖  (𝑖=1,2,…,𝑛) is a collection of IVIHFEs, and 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 is their weight vector with 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1; then one has 𝐼𝑉𝐼𝐻𝐹𝑊𝐺ℎ₁,ℎ₂,…,ℎ_𝑛≤𝐼𝑉𝐼𝐻𝐹𝑊𝐴ℎ₁,ℎ₂,…,ℎ_𝑛. (44)

Corollary 21 shows that the values obtained by the IVIHFWG operator are not bigger than the ones obtained by the IVIHFWA operator.

Theorem 22. Let ℎ_𝑖  (𝑖=1,2,…,𝑛) be a collection of IVIHFEs having the weight vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 such that 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1,  𝜆>0; then 𝐺𝐼𝑉𝐼𝐻𝐹𝑊𝐺_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛≤𝐼𝑉𝐼𝐻𝐹𝑊𝐴ℎ₁,ℎ₂,…,ℎ_𝑛. (45)

Proof. For any 𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛, by Lemma 19, we have 1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≤1−1−^𝑛_𝑖=1𝑤_𝑖1−1−𝜇⁻_𝛼𝑖^𝜆1/𝜆=1−^𝑛_𝑖=1𝑤_𝑖1−𝜇⁻_𝛼𝑖^𝜆1/𝜆≤1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^𝑤𝑖,1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≤1−1−^𝑛_𝑖=1𝑤_𝑖1−1−𝜇⁺_𝛼𝑖^𝜆1/𝜆=1−^𝑛_𝑖=1𝑤_𝑖1−𝜇⁺_𝛼𝑖^𝜆1/𝜆≤1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^𝑤𝑖,1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≥1−^𝑛_𝑖=1𝑤_𝑖1−𝜈⁻_𝛼𝑖^𝜆1/𝜆=^𝑛_𝑖=1𝑤_𝑖𝜈⁻_𝛼𝑖^𝜆1/𝜆≥^𝑛_𝑖=1𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=^𝑛_𝑖=1𝜈⁻_𝛼𝑖^𝑤𝑖,1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≥1−^𝑛_𝑖=1𝑤_𝑖1−𝜈⁺_𝛼𝑖^𝜆1/𝜆=^𝑛_𝑖=1𝑤_𝑖𝜈⁺_𝛼𝑖^𝜆1/𝜆≥^𝑛_𝑖=1𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=^𝑛_𝑖=1𝜈⁺_𝛼𝑖^𝑤𝑖. (46)
By Definition 10, we have that 𝑠GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}×#GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛⁻¹≤12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^𝑤𝑖−^𝑛_𝑖=1𝜈⁻_𝛼𝑖^𝑤𝑖+1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^𝑤𝑖−^𝑛_𝑖=1𝜈⁺_𝛼𝑖^𝑤𝑖×#IVIHFWAℎ₁,ℎ₂,…,ℎ_𝑛⁻¹=𝑠IVIHFWAℎ₁,ℎ₂,…,ℎ_𝑛. (47)
If 𝑠(GIVIHFWG_𝜆(ℎ₁,ℎ₂,…,ℎ_𝑛))<𝑠(IVIHFWA(ℎ₁,ℎ₂,…,ℎ_𝑛)), then by Definition 10, we have GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛<IVIHFWAℎ₁,ℎ₂,…,ℎ_𝑛. (48)
If 𝑠(GIVIHFWG_𝜆(ℎ₁,ℎ₂,…,ℎ_𝑛))=𝑠(IVIHFWA(ℎ₁,ℎ₂,…,ℎ_𝑛)), that is, 12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}×#GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛⁻¹=12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^𝑤𝑖−^𝑛_𝑖=1𝜈⁻_𝛼𝑖^𝑤𝑖+1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^𝑤𝑖−^𝑛_𝑖=1𝜈⁺_𝛼𝑖^𝑤𝑖×#IVIHFWAℎ₁,ℎ₂,…,ℎ_𝑛⁻¹, (49)
then, by the conditions that 1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≤1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^𝑤𝑖−^𝑛_𝑖=1𝜈⁻_𝛼𝑖^𝑤𝑖,1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≤1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^𝑤𝑖−^𝑛_𝑖=1𝜈⁺_𝛼𝑖^𝑤𝑖,∀𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛, (50)
we have 1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^𝑤𝑖−^𝑛_𝑖=1𝜈⁻_𝛼𝑖^𝑤𝑖,1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^𝑤𝑖−^𝑛_𝑖=1𝜈⁺_𝛼𝑖^𝑤𝑖. (51)
Furthermore, by the conditions that 1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≤1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^𝑤𝑖,1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≤1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^𝑤𝑖,1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≥^𝑛_𝑖=1𝜈⁻_𝛼𝑖^𝑤𝑖,1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≥^𝑛_𝑖=1𝜈⁺_𝛼𝑖^𝑤𝑖,∀𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛, (52) we have 1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^𝑤𝑖,1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^𝑤𝑖,1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=^𝑛_𝑖=1𝜈⁻_𝛼𝑖^𝑤𝑖,1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=^𝑛_𝑖=1𝜈⁺_𝛼𝑖^𝑤𝑖. (53)
Thus, ℎGIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}×#GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛⁻¹=12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^𝑤𝑖+^𝑛_𝑖=1𝜈⁻_𝛼𝑖^𝑤𝑖+1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^𝑤𝑖+^𝑛_𝑖=1𝜈⁺_𝛼𝑖^𝑤𝑖×#IVIHFWAℎ₁,ℎ₂,…,ℎ_𝑛⁻¹=ℎIVIHFWAℎ₁,ℎ₂,…,ℎ_𝑛, (54)
which implies that GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=IVIHFWAℎ₁,ℎ₂,…,ℎ_𝑛. (55)
Based on the previous analysis, we can conclude that (56) always holds.

Theorem 22 tells us that the values obtained by the GIVIHFWG operator are not bigger than the ones obtained by the IVIHFWA operator for any 𝜆>0.

Theorem 23. Let ℎ_𝑖  (𝑖=1,2,…,𝑛) be a collection of IVIHFEs having the weight vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 such that 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1,  𝜆>0; then 𝐺𝐼𝑉𝐼𝐻𝐹𝑊𝐺_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛≤𝐺𝐼𝑉𝐼𝐻𝐹𝑊𝐴_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛. (56)

Proof. According to the proof of Theorem 3.10 in [13], we can obtain 1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≥1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≥1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≤1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≤1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}. (57)
By Definition 10, we have 𝑠GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}×#GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛⁻¹≤12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}×#GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛⁻¹=𝑠GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛. (58)
If 𝑠(GIVIHFWG_𝜆(ℎ₁,ℎ₂,…,ℎ_𝑛))<𝑠(GIVIHFWA_𝜆(ℎ₁,ℎ₂,…,ℎ_𝑛)), then by Definition 10, we have GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛<GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛. (59) If 𝑠(GIVIHFWG_𝜆(ℎ₁,ℎ₂,…,ℎ_𝑛))=𝑠(GIVIHFWA_𝜆(ℎ₁,ℎ₂,…,ℎ_𝑛)), that is, 12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}×#GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛⁻¹=12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}×#GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛⁻¹, (60)
then, by the conditions that 1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≤1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≤1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},∀𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛, (61) we have 1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}−1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}. (62)
Furthermore, by the conditions that 1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≥1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≥1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≤1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}≤1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},∀𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛, (63) we have 1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}=1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}. (64)
Thus, ℎGIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−1−^𝑛_𝑖=11−1−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−^𝑛_𝑖=11−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−1−^𝑛_𝑖=11−1−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−^𝑛_𝑖=11−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}×#GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛⁻¹=12_𝛼1∈ℎ_1,𝛼2∈ℎ_{2,…,𝛼𝑛∈}ℎ_𝑛1−^𝑛_𝑖=11−𝜇⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−1−^𝑛_𝑖=11−1−𝜈⁻_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−^𝑛_𝑖=11−𝜇⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}+1−1−^𝑛_𝑖=11−1−𝜈⁺_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}×#GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛⁻¹=ℎGIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛, (65)
which implies that GIVIHFWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=GIVIHFWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛. (66)
Based on the previous analysis, we can conclude that (56) always holds.

Theorem 23 shows us that, for the same value of the parameter 𝜆>0 and the same aggregation values, the values obtained by the GIVIHFWA operator are always greater than the ones obtained by the GIVIHFWG operator.

We now look at some special cases of the IVIHFWA, IVIHFWG, GIVIHFWA, and GIVIHFWG operators obtained by using different choices of the input arguments and the weight vector.

(1) If ℎ_𝑖 (𝑖=1,2,…,𝑛) is a collection of HFEs, then the IVIHFWA operator reduces to the hesitant fuzzy weighted averaging (HFWA) operator [9], the IVIHFWG operator reduces to the hesitant fuzzy weighted geometric (HFWG) operator [9], the GIVIHFWA operator reduces to the generalized hesitant fuzzy weighted averaging (GHFWA) operator [9], and the GIVIHFWG operator reduces to the generalized hesitant fuzzy weighted geometric (GHFWG) operator [9].

(2) If ℎ_𝑖 (𝑖=1,2,…,𝑛) is a collection of IVHFEs, then the IVIHFWA operator reduces to the interval-valued hesitant fuzzy weighted averaging (IVHFWA) operator [6], the IVIHFWG operator reduces to the interval-valued hesitant fuzzy weighted geometric (IVHFWG) operator [6], the GIVIHFWA operator reduces to the generalized interval-valued hesitant fuzzy weighted averaging (GIVHFWA) operator [6], and the GIVIHFWG operator reduces to the generalized interval-valued hesitant fuzzy weighted geometric (GIVHFWG) operator [6].

(3) If ℎ_𝑖 (𝑖=1,2,…,𝑛) are a collection of IHFEs, then the IVIHFWA operator reduces to the intuitionistic hesitant fuzzy weighted averaging (IHFWA) operator: IHFWAℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝑤_𝑖ℎ_𝑖=1−^𝑛_𝑖=11−𝜇_𝛼𝑖^𝑤𝑖,^𝑛_𝑖=1𝜈_𝛼𝑖^𝑤𝑖∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛. (67)

If ℎ_𝑖 (𝑖=1,2,…,𝑛) are a collection of IHFEs, then the IVIHFWG operator reduces to the intuitionistic hesitant fuzzy weighted geometric (IHFWG) operator: IHFWGℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1ℎ_𝑖^𝑤𝑖=^𝑛_𝑖=1𝜇_𝛼𝑖^𝑤𝑖,1−^𝑛_𝑖=11−𝜈_𝛼𝑖^𝑤𝑖∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛. (68)

If ℎ_𝑖 (𝑖=1,2,…,𝑛) are a collection of IHFEs, then the GIVIHFWA operator reduces to the generalized intuitionistic hesitant fuzzy weighted averaging (GIHFWA) operator: GIHFWAℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝑤_𝑖ℎ^𝜆_𝑖^1/𝜆=1−1−^𝑛_𝑖=11−1−𝜇_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛. (69)

If ℎ_𝑖 (𝑖=1,2,…,𝑛) are a collection of IHFEs, then the GIVIHFWG operator reduces to the generalized intuitionistic hesitant fuzzy weighted geometric (GIHFWG) operator: GIHFWGℎ₁,ℎ₂,…,ℎ_𝑛=1𝜆^𝑛_𝑖=1𝜆ℎ_𝑖^𝑤𝑖=1−1−^𝑛_𝑖=11−1−𝜇_𝛼𝑖^{𝜆𝑤𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈_𝛼𝑖^{𝜆𝑤𝑖1/𝜆}∣𝛼₁∈ℎ₁,𝛼₂∈ℎ₂,…,𝛼_𝑛∈ℎ_𝑛. (70)

3.2. The IVIHFOWA, IVIHFOWG, GIVIHFOWA, and GIVIHFOWG Operators

Definition 24. Let ℎ_𝑖 (𝑖=1,2,…,𝑛) be a collection of IVIHFEs, ℎ_𝜎(𝑖) the 𝑖th largest one of them, and 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 the aggregation-associated vector such that 𝜔_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝜔_𝑖=1, then the following hold. (1) An interval-valued intuitionistic hesitant fuzzy ordered weighted averaging (IVIHFOWA) operator is a mapping 𝐻^𝑛→𝐻, where IVIHFOWAℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝜔_𝑖ℎ_𝜎(𝑖). (71) (2) An interval-valued intuitionistic hesitant fuzzy ordered weighted geometric (IVIHFOWG) operator is a mapping 𝐻^𝑛→𝐻, where IVIHFOWGℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1ℎ^𝜔𝑖_𝜎(𝑖). (72) (3) A generalized interval-valued intuitionistic hesitant fuzzy ordered weighted averaging (GIVIHFOWA) operator is a mapping 𝐻^𝑛→𝐻, where GIVIHFOWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝜔_𝑖ℎ^𝜆_𝜎(𝑖)^1/𝜆 (73) with 𝜆>0. (4) A generalized interval-valued intuitionistic hesitant fuzzy ordered weighted geometric (GIVIHFOWG) operator is a mapping 𝐻^𝑛→𝐻, where GIVIHFOWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=1𝜆^𝑛_𝑖=1𝜆ℎ_𝜎(𝑖)^𝜔𝑖 (74) with 𝜆>0.

In Definition 24, if 𝜔=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇, then the IVIHFOWA operator degenerates to the IVIHFA operator and the IVIHFOWG operator becomes the IVIHFG operator. If 𝜆=1, then the GIVIHFOWA operator reduces to the IVIHFOWA operator and the GIVIHFOWG operator reduces to the IVIHFOWG operator.

Using IVIHFE operations and mathematical induction on 𝑛, (71), (72), (73), and (74) can be transformed into the following forms: IVIHFOWAℎ₁,ℎ₂,…,ℎ_𝑛=1−^𝑛_𝑖=11−𝜇⁻_{𝛼𝜎(𝑖)}^𝜔𝑖,1−^𝑛_𝑖=11−𝜇⁺_{𝛼𝜎(𝑖)}^𝜔𝑖,_𝑛𝑖=1𝜈⁻_{𝛼𝜎(𝑖)}^𝜔𝑖,^𝑛_𝑖=1𝜈⁺_{𝛼𝜎(𝑖)}^𝜔𝑖∣𝛼_𝜎(1)∈ℎ_𝜎(1),𝛼_𝜎(2)∈ℎ_𝜎(2),…,𝛼_𝜎(𝑛)∈ℎ_𝜎(𝑛),GIVIHFOWA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=1−^𝑛_𝑖=11−𝜇⁻_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−^𝑛_𝑖=11−𝜇⁺_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜈⁻_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜈⁺_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆}∣𝛼_𝜎(1)∈ℎ_𝜎(1),𝛼_𝜎(2)∈ℎ_𝜎(2),…,𝛼_𝜎(𝑛)∈ℎ_𝜎(𝑛),IVIHFOWGℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝜇⁻_{𝛼𝜎(𝑖)}^𝜔𝑖,^𝑛_𝑖=1𝜇⁺_{𝛼𝜎(𝑖)}^𝜔𝑖,1−^𝑛_𝑖=11−𝜈⁻_{𝛼𝜎(𝑖)}^𝜔𝑖,1−^𝑛_𝑖=11−𝜈⁺_{𝛼𝜎(𝑖)}^𝜔𝑖∣𝛼_𝜎(1)∈ℎ_𝜎(1),𝛼_𝜎(2)∈ℎ_𝜎(2),…,𝛼_𝜎(𝑛)∈ℎ_𝜎(𝑛),GIVIHFOWG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=1−1−^𝑛_𝑖=11−1−𝜇⁻_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜇⁺_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈⁻_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈⁺_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆}∣𝛼_𝜎(1)∈ℎ_𝜎(1),𝛼_𝜎(2)∈ℎ_𝜎(2),…,𝛼_𝜎(𝑛)∈ℎ_𝜎(𝑛). (75)

Remark 25. The IVIHFWA, IVIHFWG, GIVIHFWA, and GIVIHFWG operators weight only the interval-valued intuitionistic hesitant fuzzy arguments. However, by Definition 24, the IVIHFOWA, IVIHFOWG, GIVIHFOWA, and GIVIHFOWG operators weight the ordered positions of the interval-valued intuitionistic hesitant fuzzy arguments instead of weighting the interval-valued intuitionistic hesitant fuzzy arguments themselves. The prominent characteristic of the IVIHFOWA, IVIHFOWG, GIVIHFOWA, and GIVIHFOWG operators is the reordering step in which the input arguments are rearranged in descending order; in particular, an interval-valued intuitionistic hesitant fuzzy argument ℎ_𝑖 is not associated with a particular weight 𝑤_𝑖, but rather a weight 𝑤_𝑖 is associated with a particular ordered position 𝑖 of the interval-valued intuitionistic hesitant fuzzy arguments.

Example 26. Assume that ℎ₁={([0.5,0.6],[0.3,0.4]),([0.7,0.8],[0.1,0.2]),([0.1,0.4],[0.5,0.6])}, ℎ₂={([0.3,0.5],[0.3,0.5]),([0.7,0.9],[0.1,0.1])}, and ℎ₃={([0.2,0.4],[0.1,0.3]),([0.6,0.8],[0.1,0.2])} are three IVIHFEs, and the aggregation-associated vector is 𝜔=(0.2,0.5,0.3)^𝑇. From Definition 24, we can calculate the score values of ℎ₁, ℎ₂, and ℎ₃ as follows: 𝑠ℎ₁=(0.2+0.2)+(0.6+0.6)+(−0.2−0.4)6𝑠ℎ=0.1667,₂=(0+0)+(0.6+0.8)4𝑠ℎ=0.3500,₃=(0.1+0.1)+(0.5+0.6)4=0.3250. (76)
Since 𝑠(ℎ₂)>𝑠(ℎ₃)>𝑠(ℎ₁), then ℎ_𝜎(1)=ℎ₂=],[,],[,ℎ{([0.3,0.50.3,0.5])([0.7,0.90.1,0.1])}_𝜎(2)=ℎ₃=],[,],[,ℎ{([0.2,0.40.1,0.3])([0.6,0.80.1,0.2])}_𝜎(3)=ℎ₁=],[,],[,],[,{([0.5,0.60.3,0.4])([0.7,0.80.1,0.2])([0.1,0.40.5,0.6])} (77)GIVIHFOWA₁ℎ₁,ℎ₂,ℎ₃=IVIHFOWAℎ₁,ℎ₂,ℎ₃=],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,([0.3235,0.48770.1732,0.3622])([0.4196,0.58390.1246,0.2942])([0.1931,0.42150.2019,0.4091])([0.5217,0.70420.1732,0.2958])([0.5896,0.75980.1246,0.2402])([0.4294,0.66600.2019,0.3340])([0.4290,0.62870.1390,0.2625])([0.5101,0.69840.1000,0.2132])([0.3189,0.58070.1621,0.2965])([0.5962,0.78560.1390,0.2144])([0.6536,0.82590.1000,0.1741])([0.5184,0.75790.1333,0.2421])GIVIHFOWA₇ℎ₁,ℎ₂,ℎ₃=],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,([0.4224,0.52400.1548,0.3451])([0.5921,0.68450.1178,0.2716])([0.2431,0.43350.1614,0.3599])([0.5561,0.73860.1548,0.2614])([0.6296,0.77830.1178,0.2217])([0.5442,0.73220.1614,0.2678])([0.5692,0.74730.1284,0.2262])([0.6359,0.78430.1000,0.1965])([0.5590,0.74140.1333,0.2308])([0.6147,0.80870.1284,0.1913])([0.6617,0.83060.1000,0.1694])([0.6085,0.80540.1333,0.1946])GIVIHFOWG₁ℎ₁,ℎ₂,ℎ₃=IVIHFOWGℎ₁,ℎ₂,ℎ₃=],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,([0.2855,0.47240.2063,0.3751])([0.3158,0.51490.1441,0.3188])([0.1762,0.41830.2825,0.4467])([0.4945,0.66800.2063,0.3320])([0.5471,0.72820.1441,0.2718])([0.3051,0.59150.2825,0.4085])([0.3382,0.53130.1654,0.2972])([0.3742,0.57920.1000,0.2338])([0.2087,0.47040.2455,0.3777])([0.5858,0.75130.1654,0.2487])([0.6481,0.81910.1000,0.1809])([0.3615,0.66530.2455,0.3347])GIVIHFOWG₇ℎ₁,ℎ₂,ℎ₃=],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[.([0.2543,0.44550.2717,0.4163])([0.2560,0.44790.2384,0.4014])([0.1658,0.41310.4223,0.5190])([0.4275,0.58660.2717,0.4134])([0.4371,0.60220.2384,0.3978])([0.2100,0.48170.4223,0.5183])([0.2675,0.45340.2526,0.3466])([0.2694,0.45600.1000,0.2731])([0.1712,0.41860.4212,0.5068])([0.5596,0.66260.2526,0.3374])([0.6312,0.80620.1000,0.1938])([0.2185,0.49400.4212,0.5060]) (78)

Similar to Theorem 17, we have the following result.

Theorem 27. Let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of IVIHFEs, 𝜆>0, and let 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 be the aggregation-associated vector such that 𝜔_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝜔_𝑖=1; then the GIVIHFOWA operator is monotonically increasing with respect to the parameter 𝜆.

Similar to Theorem 18, we have the following result.

Theorem 28. Let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of IVIHFEs, 𝜆>0, and let 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 be the aggregation-associated vector such that 𝜔_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝜔_𝑖=1; then the GIVIHFOWG operator is monotonically decreasing with respect to the parameter 𝜆.

Similar to Theorems 20, 22, and 23 and Corollary 21, we can obtain the following theorem.

Theorem 29. Let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of IVIHFEs, 𝜆>0, and let 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 be the aggregation-associated vector such that 𝜔_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝜔_𝑖=1; then 𝐼𝑉𝐼𝐻𝐹𝑂𝑊𝐺ℎ₁,ℎ₂,…,ℎ_𝑛≤𝐼𝑉𝐼𝐻𝐹𝑂𝑊𝐴ℎ₁,ℎ₂,…,ℎ_𝑛,𝐼𝑉𝐼𝐻𝐹𝑂𝑊𝐺ℎ₁,ℎ₂,…,ℎ_𝑛≤𝐺𝐼𝑉𝐼𝐻𝐹𝑂𝑊𝐴_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛,𝐺𝐼𝑉𝐼𝐻𝐹𝑂𝑊𝐺_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛≤𝐼𝑉𝐼𝐻𝐹𝑂𝑊𝐴ℎ₁,ℎ₂,…,ℎ_𝑛,𝐺𝐼𝑉𝐼𝐻𝐹𝑂𝑊𝐺_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛≤𝐺𝐼𝑉𝐼𝐻𝐹𝑂𝑊𝐴_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛. (79)

We now look at some special cases of the IVIHFOWA, IVIHFOWG, GIVIHFOWA, and GIVIHFOWG operators obtained by using different choices of the input arguments and the weight vector.

(1) If ℎ_𝑖 (𝑖=1,2,…,𝑛) is a collection of HFEs, then the IVIHFOWA operator reduces to the hesitant fuzzy ordered weighted averaging (HFOWA) operator [9], the IVIHFOWG operator reduces to the hesitant fuzzy ordered weighted geometric (HFOWG) operator [9], the GIVIHFOWA operator reduces to the generalized hesitant fuzzy ordered weighted averaging (GHFOWA) operator [9], and the GIVIHFOWG operator reduces to the generalized hesitant fuzzy ordered weighted geometric (GHFOWG) operator [9].

(2) If ℎ_𝑖 (𝑖=1,2,…,𝑛) is a collection of IVHFEs, then the IVIHFOWA operator reduces to the interval-valued hesitant fuzzy ordered weighted averaging (IVHFOWA) operator [6], the IVIHFOWG operator reduces to the interval-valued hesitant fuzzy ordered weighted geometric (IVHFOWG) operator [6], the GIVIHFOWA operator reduces to the generalized interval-valued hesitant fuzzy ordered weighted averaging (GIVHFOWA) operator [6], and the GIVIHFOWG operator reduces to the generalized interval-valued hesitant fuzzy ordered weighted geometric (GIVHFOWG) operator [6].

(3) If ℎ_𝑖 (𝑖=1,2,…,𝑛) is a collection of IHFEs, then the IVIHFOWA operator reduces to the intuitionistic hesitant fuzzy ordered weighted averaging (IHFOWA) operator: IHFOWAℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝜔_𝑖ℎ_𝜎(𝑖)=1−^𝑛_𝑖=11−𝜇_{𝛼𝜎(𝑖)}^𝜔𝑖,^𝑛_𝑖=1𝜈_{𝛼𝜎(𝑖)}^𝜔𝑖∣𝛼_𝜎(1)∈ℎ_𝜎(1),𝛼_𝜎(2)∈ℎ_𝜎(2),…,𝛼_𝜎(𝑛)∈ℎ_𝜎(𝑛). (80) If ℎ_𝑖 (𝑖=1,2,…,𝑛) is a collection of IHFEs, then the IVIHFOWG operator reduces to the intuitionistic hesitant fuzzy ordered weighted geometric (IHFOWG) operator: IHFOWGℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1ℎ^𝜔𝑖_𝜎(𝑖)=^𝑛_𝑖=1𝜇_{𝛼𝜎(𝑖)}^𝜔𝑖,1−^𝑛_𝑖=11−𝜈_{𝛼𝜎(𝑖)}^𝜔𝑖∣𝛼_𝜎(1)∈ℎ_𝜎(1),𝛼_𝜎(2)∈ℎ_𝜎(2),…,𝛼_𝜎(𝑛)∈ℎ_𝜎(𝑛). (81) If ℎ_𝑖 (𝑖=1,2,…,𝑛) is a collection of IHFEs, then the GIVIHFOWA operator reduces to the generalized intuitionistic hesitant fuzzy ordered weighted averaging (GIHFOWA) operator: GIHFOWAℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝜔_𝑖ℎ^𝜆_𝜎(𝑖)^1/𝜆=1−1−^𝑛_𝑖=11−1−𝜇_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆}∣𝛼_𝜎(1)∈ℎ_𝜎(1),𝛼_𝜎(2)∈ℎ_𝜎(2),…,𝛼_𝜎(𝑛)∈ℎ_𝜎(𝑛). (82) If ℎ_𝑖 (𝑖=1,2,…,𝑛) is a collection of IHFEs, then the GIVIHFOWG operator reduces to the generalized intuitionistic hesitant fuzzy ordered weighted geometric (GIHFOWG) operator: GIHFOWGℎ₁,ℎ₂,…,ℎ_𝑛=1𝜆^𝑛_𝑖=1𝜆ℎ_𝜎(𝑖)^𝜔𝑖=1−1−^𝑛_𝑖=11−1−𝜇_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆}∣𝛼_𝜎(1)∈ℎ_𝜎(1),𝛼_𝜎(2)∈ℎ_𝜎(2),…,𝛼_𝜎(𝑛)∈ℎ_𝜎(𝑛). (83)

By Definitions 11, 13, 15, and 24, it is worth noting that all the operators mentioned previously have some inherent limitations. Concretely, IVIHFWA, IVIHFWG, GIVIHFWA, and GIVIHFWG operators only weight the interval-valued intuitionistic hesitant fuzzy argument itself but ignore the importance of the ordered position of the arguments, whereas the IVIHFOWA, IVIHFOWG, GIVIHFOWA, and GIVIHFOWG operators only weight the ordered position of each given argument but ignore the importance of the argument. To overcome this drawback, we present some hybrid aggregation operators for interval-valued intuitionistic hesitant fuzzy arguments, which weight not only all the given arguments but also their ordered positions.

3.3. The IVIHFHA, IVIHFHG, GIVIHFHA, and GIVIHFHG Operators

Definition 30. For a collection of IVIHFEs ℎ_𝑖  (𝑖=1,2,…,𝑛), 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 is the weight vector of them with 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1, and 𝑛 is the balancing coefficient which plays a role of balance; then we define the following aggregation operators, which are all based on the mapping 𝐻^𝑛→𝐻 with an aggregation-associated vector 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 such that 𝜔_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝜔_𝑖=1. (1) The interval-valued intuitionistic hesitant fuzzy hybrid averaging (IVIHFHA) operator is IVIHFHAℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝜔ℎ_𝑖^⋅_𝜎(𝑖), (84) where ℎ^⋅_𝜎(𝑖) is the largest 𝑖th of ℎ^⋅_𝑘=𝑛𝑤_𝑘ℎ_𝑘 (𝑘=1,2,…,𝑛). (2) The interval-valued intuitionistic hesitant fuzzy hybrid geometric (IVIHFHG) operator is IVIHFHGℎ₁,ℎ₂,…,ℎ_𝑛=ℎ^𝑛_𝑖=1^⋅⋅_𝜎(𝑖)^𝜔𝑖, (85) where ℎ^⋅⋅_𝜎(𝑖) is the largest 𝑖th of ℎ^⋅⋅_𝑘=(ℎ_𝑘)^𝑛𝑤𝑘 (𝑘=1,2,…,𝑛). (3) The generalized interval-valued intuitionistic hesitant fuzzy hybrid averaging (GIVIHFHA) operator is GIVIHFHA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝜔ℎ_𝑖^⋅𝜆_𝜎(𝑖)^1/𝜆, (86) where 𝜆>0 and ℎ^⋅_𝜎(𝑖) is the largest 𝑖th of ℎ^⋅_𝑘=𝑛𝑤_𝑘ℎ_𝑘 (𝑘=1,2,…,𝑛). (4) The generalized interval-valued intuitionistic hesitant fuzzy hybrid geometric (GIVIHFHG) operator is GIVIHFHG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=1𝜆^𝑛_𝑖=1𝜆ℎ^⋅⋅_𝜎(𝑖)^𝜔𝑖, (87) where 𝜆>0 and ℎ^⋅⋅_𝜎(𝑖) is the largest 𝑖th of ℎ^⋅⋅_𝑘=(ℎ_𝑘)^𝑛𝑤𝑘 (𝑘=1,2,…,𝑛).
If 𝑤=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇 especially, then the IVIHFHA operator reduces to the IVIHFOWA operator, the IVIHFHG operator reduces to the IVIHFOWG operator, the GIVIHFHA operator reduces to the GIVIHFOWA operator, and the GIVIHFHG operator reduces to the GIVIHFOWG operator; if 𝜔=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇, then the IVIHFHA operator reduces to the IVIHFWA operator, the IVIHFHG operator reduces to the IVIHFWG operator, the GIVIHFHA operator reduces to the GIVIHFWA operator, and the GIVIHFHG operator reduces to the GIVIHFWG operator; if 𝜆=1, then the GIVIHFHA operator reduces to the IVIHFHA operator and the GIVIHFHG operator reduces to the IVIHFHG operator.

Using IVIHFE operations and mathematical induction on 𝑛, (84), (85), (86), and (87) can be transformed into the following forms: IVIHFHAℎ₁,ℎ₂,…,ℎ_𝑛=1−^𝑛_𝑖=11−𝜇^−⋅_{𝛼𝜎(𝑖)}^𝜔𝑖,1−^𝑛_𝑖=11−𝜇^+⋅_{𝛼𝜎(𝑖)}^𝜔𝑖,_𝑛𝑖=1𝜈^−⋅_{𝛼𝜎(𝑖)}^𝜔𝑖,^𝑛_𝑖=1𝜈^+⋅_{𝛼𝜎(𝑖)}^𝜔𝑖∣_⋅𝛼_𝜎(1)∈ℎ^⋅_𝜎(1),𝛼^⋅_𝜎(2)∈ℎ^⋅_𝜎(2),…,𝛼^⋅_𝜎(𝑛)∈ℎ^⋅_𝜎(𝑛),IVIHFHGℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝜇^−⋅⋅_{𝛼𝜎(𝑖)}^𝜔𝑖,^𝑛_𝑖=1𝜇^+⋅⋅_{𝛼𝜎(𝑖)}^𝜔𝑖,1−^𝑛_𝑖=11−𝜈^−⋅⋅_{𝛼𝜎(𝑖)}^𝜔𝑖,1−^𝑛_𝑖=11−𝜈^+⋅⋅_{𝛼𝜎(𝑖)}^𝜔𝑖∣_⋅⋅𝛼_𝜎(1)∈ℎ^⋅⋅_𝜎(1),𝛼^⋅⋅_𝜎(2)∈ℎ^⋅⋅_𝜎(2),…,𝛼^⋅⋅_𝜎(𝑛)∈ℎ^⋅⋅_𝜎(𝑛),GIVIHFHA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=1−^𝑛_𝑖=11−𝜇^−⋅_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−^𝑛_𝑖=11−𝜇^+⋅_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜈^−⋅_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜈^+⋅_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆}∣_⋅𝛼_𝜎(1)∈ℎ^⋅_𝜎(1),𝛼^⋅_𝜎(2)∈ℎ^⋅_𝜎(2),…,𝛼^⋅_𝜎(𝑛)∈ℎ^⋅_𝜎(𝑛),GIVIHFHG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=1−1−^𝑛_𝑖=11−1−𝜇^−⋅⋅_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜇^+⋅⋅_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈^−⋅⋅_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈^+⋅⋅_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆}∣_⋅⋅𝛼_𝜎(1)∈ℎ^⋅⋅_𝜎(1),𝛼^⋅⋅_𝜎(2)∈ℎ^⋅⋅_𝜎(2),…,𝛼^⋅⋅_𝜎(𝑛)∈ℎ^⋅⋅_𝜎(𝑛). (88)(89)(90)(91)

Example 31. Let ℎ₁={([0.1,0.2],[0.5,0.7]),([0.6,0.9],[0.1,0.1])}, ℎ₂={([0.5,0.6],[0.4,0.4]),([0.7,0.8],[0.1,0.2])}, and ℎ₃={([0.6,0.7],[0.2,0.3]),([0.2,0.5],[0.3,0.4]),([0.4,0.6],[0.2,0.3])} be three IVIHFEs, whose weight vector is  𝑤=(0.1,0.5,0.4)^𝑇, and the aggregation-associated vector is  𝜔=(0.3,0.5,0.2)^𝑇. Then we can obtain ℎ^⋅₁=3⋅0.1⋅ℎ₁=1−1−𝜇⁻_𝛼^0.3,1−1−𝜇⁺_𝛼^0.3,𝜈⁻_𝛼^0.3,𝜈⁺_𝛼^0.3∣𝛼∈ℎ₁=],[,],[,{([0.0311,0.06480.8123,0.8985])([0.2403,0.49880.5012,0.5012])}_⋅ℎ₂=3⋅0.5⋅ℎ₂=1−1−𝜇⁻_𝛼^1.5,1−1−𝜇⁺_𝛼^1.5,𝜈⁻_𝛼^1.5,𝜈⁺_𝛼^1.5∣𝛼∈ℎ₂=],[,],[,{([0.6464,0.74700.2530,0.2530])([0.8357,0.91060.0316,0.0894])}_⋅ℎ₃=3⋅0.4⋅ℎ₃=1−1−𝜇⁻_𝛼^1.2,1−1−𝜇⁺_𝛼^1.2,𝜈⁻_𝛼^1.2,𝜈⁺_𝛼^1.2∣𝛼∈ℎ₃=],[,],[,],[,𝑠{([0.6670,0.76420.1450,0.2358])([0.2349,0.56470.2358,0.3330])([0.4583,0.66700.1450,0.2358])}_⋅ℎ₁=−0.4695,𝑠ℎ^⋅₂𝑠=0.6282,_⋅ℎ₃=0.3376. (92)
Since 𝑠(ℎ^⋅₂)>𝑠(ℎ^⋅₃)>𝑠(ℎ^⋅₁), then we have ℎ^⋅_𝜎(1)=ℎ^⋅₂=],[,],[,{([0.6464,0.74700.2530,0.2530])([0.8357,0.91060.0316,0.0894])}_⋅ℎ_𝜎(2)=ℎ^⋅₃=],[,],[,],[,{([0.6670,0.76420.1450,0.2358])([0.2349,0.56470.2358,0.3330])([0.4583,0.66700.1450,0.2358])}_⋅ℎ_𝜎(3)=ℎ^⋅₁=],[,],[.{([0.0311,0.06480.8123,0.8985])([0.2403,0.49880.5012,0.5012])} (93)

According to Definition 30, we can obtain GIVIHFHA₁ℎ₁,ℎ₂,ℎ₃=IVIHFHAℎ₁,ℎ₂,ℎ₃=],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,([0.3579,0.46340.3937,0.4931])([0.5387,0.69690.2628,0.3031])([0.2124,0.39830.4268,0.5265])([0.3197,0.59910.3031,0.3489])([0.2967,0.43290.3937,0.4931])([0.4466,0.65110.2628,0.3031])([0.3866,0.49170.3446,0.4621])([0.5819,0.73960.2031,0.2604])([0.2294,0.42270.3804,0.4975])([0.3453,0.63580.2466,0.3091])([0.3204,0.45940.3446,0.4621])([0.4823,0.69090.2031,0.2604])GIVIHFHA₆ℎ₁,ℎ₂,ℎ₃=],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[.([0.1831,0.23620.6350,0.7196])([0.4081,0.61310.3853,0.3869])([0.1523,0.23450.6351,0.7201])([0.2762,0.56620.3869,0.3976])([0.1793,0.23590.6350,0.7196])([0.3842,0.60240.3853,0.3869])([0.1833,0.23630.6349,0.7196])([0.4094,0.61460.3838,0.3854])([0.1524,0.23460.6350,0.7200])([0.2766,0.56700.3854,0.3962])([0.1795,0.23600.6349,0.7196])([0.3852,0.60380.3838,0.3854]) (94)

If we utilize the GIVIHFHG operator to aggregate the three IVIHFEs ℎ₁, ℎ₂, and ℎ₃, then we have ℎ^⋅⋅₁=ℎ₁^3⋅0.1=𝜇⁻_𝛼^0.3,𝜇⁺_𝛼^0.3,1−1−𝜈⁻_𝛼^0.3,1−1−𝜈⁺_𝛼^0.3∣𝛼∈ℎ₁=],[,],[,{([0.5012,0.61700.1877,0.3032])([0.8579,0.96890.0311,0.0311])}_⋅⋅ℎ₂=ℎ₂^3⋅0.5=𝜇⁻_𝛼^1.5,𝜇⁺_𝛼^1.5,1−1−𝜈⁻_𝛼^1.5,1−1−𝜈⁺_𝛼^1.5∣𝛼∈ℎ₂=],[,],[,{([0.3536,0.46480.5352,0.5352])([0.5857,0.71550.1462,0.2845])}_⋅⋅ℎ₃=ℎ₃^3⋅0.4=𝜇⁻_𝛼^1.2,𝜇⁺_𝛼^1.2,1−1−𝜈⁻_𝛼^1.2,1−1−𝜈⁺_𝛼^1.2∣𝛼∈ℎ₃=],[,],[,],[,𝑠{([0.5417,0.65180.2349,0.3482])([0.1450,0.43530.3482,0.4583])([0.3330,0.54170.2349,0.3482])}_⋅⋅ℎ₁=0.5980,𝑠ℎ^⋅⋅₂𝑠=0.1546,_⋅⋅ℎ₃=0.1126. (95)

Since 𝑠(ℎ^⋅⋅₁)>𝑠(ℎ^⋅⋅₂)>𝑠(ℎ^⋅⋅₃), then we have ℎ^⋅⋅_𝜎(1)=ℎ^⋅⋅₁=],[,],[,{([0.5012,0.61700.1877,0.3032])([0.8579,0.96890.0311,0.0311])}_⋅⋅ℎ_𝜎(2)=ℎ^⋅⋅₂=],[,],[,{([0.3536,0.46480.5352,0.5352])([0.5857,0.71550.1462,0.2845])}_⋅⋅ℎ_𝜎(3)=ℎ^⋅⋅₃=],[,],[,],[.{([0.5417,0.65180.2349,0.3482])([0.1450,0.43530.3482,0.4583])([0.3330,0.54170.2349,0.3482])} (96)

According to Definition 30, we can obtain GIVIHFHG₁ℎ₁,ℎ₂,ℎ₃=IVIHFHGℎ₁,ℎ₂,ℎ₃=],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,([0.4417,0.55580.3315,0.4141])([0.3676,0.51070.3586,0.4375])([0.3982,0.53070.3315,0.4141])([0.5530,0.67610.1733,0.3019])([0.4937,0.64320.1874,0.3190])([0.5182,0.65780.1733,0.3019])([0.6169,0.79090.1933,0.2091])([0.5661,0.76960.2091,0.2210])([0.5871,0.77900.1933,0.2091])([0.6933,0.84750.1010,0.1525])([0.6526,0.83200.1093,0.1611])([0.6694,0.83890.1010,0.1525])GIVIHFHG₆ℎ₁,ℎ₂,ℎ₃=],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[,],[.([0.4704,0.57810.2790,0.3817])([0.4233,0.53400.3022,0.4023])([0.4266,0.54720.2790,0.3817])([0.5576,0.68090.1713,0.3006])([0.5410,0.66500.1800,0.3106])([0.5420,0.66930.1713,0.3006])([0.7263,0.86600.1285,0.1340])([0.7226,0.86310.1340,0.1362])([0.7228,0.86380.1285,0.1340])([0.7393,0.87840.0926,0.1216])([0.7361,0.87580.0962,0.1235])([0.7362,0.87650.0926,0.1216]) (97)

We now look at some special cases of the IVIHFHA, IVIHFHG, GIVIHFHA, and GIVIHFHG operators obtained by using different choices of the input arguments and the weight vector.

(1) If ℎ_𝑖 (𝑖=1,2,…,𝑛) is a collection of HFEs, then the IVIHFHA operator reduces to the hesitant fuzzy hybrid averaging (HFHA) operator [9], the IVIHFHG operator reduces to the hesitant fuzzy hybrid geometric (HFHG) operator [9], the GIVIHFHA operator reduces to the generalized hesitant fuzzy hybrid averaging (GHFHA) operator [9], and the GIVIHFHG operator reduces to the generalized hesitant fuzzy hybrid geometric (GHFHG) operator [9].

(2) If ℎ_𝑖 (𝑖=1,2,…,𝑛) is a collection of IVHFEs, then the IVIHFHA operator reduces to the interval-valued hesitant fuzzy hybrid averaging (IVHFHA) operator [6], the IVIHFHG operator reduces to the interval-valued hesitant fuzzy hybrid geometric (IVHFHG) operator [6], the GIVIHFHA operator reduces to the generalized interval-valued hesitant fuzzy hybrid averaging (GIVHFHA) operator [6], and the GIVIHFHG operator reduces to the generalized interval-valued hesitant fuzzy hybrid geometric (GIVHFHG) operator [6].

(3) If ℎ_𝑖 (𝑖=1,2,…,𝑛) is a collection of IHFEs, then the IVIHFHA operator reduces to the intuitionistic hesitant fuzzy hybrid averaging (IHFHA) operator: IHFHAℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝜔ℎ_𝑖^⋅_𝜎(𝑖)=1−^𝑛_𝑖=11−𝜇^⋅_{𝛼𝜎(𝑖)}^𝜔𝑖,^𝑛_𝑖=1𝜈^⋅_{𝛼𝜎(𝑖)}^𝜔𝑖∣_⋅𝛼_𝜎(1)∈ℎ^⋅_𝜎(1),𝛼^⋅_𝜎(2)∈ℎ^⋅_𝜎(2),…,𝛼^⋅_𝜎(𝑛)∈ℎ^⋅_𝜎(𝑛). (98) If ℎ_𝑖 (𝑖=1,2,…,𝑛) is a collection of IHFEs, then the IVIHFHG operator reduces to the intuitionistic hesitant fuzzy hybrid geometric (IHFHG) operator: IHFHGℎ₁,ℎ₂,…,ℎ_𝑛=ℎ^𝑛_𝑖=1^⋅⋅_𝜎(𝑖)^𝜔𝑖=^𝑛_𝑖=1𝜇^⋅⋅_{𝛼𝜎(𝑖)}^𝜔𝑖,1−^𝑛_𝑖=11−𝜈^⋅⋅_{𝛼𝜎(𝑖)}^𝜔𝑖∣_⋅⋅𝛼_𝜎(1)∈ℎ^⋅⋅_𝜎(1),𝛼^⋅⋅_𝜎(2)∈ℎ^⋅⋅_𝜎(2),…,𝛼^⋅⋅_𝜎(𝑛)∈ℎ^⋅⋅_𝜎(𝑛). (99) If ℎ_𝑖 (𝑖=1,2,…,𝑛) is a collection of IHFEs, then the GIVIHFHA operator reduces to the generalized intuitionistic hesitant fuzzy hybrid averaging (GIHFHA) operator: GIHFHA_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=^𝑛_𝑖=1𝜔ℎ_𝑖^⋅𝜆_𝜎(𝑖)^1/𝜆=1−^𝑛_𝑖=11−𝜇^⋅_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−1−^𝑛_𝑖=11−1−𝜈^⋅_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆}∣_⋅𝛼_𝜎(1)∈ℎ^⋅_𝜎(1),𝛼^⋅_𝜎(2)∈ℎ^⋅_𝜎(2),…,𝛼^⋅_𝜎(𝑛)∈ℎ^⋅_𝜎(𝑛). (100) If ℎ_𝑖 (𝑖=1,2,…,𝑛) is a collection of IHFEs, then the GIVIHFHG operator reduces to the generalized intuitionistic hesitant fuzzy hybrid geometric (GIHFHG) operator: GIHFHG_𝜆ℎ₁,ℎ₂,…,ℎ_𝑛=1𝜆^𝑛_𝑖=1𝜆ℎ^⋅⋅_𝜎(𝑖)^𝜔𝑖=1−1−^𝑛_𝑖=11−1−𝜇^⋅⋅_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆},1−^𝑛_𝑖=11−𝜈^⋅⋅_{𝛼𝜎(𝑖)}^{𝜆𝜔𝑖1/𝜆}∣_⋅⋅𝛼_𝜎(1)∈ℎ^⋅⋅_𝜎(1),𝛼^⋅⋅_𝜎(2)∈ℎ^⋅⋅_𝜎(2),…,𝛼^⋅⋅_𝜎(𝑛)∈ℎ^⋅⋅_𝜎(𝑛). (101)

In the present section, we introduce twelve kinds of aggregation operators for aggregating the interval-valued intuitionistic hesitant fuzzy information. To exhibit these operators more clearly, we list all of them in Table 1.

Table 1 

Interval-valued intuitionistic hesitant fuzzy aggregation operators.

4. An Approach to Multiple Attribute Group Decision-Making with Interval-Valued Intuitionistic Hesitant Fuzzy Information

In this section, we utilize the proposed interval-valued intuitionistic hesitant fuzzy aggregation operators to develop an approach to multiple attribute group decision-making with interval-valued intuitionistic hesitant fuzzy information. First, a multiple attribute group decision-making with interval-valued intuitionistic hesitant fuzzy information can be described as follows.

Let 𝑌={𝑌₁,𝑌₂,…,𝑌_𝑚} be a set of 𝑚 alternatives, 𝐺={𝐺₁,𝐺₂,…,𝐺_𝑛} a collection of 𝑛 attributes, whose weight vector is 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇, with 𝑤_𝑖∈[0,1], 𝑖=1,2,…,𝑛, and ∑^𝑛_𝑖=1𝑤_𝑖=1, and let 𝐷={𝐷₁,𝐷₂,…,𝐷_𝑙} be a set of 𝑙 decision makers, whose weight vector is 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑙)^𝑇, with 𝜔_𝑘∈[0,1], 𝑘=1,2,…,𝑙, and ∑^𝑙_𝑘=1𝜔_𝑘=1. Let 𝑅^(𝑘)=(𝑟̃^(𝑘)_𝑖𝑗)_𝑚×𝑛 be an interval-valued intuitionistic hesitant fuzzy decision matrix, where 𝑟̃^(𝑘)_𝑖𝑗={𝛾^(𝑘)_𝑖𝑗∣𝛾^(𝑘)_𝑖𝑗∈𝑟̃^(𝑘)_𝑖𝑗}={([𝜇⁻_{𝛾(𝑘)𝑖𝑗},𝜇⁺_{𝛾(𝑘)𝑖𝑗}],[𝜈⁻_{𝛾(𝑘)𝑖𝑗},𝜈⁺_{𝛾(𝑘)𝑖𝑗}])∣𝛾^(𝑘)_𝑖𝑗∈𝑟̃^(𝑘)_𝑖𝑗}∈𝐻 is an IVIHFE given by the decision maker 𝐷_𝑘∈𝐷, where [𝜇⁻_{𝛾(𝑘)𝑖𝑗},𝜇⁺_{𝛾(𝑘)𝑖𝑗}] indicates the possible degree range that the alternative 𝑌_𝑖∈𝑌 satisfies the attribute 𝐺_𝑗∈𝐺, while [𝜈⁻_{𝛾(𝑘)𝑖𝑗},𝜈⁺_{𝛾(𝑘)𝑖𝑗}] indicates the possible degree range that the alternative 𝑌_𝑖∈𝑌 does not satisfy the attribute 𝐺_𝑗∈𝐺.

In general, there are benefit attributes (i.e., the bigger the attribute values, the better) and cost attributes (i.e., the smaller the attribute values, the better) in a multiple attribute group decision-making problem. In such cases, we transform the attribute values of cost type into the attribute values of benefit type; that is, normalize the interval-valued intuitionistic hesitant fuzzy decision matrix 𝑅^(𝑘)=(𝑟̃^(𝑘)_𝑖𝑗)_𝑚×𝑛 into a corresponding interval-valued intuitionistic hesitant fuzzy decision matrix 𝐴^(𝑘)=(𝑎^(𝑘)_𝑖𝑗)_𝑚×𝑛=({𝛼^(𝑘)_𝑖𝑗∣𝛼^(𝑘)_𝑖𝑗∈𝑎^(𝑘)_𝑖𝑗})_𝑚×𝑛=({([𝜇⁻_{𝛼(𝑘)𝑖𝑗},𝜇⁺_{𝛼(𝑘)𝑖𝑗}],[𝜈⁻_{𝛼(𝑘)𝑖𝑗},𝜈⁺_{𝛼(𝑘)𝑖𝑗}])∣𝛼^(𝑘)_𝑖𝑗∈𝑎^(𝑘)_𝑖𝑗})_𝑚×𝑛 by the method given in [16, 17], where 𝑎^(𝑘)_𝑖𝑗=𝑟̃^(𝑘)_𝑖𝑗,forbenefitattribute𝐺_𝑗,𝑟̃^(𝑘)_𝑖𝑗^𝑐,forcostattribute𝐺_𝑗,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛,𝑘=1,2,…,𝑙, (102)

where (𝑟̃^(𝑘)_𝑖𝑗)^𝑐 is the complement of 𝑟̃^(𝑘)_𝑖𝑗 such that 𝑟̃^(𝑘)_𝑖𝑗^𝑐=𝛾^(𝑘)_𝑖𝑗^𝑐∣𝛾^(𝑘)_𝑖𝑗∈𝑟̃^(𝑘)_𝑖𝑗=𝜈⁻_{𝛾(𝑘)𝑖𝑗},𝜈⁺_{𝛾(𝑘)𝑖𝑗},𝜇⁻_{𝛾(𝑘)𝑖𝑗},𝜇⁺_{𝛾(𝑘)𝑖𝑗}∣𝛾^(𝑘)_𝑖𝑗∈𝑟̃^(𝑘)_𝑖𝑗. (103)

In the following, we utilize the proposed operators to develop an approach to multiple attribute group decision-making with interval-valued intuitionistic hesitant fuzzy information, which involves the following steps.

Algorithm 32. Consider the following.
Step  1. Transform the decision matrix 𝑅^(𝑘)=(𝑟̃^(𝑘)_𝑖𝑗)_𝑚×𝑛 into a normalized matrix 𝐴^(𝑘)=(𝑎^(𝑘)_𝑖𝑗)_𝑚×𝑛=({𝛼^(𝑘)_𝑖𝑗∣𝛼^(𝑘)_𝑖𝑗∈𝑎^(𝑘)_𝑖𝑗})_𝑚×𝑛=({([𝜇⁻_{𝛼(𝑘)𝑖𝑗},𝜇⁺_{𝛼(𝑘)𝑖𝑗}],[𝜈⁻_{𝛼(𝑘)𝑖𝑗},𝜈⁺_{𝛼(𝑘)𝑖𝑗}])∣𝛼^(𝑘)_𝑖𝑗∈𝑎^(𝑘)_𝑖𝑗})_𝑚×𝑛 based on (102).
Step  2. Utilize the GIVIHFWA operator (26): 𝑎_𝑖𝑗=GIVIHFWA_𝜆𝑎⁽¹⁾_𝑖𝑗,𝑎⁽²⁾_𝑖𝑗,…,𝑎^(𝑙)_𝑖𝑗=1−^𝑙_𝑘=11−𝜇⁻_{𝛼(𝑘)𝑖𝑗}^{𝜆𝜔𝑘1/𝜆},1−^𝑙_𝑘=11−𝜇⁺_{𝛼(𝑘)𝑖𝑗}^{𝜆𝜔𝑘1/𝜆},1−1−^𝑙_𝑘=11−1−𝜈⁻_{𝛼(𝑘)𝑖𝑗}^{𝜆𝜔𝑘1/𝜆},1−1−^𝑙_𝑘=11−1−𝜈⁺_{𝛼(𝑘)𝑖𝑗}^{𝜆𝜔𝑘1/𝜆}∣𝛼⁽¹⁾_𝑖𝑗∈𝑎⁽¹⁾_𝑖𝑗,𝛼⁽²⁾_𝑖𝑗∈𝑎⁽²⁾_𝑖𝑗,…,𝛼^(𝑙)_𝑖𝑗∈𝑎^(𝑙)_𝑖𝑗,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛, (104) or the GIVIHFWG operator (27): 𝑎_𝑖𝑗=GIVIHFWG_𝜆𝑎⁽¹⁾_𝑖𝑗,𝑎⁽²⁾_𝑖𝑗,…,𝑎^(𝑙)_𝑖𝑗=1−1−^𝑙_𝑘=11−1−𝜇⁻_{𝛼(𝑘)𝑖𝑗}^{𝜆𝜔𝑘1/𝜆},1−1−^𝑙_𝑘=11−1−𝜇⁺_{𝛼(𝑘)𝑖𝑗}^{𝜆𝜔𝑘1/𝜆},1−^𝑙_𝑘=11−𝜈⁻_{𝛼(𝑘)𝑖𝑗}^{𝜆𝜔𝑘1/𝜆},1−^𝑙_𝑘=11−𝜈⁺_{𝛼(𝑘)𝑖𝑗}^{𝜆𝜔𝑘1/𝜆}∣𝛼⁽¹⁾_𝑖𝑗∈𝑎⁽¹⁾_𝑖𝑗,𝛼⁽²⁾_𝑖𝑗∈𝑎⁽²⁾_𝑖𝑗,…,𝛼^(𝑙)_𝑖𝑗∈𝑎^(𝑙)_𝑖𝑗,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛, (105)
to aggregate all the individual interval-valued intuitionistic hesitant fuzzy decision matrix 𝐴^(𝑘)=(𝑎^(𝑘)_𝑖𝑗)_𝑚×𝑛  (𝑘=1,2,…,𝑙) into the collective interval-valued intuitionistic hesitant fuzzy decision matrix 𝐴=(𝑎_𝑖𝑗)_𝑚×𝑛=({𝛼_𝑖𝑗∣𝛼_𝑖𝑗∈𝑎_𝑖𝑗})_𝑚×𝑛=({([𝜇⁻_𝛼𝑖𝑗,𝜇⁺_𝛼𝑖𝑗],[𝜈⁻_𝛼𝑖𝑗,𝜈⁺_𝛼𝑖𝑗])∣𝛼_𝑖𝑗∈𝑎_𝑖𝑗})_𝑚×𝑛.
Step  3. Utilize the GIVIHFHA operator (90): 𝑎_𝑖=GIVIHFHA_𝜆𝑎_𝑖1,𝑎_𝑖2,…,𝑎_𝑖𝑛=1−^𝑛_𝑗=11−𝜇^−⋅_{𝛼𝑖𝜎(𝑗)}^{𝜆𝜉𝑗1/𝜆},1−^𝑛_𝑗=11−𝜇^+⋅_{𝛼𝑖𝜎(𝑗)}^{𝜆𝜉𝑗1/𝜆},1−1−^𝑛_𝑗=11−1−𝜈^−⋅_{𝛼𝑖𝜎(𝑗)}^{𝜆𝜉𝑗1/𝜆},1−1−^𝑛_𝑗=11−1−𝜈^+⋅_{𝛼𝑖𝜎(𝑗)}^{𝜆𝜉𝑗1/𝜆}∣_⋅𝛼_𝑖𝜎(1)∈𝑎^⋅_𝑖𝜎(1),𝛼^⋅_𝑖𝜎(2)∈𝑎^⋅_𝑖𝜎(2),…,𝛼^⋅_{𝑖𝜎(𝑛)}∈𝑎^⋅_{𝑖𝜎(𝑛)},𝑖=1,2,…,𝑚, (106) or the GIVIHFHG operator (91): 𝑎_𝑖=GIVIHFHG_𝜆𝑎_𝑖1,𝑎_𝑖2,…,𝑎_𝑖𝑛=1−1−^𝑛_𝑗=11−1−𝜇^−⋅⋅_{𝛼𝑖𝜎(𝑗)}^{𝜆𝜉𝑗1/𝜆},1−1−^𝑛_𝑗=11−1−𝜇^+⋅⋅_{𝛼𝑖𝜎(𝑗)}^{𝜆𝜉𝑗1/𝜆},1−^𝑛_𝑗=11−𝜈^−⋅⋅_{𝛼𝑖𝜎(𝑗)}^{𝜆𝜉𝑗1/𝜆},1−^𝑛_𝑗=11−𝜈^+⋅⋅_{𝛼𝑖𝜎(𝑗)}^{𝜆𝜉𝑗1/𝜆}∣_⋅⋅𝛼_𝑖𝜎(1)∈𝑎^⋅⋅_𝑖𝜎(1),𝛼^⋅⋅_𝑖𝜎(2)∈𝑎^⋅⋅_𝑖𝜎(2),…,𝛼^⋅⋅_{𝑖𝜎(𝑛)}∈𝑎^⋅⋅_{𝑖𝜎(𝑛)},𝑖=1,2,…,𝑚, (107)
to aggregate all the preference values 𝑎_𝑖𝑗  (𝑗=1,2,…,𝑛) in the 𝑖th line of 𝐴, and then derive the collective overall preference value 𝑎_𝑖={𝛼_𝑖∣𝛼_𝑖∈𝑎_𝑖}={([𝜇⁻_𝛼𝑖,𝜇⁺_𝛼𝑖],[𝜈⁻_𝛼𝑖,𝜈⁺_𝛼𝑖])∣𝛼_𝑖∈𝑎_𝑖}  (𝑖=1,2,…,𝑚) of the alternative 𝑌_𝑖  (𝑖=1,2,…,𝑚), where 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)^𝑇 is the associated weight vector of the GIVIHFHA (or GIVIHFHG) operator, with 𝜉_𝑗∈[0,1], 𝑗=1,2,…,𝑛, and ∑^𝑛_𝑗=1𝜉_𝑗=1.
Step  4. According to Definition 10, we calculate the score values 𝑠(𝑎_𝑖)  (𝑖=1,2,…,𝑚) of 𝑎_𝑖  (𝑖=1,2,…,𝑚): 𝑠𝑎_𝑖=(1/2)∑_𝛼𝑖∈𝑎_𝑖𝜇⁻_𝛼𝑖−𝜈⁻_𝛼𝑖+𝜇⁺_𝛼𝑖−𝜈⁺_𝛼𝑖#𝑎_𝑖. (108)
Step  5. Get the priority of the alternatives 𝑌_𝑖  (𝑖=1,2,…,𝑚) by ranking 𝑠(𝑎_𝑖)   (𝑖=1,2,…,𝑚).
Step  6. End.

5. The Application of the Developed Approach in Group Decision-Making Problems

5.1. An Illustrative Example

In the following, we use a practical example to illustrate the application of the approach proposed in Section 4.

Example 1 (see [18]). Let us consider a factory which intends to select a new site for new buildings. Three alternatives 𝑌_𝑖 (𝑖=1,2,3) are available, and the three decision makers 𝐷_𝑘(𝑘=1,2,3) consider three criteria to decide which site to choose: (1)𝐺₁ (price); (2)𝐺₂ (location); and (3)𝐺₃ (environment). Among the considered attributes, 𝐺₁ is of cost type, and 𝐺_𝑗(𝑗=2,3) are of benefit type. The weight vector of the decision makers 𝐷_𝑘(𝑘=1,2,3) is 𝜔=(0.1,0.6,0.3)^𝑇. The weight vector of the criteria 𝐺_𝑗(𝑗=1,2,3) is 𝑤=(0.5,0.2,0.3)^𝑇. Suppose that each decision maker provides his/her own interval-valued intuitionistic hesitant fuzzy decision matrix 𝑅^(𝑘)=(𝑟̃^(𝑘)_𝑖𝑗)_3×3(𝑘=1,2,3) as listed in Tables 2, 3, and 4, where 𝑟̃^(𝑘)_𝑖𝑗={𝛾^(𝑘)_𝑖𝑗∣𝛾^(𝑘)_𝑖𝑗∈𝑟̃^(𝑘)_𝑖𝑗} is an IVIHFE given by the decision maker 𝐷_𝑘.
Step  1. Based on (102), we transform the interval-valued intuitionistic hesitant fuzzy decision matrices 𝑅^(𝑘)=(𝑟̃^(𝑘)_𝑖𝑗)_3×3(𝑘=1,2,3) into the normalized matrices 𝐴^(𝑘)=(𝑎^(𝑘)_𝑖𝑗)_3×3(𝑘=1,2,3) (see Tables 5, 6, and 7).
Step  2. Let 𝜆=6, and utilize the GIVIHFWA operator (104) to aggregate all the individual interval-valued intuitionistic hesitant fuzzy decision matrices 𝐴^(𝑘)=(𝑎^(𝑘)_𝑖𝑗)_3×3 (𝑘=1,2,3) into the collective interval-valued intuitionistic hesitant fuzzy decision matrix 𝐴=(𝑎_𝑖𝑗)_3×3 (see Table 8).
Step  3. Utilize the GIVIHFHA operator (106) (whose associated weighting vector is 𝜉=(0.25,0.65,0.1)^𝑇 and 𝜆=6) to aggregate all the preference values 𝑎_𝑖𝑗 (𝑗=1,2,3) in the 𝑖th line of 𝐴, and then derive the collective overall preference value 𝑎_𝑖  (𝑖=1,2,3) of the alternative 𝑌_𝑖(𝑖=1,2,3). We will not list the collective overall preference values here because of space limitations.
Step 4. According to (108), we calculate the score values 𝑠(𝑎_𝑖) (𝑖=1,2,3) of 𝑎_𝑖 (𝑖=1,2,3): 𝑠𝑎₁=0.4314,𝑠𝑎₂𝑠𝑎=0.3990,₃=0.4852. (109)
Step  5. By (𝑎_𝑖)  (𝑖=1,2,3), rank all the alternatives 𝑌_𝑖(𝑖=1,2,3) in descending order: 𝑌₃>𝑌₁>𝑌₂. (110)
Thus, the best alternative is 𝑌₃.

Table 2 

The interval-valued intuitionistic hesitant fuzzy decision matrix 𝑅⁽¹⁾ provided by 𝐷₁.

Table 3 

The interval-valued intuitionistic hesitant fuzzy decision matrix 𝑅⁽²⁾ provided by 𝐷₂.

Table 4 

The interval-valued intuitionistic hesitant fuzzy decision matrix 𝑅⁽³⁾ provided by 𝐷₃.

Table 5 

The interval-valued intuitionistic hesitant fuzzy decision matrix 𝐴⁽¹⁾.

Table 6 

The interval-valued intuitionistic hesitant fuzzy decision matrix 𝐴⁽²⁾.

Table 7 

The interval-valued intuitionistic hesitant fuzzy decision matrix 𝐴⁽³⁾.

Table 8 

The collective interval-valued intuitionistic hesitant fuzzy decision matrix 𝐴.

As the parameter 𝜆 changes, we can get different results (see Table 9). From Table 9, we can find that the score values obtained by the GIVIHFWA and GIVIHFHA operators become bigger as the parameter 𝜆 increases for the same aggregation arguments, and the decision makers can choose the values of 𝜆 according to their preferences.

Table 9 

Score values obtained by the GIVIHFWA and GIVIHFHA operators and the rankings of alternatives.

Furthermore, it is possible to analyze how the different attitudinal character 𝜆 plays a role in the aggregation results; in this case, we consider different values of 𝜆: 0.0001,0.0002,0.0003,0.0004,0.0005,…,50, which are provided by the decision maker. The score functions 𝑠(𝑎_𝑖)  (𝑖=1,2,3) of the collective overall preference values 𝑎_𝑖  (𝑖=1,2,3) of the alternatives 𝑌_𝑖  (𝑖=1,2,3) are shown in Figure 1.

Figure 1 

Variation of the score functions with respect to the parameter 𝜆.

It is observed from Figure 1 that all of 𝑠(𝑎_𝑖)  (𝑖=1,2,3) increase as 𝜆 increases. From Figure 1, we can find that(1)when 𝜆∈(0,1.4234), the ranking of the three alternatives is 𝑌₃>𝑌₂>𝑌₁ and the best choice is 𝑌₃;(2)when 𝜆∈[1.4234,22.2700], the ranking of the three alternatives is 𝑌₃>𝑌₁>𝑌₂ and the best choice is 𝑌₃;(3)when 𝜆∈(22.2700,37.7207), the ranking of the three alternatives is 𝑌₃>𝑌₂>𝑌₁ and the best choice is 𝑌₃;(4)when 𝜆∈[37.7207,50], the ranking of the three alternatives is 𝑌₂>𝑌₃>𝑌₁ and the best choice is 𝑌₂.

In Steps 2 and 3, instead of the GIVIHFWA and GIVIHFHA operators, if we use the GIVIHFWG and GIVIHFHG operators to aggregate the values of the alternatives, the score values and the rankings of the alternatives are listed in Table 10. By Table 10, we can find that the score values obtained by the GIVIHFWG and GIVIHFHG operators become smaller as the parameter 𝜆 increases for the same aggregation arguments, and the decision makers can choose the values of 𝜆 according to their preferences.

Table 10 

Score values obtained by the GIVIHFWG and GIVIHFHG operators and the rankings of alternatives.

Furthermore, it is possible to analyze how the different attitudinal character 𝜆 plays a role in the aggregation results; in this case, we consider different values of 𝜆: 0.0001,0.0002,0.0003,0.0004,0.0005,…,50, which are provided by the decision maker. The score functions 𝑠(𝑎_𝑖)  (𝑖=1,2,3) of the collective overall preference values 𝑎_𝑖  (𝑖=1,2,3) of the alternatives 𝑌_𝑖  (𝑖=1,2,3) are shown in Figure 2.

Figure 2 

Variation of the score functions with respect to the parameter 𝜆.

It is observed from Figure 2 that all of 𝑠(𝑎_𝑖)  (𝑖=1,2,3) decrease as 𝜆 increases. From Figure 2, we can find that(1)when 𝜆∈(0,6.1065), the ranking of the three alternatives is 𝑌₂>𝑌₁>𝑌₃ and the best choice is 𝑌₂;(2)when 𝜆∈[6.1065,7.1504], the ranking of the three alternatives is 𝑌₂>𝑌₃>𝑌₁ and the best choice is 𝑌₂;(3)when 𝜆∈(7.1504,50], the ranking of the three alternatives is 𝑌₃>𝑌₂>𝑌₁ and the best choice is 𝑌₃.

5.2. Comparison with the Existing Hesitant Fuzzy Aggregation Operators and MAGDM Methods

In the following, we compare our operators and methods with the existing aggregation operators and methods to demonstrate the advantages of the operators and methods proposed here.

Because interval-valued intuitionistic hesitant fuzzy set is a substantial and important generalization of hesitant fuzzy set, interval-valued hesitant fuzzy set, and intuitionistic hesitant fuzzy set, the developed interval-valued intuitionistic hesitant fuzzy aggregation operators are substantial and important generalizations of the existing hesitant fuzzy aggregation operators [9], interval-valued hesitant fuzzy aggregation operators [6], and intuitionistic hesitant fuzzy aggregation operators ((67), (68), (69), (70), (80), (81), (82), (83), (98), (99), (100), and (101)). Thus, the developed interval-valued intuitionistic hesitant fuzzy MAGDM method is a substantial and important generalization of the existing hesitant fuzzy MAGDM method [9], interval-valued hesitant fuzzy MAGDM method [6], and generalized hesitant fuzzy MAGDM method [7]. Concretely, our operators and methods can be applied to decision-making problems in which the attribute values take the form of hesitant fuzzy elements, interval-valued hesitant fuzzy elements, and intuitionistic hesitant fuzzy elements. In contrast, the existing hesitant fuzzy aggregation operators and MAGDM method, interval-valued hesitant fuzzy aggregation operators and MAGDM method, and intuitionistic hesitant fuzzy aggregation operators and MAGDM method cannot be applied to decision-making problems in which the attribute values are given in the form of interval-valued intuitionistic hesitant fuzzy elements. In other words, our operators and methods have much wider applications than the existing hesitant fuzzy aggregation operators and MAGDM method, interval-valued hesitant fuzzy aggregation operators and MAGDM method, and intuitionistic hesitant fuzzy aggregation operators and MAGDM method. For example, Example 1 cannot be handled by the existing hesitant fuzzy aggregation operators and MAGDM method, interval-valued hesitant fuzzy aggregation operators and MAGDM method, and intuitionistic hesitant fuzzy aggregation operators and MAGDM method, whereas our operators and methods can deal with Example 4 in [9], Example 6 in [6], and Example 4 in [7].

5.3. Comparison with the Existing Intuitionistic Fuzzy Aggregation Operators and MAGDM Methods

Considering that interval-valued intuitionistic hesitant fuzzy set is a substantial and important generalization of intuitionistic fuzzy set and interval-valued intuitionistic fuzzy set, the developed interval-valued intuitionistic hesitant fuzzy aggregation operators and MAGDM method are substantial and important generalizations of the existing intuitionistic fuzzy aggregation operators and MAGDM methods and the existing interval-valued intuitionistic fuzzy aggregation operators and MAGDM methods. In other words, our operators and methods can be applied to decision-making problems in which the attribute values are in the form of intuitionistic fuzzy numbers and interval-valued intuitionistic fuzzy numbers, whereas the existing intuitionistic fuzzy aggregation operators and MAGDM methods and the existing interval-valued intuitionistic fuzzy aggregation operators and MAGDM methods have difficulty in dealing with decision-making problems in which the attribute values are given in the form of interval-valued intuitionistic hesitant fuzzy elements. In short, our operators and methods have much wider applications than the existing intuitionistic fuzzy aggregation operators and MAGDM methods and the existing interval-valued intuitionistic fuzzy aggregation operators and MAGDM methods.

6. Conclusions

In this paper, we first propose the concept of interval-valued intuitionistic hesitant fuzzy sets, discuss their some basic properties, and develop some operational rules for interval-valued intuitionistic hesitant fuzzy elements. Then, we focus on interval-valued intuitionistic hesitant fuzzy information aggregation techniques and propose a series of interval-valued intuitionistic hesitant fuzzy aggregation operators. Moreover, we apply the developed aggregation operators to multiple attribute group decision-making with interval-valued intuitionistic hesitant fuzzy information. Finally, a numerical example is used to illustrate the validity of the proposed approach in group decision-making problems.

In group decision-making problems, because the experts usually come from different specialty fields and have different backgrounds and levels of knowledge, they usually have diverging opinions. Therefore, how to obtain the maximum degree of consensus or agreement from these experts for the given alternatives is an interesting and important research topic which has been receiving more and more attention in recent years. However, there are not similar studies completed for group decision-makings with interval-valued intuitionistic hesitant fuzzy information. Thus, in future work, we will present a consensus model for group decision-making with interval-valued intuitionistic hesitant fuzzy information.

Acknowledgments

The authors thank the anonymous referees for their valuable suggestions for improving this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 61073121) and the Natural Science Foundation of the Hebei Province of China (Grant nos. F2012201014, F2013201060, and A2012201033).