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

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.


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 intervalvalued intuitionistic fuzzy environments and permits the membership of an element to be a set of several possible interval-valued intuitionistic fuzzy numbers.Thus, intervalvalued 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

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.

Hesitant Fuzzy Sets and Hesitant Fuzzy Elements
Definition 1 (see [4]).Let  be a fixed set; a hesitant fuzzy set (HFS) on  is given in terms of a function that when applied to  returns a subset of [0, 1].
To be easily understood, we express the HFS by a mathematical symbol where ℎ  () is a set of some values in [0, 1], denoting the possible membership degrees of the element  ∈  to the set .For convenience, Xia and Xu [9] called ℎ = ℎ  () a hesitant fuzzy element (HFE) and  the set of all HFEs.
Given three HFEs represented by ℎ, ℎ 1 , and ℎ 2 , Torra [4] defined some operations on them, which can be described as (2) Furthermore, in order to aggregate hesitant fuzzy information, Xia and Xu [9] defined some new operations on the HFEs ℎ, ℎ 1 , and ℎ 2 : To compare the HFEs, Xia and Xu [9] defined the following comparison laws.

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  be a fixed set; an interval-valued intuitionistic hesitant fuzzy set (IVIHFS) on  is given in terms of a function that when applied to  returns a subset of Ω.
To be easily understood, we express the IVIHFS by a mathematical symbol where ℎ Ẽ() is a set of some IVIFNs in Ω, denoting the possible membership degree intervals and nonmembership degree intervals of the element  ∈  to the set Ẽ.
For convenience, we call h = ℎ Ẽ() an interval-valued intuitionistic hesitant fuzzy element (IVIHFE) and H the set of all IVIHFEs.If  ∈ h, then  is an IVIFN, and it can be denoted by 1], then h reduces to a hesitant fuzzy element (HFE) [9]; if  is a closed subinterval of the unit interval, then h reduces to an interval-valued hesitant fuzzy element (IVHFE) [6]; if  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 Theorem 8. Let h, h1 , and h2 be three IVIHFEs, and  ≥ 0.
To compare the IVIHFEs, we define the following comparison laws.

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 If  = (1/, 1/, . . ., 1/)  especially, then the IVIHFWA operator reduces to the interval-valued intuitionistic hesitant fuzzy averaging (IVIHFA) operator: Theorem 12. Let h ( = 1, 2, . . ., ) be a collection of IVIHFEs.Then, their aggregated value calculated using the IVIHFWA operator is an IVIHFE and

Journal of Applied Mathematics
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 .First, we show that (16) holds for  = 2.Because we have If ( 16) holds for  = , in other words, then, when  =  + 1, IVIHFE operations yield IVIHFWA ( h1 , h2 , . . ., h , h+1 ) In other words, ( 16) holds for  =  + 1. Equation ( 16) therefore holds for all .
This completes the proof of Theorem 12.
If  = 1, then the GIVIHFWA operator reduces to the IVIHFWA operator and the GIVIHFWG operator reduces to the IVIHFWG operator.
Proof.According to the proof of Theorem 3.8 in [13], we can obtain that are monotonically decreasing with respect to the parameter .Therefore, by Definition 10, is monotonically increasing with respect to the parameter , which implies that GIVIHFWA  ( h1 , h2 , . . ., h ) is monotonically increasing with respect to the parameter .
Proof.According to the proof of Theorem 17, we can obtain that are monotonically decreasing with respect to the parameter .Furthermore, are monotonically increasing with respect to the parameter .Therefore, by Definition 10, is monotonically decreasing with respect to the parameter , which implies that GIVIHFWG  ( h1 , h2 , . . ., h ) is monotonically decreasing with respect to the parameter .
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  > 0.
If we let  = 1 in Theorem 20, then we can obtain the following result.
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  > 0.
Theorem 23 shows us that, for the same value of the parameter  > 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, IVI-HFWG, GIVIHFWA, and GIVIHFWG operators obtained by using different choices of the input arguments and the weight vector.
Using IVIHFE operations and mathematical induction on , (71), ( 72), (73), and (74) can be transformed into the following forms: (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 is not associated with a particular weight   , but rather a weight   is associated with a particular ordered position  of the interval-valued intuitionistic hesitant fuzzy arguments.
Similar to Theorem 17, we have the following result.
Similar to Theorem 18, we have the following result.
Similar to Theorems 20, 22, and 23 and Corollary 21, we can obtain the following theorem.
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.
(2) If h ( = 1, 2, . . ., ) 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]. ( 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 GIVIH-FOWG 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.If  = (1/, 1/, . . ., 1/)  especially, then the IVIHFHA operator reduces to the IVIHFOWA operator, the IVIHFHG operator reduces to the IVIHFOWG operator, the GIVI-HFHA operator reduces to the GIVIHFOWA operator, and the GIVIHFHG operator reduces to the GIVIHFOWG operator; if  = (1/, 1/, . . ., 1/)  , 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  = 1, then the GIVIHFHA operator reduces to the IVIHFHA operator and the GIVIHFHG operator reduces to the IVIHFHG operator.

An Illustrative Example.
In the following, we use a practical example to illustrate the application of the approach proposed in Section 4.
As the parameter  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  increases for the same aggregation arguments, and the decision makers can choose the values of  according to their preferences.

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 intervalvalued 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, intervalvalued hesitant fuzzy aggregation operators and MAGDM method, and intuitionistic hesitant fuzzy aggregation operators and MAGDM method cannot be applied to decisionmaking 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].

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.

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 intervalvalued intuitionistic hesitant fuzzy elements.Then, we focus on interval-valued intuitionistic hesitant fuzzy information aggregation techniques and propose a series of intervalvalued intuitionistic hesitant fuzzy aggregation operators.Moreover, we apply the developed aggregation operators to multiple attribute group decision-making with intervalvalued 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.

Figure 2 :
Figure 2: Variation of the score functions with respect to the parameter .