#### Abstract

The definition of hesitant interval-valued intuitionistic fuzzy sets (HIVIFSs) is developed based on interval-valued intuitionistic fuzzy sets (IVIFSs) and hesitant fuzzy sets (HFSs). Then, some operations on HIVIFSs are introduced in detail, and their properties are further discussed. In addition, some hesitant interval-valued intuitionistic fuzzy number aggregation operators based on *t*-conorms and *t*-norms are proposed, which can be used to aggregate decision-makers' information in multicriteria decision-making (MCDM) problems. Some valuable proposals of these operators are studied. In particular, based on algebraic and Einstein *t*-conorms and *t*-norms, some hesitant interval-valued intuitionistic fuzzy algebraic aggregation operators and Einstein aggregation operators can be obtained, respectively. Furthermore, an approach of MCDM problems based on the proposed aggregation operators is given using hesitant interval-valued intuitionistic fuzzy information. Finally, an illustrative example is provided to demonstrate the applicability and effectiveness of the developed approach, and the study is supported by a sensitivity analysis and a comparison analysis.

#### 1. Introduction

Since fuzzy sets were proposed by Zadeh [1], the studies on multicriteria decision-making (MCDM) problems have made great progress. Further, fuzzy sets were generalized to intuitionistic fuzzy sets (IFSs) by Atanassov [2, 3], where each element in an IFS has a membership degree and a nonmembership degree between 0 and 1, respectively. Then, Atanassov and Gargov [4] proposed the notion of interval-valued intuitionistic fuzzy sets (IVIFSs) which are the extension of IFSs, where the membership degree and nonmembership degree of an element in an IVIFS are, respectively, represented by intervals in rather than crisp values between 0 and 1. In recent years, many researchers have studied the theory of IVIFSs and applied it to various fields [5–8]. For instance, Atanassov [9] introduced the operators of IVIFSs. Lee [10] proposed a method for ranking interval-valued intuitionistic fuzzy numbers (IVIFNs) for fuzzy decision-making problems. Lee [11] provided an enhanced MCDM method of machine design schemes under the interval-valued intuitionistic fuzzy environment. Li [12] proposed a TOPSIS based nonlinear-programming method for MCDM problems with IVIFSs. Park et al. [13] extended the TOPSIS method to solve group MCDM problems in interval-valued intuitionistic fuzzy environment in which all the preference information provided by decision-makers is presented as IVIFNs. Chen et al. [14] developed an approach to tackle group MCDM problems in the context of IVIFSs. Nayagam and Sivaraman [15] introduced a method for ranking IVIFSs and compared it to other methods by means of numerical examples. Chen et al. [16] presented a MCDM method based on the proposed interval-valued intuitionistic fuzzy weighted average (IVIFWA) operator. Meng et al. [17] developed an induced generalized interval-valued intuitionistic fuzzy hybrid Shapley averaging (GIVIFHSA) operator and applied it to MCDM problems.

Hesitant fuzzy sets (HFSs), another extension of traditional fuzzy sets, provide a useful reference for our study under hesitant fuzzy environment. HFSs were first introduced by Torra and Narukawa [18], and they permit the membership degrees of an element to be a set of several possible values between 0 and 1. HFSs are highly useful in handling the situations where people have hesitancy in providing their preferences over objects in the decision-making process. Some aggregation operators of HFSs were studied and applied to decision-making problems [19–21]. Then, the correlation coefficients of HFSs, the distance measures, and correlation measures of HFSs were discussed [22–24], based on which Peng et al. [25] presented a generalized hesitant fuzzy synergetic weighted distance measure. Zhang and Wei [26] developed the E-VIKOR method and TOPSIS method to solve MCDM problems with hesitant fuzzy information. Zhang [27] developed a wide range of hesitant fuzzy power aggregation operators for hesitant fuzzy information. Chen et al. [28] generalized the concept of HFSs to hesitant interval-valued fuzzy sets (HIVFSs) in which the membership degrees of an element to a given set are not exactly defined but denoted by several possible interval values. Wei [29] defined HIVFSs and some hesitant interval-valued fuzzy aggregation operators. Wei and Zhao [30] developed some Einstein operations on HIVFSs and the induced hesitant interval-valued fuzzy Einstein aggregation (HIVFEA) operators and applied them to MCDM problems. Zhu et al. [31] defined dual HFSs (DHFSs) in terms of two functions that return two sets of membership degrees and nonmembership degrees rather than crisp numbers in HFSs. If the idea of dual HFSs is used from a new perspective, then another extension of HFSs may be defined in terms of one function that the element of HFSs returns a set of IFSs, which are called hesitant intuitionistic fuzzy sets (HIFSs). But decision-makers usually cannot estimate criteria values of alternatives with exact numerical values when the information is not known precisely. Therefore, interval values in fuzzy sets can represent it better than specific numbers, such as interval-valued fuzzy sets (IVFSs) and IVIFSs. Furthermore, although the theories of IVIFSs and HFSs have been developed and generalized, they cannot deal with all sorts of uncertainties in different real problems. For example, when we ask the opinion of an expert about a certain statement, he or she may answer that the possibility that the statement is true is [0.1, 0.2] and that the statement is false is [0.4, 0.5], or the possibility that the statement is true is [0.5, 0.6] and that the statement is false is [0.3, 0.5]. This issue is beyond the scope of IVFSs and IVIFSs. Therefore, some new theories are required.

So the concept of hesitant interval-valued intuitionistic fuzzy sets (HIVIFSs) is developed in this paper. Comparing to the existing fuzzy sets mentioned above, HIVIFSs are a new extension of HFSs, which support a more flexible and simpler approach when decision-makers provide their decision information in a hesitant interval-valued intuitionistic fuzzy environment. Furthermore, IVIFSs, HFSs, HIVFSs, and HIFSs are all the special cases of HIVIFSs.

In this paper, HFSs are extended based on IVIFSs. HIVIFSs are defined, and their properties and applications are also discussed. Thus, the rest of this paper is organized as follows. In Section 2, the definitions and properties of IVIFSs and HFSs are briefly reviewed. In Section 3, the notion of HIVIFSs is proposed, and the operations and properties of HIVIFSs based on -conorms and -norms are discussed. In Section 4, some hesitant interval-valued intuitionistic fuzzy number aggregation operators are developed and applied to MCDM problems. Section 5 gives an example to illustrate the application of the developed method. Finally, the conclusions are drawn in Section 6.

#### 2. Preliminaries

In this section, some basic concepts and definitions related to HIVIFSs are introduced, including interval numbers, IVIFSs, and HFSs. These will be utilized in the subsequent analysis.

##### 2.1. Interval Numbers and Their Operations

*Definition 1 (see [32–34]). *Let ; then is called an interval number. In particular, if , then is reduced to a positive interval number.

Consider any two interval fuzzy numbers and , and their operations are defined as follows:(1);
(2);
(3);(4), ;(5).

##### 2.2. IVIFSs

Atanassov first proposed IFSs, being enlargement and development of Zadeh’s fuzzy sets. IFSs contain the degree of nonmembership, which makes it possible for us to model unknown information. The definition of IVIFSs given by Atanassov and Gargov [4] is shown as follows.

*Definition 2 (see [4]). *Let be the set of all closed subintervals of the interval . Let be a given set and . An IVIFS in is an expression given by , where , with the condition . The intervals and denote the degree of belongingness and nonbelongingness of the element to the set , respectively. Thus, for each , and are closed intervals whose lower and upper boundaries are denoted by and , respectively, and then
where , . For each element , the hesitancy degree can be calculated as follows: . The set of all IVIFSs in is denoted by IVIFS(). An interval-valued intuitionistic fuzzy number (IVIFN) is denoted by and the degree of hesitance is denoted by for convenience.

*Definition 3 (see [16]). *Let be a collection of IVIFNs and let be the crisp values, where , , , , and , and then the interval-valued intuitionistic fuzzy weighted average operator can be defined as follows:
where is an interval-valued intuitionistic fuzzy value;, , , and are calculated by the Karnik-Mendel algorithms [35].

*Example 4. *Let and be two IVIFNs, and . According to (2),

*Definition 5 (see [36]). *Let be an IVIFN, and then an accuracy function can be defined as follows:
where and .

*Definition 6 (see [36]). *Let and be two IVIFNs, and then the following comparison method must exist.(1)If , then .(2)If , then .

*Example 7. *Let and be two IVIFNs. According to (4), and . can be obtained, so the optimal one(s) is .

*Definition 8 (see [37–39]). *A function is called -norm if it satisfies the following conditions:(1)for all ;(2)for all ;(3)for all ;(4)if , then .

*Definition 9 (see [37–39]). *A function is called -conorm if it satisfies the following conditions:(1)for all ;(2)for all ;(3)for all ;(4)if , then .There are some well-known Archimedean -conorms and -norms [39, 40].(1)Let , , , , and then algebraic -conorms and -norms are obtained as follows: , .(2)Let , , , , and then Einstein -conorms and -norms are obtained as follows: , .(3)Let , , , , , and then Hamacher -conorms and -norms are obtained as follows:
Based on the Archimedean -conorms and -norms, some operations of IVIFSs are discussed as follows.

*Definition 10. *Let , , be three IVIFNs, , and then their operations could be defined as follows [19, 41–43]: (1) ;(2), , ;(3);(4).Here, , and is a strictly decreasing function.

##### 2.3. HFSs

*Definition 11 (see [44]). *Let be a universal set, and a HFS on is in terms of a function that when applied to will return a subset of , which can be represented as follows:
where is a set of values in , denoting the possible membership degrees of the element to the set . is called a hesitant fuzzy element (HFE) [23], and is the set of all HFEs. It is noteworthy that if contains only one element, then is called a hesitant fuzzy number (HFN), briefly denoted by . The set of all hesitant fuzzy numbers is represented as HFNS.

Torra [44] defined some operations on HFNs, and Xia and Xu [19, 22] defined some new operations on HFNs and the score function.

*Definition 12 (see [43]). *Let , and be three HFNs, , and then four operations are defined as follows:(1);(2);(3);(4).Here, , and is a strictly decreasing function.

*Definition 13 (see [19]). *Let , and is called the score function of , where is the number of elements in . For two HFNs and , if , then ; if , then .

*Example 14. *Let be two HFNs. According to Definition 13, , , so .

Furthermore, Torra and Narukawa [18, 44] proposed an aggregation principle for HFEs.

*Definition 15 (see [18, 44]). *Let be a set of HFEs, let be a function on , and let , and then

#### 3. HIVIFSs and Their Operations

HFSs are the extension of traditional fuzzy sets, and their membership degree of an element is a set of several possible values between 0 and 1. In some cases, decision-makers usually cannot estimate criteria values of alternatives with an exact numerical value when the information is not precisely known. Therefore, interval values in fuzzy sets can represent it better than specific numbers, such as IVFSs and IVIFSs. Furthermore, IVIFSs could describe the object being “neither this nor that,” and the membership degree and nonmembership degree of IVIFSs are interval values, respectively. Thus, precise numerical values in HFSs can be replaced by IVIFSs, which are more flexible in the real world, and this is what this section will solve.

*Definition 16. *Assume that is a finite universal set. A HIVIFS in is an object in the following form:
where is a finite set of values in IVIFSs, denoting the possible membership degrees and nonmembership degrees of the element to the set .

Based on the definition given above, where , , , , and . Actually, HIVIFSs have several possible membership degrees taking the form of IVIFSs instead of FSs in HFSs. If , then the HIVIFS is reduced to an IVIFS; if and , then the HIVIFS is reduced to a HFS; if or , then the HIVIFS is reduced to a HIVFS; if and , then the HIVIFS is reduced to a HIFS. Furthermore, is called a hesitant interval-valued intuitionistic fuzzy element (HIVIFE), and is the set of all HIVIFEs. In particular, if has only one element, is called a hesitant interval-valued intuitionistic fuzzy number (HIVIFN), briefly denoted by The set of all HIVIFNs is denoted by HIVIFNS.

*Definition 17. *Let , , and for all . Then, is called the hesitant interval-valued intuitionistic index of .

*Example 18. *Let , and let , , be a HIVIFS, and then , . Thus, .

The operations of HIVIFNs are defined as follows.

*Definition 19. *Let and be two HIVIFNs, , and four operations are defined as follows:(1), ;(2), ;(3) ;(4) .Here, , and is a strictly decreasing function.

*Example 20. *Let and be two HIVIFNs, and , , , and . The following can be calculated:(1), ,, , ,, ;(2),;(3),;(4),.

Theorem 21. *Let , , and then*(1)*;*(2)*;*(3)*;*(4)*;*(5)*;*(6)*;*(7)*.*

*Proof. *According to Definition 19, it is clear that , , , and are obvious. , , and will be proved as follows:
The proof is completed.

Based on Definitions 5, 6, and 13, the ranking method for HIVIFNs is defined as follows.

*Definition 22. *Let , is called the score function of , where is the number of the interval-valued intuitionistic fuzzy values in . For two HIVIFNs and , if , then ; if , then .

Note that and could be compared by utilizing Definitions 5 and 6.

*Example 23. *Let and be two HIVIFNs, and then
According to Definitions 5 and 6,
Hence, , which indicates that is preferred to .

#### 4. HIVIFN Aggregation Operators and Their Applications in MCDM Problems

In this section, HIVIFN aggregation operators are proposed, and some properties of these operators are discussed. In particular, some hesitant interval-valued intuitionistic fuzzy algebraic aggregation operators are proposed based on algebraic -conorms and -norms. Then, how to utilize these operators to MCDM problems is discussed as well.

##### 4.1. HIVIFN Aggregation Operators

*Definition 24. *Let () be a collection of HIVIFNs, and HIVIFNWA: , and then
The HIVIFNWA operator is called the HIVIFN weighted averaging operator of dimension , where is the weight vector of , with and .

Theorem 25. *Let be a collection of HIVIFNs and let be the weight vector of , with and . Then, the aggregated result using the HIVIFNWA operator is also a HIVIFN, and
*

*Proof. *By using mathematical induction on , we have the following.(1)For , since
the following can be obtained:

If (15) holds for , then

When , in terms of (1) and (3) in Definition 19,
that is, (15) holds for ; thus, (15) holds for all . Then,

*Definition 26. *Let be a collection of HIVIFNs, HIVIFNWG: , and then
The HIVIFNWG operator is called the HIVIFN weighted geometric operator of dimension , and is the weight vector of , with and .

Similarly, the following theorems can be obtained.

Theorem 27. *Let be a collection of HIVIFNs and let be the weight vector of , with and . Then, the aggregated result using the HIVIFNWG operator is also a HIVIFN, and
*

*Definition 28. *Let ) be a collection of HIVIFNs, HIVIFNWAA: , and then
The HIVIFNWAA operator is called the HIVIFN weighted arithmetic averaging operator of dimension , where is the weight vector of , with and .

Theorem 29. *Let be a collection of HIVIFNs and let be the weight vector of , with and . Then, the aggregated result using the HIVIFNWAA operator is also a HIVIFN, and*

*Definition 30. *Let ) be a collection of HIVIFNs, HIVIFNWAG: , and then
The HIVIFNWAG operator is called the HIVIFN weighted arithmetic geometric operator of dimension , where is the weight vector of , with and .

Theorem 31. *Let be a collection of HIVIFNs and let be the weight vector of , with and . Then, the aggregated result using the HIVIFNWAG operator is also a HIVIFN, and*

*Definition 32. *Let ) be a collection of HIVIFNs, GHIVIFNWA: , and then
The GHIVIFNWA operator is called the generalized HIVIFN weighted averaging operator of dimension , where is the weight vector of , with and . If , the GHIVIFNWA operator is reduced to the HIVIFNWA operator. If , the GHIVIFNWA operator is reduced to the HIVIFNWAA operator.

Theorem 33. *Let be a collection of HIVIFNs and let be the weight vector of , with , , and . Then, the aggregated result using the GHIVIFNWA operator is also a HIVIFN, and*

*Definition 34. *Let () be a collection of HIVIFNs, GHIVIFNWG: