Multicriteria Decision-Making Approach with Hesitant Interval-Valued Intuitionistic Fuzzy Sets

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.


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 [0, 1] 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][6][7][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 decisionmaking problems. Lee [11] provided an enhanced MCDM method of machine design schemes under the intervalvalued 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 2 The Scientific World Journal 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][20][21]. Then, the correlation coefficients of HFSs, the distance measures, and correlation measures of HFSs were discussed [22][23][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 intervalvalued 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 intervalvalued 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.

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.
Based on the Archimedean -conorms and -norms, some operations of IVIFSs are discussed as follows.

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 [0, 1], which can be represented as follows: where ℎ ( ) is a set of values in [0, 1], denoting the possible membership degrees of the element ∈ to the set . ℎ ( ) 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 = {ℎ ( )}. 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.

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 .
The Scientific World Journal 5 The operations of HIVIFNs are defined as follows.
Based on Definitions 5, 6, and 13, the ranking method for HIVIFNs is defined as follows.

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.
Similarly, the following theorems can be obtained.

HIVIFN Algebraic Aggregation Operators and HIV-IFN Einstein Aggregation Operators.
Obviously, differentconorms and -norms may lead to different aggregation operators. In the following, HIVIFN algebraic aggregation operators and Einstein aggregation operators are presented based on algebraic norms and Einstein norms. (1) Hesitant interval-valued intuitionistic fuzzy number algebraic weighted averaging operator is as follows: (2) Hesitant interval-valued intuitionistic fuzzy number algebraic weighted geometric operator is as follows:

16
The Scientific World Journal ) .

(61)
In particular, if = 1, then (58) is reduced to (52) and (60) is reduced to (53); if = 2, then (58) is reduced to (54) and (60) is reduced to (56). In the following, we propose one approach to rank and select the most desirable alternative(s). The procedure of this approach is shown as follows.
Step 4. Select the optimal one(s).

Illustrative Example
In this section, the proposed approach and one existing method are utilized to evaluate four companies with hesitant interval-valued intuitionistic fuzzy information. The enterprise's board of directors intends to find an automobile company and establish a foundation for deeper and more extensive cooperation with it in the following five years. Suppose there are four possible projects ( = 1, 2, 3, 4) to be evaluated. It is necessary to compare these companies and rank them in terms of their importance. Four criteria, suggested by the Balanced Scorecard methodology, could be taken into account (it should be noted that all of them are of the maximization type): 1 : economy, 2 : comfort, 3 : design, and 4 : safety. And suppose that the weight vector of the criteria is = (0.2, 0.3, 0.15, 0.35). The decision-makers are required to provide their evaluation of the company under the criterion ( = 1, 2, 3, 4, = 1, 2, 3, 4). The hesitant interval-valued intuitionistic fuzzy decision matrix = (̃) 4×4 is shown in Table 1, wherẽ( = 1, 2, 3, 4, = 1, 2, 3, 4) are in the form of HIVIFNs.

Illustration of the Proposed Approach.
In order to get the optimal alternative(s), the following steps are involved. Step (1 − ( ) ) ) Let = 2, and according to the formula listed above, the overall HIVIFNs of the alternatives ( = 1, 2, 3, 4) could be obtained and shown in Table 2.
Step 4. Select the best one(s). In Step 3, if the GHIVIFNAWA operator is utilized, then the optimal alternative is 2 while the worst alternative is 1 ; if the GHIVIFNAWG operator is used, then the optimal alternative is 4 while the worst alternative is 1 .

Sensitivity Analysis. In
Step 1, two aggregation operators can be used and the sensitivity analysis will be conducted in these following cases.
(1) The hesitant interval-valued intuitionistic fuzzy algebraic aggregation operators in Step 1 are illustrated as follows.
In order to investigate the influence of on the ranking of alternatives, different are utilized. The ranking results are shown in Tables 4 and 5.
From Tables 4 and 5, the GHIVIFNAWA and GHIV-IFNAWG operators have produced different rankings of the alternatives. However, for each operator, the rankings obtained are consistent as changes. Moreover, 4 or 2 is always the optimal one while the worst one is always 1 .
(2) The hesitant interval-valued intuitionistic fuzzy Einstein aggregation operators in Step 1 are illustrated as follows.
From Tables 6 and 7, the GHIVIFNEWA and GHIV-IFNEWG operators have produced different rankings of the alternatives. Furthermore, for each operator, the aggregation parameter also leads to different aggregation results, but the final rankings of alternatives are the same as the parameter changes. What is more, regardless of using the GHIVIFNEWA and GHIVIFNEWG operators, is that 4 or 2 is always the optimal one while the worst one is always 1 .
It can be concluded from the sensitivity analysis that different -conorms and -norms could lead to different aggregation results. However, the rankings using each operator are consistent.

Comparison Analysis.
Based on the same decisionmaking problem, if the method of Chen et al. [16] is employed, HIVIFNs are transformed to IVIFNs by using the score function firstly, and then IVIFNs could be aggregated by the interval-valued intuitionistic fuzzy weighted aggregation operators, proposed by Chen et al. [16].
So 4 ≻ 2 ≻ 3 ≻ 1 and the best optimal one is

Conclusion
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. IVIFSs can describe the fuzzy concept "neither this nor that, " and the membership degrees and nonmembership degrees of IVIFSs are not only real numbers but interval values, respectively. Precise numerical values in HFSs can be replaced by IVIFSs, which provide more preference information for decision-makers. In this paper, the definition of HIVIFSs was developed and applied to the MCDM problems, in which the evaluation values of alternatives on criteria were expressed with HIVIFNs. Furthermore, based on -conorms and -norms, some aggregation operators, namely, the HIVIFNWA and HIVIFNWG, HIVIFNWAA and HIVIFNWGA, and GHIVIFNWA and GHIVIFNWG operators, were proposed, respectively. Their properties were discussed in detail as well. In particular, the corresponding hesitant interval-valued intuitionistic fuzzy algebraic aggregation operators based on algebraic -conorm and -norm and hesitant interval-valued intuitionistic fuzzy Einstein aggregation operators based on Einstein -conorm and -norm were presented. In addition, different aggregation operators were utilized to fuse the hesitant interval-valued intuitionistic fuzzy information to get the overall HIVIFNs of alternatives and the ranking of all given alternatives. At last, the example was presented to illustrate the fuzzy decisionmaking process, and the sensitivity analysis and comparison analysis were conducted to enrich the paper. The prominent feature of the proposed method is that it could provide a useful and flexible way to efficiently facilitate decisionmakers under a hesitant interval-valued intuitionistic fuzzy environment, and the related calculations are simple. Hence, it has enriched and developed the theories and methods of MCDM problems and also has provided a new idea for solving MCDM problems. In the future research, the distance and similarity measure of HIVIFSs will be studied to solve MCDM problems.