Interval-Valued Hesitant Fuzzy Hamacher Synergetic Weighted Aggregation Operators and Their Application to Shale Gas Areas Selection

We investigate the multiple criteria decision making (MCDM) problem concerns on the selection of shale gas areas with intervalvalued hesitant fuzzy information. First, some Hamacher operations of interval-valued hesitant fuzzy information are introduced, which generalize and extend the existing ones.Then some interval-valued hesitant fuzzyHamacherweighted aggregation operators, especially, the interval-valued hesitant fuzzyHamacher synergetic weighted averaging (IVHFHSWA) operators and their geometric version (IVHFHSWG) operators that weight simultaneously the argument variables themselves and their position orders and thus generalize the ideas of the weighted averaging and the ordered weighted averaging, are proposed. The distinct advantages of these operators are that they can provide more choices for the decision makers and considerably enhance or deteriorate the performance of aggregation. The essential properties of these operators are studied and their specific cases are discussed. Based on the IVHFHSWA operator, we propose a practical approach to shale gas areas selection with interval-valued hesitant fuzzy information. Finally, an illustrative example for selecting the shale gas areas is used to demonstrate the practicality and effectiveness of the proposed approach and a comparative analysis is performed with other approaches to highlight the distinctive advantages of the proposed operators.


Instruction
As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [1,2] introduced by Torra and Narukawa have been successfully used in the decision making field as a powerful tool for processing with uncertain and vague information.Unlike the other generalizations of fuzzy sets, HFSs, which permit the membership degree of an element to a set to be represented as several possible values between 0 and 1, are quite suited for describing the situation where we have a set of possible values, rather than a margin of error or some possibility distribution on the possible values, and thus HFSs are very useful in dealing with the practical decision making situations where people hesitate among several values to express their opinions [3][4][5] or their opinions with incongruity [6][7][8], especially, the group decision making with anonymity [9][10][11][12].Moreover, HFSs could also avoid performing information aggregation and can directly reflect the differences of the opinions of different experts [1,13,14].In addition, it is proven that the envelope of hesitant fuzzy set is an intuitionistic fuzzy set (IFS); all HFSs are type-2 fuzzy set and hesitant fuzzy set and fuzzy multiset have the same form, but their operations are different [2].Thus, HFSs open new perfectives for further research on decision making under hesitant environments and have received much attention from many authors.Torra and Narukawa [1,2] proposed some set theoretic operations such as union, intersection, and complement on HFSs.Subsequently, Xia and Xu [6] defined some new operations on HFSs based on the interconnection between HFSs and the IFSs and then made an intensive study of hesitant fuzzy information aggregation techniques and their applications in decision making.Xu and Xia [7] 2 Mathematical Problems in Engineering investigated some distance measures for HFSs drawing on the well-known Hamming distance, the Euclidean distance, the Hausdorff metric, and their generalizations.Following these pioneering studies, many subsequent studies on the aggregation operators [8,9,[12][13][14][15], the discrimination measures [16] (including distance measures [3][4][5][13][14][15], similarity measures [3,7], correlation measures [3,17], entropy, and cross-entropy [18]) for hesitant fuzzy sets (HFSs), and the further extensions of the HFSs, such as the interval-values HFSs (IVHFSs) [11,17], the dual (or generalized) HFSs (DHFSs) [10,19], and the hesitant fuzzy linguistic term sets (HFLTSs) [20,21], have been conducted.
In some practical decision making problems, however, the precise membership degrees of an element to a set are sometimes hard to be specified.To overcome the barrier, Chen et al. [11,17] proposed the concept of interval-valued hesitant fuzzy sets (IVHFSs) that represent the membership degrees of an element to a set with several possible interval values and then presented some interval-valued hesitant fuzzy aggregation operators.Wei [22] developed some hesitant interval-valued fuzzy aggregation operators (which are essential interval-valued hesitant fuzzy aggregation operators), such as the hesitant interval-valued fuzzy weighted aggregation operators (HIVFWA and HIVFWG), the hesitant interval-valued fuzzy ordered weighted aggregation operators (HIVFOWA and HIVFOWG), the hesitant intervalvalued fuzzy choquet ordered aggregation operators (HIVF-COA and HIVFCOG), the hesitant interval-valued fuzzy prioritized aggregation operator and the hesitant intervalvalued fuzzy power aggregation operator.
It is well known that the aggregation operators introduced above are based on the basic algebraic -norms (-norm and -conorm) of HFSs (or IVHFSs) for carrying the combination process, which are not the unique -norms that can be chosen to model the intersection and union of HFSs (or IVHFSs).For instance, Wei and Zhao [23] presented the Einstein operations of interval-valued hesitant fuzzy sets based the Einstein -norms and then developed some interval-valued hesitant fuzzy Einstein aggregation operators and induced intervalvalued hesitant fuzzy Einstein aggregation operators.Besides, there are a lot of -norms that can be used to construct the operations of HFSs (or IVHFSs); one of them is the Hamacher -norms [24][25][26][27], which have proven that the basic algebraic -norms and the Einstein -norms are the special cases of the Hamacher -norms and can supply a wide class of norm operators ranging from the probabilistic product to the weakest -norm by the choice of a parameter.Thus the Hamacher -norms can considerably enhance or deteriorate the performance of aggregation.Given the advantages of the Hamacher -norms, in this paper, we will investigate the interval-valued hesitant fuzzy aggregation operators based on the Hamacher -norms and apply them to the multiple criteria decision making.
To do so, the remainder of this paper is organized as follows.Section 2 introduces some preliminary concepts related to the interval-valued hesitant fuzzy sets and their operations based on the Hamacher -norms.In Section 3, based on the defined operations, we first develop the interval-valued hesitant fuzzy Hamacher weighted averaging operators and the interval-valued hesitant fuzzy Hamacher ordered weighted averaging operators, then, based on which, we further propose the interval-valued hesitant fuzzy Hamacher synergetic weighted aggregation operators that simultaneously consider the weights of argument variables themselves and their position orders.Moreover, some essential properties and special cases of these operators are studied.In Section 4, we develop a practical approach based on the IVHFHSWA operators to multicriteria decision making under interval-valued hesitant fuzzy environments.Section 5 an illustrative example for selecting the shale gas areas is used to demonstrate the practicality and effectiveness of the proposed approach and Section 6 a comparative analysis is performed with other approaches to highlight the distinctive advantages of the proposed operators.Finally, we summarize the main conclusions of the paper in Section 7.

Preliminaries
To overcome the barrier that the precise membership degrees of an element to a set are sometimes hard to be specified, Chen et al. [11,17] introduce interval-valued hesitant fuzzy set (IVHFS), that represents the membership degrees of an element to a set with several possible interval values.
Definition 1 (see [11]).Let  be a reference set and let [0, 1] be the set of all closed subintervals of [0, 1].Then an IVHFS on  is defined as where h Ẽ() :  → [0, 1] denotes all possible intervalvalued membership degrees of the element  ∈  to the set Ẽ.For convenience, we call h Ẽ() an interval-valued hesitant fuzzy element (IVHFE), which reads The operational laws of IVHFSs can be constructed by -norms (-norm and -conorm), which satisfy the requirements of the conjunction and disjunction operators, respectively.The existing interval-valued hesitant fuzzy operational laws include the ones based on the algebraic -norms [6,11,[13][14][15]22] and the ones based on the Einstein -norms [23].It is well known that the Hamacher -norms are more generalized and flexible than the algebraic -norms and the Einstein -norms, and they are defined as follows.
Definition 2 (see [24,25]).The Hamacher -norm   and its conorm   are defined as Similar to the existing operations of IVHFEs, based on the Hamacher -norms, we can establish some fundamental Hamacher operations of IVHFEs.Definition 3. Let h, h1 , and h2 be three IVHFEs; then the Hamacher operations of IVHFSs are defined as follows: (1) (2) (4) The Hamacher operations of IVHFEs contain a wide class of special cases.Especially, if  = 1, then we have (1)-( 4) reduced to the following forms, which are presented by Chen et al. [11]: If  = 2, then we have (1)-( 4) reduced to the following forms, which are presented by Wei and Zhao [23]: , γ Mathematical Problems in Engineering Based on Definition 3, the following Theorem 4 can be easily proven.Theorem 4. Let h1 and h2 be two IVHFEs; then The proofs of Theorem 4 are straightforward and omitted here for saving space.
Chen et al. [11] defined the score function of IVHFE, and gave a comparison approach of the score values of two IVHFEs with the possibility degree.

Interval-Valued Hesitant Fuzzy Hamacher Synergetic Weighted Aggregation Operators
The weighted averaging operator and the ordered weighted averaging operator are the most common and basic aggregation operators.
Because the algebraic -norms and Einstein -norms are the special cases of the Hamacher -norms, the following theorems hold.
From the above analysis, we know that the IVHFWA operator [11] and the IVHFEWA operator [23] are the special cases of the IVHFHWA operator, and the IVHFHWA operator can provide more special cases by selecting different values of parameter , which can provide more choices for the decision makers and considerably enhance or deteriorate the performance of aggregation.Thus, the IVHFHWA operator is more general and more flexible.Similar to the IVHFWA operator and the IVHFEWA operator, the IVHFHWA operator is also monotonic, bounded, and idempotent. where where  : {1, 2, . . ., } → {1, 2, . . ., } is a permutation function such that h is the ()th largest element of the collection of h ( = 1, 2, . . ., ).Analogously, because the algebraic -norms and Einstein -norms are the special cases of the Hamacher -norms, the following theorems hold.Theorem 14.The IVHFOWA operator proposed by Chen et al. [11] is a special case of the IVHFHOWA operator; that is, if  = 1, then  ℎ ( h1 , h2 , . . ., h ) Theorem 15.The IVHFEOWA operator proposed by Wei and Zhao [23] Analogously, compared with IVHFOWA operator [11] and the IVHFEOWA operator [23], the IVHFHOWA operator can provide more special cases by selecting different values of parameter , which can provide more choices for the decision makers and considerably enhance or deteriorate the performance of aggregation.Thus, the IVHFHOWA operator is more general and more flexible.Similar to the IVHFOWA operator and the IVHFEOWA operator, the IVHFHOWA operator is also commutative, monotonic, bounded, and idempotent.
we have h1 < h2 and the assigned associated weights of h1 and h2 are  (1) = 0.4 and  (2) = 0.6, respectively.Then According to the above analysis, we know that the IVHFHWA operator weights only the interval-valued hesitant fuzzy argument variables; its weights are the relative weights and represent the differential importance (salience, significance) of argument variables themselves, while the IVHFHOWA operator weights only the ordered positions of the intervalvalued hesitant fuzzy argument variables (or the magnitudes of the interval-valued hesitant fuzzy argument values); its weights are the associated weights and depend on the corresponding satisfaction values of argument variables.According to the associated weights are derived in view of the satisfaction values of argument variables; the IVHFHOWA operator can relieve (or intensify) the influence of unduly large or unduly small deviations on the aggregation results [31] or decide the portion of the criteria they feel is necessary for a good solution [28,32,33].Therefore, weights represent different aspects in both the IVHFHWA and IVHFHOWA operators.However, in general, we need to consider the two weights because they represent different aspects of decision making problems.Obviously, both the IVHFHWA and IVHFHOWA operators have drawbacks.In order to solve these drawbacks, according to Theorem 8 and inspired by the idea of twofold weighting [4,34] where we have h1 < h2 , and the associated weights of h1 and h2 are  (1) = 0.4 and  (2) = 0.6, respectively.Thus By comparing Examples 12, 16, and 18, we know that the results derived from the IVHFHWA, IVHFHOWA, and IVHFHSWA operators are different; the reason is intuitive; the IVHFHWA operator focuses solely on the relative weights and ignores the associated weights in the process of aggregation, while the IVHFHOWA operator focuses only on the associated weights and ignores the relative weights in the process of aggregation; the IVHFHSWA operator comprehensively and simultaneously considers the relative weights and the associated weights and then its aggregated results are more feasible and effective.
Inevitably, we can see that the aggregation procedure of the interval-valued hesitant fuzzy information is complex, especially, with the increases of the number of argument variables, but it can surely better describe the situations where people have hesitancy in providing their preferences in the process of decision making, and the proposed operators can be easily solved using Microsoft Excel Solver or integrated in a decision support system [4,10].
IVHFHSWA operator not only integrates the relative weights and the associated weights into the weighted averaging operation and then generalizes the IVHFHWA operator and IVHFHOWA operator but also satisfies the properties of idempotency, boundary and monotonicity, and so on.

Interval-Valued
Furthermore, we can obtain the aggregated results corresponding to some special cases of the parameter , which are shown in Table 1.
From Table 1, we know that the aggregated results derived from the IVHFHSWG steadily increases as the parameter  increases, which implies that the IVHFHSWG operator with parameters can provide the decision makers more choices and thus the aggregated results are more flexible than the existing ones, because we can choose different values of the parameter according to the different situations.
The IVHFHSWG operator has some essential properties, such as idempotency, boundary, and monotonicity.
In the following, we discuss the special cases of the IVHFHSWG operator.
(1) From the perspective of parameter .
(2) From the perspective of the types of weights.
Furthermore, we use the IVHFHSWG operator to the same decision problem, we can obtain the ranking results that consist with the ones above, which are validate each other.

Comparative Analysis
6.1.Performance Analysis of the IVHFHSWA Operator.From the definition of IVHFHSWA operator, we know that the IVHFHSWA operator provides a wide class of interval-valued hesitant fuzzy aggregation operators via the parameters .To understand the performance of aggregation in depth, we adopt the parameter  = 1 to 10 for the numerical example above.When the  takes the different values, the scores values are obtained based on IVHFHSWA operator as shown in Table 5 and represented graphically in Figure 1.
From Table 5, it is obvious that the score values derived by using the IVHFHSWA operator are nonincreasing with respect to , which implies that decision makers can utilize their preferences to give the preferred values of  according to practical decision situations.On the basis of the score values, we can obtain the precisely ranking results of the alternatives by computing their relative possibility degrees.Moreover, from Figure 1, we find that the ranking results of the alternatives are the same when the values of  are different in the example, and the consistent ranking results demonstrate the stability of the proposed operators.

Comparison With Other
Operators.Similar to the IVHFHSWA operator, in order to integrate simultaneously  the ranking results of the alternatives derived from them are the same; that is,  2 ≻  1 ≻  5 ≻  3 ≻  4 .
The reasons about the difference of score values is intuitive that, as discussed above, the IVHFHWA operator focuses solely on the relative weights and ignores the associated weights, while the IVHFHOWA operator focuses only on the associated weights and ignores the relative weights.The IVHFHSWA operator comprehensively considers both the associated weights and the relative weights.Hence, the results derived by IVHFHSWA operator are more feasible and effective.On the other hand, the identical ranking results imply that the IVHFHSWA, IVHFHWA, and IVHFHOWA operators all are effective.Finally, our interval-valued hesitant fuzzy aggregation (IVHFHWA, IVHFHOWA, IVHFHSWA, IVHFHWG, IVHFHOWG, and IVHFHSWG) operators can also be applied to deal with the situations when the interval-valued hesitant fuzzy sets are reduced to hesitant fuzzy sets.In contrast, the existing hesitant fuzzy aggregation operators (HFWA, HFOWA, HFHA, HFWG, HFOWG, and HFHG) cannot be applied to deal with the interval-valued hesitant fuzzy situation.In other words, our interval-valued hesitant fuzzy aggregation operators have much wider applications than the existing hesitant fuzzy aggregation operators.

Conclusion
In this paper, we first introduce the Hamacher operations of interval-valued hesitant fuzzy sets and developed some interval-valued hesitant fuzzy Hamacher operators, including the interval-valued hesitant fuzzy Hamacher weighted averaging (IVHFHWA) operator and interval-valued hesitant fuzzy Hamacher ordered weighted averaging (IVHFHOWA) operator.The prominent advantages of the developed operators are that they provide a family of interval-valued hesitant fuzzy aggregation operators that include the existing intervalvalued hesitant fuzzy operators and interval-valued hesitant fuzzy Einstein operators as special cases and then provide more choices for the decision makers.Then, we proposed an interval-valued hesitant fuzzy Hamacher synergetic weighted averaging operator to generalize further the IVHFHWA and IVHFHOWA operator.Some essential properties of the proposed operators are studied and their special cases are discussed.Based on the IVHFHSWA operator, we develop a practical approach to multiple criteria decision making with interval-valued hesitant fuzzy information.Finally, an illustrative example for selecting the shale gas areas is used to illustrate the proposed approach and a comparative analysis is performed with other approaches to highlight the distinctive advantages of the proposed operators.

Table 1 :
Aggregated results for the different .

Table 4 :
Rankings and the assigned associated weights of criteria values against alternatives.

Table 5 :
Score values obtained by the IVHFHSWA operator with different values of .