An Approach to Interval-Valued Hesitant Fuzzy Multiattribute Group Decision Making Based on the Generalized Shapley-Choquet Integral

The purpose of this paper is to develop an approach to multiattribute group decision making under interval-valued hesitant fuzzy environment. To do this, this paper defines some new operations on interval-valued hesitant fuzzy elements, which eliminate the disadvantages of the existing operations. Considering the fact that elements in a setmay be interdependent, two generalized intervalvalued hesitant fuzzy operators based on the generalized Shapley function and the Choquet integral are defined.Then, somemodels for calculating the optimal fuzzy measures on the expert set and the ordered position set are established. Because fuzzy measures are defined on the power set, it makes the problem exponentially complex. To simplify the complexity of solving a fuzzy measure, models for the optimal 2-additive measures are constructed. Finally, an investment problem is offered to show the practicality and efficiency of the new method.

Although there are several families of fuzzy sets, all of the above-mentioned fuzzy sets only consider the membership information.As Atanassov [18] noted, in some situations, it is insufficient to only know the membership degree for a certain fuzzy concept.Thus, Atanassov [18] introduced the concept of intuitionistic fuzzy sets (IFSs), which are characterized by a membership degree, a nonmembership degree, and a hesitancy degree.Since then, many intuitionistic fuzzy decisionmaking methods are proposed [19][20][21].To further extend the application of IFSs, Atanassov and Gargov [22] introduced the concept of interval-valued intuitionistic fuzzy sets (IVIFSs), which are characterized by an interval membership function and an interval nonmembership function rather than real numbers.Such a generalization is further facilitated effectively to represent inherent imprecision and uncertainty in the human decision-making analysis.Many theories and methods on IVIFSs have been put forward and used to solve decision-making problems [23][24][25][26][27].
Recently, Torra and Narukawa [28] noted when an expert makes a decision, there may be several possible values for one thing.To deal with this situation, Torra [29] introduced the concept of hesitant fuzzy sets (HFSs) that permit the membership to have a set of possible values.Later, Xia and Xu [30] 2 Complexity defined some operational laws on HFSs and presented some aggregation operators for hesitant fuzzy elements.Furthermore, Xia et al. [31] defined a series of hesitant fuzzy aggregation operators with the aid of quasi-arithmetic means and developed an approach to hesitant fuzzy multiple attribute decision making.Motivated by the ideal of prioritized aggregation operators, Wei [32] developed the hesitant fuzzy prioritized weighted average (HFPWA) operator and the hesitant fuzzy prioritized weighted geometric (HFPWG) operator, whilst Zhu et al. [33] introduced the weighted hesitant fuzzy geometric Bonferroni mean (WHFGBM) operator.More researches can be seen in the literature [34][35][36][37].Just as interval type-2 fuzzy sets and IVIFSs, in some situations, it is still difficult to require an expert to give the exact possible values for one thing.Very recently, Chen et al. [38] introduced the concept of interval-valued hesitant fuzzy sets (IVHFSs) and defined some aggregation operators.Farhadinia [39] investigated the relationship between the entropy, the similarity measure, and the distance measure for HFSs and IVHFSs.Wei and Zhao [40] presented several induced hesitant interval-valued fuzzy Einstein aggregation operators and applied them to multiattribute decision making.Meanwhile, Wei et al. [41] defined two hesitant intervalvalued fuzzy Choquet operators and studied their application in interval-valued hesitant multiattribute decision making.Meng and Chen [42] introduced two induced generalized interval-valued hesitant fuzzy hybrid Shapley operators that globally consider the interactions between the weights of elements in a set.It is noteworthy that all these aggregation operators are based on the operational laws presented by Chen et al. [38].These operations cannot preserve the order relationship under multiplication by a scalar.It means that monotonicity is not always true.Thus, when these operators are used in decision making, it cannot guarantee to obtain the best choice.Furthermore, Meng et al. [43] researched the correlation coefficients of IVHFSs that need not consider the lengths of interval-valued hesitant fuzzy elements (IVH-FEs).However, the correlation coefficients only consider the weights of attributes and disregard that of orders.
To address the above-mentioned issues for decision making with IVHFSs, this paper continues to study group decision making under interval-valued hesitant fuzzy environment.First, some new operations that eliminate the existing issues are defined.To deal with the situation where the elements in a set are correlative, two generalized intervalvalued hesitant fuzzy dependent operators are defined, which can be seen as an extension of some hesitant fuzzy operators.Then, a distance measure on IVHFSs is defined, which does not consider the length of IVHFEs and the arrangement of their possible interval membership degrees.Based on the Shapley function and the defined distance measure, models for the optimal fuzzy measures and the optimal 2-additive measures are constructed, respectively.Finally, approach to interval-valued hesitant fuzzy multiattribute group decision making is developed.Comparing the existing methods, the new approach includes the following four features: (i) it uses the new defined operations that avoid the nonmonotonic problem; (ii) it applies the aggregation operator based on fuzzy measures that can address the interactive situations; (iii) when the weighting vector is partly known, models for the optimal fuzzy measure and the optimal 2-additive measure are built; (iv) because the experts' knowledge, skills, and experiences are different, the new method gives the experts' weights with respect to each attribute.
The paper is organized as follows: In Section 2, some basic concepts related to IVHFSs are reviewed, and some new operations on IVHFSs are defined.In Section 3, some generalized interval-valued hesitant fuzzy Choquet operators are defined, and some special cases are examined.Meanwhile, to simplify the complexity of solving a fuzzy measure, a generalized interval-valued hesitant fuzzy operator based on 2-additive measures is introduced.In Section 4, a new distance measure is defined, and then models for the optimal fuzzy measure and the optimal 2-additive measure on the associated set are built, respectively.After that, an approach to multiattribute group decision making under intervalvalued hesitant fuzzy environment is developed.In Section 5, an illustrative example is provided to show the concrete application of the proposed procedure.Conclusions are made in the last section.

Some Basic Concepts
To address the situation where the membership degree of an element has several possible interval values, Chen et al. [38] presented the concept of interval-valued hesitant fuzzy sets (IVHFSs), which is an extension of hesitant fuzzy sets (HFSs) [29].
Definition 1 (see [38]).Let  = { 1 ,  2 , . . .,   } be a finite set, and IVHFS in  is in terms of a function that when applied to  returns a subset of [0, 1], denoted by where ℎ  (  ) is a finite set of all possible interval-valued membership degrees of the element   ∈  to the set  with [0, 1] being the set of all closed subintervals in [0, 1].For convenience, Chen et al. [38] called ℎ = ℎ  (  ) an intervalvalued hesitant fuzzy element (IVHFE) and  is the set of all IVHFEs.
If all possible interval-valued membership degrees of each element   ∈  degenerate to real numbers, it derives an HFS [29].
Based on this possible degree formula on intervals, Chen et al. [38] introduced the following order relationship on IVHFEs.
To avoid these disadvantages, we adopt the following operations on IVHFEs.Let ℎ, ℎ 1 , and ℎ 2 be any three IVHFEs in , It is easy to verify that the new defined operations can eliminate the issues listed above.Without special explanation, this paper adopts the operations on IVHFEs defined by (I)-(IV).

Several Generalized Interval-Valued Hesitant Fuzzy Dependent Aggregation Operators
Let us consider the following example: "We are to evaluate three companies according to three attributes: {economic benefits, environment benefits, social benefits}, we want to give more importance to environment benefits than to economic benefits or social benefits, but on the other hand we want to give some advantage to companies that are good in environment benefits and in any of economic benefits and social benefits".In this situation, the aggregation operator based on additive measures seems to be insufficient.
To obtain the comprehensive attribute values and reflect the interactions between attributes as well as the ordered positions, this section introduces several interval-valued hesitant fuzzy operators based on the Choquet integral and the generalized Shapley function.First, let us review the following definitions.
From the definition of fuzzy measures, we know that the fuzzy measure does not only give the importance of every element but also consider the importance of all their combinations.Corresponding to fuzzy measures, fuzzy integrals are important tools to aggregate information with interactive characteristics.The Choquet integral is one of the most important fuzzy integrals, which can be seen as an extension the ordered weighted averaging (OWA) operator.Grabisch [62] gave the following expression of the Choquet integral on discrete sets.Definition 6 (see [62]).Let  be a positive real-valued function on  = { 1 ,  2 , . . .,   } and  be a fuzzy measure on  = {1, 2, . . ., }.The discrete Choquet integral of  for  is defined as where (⋅) indicates a permutation on  such that ( ), and   = {, . . ., } with  (+1) = ⌀.

Remark 7.
From Definition 6, one can see that the fuzzy measure  only relates to the positions.It does not consider which element in the position.
From Definition 6, we know that the Choquet integral only considers the correlations between the ordered subsets   and  +1 ( = 1, 2, . . ., ).If there are interdependences, it seems to be insufficient.To globally reflect the interactions between the ordered subsets, the generalized Shapley function [63] seems to be a good choice, denoted as where  is a fuzzy measure on  = {1, 2, . . ., }, and , , and  denote the cardinalities of the coalitions , , and , respectively.Form (6), we know that it is an expect value of the overall marginal contributions between the coalition  ⊆  and any coalition in  \ .When  = {}, it degenerates to the famous Shapley function [53]: From ( 7), we know that when the elements in  are uncorrelated, then their Shapley values equal to their own importance, namely,   (, ) = () for all  = 1, 2, . . ., .
From Definition 8, one can see that the generalized Shapley-Choquet integral overall considers the interactions between any two adjacent coalitions.Now, let us introduce the generalized interval-valued hesitant fuzzy Shapley-Choquet weighted averaging (G-IVHFSCWA) operator as follows.
Definition 9. Let ℎ  ( = 1, 2, . . ., ) be a collection of IVHFEs in  and  be a fuzzy measure on the ordered set  = {1, 2, . . ., }.The generalized interval-valued hesitant fuzzy Shapley-Choquet weighted averaging (G-IVHFSCWA) operator is defined as where  > 0, (⋅) indicates a permutation on  such that ℎ (1) ≤ ℎ From Definition 9, we know that the G-IVHFSCWA operator only gives the importance of the ordered positions.To further consider the importance of elements and reflect their correlations, we introduce the interval-valued hesitant fuzzy Shapley-Choquet hybrid operator that considers the importance of the attributes (or experts) and their ordered positions as well as reflects their interactions.
Definition 12. Let ℎ  ( = 1, 2, . . ., ) be a collection of IVHFEs in , V be a fuzzy measure on  = {ℎ 1 , ℎ 2 , . . ., ℎ  }, and  be a fuzzy measure on the ordered set  = {1, 2, . . ., }.The generalized interval-valued hesitant fuzzy Shapley-Choquet hybrid weighted averaging (G-IVHFSCHWA) operator is defined as where  > 0, (⋅) indicates a permutation on  such that Remark 15.If  = 2, then the G-IVHFSCHWA operator degenerates to the interval-valued hesitant fuzzy Shapley-Choquet quadratic hybrid weighted averaging (IVHFSC-QHWA) operator 2 ) Although the fuzzy measure can address the situation where the elements in a set are correlative, they define the power set.It makes the problem exponentially complex.Thus, it is not easy to solve a fuzzy measure when the set is large.
To reflect the interactions between elements and simplify the complexity of solving a fuzzy measure, we introduce a special case of the G-IVHFSCHWA operator using 2-additive measures.
Definition 16 (see [64]).A fuzzy measure  on  = {1, 2, . . ., } is said to be k-additive if its corresponding pseudo-Boolean function is a multilinear polynomial of degree , i.e.,   = 0 for all  such that  > , and there exists at least one subset  with  elements such that   ̸ = 0.
For a 2-additive measure, we only need ( + 1)/2 coefficients to determine it for a set with  elements.

An Approach to Multiattribute Group Decision Making
Because of various reasons, the weighting information may be incompletely known.To solve this situation, this section first establishes models for the optimal fuzzy measure and the optimal 2-additive measure on the associated sets.Then, an approach to multiattribute group decision making under interval-valued hesitant fuzzy environment with incomplete weighted information and interactive characteristics is developed.
Different from this distance measure, we define another one that need not consider the length of IVHFEs.Definition 19.Let ℎ 1 and ℎ 2 be any two IVHFEs, then the generalized distance measure between ℎ 1 and ℎ 2 is defined as where  > 0 and #ℎ 1 and #ℎ 2 denote the number of the possible interval value in ℎ 1 and ℎ 2 , respectively.
where   and   are the coefficient matrices,   and   are the constant vectors,   (V   ( 1 ), V   ( 2 ), . . ., V   (  1 )) ≤   and   (V   ( 1 ), V   ( 2 ), . . ., V   (  2 )) =   are the known constraints, V   is the fuzzy measure on the expert set  with respect to the attribute   ,    (V   , ) is the Shapley value of the expert   , and     is the known weighting information.
The optimal fuzzy measure obtained from this model has the following desirable characteristics: the closer an expert's evaluation values are to the other experts' , the larger the fuzzy measure will be.This can decrease the influence of the unduly high or low evaluation value induced by the experts' limited knowledge or expertise.

Models for the Optimal Fuzzy Measure on the Ordered
Set .To construct the model for the optimal fuzzy measure on the ordered set , the following procedure is needed.
Step 1. Calculate the interval-valued hesitant fuzzy Shapley weighted decision matrices      (  ,) = (ℎ   ) × ,  ∈ , where Step 2. Calculate the score matrices ( Step 3. Calculate the mid-width matrices   = (   ) × ,  ∈ , where Step 4. For each pair (, ), we rearrange each    ,  ∈ , such that  (1)   ≤  (2)   ≤ ⋅ ⋅ ⋅ ≤ Because there is no inferior for the ordered positions with respect to the different attributes, if the weighting information on the ordered set  is not exactly known, the following model for the optimal fuzzy measure is built: where  and  are the coefficient matrices,  and  are the constant vectors, (  ( 1 ), . . .,   (  1 )) ≤  and (  ( 1 ), . . .,   (  2 )) =  are the known constraints,   is the fuzzy measure on the ordered set ,   (  , ) is the Shapley value of the th position, and   is the known weighting information.

Models for the Optimal Fuzzy
where   (ℎ  , ℎ +  ) and   (ℎ  , ℎ −  ) are defined in Definition 19,  and  are the coefficient matrices,  and  are the constant vectors, (V  ( 1 ), . . ., V  (  1 )) ≤  and (V  ( 1 ), . . ., V  (  2 )) =  are the known constraints, V  is the fuzzy measure on the attribute set ,    (V  , ) is the Shapley value of the attribute   , and    is the known weighting information.
If   is a 2-additive measure, then it derives the following model: where W and P are the coefficient matrices, π and τ are the constant vectors, and W(  (),   (, ), ,  = 1, . . ., ,  ̸ = ) ≤ π, and P(  (),   (, ), ,  = 1, . . ., ,  ̸ = ) = τ, are the equivalent expressions of the known constraints given in model (34) with respect to 2-additive measure   .Remark 20.In built models, we apply the elements' Shapley values as their weights that overall consider their interactions.Furthermore, if the elements in a set are independent, the built models degenerate to models for the optimal additive measure vector on the associated sets.

An Approach to Multiattribute Group Decision Making.
Based on the analysis above, this section introduces an approach to interval-valued hesitant fuzzy multiattribute group decision making with incomplete weighting information and interactive characteristics.The main decision procedure to obtain the most desirable alternative(s) can be described as follows.
Step 1.If all attributes are benefits (i.e., the bigger the better), then the attribute values need not transformation. with Step 2. Use model (22) to calculate the optimal fuzzy measure on the expert set  with respect to each attribute.
Step 3. Use model (27) to calculate the optimal fuzzy measure on the ordered set .
Step 4. Utilize the G-IVHFSCHWA operator to calculate the interval-valued hesitant fuzzy element ℎ  ; it derives the comprehensive interval-valued hesitant fuzzy matrix  = (ℎ  ) × .
Step 5. Use model (31) to calculate the optimal fuzzy measure on the attribute set .
Step 6. Use model (34) to calculate the optimal fuzzy measure on the ordered set .
Step 8.According to the comprehensive value ℎ  of the alternative   , we calculate the score Then, we rank the comprehensive IVHFEs ℎ  ,  = 1, 2, . . ., , and select the best alternative(s).
Step 9. End. the importance of the expert  1 is no smaller than that of the expert  2 or  3 ; their differences belong to the intervals [0.1, 0.2] and [0, 0.1], respectively.Moreover, the importance of the combination of the experts  1 and  3 is no less than that of the combination of the experts  1 and  2 as well as the combination of the experts  2 and  3 .Furthermore, the weighting information on the ordered set  is defined as follows: From the weighting information above, it indicates that the importance is increasing with respect to the ordered positions.The range of their individual weights is [0.2, 0.5], and the range of the combinations of any two ordered positions' weights is [0.5, 0.9].
Considering the following facts: "These four companies belong to one state that has a stable social-political environment.Its government always attaches great importance to environmental protection.In addition, with the help of the government, they have a certain antirisk ability".The weighting information on the attribute set  is given as follows: V  ( 1 ,  2 ,  4 ) ≥ 0.8.
In the following, we can utilize the proposed procedure to obtain the most desirable alternative(s).
Step 1.Because the attributes  1 ,  3 , and  4 are cost and the attribute  2 is benefit, it needs to transform the decision matrix   into   ,  = 1, 2, 3. Take  1 , for example; the decision matrix  1 is given as shown in Table 4.
From Table 5, the experts' Shapley values with respect to each attribute are obtained as shown in Table 6.
According to model (27), the following linear programming is constructed:  Step 4. Let  = 2, by the G-IVHFSCHWA operator the comprehensive interval-valued hesitant fuzzy matrix is obtained as shown in Table 8.

Table 5 :
The optimal fuzzy measures.