Generalized Shapley Choquet Integral Operator Based Method for Interactive Interval-Valued Hesitant Fuzzy Uncertain Linguistic Multi-Criteria Group Decision Making

This paper develops a new method based on generalized hybrid Shapley Choquet integral for interval-valued hesitant fuzzy uncertain linguistic multi-criteria group decision making (MCGDM), where interactive characteristics of criteria is taken into account. First, a score function and a new accuracy function are defined to rank the interval-valued hesitant fuzzy uncertain linguistic elements (IVHFULEs). Based on the Shapley Choquet integral, some generalized Shapley Choquet operator for IVHFULEs are proposed including an interval-valued hesitant fuzzy uncertain linguistic generalized Shapley Choquet (IVHFULGSC) operator and an interval-valued hesitant fuzzy uncertain linguistic generalized hybrid Shapley Choquet (IVHFULGHSC) operator and some attractive properties for these two operators are investigated in detail. Then a Shapley Choquet integral-based Minkowski distance between interval-valued hesitant fuzzy uncertain linguistic sets (IVHFULSs) is defined. Subsequently, based on the maximization deviation method, a linear programming model is built to determine the individual fuzzy measure on criteria set for each decision maker (DM). By utilizing the individual closeness coefficients matrices, a new multi-objective fractional programming model which maximizes the collective relative closeness degrees of all alternatives is constructed to obtain the fuzzy measure on DMs set. The individual decision matrices are aggregated into the collective one by using the IVHFULGHSC operator. To obtain the collective Shapely fuzzy measure on criteria set, a linear programming model is established by maximizing the deviation of the evaluations of different alternatives on each criterion. Then, the collective closeness coefficients of alternatives are calculated to rank the alternatives. Thus, a new method based on generalized hybrid Shapley Choquet integral operator is brought forward for MCGDM with IVHFULEs. Finally, an example of investment on shared bikes is provided to illustrate the applicability of the proposed method and a validity test is conducted to demonstrate the effectiveness of the proposed method.


I. INTRODUCTION
In group decision making (GDM), multi-criteria group decision making (MCGDM) is an important branch of decision The associate editor coordinating the review of this manuscript and approving it for publication was Zeshui Xu . analysis. Aim of MCGDM is to rank a finite set of alternatives and select an optimal alternative by the evaluations given by decision makers (DMs) on multiple criteria. At present, MCGDM has been widely utilized to solve the social, economic and management problems, such as bank recruitment [63], potential evaluation of emerging technology commercialization [64], the investment of shared bikes [65] and selecting an agricultural socialization service provider [66], name to just a few.
Various kinds of uncertain and ambiguous information appear in decision making process due to the complexity of the decision environment and the limited ability of human beings. The fuzzy set (FS) proposed by Zadeh [1] is a useful tool to describe vagueness and uncertainty in decision making process. Then FSs have been extended to intuitionistic fuzzy sets [2], interval-valued intuitionistic fuzzy sets [3], fuzzy multisets [4], type-2 fuzzy sets [5] and type-n fuzzy sets [6], etc. In order to capture uncertainty and fuzziness better, Torra [7] proposed hesitant fuzzy set (HFS), which represents an element of a given set by several membership values. Based on the concept of HFS, lots of research has been investigated. Xia and Xu [8] presented the mathematical symbol for HFS. Xia and Xu [8] introduced some aggregation operators for hesitant fuzzy information. Torra and Narukawa [9] generalized existing operations of FSs to HFSs by gave an extension principle. Xu and Xia [10] proposed a variety of distance measures of HFSs for GDM.
Considering the expression habits of human beings, it may be more suitable to describe uncertain information through linguistic term for DMs. Linguistic term is an effective tool to evaluate objective than numerical values in humans' cognitive process. Linguistic term sets have been studied in various fields [11]- [14], [61], [62] and there exist different extended forms of linguistic terms. In order to improve the uncertain and fuzzy of linguistic description, Xu [22] defined an uncertain linguistic variable. Rodriguez et al. [15] presented hesitant fuzzy linguistic term sets (HFLTSs) and there are many studies about HFLTSs [16]- [20], [67].
However, it is still difficult for DMs to evaluate an alternative by a single linguistic term when time is pressure and DMs' expertise is limited. Thus, Lin [21] defined hesitant fuzzy uncertain linguistic sets (HFULSs) which can express DMs' hesitance that exists in giving the associated membership degree of an uncertain linguistic term. Wang et al. [69] proposed the interval-valued hesitant fuzzy linguistic term sets (IVHFLSs) which permit the membership degrees having a set of possible interval-values to represent the degree that an element belongs to a linguistic variable. Combining both uncertain linguistic term and the interval-values degree, Ju and Liu [70] proposed the concept of interval-valued hesitant uncertain linguistic set (IVHULS) which can represent the uncertain and fuzzy information better.
With the wide applications of IVHFLSs, HFULSs and IVHULSs, a lot of scholars researched multi-criteria decision making (MCDM) problems under hesitant fuzzy uncertain linguistic environment. Zheng [23] defined the hesitant fuzzy uncertain linguistic power weighted average (HFULPWA) operator and applied it to evaluate the college English teachers' professional development competence. Zhao et al. [56] developed hesitant fuzzy uncertain linguistic power weighted geometric (HFULPWG) operator to evaluate the resource integration capability of textile enterprise. Shao [59], Yang and Xiang [60] investigated HFULPWA operator and HFULPWG operator for MCDM. Considering the interactive characteristics between criteria, Huo and Zhou [24] proposed a hesitant fuzzy uncertain linguistic correlated averaging operator (HFULCA) and a hesitant fuzzy uncertain linguistic correlated geometric (HFULCG) operator for MCDM. Sun and Xia [57] defined induced hesitant fuzzy uncertain linguistic correlated geometric (IHFULCG) operator. Yu [58] introduced the induced hesitant fuzzy uncertain linguistic Einstein correlated average operator (IHFULECA). Liu et al. [71] proposed the generalized IVHUL weighted aggregation operators, the generalized IVHUL ordered weighted aggregation operators and the generalized IVHUL hybrid aggregation operators.
For MCDM and MCGDM, it is critical to determine the criteria weights. Traditionally, criteria are considered to be independent of each other. However, there may exists some interactive characteristics between criteria in real-world decision problems. To overcome the limitations, fuzzy measure proposed by Sugeno [34] is a useful tool to measure the interactions between criteria. Then, Choquet [35] introduced Choquet integral as a concept associated with fuzzy measures to aggregate fuzzy information. Subsequently, the Choquet integral is widely applied to the decision making problems. Xu [36] proposed some intuitionistic fuzzy aggregation operators based on Choquet integral. Joshi and Kumar [33] defined interval-valued intuitionistic hesitant fuzzy Choquet integral. Tan [37] presented a distance measure by Choquet integral and applied it to multi-criteria intervalvalued intuitionistic fuzzy group decision making problems. Zhang et al. [72] developed two kinds of generalized IVHFLs Shapley Choquet integrals to globally characterize interactions between criteria combinations. Wei [73] further proposed some aggregation operators of IVHFULSs, such as IVHFUL correlated averaging aggregation operators and IVHFUL correlated geometric aggregation operators.
The interactive characteristics of criteria can be effectively described by the Choquet integral. However, it only considers the interactions between two ''adjacent'' coalitions, which seems to be unreasonable. It is should be take into account the overall interaction phenomenon among combinations in MCGDM. Marichal [38] gave a generalized Shapely index. Combining the Shapely index and Choquet integral, Meng [39] defined Shapley Choquet integral to illustrate the interaction among criteria. Then the arithmetical generalized λ-Shapley Choquet integral operator and the geometric generalized λ-Shapley Choquet integral operator are proposed for interval-valued intuitionistic fuzzy elements.
Although the aforesaid methods have made great contributions to solve MCDM problems or MCGDM problems with IVHFLSs, HFULSs and IVHULSs, there still exist some limitations which are summarized as follows: (1) Zheng [23], Huo and Zhou [24], Zhang et al. [72], Wei [73] only investigated single person decision making methods for the HFUL, IVFL and IVHULS MCDM problems respectively. With increasing competition and development VOLUME 8, 2020 of modern economy and technology, single DM cannot evaluate the complicated decision making problems comprehensively. Thus, MCGDM problems emerge. It is necessary to develop some new methods for solving MCGDM problems.
(2) Though some aggregation operators of hesitant fuzzy uncertain linguistic elements (HFULEs) have been proposed in [23], [56], [59], [60], these operators neglect the interactions between elements in a set. Although, [24], [57], [58] considered the interactions between elements, it only considers the interactions between two ''adjacent'' elements, which is not comprehensive. Similarity, the aggregation operators of IVHUL proposed in [71] do not consider the interactions between elements in a set. Although, the aggregation operators of IVHUL consider the interactions between elements, Wei [73] only focused on the interactions between two ''adjacent'' elements too.
(4) For MCDM, it is critical to determine the criteria weights. However, the criteria weights in methods [23], [24], [33] are given in a priori, which is not easy to avoid subjective randomness.
Considering HFULSs and IVHFLSs are both the special cases of IVHULS, this paper mainly focuses on the investigation of IVHULS. To overcome the above limitations, this paper investigates a generalized hybrid Shapley Choquet integral operator based method for MCGDM with IVHULEs. Firstly, a new accuracy function is defined to rank IVHULEs. Based on the Shapley Choquet integral, two generalized Shapley Choquet operators are developed including an interval-valued hesitant uncertain linguistic generalized Shapley Choquet (IVHULGSC) operator and an interval-valued hesitant uncertain linguistic generalized hybrid Shapley Choquet (IVHULGHSC) operator. Then a Shapely Choquet integral-based Minkowski distance between IVHULSs is defined. Meanwhile, a maximization deviation method is used to determine the individual Shapley fuzzy measure on criteria set for each DM. Through maximizing collective relative closeness degrees of all alternatives, a new multi-objective fractional programming model is constructed to obtain Shapley fuzzy measure on DMs set. Consequently, the individual decision matrices are aggregated into the collective one by using the IVHULGHSC operator. To obtain the collective Shapley fuzzy measure on criteria set, a linear programming model is established by maximizing the deviation between the evaluations on different alternatives. The closeness coefficient of each alternative is calculated to rank the alternatives. Thus, a new method using the IVHUL-GHSC operator is proposed for MCGDM with IVHULEs. The major contributions of this paper are highlighted from four aspects: (1) A new accuracy function of IVHULE is defined. Based on the score function and new accuracy function, the order relation of the IVHULEs is presented to rank IVHULEs.
(2) Using the Shapley Choquet integral, the IVHULGSC operator and IVHULGHSC operator are developed. The prominent characteristic of these operators is that they take the overall interaction phenomenon among IVHULEs into account. Then, the generalized Shapley Choquet integralbased Minkowski distance for IVHULSs is defined.
(3) A linear programming model based on maximization deviation method is established to determine the individual Shapley fuzzy measure on criteria set for each DM. Based on the individual closeness coefficients matrices, a new multi-objective fractional programming model is constructed to obtain the Shapley fuzzy measure on DMs set. Then the individual decision matrices are aggregated into the comprehensive one by using the IVHULGHSC operator.
(4) A linear programming model is built to obtain the collective Shapley fuzzy measure on criteria set. Then the closeness coefficients of alternatives are obtained to rank the alternatives.
The rest of this paper is organized as follows. In Section II, some basic concepts related to HFSs, uncertain linguistic variables, IVHULSs, fuzzy measure and Shapley index are reviewed. In Section III, the IVHULGSC operator and IVHULGHSC operator are proposed and some desirable properties are discussed. Then a Shapely Choquet integral-based Minkowski distance between IVHULSs is proposed. In Section IV, the Shapley fuzzy measure on DMs set and criteria set are determined by maximization deviation method. Thus a new method using the IVHULGHSC operator is brought forward for MCGDM with IVHULEs. An example of investment on shared bikes is provided to illustrate the effectiveness of the proposed method and a validity test of the proposed method is conducted in Section V. Finally, conclusions are made in Section VI.

II. PRELIMINARIES
This section reviews some basic concepts of HFSs, linguistic variables, uncertain linguistic variables, HFULSs, IVHFLS, IVHULS, fuzzy measure, Choquet integral and Shapley index to facilitate future discussions.

A. HESITANT FUZZY SETS
Definition 1 [7], [9]: Given a non-empty set of the universe, X = {x 1 , x 2 , . . . , x n } then a HFS on X is defined in terms of a function that when applied to X returns a subset of [0, 1].
To be easily understood, Xia and Xu [8] gave the mathematical symbol of HFS as: where h E (x) is a set of some values in [0, 1], denoting the possible membership degree of the element x ∈ X to the set E. h E (x) is called a hesitant fuzzy element (HFE).
Definition 2 [7], [9]: Given a HFE h, its lower bound h − (x) and upper bound h + (x) are defined as follows: Definition 3 [7], [9]: Let h be a HFE. Its complement h c is defined as Definition 4 [7], [9]: Let h 1 and h 2 be two HFEs. The union between HFSs h 1 and h 2 is defined as [7], [9]: Let h 1 and h 2 be two HFEs. The intersection between HFEs h 1 and h 2 is defined as [8]: Given three HFEs h 1 , h 2 and h, some operations of them are defined by: (

B. LINGUISTIC VARIABLES AND UNCERTAIN LINGUISTIC VARIABLES
Let S = {s i |i = 0, 1, . . . , 2t} be a linguistic term set, where s i expresses a possible value for a linguistic variable, t is a positive integer. For example, a set of seven terms S can be defined as, S = {s 0 = very poor, s 1 = poor, s 2 = slightly poor, s 3 = average, s 4 = slightly good, s 5 = good, s 6 = very good} Moreover, linguistic term set S = {s i |i = 0, 1, . . . , 2t} should satisfy the following characteristics [15], [41]: (1) The set is ordered: (2) The negation operator is defined as: To preserve all the given information, Xu [42] extended the discrete term set S to a continuous term setS = {s α |s 0 ≤ s α ≤ s t , α ∈ [0, t]}, whose elements also meet all the above characteristics. If s α ∈ S, then s α is called as the original linguistic term, otherwise, s α is called as the virtual linguistic term.
Definition 7 [22]: Lets = [s α , s β ], where s α , s β ∈S. s α and s β are the lower and the upper limits ofs, respectively, thens is named as the uncertain linguistic variable.

C. INTERVAL-VALUED HESITANT UNCERTAIN LINGUISTIC SETS
Before review the definition of IVHULSs, this paper reviews the definition of HFULSs and IVHFLS firstly. Definition 8 [21], [68]: Let X be a given domain and[s θ L (x) , s θ R (x) ] ∈S. Then a HFULS is defined as Definition 9 [69]: Let X be a given domain and s θ (x) ∈ S. An IVHFLS H in X is defined as follows: where h H (x) is a set of finite numbers of closed intervals belonging to (0, 1] and denotes the possible interval-valued membership degrees that x belong to s θ (x) . For convenience, h =< s θ(x) , h H (x) > is called an interval-valued hesitant fuzzy linguistic element (IVHFLE).

D. FUZZY MEASURE AND CHOQUET INTEGRAL
In the MCDM problems, for traditional additive aggregation operators, each element x j (j = 1, 2, . . . , n) is allocated a weight w j which represents the importance of this 202198 VOLUME 8, 2020 elements and satisfies w j ∈ [0, 1](j = 1, 2, . . . , n) and n j=1 w j = 1. In real-world decision problems, there exist inter-dependent or interactive phenomena among elements. The overall importance of an element x j is not only determined by itself, but also by all other elements.
Sugeno [34] introduced the concept of fuzzy measure which is a non-additive measure and makes monotonic property instead of additive property. Fuzzy measure defines a weight on not only each element, but also each combination of elements, and the sum of w j (j = 1, 2, . . . , n) does not equal to one. By extending the Lebesgue integral, Murofushi and Sugeno [47] proposed Choquet integral. It is an important aggregating tool for MCDM, which takes the importance of elements represented by fuzzy measure into account. The definitions of fuzzy measure and discrete Choquet integral are reviewed below.
Definition 13 [34]: . , x n } be a universe of discourse. A fuzzy measure on finite set X is a set function µ : P(X ) → [0, 1] satisfying the following properties: In the MCDM problems, let C = {c 1 , c 2 , . . . , c n } be a criteria set. The interaction among the criteria c j (j = 1, 2, . . . , n) can be described by employing µ(c j ) to express the grade of subjective importance of criteria set C. In the following, three kinds of interaction of any pair of criteria sets A, B ∈ P(X ) satisfying A ∩ B = Ø are introduced: (1) A and B are considered to be independent (or to be without interaction) if µ(A∪B) = µ(A)+µ(B). This is called an additive measure.
(2) A and B are considered to be positive synergetic interaction if µ(A ∪ B) > µ(A) + µ(B). This is called super additive measure.
(3) A and B are considered to be negative synergetic interaction if µ(A ∪ B) < µ(A) + µ(B). This is called sub additive measure.
Let N = {1, 2, . . . , n} be a finite index set. To overcome the limitation, Marichal [50] proposed the generalized Shapley index as where µ is a fuzzy measure on N . n, t and s denote the cardinalities of N , T and S, respectively. Motivated by Eq. (6), Meng [39] defined the generalized Shapley index with respect to λ-fuzzy measure g λ on N as follows: For finite set N , the λ-fuzzy measure could be expressed by Eq. (7) can be regarded as an expected value of the overall interaction between the coalition S and each coalition in N \S. As a special case, Eq. (8) is an expected value of the overall interaction between the coalition {i} and each coalition in N \i. Meng [39] proved ρ sh S (g λ , N ) is also a fuzzy measure on N which is called Shapley fuzzy measure.
For n = 1, the proof of Eq. (12) is obvious according to Definition 11 and Definition 15.
For n = 2, according to the operational laws of Definition 11, one has According to Definition 11, then it holds that , That is, for n = 2, Eq. (12) holds. Suppose that if for n = k, Eq. (12) holds, i.e., Then for n = k + 1, according to Definition 15, one has 202200 VOLUME 8, 2020 That is, for n = k + 1, Eq. (12) still holds. Therefore, for all n, Eq. (12) always holds, which completes the proof of Theorem 2.
Additionally, some special cases of the IVHFULGSC operator are shown as follows: (1) When q → 0, which is called an interval-valued hesitant fuzzy uncertain linguistic geometric ordered weighted shapely Choquet (IVHFULGOWSC) operator.

Property 4 (Commutativity):
Proof: The proof of Property 4 is similar to that of Property 1.
Proof: The proof of Property 6 is similar to that of Property 3.
Proof: The proof of Property 7 is similar to that of Property 1.
Hence, the property can be proven.
By Theorem 3, some special cases of IVHFULGHSC operator could be obtained as follows: (1) When q = 0, which is called an interval-valued hesitant fuzzy uncertain linguistic geometric hybrid weight shapely Choquet (IVH-FULGHWSC) operator.
Proof: The proof of Property 15 is similar to that of Property 6.
Proof: The proof of Property 18 is similar to that of Property 9.

Based on the Shapley Choquet integral, a Shapley Choquet integral-based Minkowski distance between IVHFULEs is defined.
Definition , hB(b) > be two IVHFULEs. The Minkowski distance between IVHFULEsã andb is defined as where q (q > 0) is a parameter that reflects the importance assigned to the largest deviation. If q = 1 the Minkowski distance betweenã andb is reduced to Hamming distance; if q = 2 the Minkowski distance betweenã andb is reduced to Euclidean distance; if q → +∞, the Minkowski distance betweenã andb is reduced to Chebyshev distance. Inspired by the Shapley Choquet integral, a Shapley Choquet integral-based Minkowski distance between IVHFULSs is further investigated.

IV. A NEW METHOD FOR MCGDM WITH IVHFULEs
In this section, a new method for MCGDM with IVHFULEs is proposed by using the IVHFULGHSC operator.

A. DESCRIPTION OF PROBLEMS
Consider a MCGDM problem with interval-valued hesitant fuzzy uncertain linguistic information, let A = {a 1 , a 2 , . . . , a m }, C = {c 1 , c 2 , . . . , c n }, and E = {e 1 , e 2 , . . . , e r } be alternatives set, criteria set and DMs set, respectively. Suppose that the evaluation of alternative a i with respect to criterion c j is expressed by an IVHFULẼ 1, 2, . . . , m; j = 1, 2, . . . , n; k = 1, 2, . . . , r). Thus, an interval-valued hesitant fuzzy uncertain linguistic decision matrixÃ k = (ã k ij ) m×n is elicited. The information about the individual λ-fuzzy measure and collective λ-fuzzy measure on criteria set are not completely unknown but partially known. Suppose that g k λ (c j ) ∈ H k c j , (k = 1, 2, · · · , r; j = 1, 2, . . . , n), where H k c j is an interval number which denotes the partially known information of individual λ-fuzzy measure g k λ (c j ) of criterion c j given by DM e k . Analogously, suppose that g λ (c j ) ∈ H c j , (j = 1, 2, · · · , n), where H c j is an interval number which denotes the partially known information of the collective λ-fuzzy measure g λ (c j ) of the criterion c j given by the decision group.

B. DETERMINATION OF THE INDIVIDUAL SHAPLEY FUZZY MEASURE ON CRITERIA SET
As mentioned previously, the Shapley Choquet integral can flexibly describe and deal with the interaction phenomena among the elements. However, there exists the exponential complexity in the Shapley Choquet integral. In order to overcome the drawback, a maximizing deviation method is developed to obtain the Shapley fuzzy measure on criteria set.
The maximizing deviation method proposed by Wang [51] is to deal with MCDM problems with numerical information. After this method is put forward, some scholars regard it as a kind of evaluation method applied to many fields so as to achieve good evaluation results [31], [52], [53]. According to Wang [51], for a MCDM problem, alternatives are ranked by the collective preference values. The larger the ranking value, the better the corresponding alternative. For a criterion, if the preference values given by a DM are small across alternatives, it shows that such a criterion is useless. Therefore, criterion should be assigned a smaller weight for the DM. On the contrary, the criterion should be assigned a bigger weight.
Based on the above analysis, in order to make deviation between each criterion bigger enough, some programming models are constructed to obtain the fuzzy measure on criteria set for each DM e k (k = 1, 2, . . . , r), i.e., different measures of criteria should be assigned by different DMs.
For criterion c j , the evaluation deviation between all alternatives for DM e k can be defined as follows: where is the Minkowski distance betweenã k ij andã k lj calculated by Eq. (15), (1), (2), · · · , (n) denotes a permutation of (1, 2, · · · , n) such that d k (1) ≤ d k (2) ≤ · · · ≤ d k (n) and C (j) = {c (j) , c (j+1) , · · · , c (n) } with C (n+1) = Ø, ρ C (j) (g k λ , C) and ρ C (j+1) (g k λ , C) are the individual Shapley fuzzy measure of criteria subsets C (j) and C (j+1) , i.e., where c and c represent the numbers of elements in set C and C , respectively. The total evaluation deviation between all criteria for DM e k could be calculated as To determine the individual λ-fuzzy measure g k λ (c j ) of criterion c j for DM e k , an optimization model is constructed VOLUME 8, 2020 as follows: where H k c j is the partially known information of individual λfuzzy measure of criterion c j given by DM e k . Solving the above model, the individual λ-fuzzy measure g k λ on criteria set C for DM e k is obtained. Then, utilizing Eq. (7), the individual generalized Shapley index ρ sh S (g k λ , C) with respect to λ-fuzzy measure g k λ on criteria set C for each DM is derived. and Apparently, the closer distance between alternative a i and individual IVHFUL-PIS and the further distance between alternative a i and individual IVHFUL-NIS, the better the alternative a i . Thus the individual closeness coefficient of alternative a i for DM e k is defined as follows: where Next, a multi-objective programming model is constructed to obtain the fuzzy measure on DMs set.
Obtain the best and worst individual closeness coefficients vectors for each DM as follows: where is the λ-fuzzy measure of DM e k and g λ (e k ) ∈ H k is the partially known information of λ-fuzzy measure of DM e k . Let (1) + , (2) + , · · · , (r) + be a permutation of (1, 2, · · · , r) such that d and (1) − , (2) − , · · · , (r) − be a permutation of (1, 2, · · · , r) such that d e (k+1) , . . . , e (r) } with E (r+1) = Ø. Let ρ E (k) (g λ , E) and ρ E (k+1) (g λ , E) be the Shapley fuzz measures of DMs subsets E (k) and E (k+1) , i.e., where e and e represent the numbers of elements in set E and E , respectively. Thus, the collective relative closeness degree S i of alternative a i can be defined as (28), shown at the bottom of the next page.
Then, the λ-fuzzy measure on DMs set can be obtained by constructing a multi-objective fractional programming model that aims at maximizing all collective relative closeness degrees of alternatives. The multi-objective fractional programming model is constructed as follows: Set S = min{S 1 , S 2 , · · · , S m }. Using the max-min method, Eq. (29) is transformed as follows: According to Eq. (28), the above model can be converted to the following programming model: The λ-fuzzy measure on DMs set can be derived by solving Eq. (30).
Thus, all the individual decision matrices A k = (ã k ij ) m×n can be aggregated to the collective decision matrix A = (ã ij ) m×n by utilizing the IVHFULGHSC operator, wherẽ

D. DETERMINATION OF THE COLLECTIVE SHAPLEY FUZZY MEASURE ON CRITERIA SET
Similarity, method obtained individual Shapely fuzzy measure on criteria set can be used to obtain the collective Shapely fuzzy measure on criteria set C For criterion c j (j = 1, 2, . . . , n), the evaluation deviation between all alternatives with regard to collective decision matrix can be defined as follows: where d j = m i=1 m l=1,l =i d(ã ij ,ã lj ). d(ã ij ,ã lj ) is the Minkowski distance betweenã ij andã lj calculated by Eq. (15).ã ij is the collective overall value of alternative a i on criterion c j derived by Eq. (31), (1), (2), · · · , (n) denotes a permutation of (1, 2, · · · , n) such that d (1) ≤ d (2) ≤ · · · ≤ d (n) and and ρ C (j+1) (g λ , C) are the collective Shapley fuzzy measure of criteria subsets C (k) and C (k+1) , i.e., where n represents the number of elements in set C . The total evaluation deviation between all alternatives on all criteria can be calculated as To determine the collective λ-fuzzy measure g λ of criterion c j , an optimization model is constructed as follows: where H c j is the partially known information of the collective λ-fuzzy measure of the criterion c j given by the decision group.
Solving the above model, the collective λ-fuzzy measure g λ on criteria set C is obtained. Then, utilizing Eq. (7), the collective Shapely fuzzy measure on criteria set C is derived.
Then, the collective closeness coefficient Cc i of alternative a i can be calculated as Clearly, the greater the value of Cc i , the better alternative a i .
Theorem 4: There exist optimal solutions for Eq. (19) and Eq. (34) Proof: Apparently, the objective functions and constraints of Eq. (19) and Eq.(34) are linear equalities or linear inequalities. Thus Eq. (19) and Eq.(34) are a linear programming model. Notice that g k λ (c j ) ∈ H k c j , g k λ (c j ) ≥ 0 and g λ (c j ) ∈ H c j , g λ (c j ) ≥ 0 in Eq. (19) and Eq.(34), the four linear equalities and inequalities make the feasible regions are bounded. According to the optimal solution existence theory [54], for any linear program with the bounded feasible region, there must exist optimal solutions in its feasible region, that is to say, Eq. (19) and Eq.(34) have optimal solutions.

E. A NEW METHOD FOR MCGDM WITH IVHFULEs
Based on the above analyses, a new method using IVHFUL-GHSC operator is proposed for MCGDM with IVHFULEs. The concrete steps of the proposed method are described below.
Step 1: Construct programming model by Eq. (19) to obtain the individual λ-fuzzy measure g k λ on criteria set C for each DM e k (k = 1, 2, · · · r).
Step 4: Calculate the individual closeness coefficient of alternative a i for DM e k by Eq. (22) and derive the individual closeness coefficient matrix by Eq.(23).
Step 5: Construct the multi-objective fractional programming model by Eq. (29) and transform it into Eq. (30) to obtain the Shapely fuzzy measure on DMs set.
Step 6: Based on IVHFULGHSC operator, aggregate all individual decision matrices to the collective decision matrix by Eq. (31).
Step 7: Determine the collective λ-fuzzy measure g λ on criteria set C by Eq. (34) and calculate the collective Shapely fuzzy measure on criteria set C by Eq. (7).

V. AN ILLUSTRATIVE EXAMPLE
In this section, an example of investment on shared bikes is provided to clarify the application of the method proposed in this paper. Then validity test of the proposed method is given to illustrate the effectiveness of the proposed method. The comparative analyses with existing MCDM method is developed to show the advantages of the proposed method.

A. BACKGROUND
With rapid development of China social economy, modern public transport is popularize, such as bus, subway, taxi and so on, which greatly makes people's travel more convenient. In recent years, shared bicycles get mushroom growth to deal with the urban traffic problem which calls ''the last kilometer'' under the environment that sharing economy gets a lot of attention. Shared bicycles could not be replaced by other public transport while solve ''the last kilometer'' problem due to its advantage of no fixed return site, environmental protection, energy saving, convenience and intelligence. There are many different types of bike-sharing brands in Chinese market, such as Hello bike, ofo bicycle, Mobike, Didi Bike and so on. It is convenient for people's outside travel.

B. AN ILLUSTRATION OF THE PROPOSED APPROACH
Assume that there exists an investment company that plans to invest one type of bike-sharing brands. There are four possible alternatives to be considered, which are listed as: Hello bike a 1 , ofo bicycle a 2 , Mobike a 3 , Didi Bike a 4 . To evaluate the four types of bike-sharing brands, the proposed method of this paper is applied to the example. After analyzing the example, the evaluation criteria includes popularity c 1 , profitability c 2 , fund management c 3 and social impact c 4 . The four criteria are benefit. Three DMs e 1 , e 2 , and e 3 are invited to evaluate the four types of bike-sharing brands from above four criteria by the IVHFULEs, which are listed in Tables 1-3.
The information about the individual λ-fuzzy measure is partially known. Suppose that the partially known λ-fuzzy measure of each criterion for each DM are respectively given by    In the following, the proposed method is used to rank the four types of bike-sharing brands.
Step 1: According to Eq. (19), a linear programming model of the individual λ-fuzzy measure on criteria set C for DM e 1 is built as follows: Step 3: According to Eq. (20) and Eq. (21), individual IVHFUL-PIS and IVHFUL-NIS are determined.
Step 4: According to Eq. (22), the individual closeness coefficients matrix is derived as follows: Step 6: Using the proposed HFULGSC operator in Definition 16 with q = 1, a comprehensive interval-valued hesitant fuzzy uncertain linguistic decision matrix A = (ã ij ) 4×4 is obtained wherẽ a 11 = HFULGSC(ã 1 11 ,ã 2 11 ,ã 3 11 ) = < [s 4 , s 5 Step 7: According to Eq. (22), a linear programming model for the collective λ-fuzzy measure on criteria set C is built as follows: Solving the above model, the collective λ-fuzzy measure on criteria set C are obtained as follows: By Eq. (7), the collective Shapley fuzzy measure on criteria set C are derived as:   According to the collective closeness coefficients, the ranking order of alternatives is generated as a 1 a 2 a 4 a 3 .
Thus Hello bike is the most valued alternative for the investment company.
Additionally, the final results may be different with the different values of parameter q when aggregating the individual decision making matrices to the comprehensive one. Thus when q takes different values, the corresponding closeness coefficients and ranking orders are shown in Table 4 and depicted in Fig. 1.   According to Table 4 and Fig 1, the optimal alternative not change with different parameter value q and the optimal alternative is always a 1 . However, the non-optimal alternative may be different when parameter value q takes different values. The ranking orders of non-optimal alternative are totally changed when q = 0 and when q → +∞. The above analysis reveals that the parameter q plays an important role in decision making. It is necessary and reasonable to consider the appropriate values of parameter q during the process of making decision.

C. VALIDITY TEST OF THE PROPOSED METHOD
To verify the validity of the decision making method, Wang [55] proposed three criteria to test the relative performance of various MCDM methods.
Test Criterion 1: An effective MCDM method should not change the indication of the best alternative when a non-optimal alternative is replaced by another worse alternative (given that the relative importance of each decision criterion remains unchanged).
Test Criterion 2: The rankings of alternatives obtained by an effective MCDM method should follow the transitivity property.
Test Criterion 3: For the same decision problem and when using the same MCDM method, after combining the rankings of the smaller problems decomposed from an MCDM problem, the new overall ranking of the alternatives should be identical to the original overall ranking of the un-decomposed problem.
The validity of the method proposed in this paper is tested by the three criteria below.

1) VALIDITY TEST OF PROPOSED METHOD USING TEST CRITERION 1
In order to test the validity of the proposed method for MCGDM under test criterion 1, we change the preference values of a non-optimal alternative a 2 . The changed individual hesitant fuzzy uncertain linguistic decision matrix provided by three DMs e 1 , e 2 and e 3 are given in Table 5-7. Then, according to the test criterion 1, the indication of the best alternative should not change when the alternatives are ranked again by the same method.
The proposed method is applied to solve the changed example. The ranking order of alternatives is a 1 a 4 a 2 a 3 . Thus Hello bike is also the most valued alternative for the investment company.
The choice of the final alternative is the same as that in the original decision problem. It shows that the proposed method do not change the indication of the best alternative when a non-optimal alternative is replaced by another worse  alternative. Hence the proposed method is valid under test criterion 1.

2) VALIDITY TEST OF PROPOSED TOPSIS METHOD USING TEST CRITERION 2 AND TEST CRITERION 3
To illustrate the proposed method is valid under test criterion 2 and test criterion 3, suppose that original decision problem is decomposed into a set of smaller problems (a 1 , a 2 , a 4 ) and (a 1 , a 3 , a 4 ). By the proposed method, a 1 a 2 a 4 and a 1 a 4 a 3 are obtained for the corresponding smaller problems. If combining the results of the smaller problems, the new overall ranking of alternatives is a 1 a 2 a 4 a 3 , which is identical to the ranking order of the original problem.
Hence, the proposed method is valid under test criterion 2 and test criterion 3.

D. COMPARATIVE ANALYSES WITH EXISTING MCDM METHOD
To demonstrate the superiority of the proposed method, this section conducts comparative analyses with existing MCDM method in literature [24].
To compare with method [24], suppose the fuzzy measure µ on criteria set C as follows: Utilizing the decision information given by DM e 1 , e 2 , e 3 in Tables 1-3 and the comprehensive matrix, respectively, the ranking order of alternatives is always obtained as a 1 a 3 a 2 a 4 by the MCDM method in [24]. The ranking order of alternatives obtained by method [24] is different from that by the proposed method. Although the optimal alternative is not changed, ranking order of a 2 , a 3 and a 4 are totally different. The primary reasons may come from three aspects: (1) The method proposed by Huo and Zhou [24] is suitable to deal with the single person decision making problems. However, it is not suitable to the MCGDM. The proposed method of this paper not only can deal with the MCGDM problems, but also can solve MCDM problems, which is reasonable and consistent with the real-world decision situations.
(2) Method [24] and method proposed in this paper both applied the fuzzy measure to MCDM. However, method [24] only considers the interactions between two ''adjacent'' coalitions A (i) and A (i+1) . Considering the interactions of each coalition among elements, this paper employs the generalized Shapley fuzzy measure to aggregate the individual decision information. Thus, aggregation method proposed in this paper is more comprehensive and can effectively avoid information loss.
(3) In method [24], the fuzzy measure on criteria set are given directly, which is somewhat subjective. In this paper, two optimization models are constructed to derive the individual λ-fuzzy measure and collective λ-fuzzy measure on criteria set. The determination of the fuzzy measure in this paper is more reliable and objective, which can make the results more believable.

VI. CONCLUSION
This paper proposes a new method for MCGDM with IVH-FULEs by using the developed IVHFULGHSC operator. The proposed method takes the interactive characteristics of criteria and DMs' preference into account. The prominent features of the proposed method are summarized as follows: (1) A score function and a new accuracy function are defined to rank IVHFULEs. Based on the Shapley Choquet integral, the IVHFULGSC operator and the IVHFUL-GHSC operator are proposed to consider the interactive characteristics among elements. Then a Shapley Choquet integral-based Minkowski distance between IVHFULSs is defined.
(2) A maximization deviation method is used to build a linear programming model to determine the individual Shapley fuzzy measure on criteria set C for each DM. By utilizing the individual closeness coefficients matrices, a new multi-objective fractional programming model which maximizes the collective relative closeness degrees of all alternatives is constructed to obtain Shapley fuzzy measure on DMs set. The individual decision matrices are aggregated into the collective one by using the IVHFULGHSC operator (3) To obtain the collective fuzzy measure on criteria set, a linear programming model is established by maximizing the deviation of the evaluations on different alternatives. Thereby, the collective closeness coefficients of alternatives are calculated and used to rank alternatives. Therefore, a new method based on the IVHFULGHSC operator is proposed for MCGDM with IVHFULEs.
The developed generalized Shapley Choquet integral operators consider the interactions among the IVHFULEs. However, they overlook the priority among the criteria. Therefore, how to investigate some prioritized aggregation operators of IVHFULEs for MCGDM is very interesting and deserved to be studied in future research.
SHU-PING WAN was born in 1974. He received the Ph.D. degree in control theory and control engineer from Nankai University, in 2005. He is currently a Professor with the College of Information Technology, Jiangxi University of Finance and Economics. He has contributed more than 100 journal articles to professional journals. His current research interests include decision analysis, fuzzy game theory, information fusion, and financial engineering.
JIA YAN was born in 1997. She is currently pursuing the master's degree in management science and engineering and decision analysis, supervised by Prof. Shu-Ping Wan, with the School of Information Management, Jiangxi University of Finance and Economics. Her research interest includes multi-attribute group decision making and its applications.
WEN-CHANG ZOU was born in 1996. He received the M.S. degree in management science and engineering from the Jiangxi University of Finance and Economics, Nanchang, China, in 2020, and is majoring in fuzzy decision and supervised by Prof. Shu-Ping Wan, where he is currently pursuing the Ph.D. degree with the School of Information Management. His current research interests include the investigation on fuzzy decision and its applications to decision making.
JIU-YING DONG received the Ph.D. degree in graph theory and combinatorial optimization from Nankai University, in 2013. She is currently an Associate Professor with the School of Statistics, Jiangxi University of Finance and Economics. She has contributed more than 30 journal articles to professional journals, such as Information Sciences, Discrete Mathematics, Graphs and Combinatorics, and Information Fusion. Her current research interests include decision analysis, graph theory, and combinatorial optimization. VOLUME 8, 2020