Profit Allocations for Restricted Coalition With Hesitation Degrees in Cooperative Game Theory

Profit allocation plays an important role in the decision-making field. In this paper, we study an allocation method on restricted coalition cooperation with intuitionistic fuzzy coalitions, in which the partners have some hesitation degree and different risk preferences when they participate in a limited communication structure game. In order to sufficiently analyze the profit allocation strategy, an average tree solution (A-T solution) with Choquet integrals and hesitation degrees is studied. In particular, a simple solving method for the A-T solution is proposed by proving that the characteristic functions of the cooperative game satisfy the monotonicity condition. Using this method, the upper and lower bounds of the A-T solution can be calculated directly from the upper and lower bounds of the interval characteristic functions. This method avoids the subtraction of interval numbers. Furthermore, the properties of the A-T solution according to an axiomatic system are proved in this paper. Finally, the applicability and superiority of the proposed approach are demonstrated through comparison with other methods.


I. INTRODUCTION
Currently, research into cooperative games is based mainly on the hypothesis of arbitrary coalitions being formed. However, due to the limitation of resources, status, and culture, cooperation is always limited in reality [1]. In this situation, direct or indirect connection among players is a necessary condition for forming an alliance, and it is a cooperative game with a limited communication structure, [2]. This game can be categorized as a limited graph cooperative game, in which the alliances are restricted. As the restricted coalition cooperation can effectively reflect complicated cooperation, the corresponding solutions have been proposed [3]- [6]. An average tree solution (A-T solution) is defined for an acyclic restricted coalition cooperative game [7], which satisfies the properties of component efficiency and component fairness. The A-T solution is analyzed by presenting a weaker condition that The associate editor coordinating the review of this manuscript and approving it for publication was Oussama Habachi . the super-additivity belongs to the core [8]. Later, the A-T solution for an acyclic restricted coalition cooperative game with communication structure was generalized [9], [10]. And a version of the A-T solution appropriate for a game based on an acyclic graph was studied [11]. The A-T solution of a restricted coalition cooperative game is concerned more and more because its good properties [12]: it must lie in the core if the game is super-additive but the Myerson value must not, and it is much simpler to find the marginal eigenvector of the allocation than the Shapley value.
In addition, fuzzy cooperative games have attracted great attention from researchers due to players often being unable to evaluate situations accurately in a real game. Fuzzy cooperative games with fuzzy coalitions were introduced to extend crisp coalitions [13], in which the degree of participation is a real number in the interval [0,1] instead of 1 or 0. This means that players may not fully participate in a coalition, but only partially participate in it. For a restricted coalition cooperative game, a fuzzy coalition Myerson value was proposed by VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ using the proportional and the Choquet model [14]. Then the Myerson value of cooperative games with communication structure and fuzzy coalitions was studied [15]. In the above, all fuzzy coalitions are described by the set of real numbers, and the non-membership degree is simply the complement of the membership degree to 1. In practice, however, players do not often express the non-membership degree of a given element as the complement of the membership degree.
In other words, players may have some hesitation degree. For example, if a player knows his/her participation degree in a coalition is at least 0.6, and the non-participation degree is 0.3, then their hesitation degree is 0.1. In order to incorporate the hesitation degree into cooperative game theory and to make it more applicable, we analyzed it using intuitionistic fuzzy information. An intuitionistic fuzzy set [16], [17] can express more abundant information by using membership degree, non-membership degree and hesitation degree, and this plays an important role in decision-making. However, as the calculation and complete ranking problems of intuitionistic fuzzy numbers, research into cooperative games is limited. Currently, only set solutions are generally studied, such as least square prenucleolus [18] and core [19]. Our other paper [20] studied A-T solution with intuitionistic fuzzy coalition based on players' risk preferences weighted form.
In this paper, we study the restricted coalition cooperative game with intuitionistic fuzzy coalitions which satisfies the monotonicity condition and proposes a corresponding A-T solution based on Choquet integrals. There are lower and upper degrees of participation of players when we introduce confidence levels to hesitation degrees, and then the characteristic functions can be integrated into interval numbers. Nowadays, interval numbers are widely used in fuzzy decision, and cooperative games are no exception. The core of interval-valued cooperative games based on an intervalvalued square dominance core and interval-valued dominance core is extended [21]. And an axiomatic characterization of the interval-valued Shapley-like value of a subclass of interval-valued cooperative games is given [22], in which the interval-valued cooperative game is called size monotonicity. In addition, [23] studied the interval Shapley function based on the extended Hukuhara difference [24], which is an interval population monotonic allocation function when the games are convex. Most of the aforementioned works used the partial subtraction operator [25], and Moore's [26] and Hukuhara [24] interval subtractions, which are irreversible. What's worse, Moore's subtraction usually increases the uncertainty of the resulted interval, and Hukuhara subtraction may not be used when the interval value does not satisfy the defined condition [27]. In view of the facts mentioned above, this paper proposes a simple method to the profit allocations for restricted coalition with hesitation degrees in cooperative game theory. There are two important contributions: (1) we combining the idea of intuitionistic fuzzy coalitions and the restricted coalition cooperative games to study the fuzzy profit allocation problem. (2) due to the interval subtraction is complex, and is not invertible in the compute process, we propose a simple method for the A-T solution by proving that the characteristic functions of the cooperative game satisfy the monotonicity condition. Then the upper and lower bounds of the A-T solution can be calculated directly from the upper and lower bounds of their interval characteristic functions, and this can effectively avoid interval subtraction or interval order relation. The method of this paper is completely different from author's another paper [20], which aggregated the characteristic functions of intuitionistic fuzzy coalitions into crisp numbers by the attitude factors of risk preferences. It means that the essence of the cooperative game in [20] is a crisp cooperative game.
The remaining content is organized as follows: Section 2 is preliminaries; Section 3 defines a restricted coalition cooperative game with intuitionistic fuzzy coalitions, and gives the characteristic function using Choquet integral form; Section 4 proposes a formula for the intuitionistic fuzzy coalition A-T solution and proves the existence and rationality of the solution according to an axiomatic system in view of the crisp cooperative game; Section 5 illustrates the proposed method with numerical examples and comparisons with other methods to show its applicability and superiority; and Section 6 shows conclusions.
It is easy to see from Definition 1 that an intuitionistic fuzzy set is defined by a pair of membership and non-membership degrees (functions), which are more or less independent of each other, and the sum of the membership degree and non-membership degree is not greater than 1.

B. INTERVALS AND THEIR ARITHMETIC OPERATIONS
where is the set of real numbers. a D ∈ and a R ∈ are called the lower bound and the upper bound of the intervalā respectively. Clearly, interval numbers are a generalization of real numbers. Conversely, real numbers are a special case of intervals.
In the following, we give some interval arithmetic operations such as the equality, the addition, and the scalar multiplication.
Definition 2 [26]: Letā = [a D , a R ] andb = [b D , b R ] be two intervals on the set¯ , and γ is any real number. The interval arithmetic operations are defined as follows: 83106 VOLUME 8, 2020 (1) Interval equality:ā =b if and only if a D = b D and a R = b R ; (2) Interval addition:ā +b = [a D + b D , a R + b R ]; (3) Interval's scalar multiplication: The above interval arithmetic operations are an extension of arithmetic operations on real numbers. However, interval subtraction is complex, and is not invertible. The common subtraction is as follows: Moore's interval subtraction [26]: Partial subtraction operator [25]: Hukuhara subtraction [24]: ifā =b +c andc =ā −b, then the subtraction ofā andb is denoted byc

III. A RESTRICTED COALITION COOPERATIVE GAME WITH INTUITIONISTIC FUZZY COALITIONS AND ITS CHARACTERISTIC FUNCTION A. FUZZY COALITION AND INTUITIONISTIC FUZZY COALITION
In a cooperative game with fuzzy coalitions, the set N = {1, 2, · · · , n} composed by all the fuzzy coalitions is denoted as F(N ), and arbitrary elementS represents a fuzzy coalition and can be demonstrated by a fuzzy vector as: where µS (i) is the degree of participation of the player i (i = 1, 2, · · ·, n) in the coalitionS, namely the ratio of the resources invested by player i to the required resources. Currently, µS (i) describes a real number in the interval [0, 1], and the cooperative game with fuzzy coalitions [0, 1] n , the essence is still a crisp number. It shows that there is no uncertain information, let alone players' hesitation degrees.

B. RESTRICTED COALITION COOPERATIVE GAMES WITH INTUITIONISTIC FUZZY COALITIONS
In a crisp cooperative game with transferable utility (a TU-game), the triad (N , v, L) constitutes a restricted coali- then players are free to choose their own cooperative partners in the game, and the corresponding graph is a complete graph (N , L). In this case, (N , v, L) is a cooperative game with complete communication structure or a complete graph cooperative game, and usually the well-known cooperative game refers to this one, which can be abbreviated as (N , v). In this game, any player in the game can form coalitions with others freely and without restriction.
is a cooperative game with a limited communication structure. In this game, players may have a cooperation if and only if they are interconnected, and coalitions forming is restrictedly. This paper discusses a cooperative game with acyclic and restricted communication structure.
Definition 3: According to the crisp cooperative game with communication structure (N , v, L), if the fuzzy payoffsṽ of (N ,ṽ, L) are mapping functions from intuitionistic fuzzy coalitionsF(N ) to a fuzzy number set˜ , namely,ṽ : IF(N ) →˜ withṽ(∅) = 0, then (N ,ṽ, L) is a restricted coalition cooperative game with intuitionistic fuzzy coalitions. For conciseness, the restricted coalition cooperative games with intuitionistic fuzzy coalitions is denotedG, and the entirety ofG is denotedG n . This paper discusses the most common cooperative game, which satisfies general properties of convexity and super additivity. In a (N ,ṽ, L), the minimum participation degrees is µS (i), and the maximum participation degrees depend on the hesitation degree πS (x) = 1−µS (x)−υS (x). If the whole hesitation degree is meant to participate in the cooperation, then the maximum participation degrees is µS (i)+πS (i) (or 1−υS (x)). Thus the participation degree of player i (i ∈ N ) can be expressed as a interval value, the lower of which is µS (i), and the upper is different for different attitudes of hesitation degree. Therefore, the participation degree ηS (i) of player i can be defined as a closed interval number with a confidence level α (α ∈ [0, 1]): It is clear that the participation degree of player i is In a restricted coalition cooperative game with intuitionistic fuzzy coalitions (N ,ṽ, L), for any coalitionS ∈ IF(N ), let ≤ 1, then the characteristic function by Choquet integral form [28] of (N ,ṽ, L) can be expressed as: where

IV. INTUITIONISTIC FUZZY COALITION A-T SOLUTION AND ITS PROPERTIES A. CRISP RESTRICTED COALITION COOPERATIVE GAME AND ITS A-T SOLUTION
For an undirected graph (N , L), a coalition of players K ⊆ N is a network of (N , L) if K is connected, i.e. between any two members of K , there is a path with in L. A network is called a component if no larger network contains it. We denote byĈ L (N ) the set of all components of (N , L). An n-tuple In a crisp restricted coalition cooperative game (N , v, L), a A-T solution is defined as follows [7]: where i = 1, 2, · · · , n, B L is the collection of all admissible n-tuple of coalitions B, B L represents the number of components of B L . It has been proved that A-T solution is characterized by efficiency, dummy, linearity, independent, and it satisfies properties of component efficiency, component fairness, and additivity [12].

B. A-T SOLUTION OF A RESTRICTED COALITION COOPERATIVE GAME WITH INTUITIONISTIC FUZZY COALITIONS
Given the characteristic functions of a restricted coalition cooperative game with intuitionistic fuzzy coalitions can be transformed into interval values from Section 3.2, the arithmetic operations of interval numbers are necessary for solving an intuitionistic fuzzy coalition A-T solution. However, the interval subtraction may result in irrational conclusions as it is not an invertible operator [24]. In this paper, we focus on developing an effective and simplified method for computing an intuitionistic fuzzy coalition A-T solution by using monotonicity, rather than the special interval subtraction operator or ranking method.
For any interval-valued characteristic functions of (N ,ṽ, L), we can convert it by introducing weighting factor as follows: whereṽ(λ)(∅) = 0. The weighting parameter λ ∈ [0, 1] is any real number, which can be interpreted as an attitude factor of players (or decision makers), i.e., it reflects the attitude of players towards uncertainty.
Therefore, for any coalition monotonicity-like (N ,ṽ, L) ∈G n , i.e., it satisfies Eq. (6), then it can be derived directly from Theorem 1 and Eq. (4), and the lower and upper bounds of the components (intervals) ψ i (ṽ) (i = 1, 2, · · · , n) of the intuitionistic fuzzy coalition A-T solution ψ(ṽ) = (ψ 1 (ṽ), ψ 2 (ṽ), · · · , ψ n (ṽ)) T are given as follows: So ψ i (ṽ) for the players i(i = 1, 2, · · · , n) in the coalition monotonicity-like (N ,ṽ, L) ∈G n are directly and explicitly expressed as follows: Therefore, the lower bounds of the intervals ψ i (ṽ) (i = 1, 2, · · · , n) of the intuitionistic fuzzy coalition A-T solution can be obtained by distributing the lower bounds of the interval-valued coalitions' payoffs to player i who are in the coalition. Similarly, we can obtain the upper bounds of ψ i (ṽ) for player i. It is obvious that Eq. (6) is an important condition which ensures that the intuitionistic fuzzy coalition A-T solution ψ i (ṽ) possesses monotonicity. It requires that the sum of all interval-valued characteristic lengths of bigger coalitions with player i to that of the coalitions without player i is monotonic It is clear that the monotonicity of Eq. (6) is a good property, and it can be satisfied more easily than traditional monotonicity of a cooperative game (N ,ṽ, L) ∈G n : len(ṽ(B)) ≥ len(ṽ(B\{i})) (i = 1, 2, · · · , n) where B is an admissible coalition as above in section IV, and len(ṽ(B)) is the interval length of characteristic function of B.
The interpretation is that the interval length of the payoffs of the bigger coalition is not necessarily bigger than that of a coalition without player i ∈ B. That is, if an (N ,ṽ, L) ∈G n satisfies Eq. (6), even if len(ῡ(B)) ≤ len(ῡ(B\{i})), we can use the method to easily compute intuitionistic fuzzy coalition A-T solution. VOLUME 8, 2020

C. PROPERTIES OF INTUITIONISTIC FUZZY COALITION A-T SOLUTION
In this section, we will discuss some useful and important properties of the intuitionistic fuzzy A-T solution of monotonicity-like restricted coalition cooperative games according to the relevant properties of crisp A-T solution.
where K h and K l express the link networks containing nodes i and j when the edge L{h, l} is deleted in component.
This is to say,

P2 (Component Fairness):
As the A-T solution of crisp restricted coalition games satisfies the fairness for any edge L{h, l} in L(K ), it holds that Additionally, with component fairness of crisp A-T solution, it directly follows that Component fairness means that if the link between two players is deleted, the changes allocation of players is the same, which implies that their marginal contributions in this component are equal.
P3 (Additivity): For any (N ,ṽ, L) ∈G n and (N ,w, L) ∈G n , the linearity of characteristic function of Eq. (7) implies that Analogously, according to Eq. (8), we can easily prove that . Combined with the aforementioned conclusion, and according to case (1) of Definition 2, we obtain additivity Therefore, there exists ψ i (ṽ) = 0 (i = 1, 2, · · · , n) whenever v(i) = 0 for all intuitionistic coalitionS ∈ N . P6 (Linearity): For any (N ,ṽ 1 , L) ∈G n , (N ,ṽ 2 , L) ∈G n , from the additivity of intuitionistic fuzzy coalition A-T solution ψ(ṽ) :G n →˜ , there have Combined with the linearity of characteristic function of Eq. (9), we can easily get ψ i (aṽ 1 + bṽ 2 ) = aψ i (ṽ 1 ) + bψ i (ṽ 2 ). P7 (Independence): In a restricted coalition game (N ,ṽ, L) ∈G n , there existsṽ T (S) = 0 when coalitionT ⊂S and otherwiseṽT (S) = 0. If a player joins a coalition in the game, it holds the independence property [12]: AT i (vT ) = AT i (vT ∪{j} ) when i ∈T , i ∈S, j / ∈T and j / ∈S. We therefore have, Independence means that if a player joins a coalition, the payoff of any player in the coalition not linked to that player remains the same, because players can form coalitions if and only if they are interconnected. And the A-T solution treats each player to the number of agents they are connected to outside of the coalition.
Theorem 3: If (N ,ṽ, L) ∈G n is a monotonicity-like restricted coalition cooperative game, then the intuitionistic fuzzy coalition A-T solution is an reasonable existence allocation of (N ,ṽ, L).
Proof: If i ∈ N , according to component efficiency of intuitionistic fuzzy coalition A-T solution, we have i∈K ψ i (ṽ) =ṽ(K ) for any K ∈Ĉ L (N ). That is to say, ψ i (ṽ) satisfies group rational.
In addition, as So it follows directly that Therefore, ψ(ṽ) satisfies the group rationality and individual rationality conditions from allocations. Hence, the intuitionistic fuzzy coalition A-T solution is an efficient allocation. Theorem 4: Ifṽ is an interval-valued complete communication graph game, then the intuitionistic fuzzy coalition A-T solution AT (ṽ) is equivalent to the interval-valued Shapley value ϕ(ṽ).
Proof It has been proven that the A-T solution is equivalent to the Shapley value when the crisp cooperative game is a complete communication graph game [9], [12]. For intervalvalued payoffs, we have Then there is ψ i (ṽ) = ϕ i (ṽ)(i = 1, 2, · · · , n). Therefore, the intuitionistic fuzzy coalition A-T solution, AT (ṽ) is equivalent to the fuzzy Shapley value ϕ(ṽ).

V. PROFIT ALLOCATION STRATEGY DECISION OF RESTRICTED COOPERATION AND COMPARISON A. PROFIT ALLOCATION STRATEGY DECISION OF RESTRICTED COOPERATION
Software cooperation is one of the important development models in the new era. Forming an alliance with appropriate partners and having a fair distribution mechanism are key issues. In a project of software cooperation, there is a knowledge value chain based on knowledge, skills and VOLUME 8, 2020 capital investors, and the coalition formation is limited to the upstream and downstream of knowledge value chain. For conciseness, upstream, middle-stream and downstream are called player 1, player 2, and player 3. Current research into software cooperation is based on the assumption that players have equal power and can form coalitions freely [29]. However, this is untrue. Restricted cooperation in software cooperation is obvious, as different players have different marginal contributions. Due to the weak marginal revenue of players 1 and 3, this means that they cannot cooperate directly, but that player 1 must deal with player 2 first, then join the coalition with player 3 via the intermediary role of player 2. In another words, player 2 is the crucial player in this cooperation. The essence of this is a cooperative game with limited communication structure, and all the potential coalitions are {1,2}, {2,3}, {1,2,3}. This phenomenon is particularly common in social and economic cooperation, such as watershed governance and supply chain management. In this cooperation, there are upper, middle and lower reaches, the upstream and downstream cannot communicate directly due to restricted communication, and they must turn to the key player, middle reach, in order to create an alliance. With restricted cooperation, the alliance structure of cooperation and players' location within the structure are incredibly different from traditional cooperation.
In other words, if three players participate fully in a restricted cooperation as above, everyone can invest 100% of the required resources into the alliance, i.e. crisp coalition, where player1 invests at least 10% into the coalitions, 50% must not be invested, while πS(1) = 1 − µS(1) − υS(1) = 40% is his/her hesitation degree. Therefore, the minimum participation degree of player1 is 0.1, and the maximum participation degree is 0.5. Similarly, the intuitionistic fuzzy coalition of players 2 and 3 can be explained.
According to Eq. (4), the characteristic functions of an intuitionistic fuzzy coalition {1, 2, 3} with α = 1 are In a similar way, we can obtain all the characteristic functions of intuitionistic fuzzy coalitions, as shown in Table 1.

B. COMPARISON WITH ALLOCATION BASED ON INTERVAL SUBTRACTION
In order to show the applicability and superiority of the proposed method, we compare the allocations based on Moore's interval subtraction [26], the partial subtraction operator [25] and Hukuhara subtraction [24].
By using the Moore's interval subtraction, i.e., And it is irrational that the lower bound is larger than the upper bound, which conflicts with the notation of intervals.
According to the rule of interval addition, the Hukuhara subtraction cannot be used to calculate the above ψ 3 (ṽ) due to the fact that the interval numbers [21,42] and [3,31] cannot satisfy the condition In this example, for clarity, the allocation values of players based on different subtraction are shown in Table 2. It can be observed that the conditions of the partial subtraction and the Hukuhara subtraction are similar to Eq. (10), and it is always useless when the conditions cannot be satisfied. But the condition given by Eq. (6) is weaker than Eq. (10). That is to say, if Eq. (10) is satisfied, then Eq. (6) is always true. Therefore, if the intuitionistic fuzzy coalition A-T solution can be calculated by using the partial subtraction operator or Hukuhara subtraction, it can then be calculated using this proposed method. In other words, both subtractions are special cases of the proposed method in this paper.

C. COMPARISON WITH SHAPLEY VALUE
To highlight the superior applicability and effectiveness of the profit allocation for restricted coalitions, it is compared with the fuzzy Shapley value. In this case, there is no coalition between player 1 and player 3, if we hypothesize that a coalition exists, and its payoff equals the sum of independent payoff of each player. Based on this hypothesis, the coalition payoff with player 1 and player 3 can be givenv({1, 3}) = v({1}) +v({3}) = [12,24]. The allocation strategy for each player can then be obtained from the Shapley value in Table 3.

TABLE 3. Allocation values of players by A-T solutions and Shapley value.
Comparing the interval-valued A-T solution with the interval-valued Shapley value, we can obtain ψ 1 (ῡ) < ϕ 1 (ῡ), ψ 2 (ῡ) > ϕ 2 (ῡ), ψ 3 (ῡ) < ϕ 3 (ῡ). That is to say, the allocation of player 2 increases while the allocations of player 1 and player 3 both decrease according to the A-T solution, relative to the Shapley value. This is a result of highlighting the special status of player 2, which is indispensable in restricted coalition cooperative games with restricted communication structure. In the Shapley value, it is not objective or scientific enough that we suppose the unrealistic coalition (e.g. the coalition {1, 3}) is existing and use its payoff value in the cooperative game. In addition, we have demonstrated that players' profitability depends not only on their marginal degree of contribution to the coalition, but also on the communication structure of the coalition and players' positions in the cooperation. Therefore, the interval-valued A-T solution is more reasonable and appropriate than the interval-valued Shapley value in cooperative games with limited communication structure.

VI. CONCLUSIONS
In reality, cooperative games with restricted coalition and fuzzy information are very common. As A-T solution is an important single-valued solution in limited communication structure cooperative games, we propose the fuzzy A-T solution of restricted coalition cooperative games with intuitionistic fuzzy coalitions. The proposed method can take into account players' judgments and hesitations in cooperation. It is worth pointing out that the intuitionistic fuzzy coalition A-T solution is a generalized form of that of a classical cooperative game, and it extends the fuzzy cooperative game theory. To solve the intuitionistic fuzzy coalition A-T solution, the upper and lower limits of characteristic functions for the cooperative game are first obtained based on Choquet integrals and confidence levels. Then, a simplified method for the A-T solution is proposed, which can effectively avoid the irreversible interval subtraction. With this method, the upper and lower bounds of the A-T solution can be directly calculated from the upper and lower bounds by proving that the restricted coalition cooperative game is monotonic. Furthermore, we prove that the intuitionistic fuzzy coalition A-T solution satisfies several important and useful properties. In addition, the applicability and superiority of the proposed approach is demonstrated by comparing it with partial subtraction operator, Moore subtraction and Hukuhara subtraction. The method and model proposed in this paper can describe the restrictiveness and fuzziness of cooperation, and they are more practical and universal for allocation decision-making.

CONFLICTS OF INTEREST
The authors declare that there is no conflict of interest regarding the publication of this article.
DENGFENG LI was born in 1965. He received the B.Sc. and M.Sc. degrees in applied mathematics from the National University of Defense Technology, Changsha, China, in 1987 and 1990, respectively, and the Ph.D. degree in system science and optimization from the Dalian University of Technology, Dalian, China, in 1995.
He is currently the Cheung Kong Scholar and a Distinguish Professor with the School of Economics and Management, Fuzhou University, Fuzhou, China. He has authored or coauthored more than 300 journal articles and eight monographs. His research interests include fuzzy decision analysis, group decision making, fuzzy game theory, fuzzy sets and system analysis, fuzzy optimization, and differential games. He is also a Professor with the Business School, Sichuan University, Chengdu, China, and also with the College of Sciences, PLA University of Science and Technology, Nanjing. He has contributed more than 560 journal articles to professional journals. His current research interests include information fusion, group decision making, computing with words, and aggregation operators. VOLUME 8, 2020