Paradigm Shift Toward Aggregation Strategies in Proportional Hesitant Fuzzy Multi-Criteria Group Decision Making Models of Advanced Practice for Selecting Electric Vehicle Battery Supplier

Since its initiation, hesitant fuzzy sets (HFSs) have gained prominence thanks to their capability to describe the hesitation of experts to assign membership degrees to objects belonging to a concept. Proportional hesitant fuzzy sets (PHFSs) are an important extension of HFSs and are characterized by the combination of possible membership degrees and their associated proportional information. PHFSs have a huge application potential for hesitant fuzzy GDM problems, because the proportional information in PHFSs can be determined objectively and the introduction of this new information dimension can effectively reduce the uncertainty. Nevertheless, PHFSs have not yet attracted sufficient attention from researchers and practitioners, which motivates us to expand the theory of PHFSs and explore its application potential. The main work comprises the following three aspects: First, we define some basis operations on PHFSs, develop aggregation operators for PHFSs, and demonstrate their properties and interrelationships to lay the theoretical foundations for the application of PHFSs. Next, we construct two multicriteria group decision making (MCGDM) models based on the proposed PHFS-based aggregation operators to bridge between theory and practice for PHFSs. In this step, we propose a method for transforming HFSs or fuzzy sets (FSs) into PHFSs, and two methods based on the maximum entropy principle are proposed for specifying criterion weights. Finally, we investigate a practical case study of the problem of selecting an electric vehicle battery (EVB) supplier to validate the outstanding advantages of PHFSs, explore the compensation characteristics and the applicability of the PHFS-based aggregation operators, and demonstrate the effectiveness and feasibility of the proposed MCGDM models. This paper provides a useful reference for MCGDM in a hesitant fuzzy context.


I. INTRODUCTION
A fuzzy set is a class of objects with a continuum of membership degrees and is represented mathematically by a membership function that assigns a membership degree in the interval [0,1] to each object [1].Starting with the original work of Zadeh [1], fuzzy sets have gained significant attention because of their outstanding ability to model uncertainty.At present,several generalizations exist to relax the requirement that only a single value between 0 and 1 be assigned to the membership of an element.Examples include type-2 fuzzy sets [2], intuitionistic fuzzy sets [3], Pythagorean fuzzy sets [4],and hesitant fuzzy sets [5].Type-2 fuzzy sets incorporate uncertainty about the membership function into fuzzy sets to address the drawback of the original fuzzy sets whereby the membership function has no uncertainty associated with it [6], [7].Type-n fuzzy sets generalize type-2 fuzzy sets by permitting the membership to be a type-(n − 1) fuzzy set.Intuitionistic fuzzy sets are characterized by membership degree and nonmembership degree which satisfy the condition that their sum is equal to or less than one.As a generalization of IFSs, PFSs accommodate more relaxed condition that the square sum of the membership degree and nonmembership degree is equal to or less than one [8]- [10].Hesitant fuzzy sets (HFSs) are introduced to model the scenario in which a set of values are possible for assigning membership degree to an object belonging to a concept.Scenarios involving hesitation can be classified into the following two types: • Scenario 1: When an expert is required to assign a membership degree to an object belonging to a concept, the expert hesitates among a set of possible values.To characterize the hesitation of the expert, all possible values are retained in a set instead of selecting a single value.
• Scenario 2: When each member of an expert team is asked to assign a membership degree to an object belonging to a concept, the experts have different opinions and fail to reach an agreement.Instead of using an aggregation operator to fuse these values into a single value, all values are retained in a set to characterize the hesitation of this expert team.
Both types of hesitant scenarios are common in real decision-making problems.Thus, since their introduction, HFSs have attracted significant research attention [11]- [17].Existing theoretical research on HFSs mainly focuses on diverse extensions, information measures, and aggregation operators.
PHFSs are an extension of HFSs that were introduced by [24] and are characterized by a predefined set of possible membership degrees for elements and the proportional information of each membership degree.These are mainly used to address group-decision-making (GDM) problems in a fuzzy hesitant context.Initially, PHFSs were introduced to model Scenario 2 with the proportion of each membership degree measurable.In this scenario, each member of an expert team is required to output individual assessment information in the form of a fuzzy set (FS), and PHFSs can be used to characterize the collective assessment information in which membership degrees comprise all values given by all the experts, and the proportion of each membership degree equals the proportion of experts who output it.In practical decisionmaking scenarios, it is hard for experts to provide assessment information in the form of PHFSs, but outputting assessment information in the form of HFSs is feasible.Scenario 1 describes a GDM context in which an expert may prefer to articulate her preferences in the form of a HFS when she hesitates between possible values.Thus, the initial condition should be relaxed to enable PHFSs to characterize the collective assessment information synthesized from the individual assessment information in the form of FSs or HFSs.In addition, the linguistic counterpart of PHFSs, which is called a proportional hesitant fuzzy linguistic term set [29], is an effective alternative when linguistic uncertainty needs to be modeled in real-life scenarios.
PHFSs are easily mistaken by the concepts of probability hesitant fuzzy sets proposed by Zhu and Xu [30] and weighted hesitant fuzzy sets proposed by Zhang and Wu [31] owing to the fact that the three concepts share the same mathematical structure, but there are obvious differences between them.Subsequently, we highlight the merits of PHFSs by differentiating these similar concepts.For probability hesitant fuzzy sets, the authors clearly state that the probability information is assigned to each possible membership degree with the sum of all the probability information equaling 1.The probability information is derived from the subjective judgement of expert to measure the likelihood of each membership degree in HFS.For weighted hesitant fuzzy sets, different weights are assigned to all the possible membership degrees and the sum of these weights is equal to 1.The weight information is also specified by experts to depict the relative importance ratings of each membership degree.Thus, both the probability information in probability hesitant fuzzy sets and the weight information in weighted hesitant fuzzy sets are subjectively given by experts, and probability hesitant fuzzy sets and weighted hesitant fuzzy sets can be used to model assessment information in hesitant fuzzy GDM settings both individually and collectively.Nevertheless, it is difficult for decision makers (DMs) to further output the corresponding probability information or weighting information besides hesitant fuzzy evaluations in practical decision-making context.For PHFSs, proportional information is introduced to depict the collective preferences on all the possible membership degrees.Thus, PHFSs are only used to model collective assessment information.More importantly, the proportional information can be objectively calculated from the assessment information given by all the DMs, thus PHFSs are more preferable than probability hesitant fuzzy sets and weighted hesitant fuzzy sets in hesitant fuzzy GDM settings.Furthermore, the introduction of a new dimension of information about proportional information can reduce the uncertainty of characterizing group evaluations while preserving the original assessment information, effectively improving the reliability of decision-making results.Thus, PHFSs has a significant application potential for hesitant fuzzy GDM problems.
Xiong et al. [24] defined a normalized Hamming distance measure and a comparison law for PHFSs and developed a multiple criterion GDM method in the context of PHFSs.However, few studies have since followed up on the theory and the application of PHFSs, which motivates us to expand the theory of PHFSs and explore its application potential.
Information aggregation functions, which are used to combine various inputs coming from different sources into a single representative value [32], [33], have been widely applied in the aspects of decision-making, expert systems, risk analysis, and image processing [33]- [36].This paper concentrates on the proposal of proportional hesitant fuzzy weighted averaging (PHFWA) operator, proportional hesitant fuzzy weighted geometric (PHFWG) operator, proportional hesitant fuzzy ordered weighted averaging (PHFOWA) operator, proportional hesitant fuzzy ordered weighted geometric (PHFOWG) operator, and further puts forward their generalized types, including the generalized PHFWA operator (GPHFWA), generalized PHFWG operator (GPHFWG), generalized PHFOWA operator (GPHFOWA), and generalized PHFOWG operator (GPHFOWG), to extend the weighted averaging and geometric aggregation operators, the ordered weighted averaging and geometric aggregations to accommodate the PHFS environments and to lay the theoretical foundations for the application of PHFSs.In addition, the PHFS-based aggregation operators are preceded by the operation laws for PHFSs to serve as the basis of developing these aggregation operators.
As an important extension of multi-criteria decision making (MCDM) [37]- [41] and one of the main application fields of aggregation functions [42]- [44], multi-criteria GDM (MCGDM) has garnered considerable attention from scholars and practitioners and been widely applied to various socioeconomic fields including management science, social science, economics, public administration, military research, etc. [45]- [52] In this paper, we further develop two MCGDM models that use the PHFS-based aggregation operators to connect theory and practice for PHFS.One model is constructed based on the GPHFWA or GPHFWG operator, and the other model is based on the GPHFOWA or GPHFOWG operator.The main reason for developing two different MCGDM models is that clear differences exist between the prerequisites of applications of the GPHFWA or GPHFWG operator and those of the GPHFOWA or GPHFOWG operator.Additionally, because PHFSs exist only as collective assessment information during hesitant fuzzy GDM processes,we propose a method to transform individual assessment information outputted by DMs in the form of FSs or HFSs to collective assessment information in the form of PHFSs during the construction of MCGDM models.Furthermore, for the proposed MCGDM model based on the GPHFWA or GPHFWG operator, we provide a method based on maximum entropy to specify the weights of the criteria.For the proposed MCGDM model based on the GPHFOWA or GPHFOWG operator, we provide a similar method based on maximum entropy and the attitudinal character given by DMs.Finally, we conduct a thorough practical case study of the selection of a strategic supplier of electric-vehicle batteries (EVBs), which involves multiple qualitative and quantitative criteria and necessitates a multifunctional expert team.Therefore, this problem can be considered as an MCGDM problem in a hesitant fuzzy context.Through this case study, we not only test the properties of the PHFS-based aggregation operators and demonstrate the effectiveness and feasibility of the proposed MCGDM models but also explore the compensation characteristics and the applicability of these aggregation operators and validate the advantages of PHFSs.
This paper is organized as follows: Section 2 briefly reviews several basic concepts of HFSs and PHFSs and presents some basic laws for operations on PHFSs.Section 3 presents a series of PHFS-based averaging operators and their properties and interrelationships.Two MCGDM models based on PHFS-based averaging operators are developed in Sec. 4, in which we propose a method for transforming FSs or HFSs into PHFSs and methods to specify criterion weights.Section 5 undertakes a practical case study to validate the effectiveness and practicality of the proposed techniques.Finally, Sec. 6 concludes this paper.

II. PRELIMINARIES
This section reviews the basic concepts of HFSs and PHFSs and gives some basic laws for operations on PHFSs.

A. HESITANT FUZZY SETS
A hesitant fuzzy set is defined by a function that returns a set of possible membership degrees for each element in the domain [5], [53].
Definition 1 [5], [53]: Let X be a reference set: a HFS on X is a function that, when applied to X , returns a subset of [0, 1].
The HFS can be mathematically expressed as [25], [26] where h E (x) is a set of values in [0, 1] that denotes the possible membership degrees of the element x ∈ X to the set E. For convenience, [26] called h = h E (x) a hesitant fuzzy element (HFE).

B. PROPORTIONAL HESITANT FUZZY SETS
A HFS and its related operations can be used to deal with GDM problems in a fuzzy, hesitant context.However, if the proportion of each membership degree is measurable, the information characterized by the HFS is incomplete.To bridge this gap, Xiong et al. [24] proposed PHFSs based on HFSs.
Definition 3 [24]: Let X be a reference set; the PHFS E on X is then represented as that represents n possible degrees of membership of the element x to the set X ; and (b) where τ i (i = 1, 2, . . ., n) is the proportion of membership degree γ i (i = 1, 2, . . ., n) and n i=1 τ i = 1.For convenience, Xiong et al. [24] called P = P E (x) a PHFE.This paper similarly uses ''PHS'' to refer to the set of all PHFEs.
Definition 4 [24]: Let X be a reference set; for any x ∈ X , (1) P E (x) = {(0, 1)} is the empty PHFS, denoted by ∅; (2) P E (x) = {(1, 1)} is the full PHSF, denoted by .Definition 5 [24]: Given a PHFS represented by its PHFE P, the complement of P is If l (P) represents the number of elements in PHFE P, it is difficult to calculate the distance measure between PHFSs A and B because l(P A (x)) is usually not equal to l(P B (x)) for any x ∈ X .Supposing l x = max {l(P A (x)), l(P B (x))}, this problem can be handled by the following two steps: (1) Ordering: Arrange the elements in l(P A (x)) and l(P B (x)) in decreasing order according to the product values of the membership degrees and their associated proportions.(2) Adding: Add several times the PHFE with smaller l ( * ) with element (0, 0) until both have the same number of elements, i.e., l x .
The distance measure of PHFS is then given as follows: Definition 6 [24]: Let A and B be two PHFSs on the reference set X = {x 1 , x 2 , . . ., x n }, then the proportional hesitant normalized Hamming distance is where l x i = max{l(P A (x i )), l(P B (x i ))}, and γ σ (j) A (x i ) are the jth largest product value in PHFEs P A (x i ) and P B (x i ), respectively.

C. OPERATIONAL LAWS FOR PROPORTIONAL HESITANT FUZZY ELEMENTS
In this section, we present some basic operations on PHFEs to lay the foundations for the development of PHFS-based aggregation operators.
Let P 1 and P 2 be two PHFEs in the reference set X and suppose the membership degree of the x ∈ X for the set ''1" and that for the set ''2" are mutually independent.Based on the operation laws for HFSs [26], the following operations are defined from the angle of probability: Definition 9: Let P, P 1 , and P 2 be three PHFEs in the fixed set X , then (1) Theorem 1: For three PHFEs β, β 1 , and β 2 in the fixed set X , we have (1 For any λ > 0, (2) For any λ > 0, (3) Theorem 2: Let P j (j = 1, 2, . . ., n) be a collection of PHFEs and let ω = (ω 1 , ω 2 , . . ., ω n ) T be the weight vector of P j (j = 1, 2, . . ., n) with w j ∈ [0, 1] and n j=1 w j = 1, λ > 0. Then .

III. AGGREGATION OPERATORS FOR PROPORTIONAL HESITANT FUZZY SETS
This section presents a series of aggregation operators for PHFSs to lay the foundation for the construction of MCGDM models, including the PHFWA, PHFWG, PHFOWA, and PHFOWG operators and their generalized forms.In addition, we validate the properties and interrelationships of these aggregation operators.

A. PROPORTIONAL HESITANT FUZZY WEIGHTED AVERAGING OPERATOR
This section defines the PHFWA operator based on PHFEs and the traditional weighted averaging operator and gives the properties of the PHFWA operator.Definition 10: Let E = {P 1 , P 2 , . . ., P n } be n PHFEs defined for a fixed set X , and let be a function of E, Following Definition 10, we begin our discussion.Definition 11: Let P j (j = 1, 2, . . ., n) be a collection of PHFEs.A PHFWA operator is a mapping PH n → PH such that where ω = (ω 1 , ω 2 , . . ., ω n ) T is the weighting vector of P j (j = 1, 2, . . ., n) with ω j ∈ [0, 1] and n j=1 ω j = 1.In particular, if ω = 1 n, 1 n, . . ., 1 n T , then the PHFWA operator reduces to the PHFA operator: Theorem 3: Let P j (j = 1, 2, . . ., n) be a collection of PHFEs and let ω = (ω 1 , ω 2 , . . ., ω n ) T be the weighting vector of P j (j = 1, 2, . . ., n) with ω j ∈ [0, 1] and n j=1 ω j = 1.The aggregated value using the PHFWA is also a PHFE, and Proof: The proof is by mathematical induction on n.First, we show that Following the operations of PHFEs, we have 2 holds for all n.
Similarly, we have the following theorem: Theorem 4: Let P j (j = 1, 2, . . ., n) be a collection of PHFEs and let ω = 1 n, 1 n, . . ., 1 n T be the weight VOLUME 7, 2019 vector of P j (j = 1, 2, . . ., n).The aggregated value using the PHFA is also a PHFE, and Proof: The proof of Theorem 4 is similar to that of Theorem 3.
Theorem 5 (Commutativity): Let P j (j = 1, 2, . . ., n) be a collection of PHFEs.If P j (j = 1, 2, . . ., n) is any permutation of P j (j = 1, 2, . . ., n), then we have Proof: Since P j (j = 1, 2, . . ., n) is any permutation of P j (j = 1, 2, . . ., n), then there exists for any P j one and only one P i such that P j = P i , and vice versa.We then have The following example shows that the PHFA operator is neither idempotent, bounded, nor monotonic.

B. PROPORTIONAL HESITANT FUZZY WEIGHTED GEOMETRIC OPERATOR
This section defines the PHFWG operator based on PHFEs and the traditional weighted geometric operator and also presents the properties of the PHFWG operator.Definition 12: Let P j (j = 1, 2, . . ., n) be a collection of PHFEs.A PHFWG operator is a mapping PH n → PH such that where ω = (ω 1 , ω 2 , . . ., ω n ) T is the weighting vector of T , the PHFWG operator reduces to the PHFG operator: Theorem 6: Let P j (j = 1, 2, . . ., n) be a collection of PHFEs and let ω = (ω 1 , ω 2 , . . ., ω n ) T be the weighting vector of P j (j = 1, 2, . . ., n) with ω j ∈ [0, 1] and n j=1 ω j = 1.The aggregated value using the PHFWG operator is also a PHFE, and Proof: The proof of Theorem 6 is similar to that of Theorem 3.
Similar to the PHFA operator, the PHFG operator is neither idempotent, bounded, nor monotonic, but it is commutative.

C. GENERALIZED PROPORTIONAL HESITANT FUZZY WEIGHTED AVERAGING OPERATOR
This section generalizes the PHFWA operator to define the GPHFWA operator and its properties.
Before proving this theorem, we first introduce the following lemma which was proven by Zhang [27]: Lemma 2: For any x ∈ [0, 1], is strictly convex.We now prove Theorem 10.
Proof: For any the proportion of the aggregated value is n j=1 τ j for any λ > 0, then we only need to show that the function f (λ) increases monotonically with λ (λ > 0).Since where By Lemma 1, we have Thus, f (λ) > 0, and f (λ) increases monotonically with λ (λ > 0).Theorem 11: Let P j (j = 1, 2, . . ., n) be a collection of PHFEs.As λ → 0 (λ > 0), the GPHFWA operator approaches the following limit: Proof: Following Theorem 8, we have , so we only need to consider the limit of the function Thus,
Theorem 15 shows that the aggregated value obtained by the PHFWA operator is no less than that obtained by the GPHFWG operator.
Proof: The proof of Theorem 19 is similar to that of Theorem 4.

IV. MULTIPLE-CRITERIA DECISION MAKING MODELS BASED ON PHFS-BASED AGGREGATION OPERATORS
In this section, we develop two models based on GPH-FWA (or GPHFWG) and GPHFOWA (or GPHFOWG) operators to solve the MCGDM problem in an uncertain context.First, we use the following notations to denote the important indices, sets, and variables in the MCGDM problem in the proportional hesitant fuzzy context.
• m: total number of alternatives; • n: total number of criteria; • t: total number of DMs involved in the decision process; • i ∈ M = {1, 2, . . ., m}: index of alternative; • j ∈ N = {1, 2, . . ., n}: index of criterion; • k ∈ T = {1, 2, . . ., t}: index of DM involved in the decision process; • A i : ith alternative; • A = {A 1 , A 2 , . . ., A m }: set of m alternatives; • C j : jth criterion; • C = {C 1 , C 2 , . . ., C n }: set of n criteria, which are considered to be independent; • D k : kth DM; • D = {D 1 , D 2 , . . ., D t }: set of t DMs; • δ k : Weight of kth DM; • ϕ = (ϕ 1 , ϕ 2 , . . ., ϕ t ) T : vector of weights of DMs, where t k=1 ϕ k = 1, 0 ≤ ϕ k ≤ 1, and k = 1, 2, . . ., t; • ω = (ω 1 , ω 2 , . . ., ω n ) T : weighting vector of criteria with respect to DM D k , where n j=1 ω j = 1, 0 ≤ ω j ≤ 1, and j = 1, 2, . . ., n; • N b : collection of benefit criteria; }: assessment information on the performance of alternative A i with respect to criterion C j that is given by decision maker D k and takes the form of a FS or a HFS, in which each element of the set represents the possible membership degree to which the alternative should satisfy the criterion and #s k ij represents the number of elements in the set; • S = S 1 , S 2 , . . ., S t T : vector of proportional hesitant fuzzy decision matrices with respect to all DMs, where In group decision making under uncertainty individuals feel easier to elicit FSs or HFSs than PHFSs.But from FSs or HFSs we propose to fuse individual assessments into collective assessments represented by PHFSs.
In addition, the accurate specification of expert weights and criterion weights is a usual core prerequisite for a MCGDM problem.The existing methods for deriving expert weights and criterion weights can be classified into three categories: subjective methods, objective methods, and methods integrating subjective methods and objective methods.In our proposals, the expert weights are predetermined by a supervisor who is familiar with all of the DMs based on the richness of his relevant knowledge and experience, and the weights of criteria can be specified based on the principle of maximum entropy principle.
Firstly, we give the basic framework of the proposed MCGDM models, which is presented in Figure1.Subsequently,we build the MCGDM model based on the GPHFWA or GPHFWG operator.
Model 1: Step 1-1: Normalize the evaluation information.In a MAGDM problem, benefit criteria frequently coexist with cost criteria.If N c = ∅, the normalization for assessment information is unnecessary.If N c = ∅, we transform the assessment values of cost type into values of benefit type.Assessment information {S k = (s k ij ) m×n |k ∈ T } is then transformed to the normalized assessment information  determined by , for cost criterionC j .
where i ∈ M , j ∈ N , and k ∈ T .Without loss of generality, the normalized values r k ij are also represented as the set }, where #r k ij is the number of elements in the set r k ij .
Step 1-2: Aggregate individual arguments into collective assessment information by transforming FSs or HFSs into PHFSs.In this paper, we propose a method to achieve this transformation.This method simultaneously accommodates the membership degrees and the corresponding proportional information, thereby more accurately characterizing the hesitancy of DMs.Given the fact that r k ij (k = 1, 2, . . ., t) is made up of a set of values, we can determine the collective assessment information . ., r t ij , and τ l ij l = 1, 2, . . ., l ij denotes the proportion of the membership degree γ l ij l = 1, 2, . . ., l ij that can be determined by where D is the set of DMs who provide the value γ l ij .
Step 1-3: Determine the weights of the criteria.An entropy measure of weight, which is called a ''weight-dispersion'' measure, is introduced to quantify the degree to which the corresponding weighted aggregation function takes into account all the inputs.For a given weighting vector ω = (ω 1 , ω 2 , . . ., ω n ) T , the entropy measure is determined by with the convention 0 • log 2 0 = 0.
Remark: According to the principle of maximum entropy, we assume that the weighting vector with the largest entropy is the best because the corresponding weighted aggregation function can make full use of the information on all the criteria.For example, given the two weighting vectors ω 1 = (0.5, 0.5) and ω 2 = (0.1, 0.9), the former is assumed to be preferable because the weighted aggregation function based on the former can use information from two sources with equal degree and is less sensitive to inaccurate input.
During the decision process, the decision team can easily express the preference information on criteria, but the team has difficulty accurately specifying the weights of all the criteria.With respect to a MAGDM problem, let ω j ∈ (j ∈ N ), where is a set of known information of criterion weights provided by the DM D k (k ∈ T ) and can be constructed by the following ranking forms [54]: For j 1 = j 2 = j 3 and k ∈ T , Form 1: a weak ranking: 1 = ω j 1 ≥ ω j 2 j 1 , j 2 ∈ N ; Form 2: a strict ranking: 2 = ω j 1 −ω j 2 ≥ σ j 1 j 2 j 1 , j 2 ∈ N ; Form 3: a ranking of differences: Form 4: a ranking with multiples: Based on the principle of maximum entropy and the incomplete information given by DMs of criterion weights, we can determine the criterion weights for each DM by constructing the following programming model: By solving the method [M1], we can determine the optimal weighting vector (ω 1 , ω 2 , . . ., ω n ) T to maximize the information contained in the original decision information.
Step 1-4: For each alternative, aggregate assessment information on all criteria into overall assessment information.This model uses the GPHFWA operator, , or the GPHFWG operator, to determine the overall performance of each alternative {r i |i ∈ M }, where ω j is the weight of criterion k, n j=1 ω k j = 1, 0 ≤ ω j ≤ 1, λ > 0, i ∈ M , and j ∈ N .
Step 1-5: Rank the r i (i ∈ M ) in descending order by Definitions 7 and 8 and select the best alternative according to the ranking.
Finally, we formulate another MAGDM model based on the GPHFOWA or GPHFOWG operator.
Model 2: Step 2-1: The information processing process is similar to Step 1 of Model 1.
Step 2-2: The information processing process is the same as Step 1 of Model 1.
Step 2-3: Determine the criterion weights.In this model, we also specify the criterion weights on the principle of maximum entropy.In addition, we introduce as constraint the desired ORness measure [42], [43] instead of the partial preference information.The ORness measure, also known as the attitudinal character (AC), is used to measure how far a given averaging function is from the max function and reflects changes in the DM's attitude.For an OWA function with a weighting vector ω = (ω 1 , ω 2 , . . ., ω n ) T , its ORness measure is If ω * = (1, 0, . . ., 0) T , the ORness measure is unity and the OWA operator is reduced to the max operator, corresponding to the fully optimistic decision.If ω * = (0, 0, . . ., 1) T , the ORness measure is zero and the OWA function is reduced to the min function, corresponding to a fully pessimistic decision.Finally, if ω A = 1 n, 1 n, . . ., 1 n T , the ORness measure is 0.5 and the OWA operator is reduced to the averaging operator, corresponding to a Laplace decision [42], [43].
During the decision process, each DM may have different attitudes towards each alternative.We can fuse individual attitudes into a collective attitude towards each alternative.If the decision team has an optimistic collective attitude on the overall performance of the alternative AC i (i ∈ M ), the team can determine the AC value AC i ∈ (0.5, 1], where a more positive team reaction corresponds to a bigger AC.Similarly, if the team expresses a pessimistic attitude regarding the overall performance of the alternative AC i (i ∈ M ), they can give an AC value AC i ∈ [0, 0.5), where a smaller AC represents a more pessimistic reaction of the DM.The AC is used to guide the OWA aggregation process by specifying the corresponding weighting vector for each alternative.
Based on the principle of maximum entropy and the perceived AC value AC i (i ∈ M ) for each alternative, we determine the weighting vector for each alternative by calculating the following programming model: By solving the method [M2], we can determine the optimal weighting vector {(ω i1 , ω i2 , . . ., ω in ) T  |i ∈ M } for each alternative to maximize the information contained in the original decision information and to characterize the AC of each DM simultaneously.
Step 2-4: For each alternative, use the GPHFOWA operator, or the GPHFOWG operator.
Step 2-5: Rank the r i (i ∈ M ) in descending order by Definitions 7 and 8, and select the best alternative in accordance with the ranking.

V. CASE STUDY
Over the last decade, due to environmental issues and the fear of peak oil prices, battery electric vehicles (BEVs), which use for vehicle propulsion the chemical energy stored in rechargeable battery packs combined with electric motors and motor controllers, have garnered considerable attention from governments around the word, including China, the European Union, and the United States.These governments have issued a series of regulations to compel original equipment manufacturers (OEMs) to produce more EVs and encourage consumers to buy EVs.To date, the price and driving range are the biggest obstacles preventing consumers from purchasing EVs, and both obstacles are directly linked to the EVB.At present, most OEMs are inclined to outsource EVB manufacturing to a battery manufacturer.Thus, selecting a suitable EVB supplier is extremely important for OEMs.
In China, an OEM has made a strategic decision to invest in the research and development of EVs.During the development of a new BEV, the OEM tries to select the best EVB supplier on the market and hope to establish a long-term, stable, and mutually beneficial strategic partnership with the selected EVB supplier.The OEM builds a decision-making team that is responsible for selecting the EVB supplier.The team consists of five experts {D 1 , D 2 , D 3 , D 4 , D 5 } who are weighted by the weighting vector ϕ = (0.15, 0.2, 0.3, 0.1, 0.25) T on the basis of their relevant knowledge and experience.The first task assigned to the decision team is to specify the assessment criteria for selecting the EVB supplier.For the strategicsupplier-selection problem, the OEM should not only focus on short-term criteria, such as cost and quality, but also focus on long-term criteria, such as technical capability, company profile, and level of risk.Based on the objectives for the EVB supplier, the decision team specifies seven criteria: cost (C 1 ), quality (C 2 ), delivery and lead time (C 3 ), service level (C 4 ), technical capability (C 5 ), company profile (C 6 ), and risk level (C 7 ).Evidently, C 1 , C 3 , and C 7 are cost criteria whereas the others are benefit criteria.Table 1 gives detailed information on the seven criteria.
Once the criteria are specified, the alternative potential suppliers are identified from upstream of the EV industry chain, which is not limited to the domestic market.The decision team identifies a list of suppliers through a variety of channels and slims down the list to five potential suppliers thorough a preliminary review.These suppliers are represented as {A 1 , A 2 , A 3 , A 4 , A 5 }.Next, each expert D k (k = 1, 2, . . ., 5) assesses the performance of each potential supplier A i (i = 1, 2, . . ., 5) with respect to each criterion C j (j = 1, 2, . . ., 7).The following rules are used to facilitate the articulation of assessment information for DMs: • The assessment value given by each DM should fall in the range of 0 to 1 and is used to embody the possible membership degree that alternatives should satisfy or to measure the performance of an alternative with respect to a specific criterion.
• The better an alternative performs with respect to a specific criterion, the larger assessment value given by DMs.
• DMs can assign only a single value to the performance of an alternative with respect to a specific criterion when she affirms her judgment.She can also give a set of possible values when she hesitates about her judgment.Thus, the assessment should be either a degree of membership of a FS or several degrees of membership of a HFS.The initial assessment information is presented in Table 2.
Next, Models 1 and 2 are used to address the problem of selecting the EVB supplier.

A. APPLICATION OF MODEL 1
Step 1-1: Transform the initial assessment information into normalized assessment information which is presented in Table 3.
Step 1-2: Aggregate individual assessment information into collective assessment information R = r ij 5×7 by transforming the FS or HFS into a PHFS, the result of which is given in Table 4.
Step 1-3: Determine the weights of criteria based on the principle of maximum entropy.After thorough discussion, the decision team gives incomplete preference information on the importance of criteria, which is presented as follows: The input of information into Method [M1], which was developed based on the maximum entropy principle, gives the optimal weighting vector for criteria, ω = (0.097, 0.147, 0.097, 0.147, 0.184, 0.184, 0.141) T .
Step 1-5: Using Definitions 7 and 8, we find that the application of the GPHFWA operator leads to A 4 A 3 A 1 A 2 A 5 , with A 4 being the best alternative, and the use of the GPHFWG operator leads to A 3 A 4 A 2 A 5 A 1 , with A 3 being the best EVB supplier.
After negotiation with the OEM management, the decision team chooses A 3 as the best EVB supplier for the following reasons:  obtain a lower score, placing them beyond the scope of tolerance.
• The alternative A 3 , which is also an overseas EVB supplier, obtains more balanced scores for each criterion.The cost (C 1 ) and the geopolitical risk (C 7 ) have moderate scores that fall within the scope of tolerance.The main reason that the decision team chooses A 3 is that the supplier has set up a factory for EVB manufacturing with a domestic manufacturer.
• The domestic EVB supplier A 1 performs well for cost (C 1 ), quality (C 2 ), and risk (C 7 ) but performs poorly on company profile (C 6 ), which plays a significant role in the establishment of a strategic partnership.The domestic alternative A 2 obtains a lower score for technical capability (C 5 ), which plays a significant role in a long-term partnership because EVB technology is changing very fast.The alternative A 5 , which is also a domestic EVB supplier, obtains balanced scores on  When the parameter λ varies, we attain different results.In this section, we assign 1/10, 1/5, 1/2, 1, 2, 5, and 10 to λ and use the GPHFWA operator to calculate the results, which are presented in Table 5 and Figure 2. Similarly, we use the GPHFWG operator with these seven values of λ to determine the results given in Table 5 and Figure 3. (1) When the input information of the model remains the same, the score determined by the GPHFWA operator becomes larger with increasing λ, and the score value obtained by the GPHFWG operator becomes smaller with increasing λ.In addition, the scores obtained by the GPHFWA operator are always greater than those obtained by the GPHFWG operator.(2) Applying the GPHFWA operator gives alternative A 4 as the best supplier, whereas applying the GPHFWG operator gives alternative A 3 as the best supplier.From Table 4, we see that the differences between the performances of A 4 with respect to each criterion are greater than those of its counterpart A 3 .In other words, the cohesiveness of the assessment information of A 3 exceeds that of A 4 .We further conclude that, upon applying the GPHFWA operator, the criteria with lower scores obtain more compensation from the criteria with higher scores, thus leading to the result that A 4 is the best supplier.However, applying the GPHFWG operator yields the result that the criteria with lower scores obtain limited compensation from the criteria with higher scores, thus leading to the result that A 3 is the best supplier.Thus, we conclude that, for PHFSs with less difference between elements, the GPHFWG operator is more friendly than the GPHFWA operator.In this case, the OEM prefers A 3 to A 4 , which indicates that, in practical applications, the GPHFWG operator is preferable when the compensation is limited between the performances with respect to criteria.(3) For the GPHFWA operator, as the parameter λ increases, the ranking of A 1 rises and the ranking of A 3 drops.For the GPHFWG operator, increasing the parameter λ decreases the ranking of A 4 and increases the ranking of A 5 .Table 4 reveals that A 1 and A 4 have a greater difference between performances for all criteria, and A 3 and A 5 have less difference.We thus infer that, upon increasing λ, the compensation of the GPHFWA operator increases, and the compensation of the GPHFWG operator decreases.

B. APPLICATION OF MODEL 2
This section describes the application of Model 2 to the case described above.
Step 2-1: The results of the information processing procedure of this step are listed in Table 3.
Step 2-2: The results of this step are given in Table 4.
Step 2-3: Determine the weights of criteria based on the maximum entropy principle and the AC of DMs.In this case, different DMs have different ACs for each alternative.We simplify the information processing procedure and, after much deliberation, let the decision team output a collective AC for each alternative:

Based on Method [M2]
, we calculate the final weights of criteria with respect to each DM.The results are given in Table 6.
Step 2-5: Applying the GPHFOWA operator gives A 1 , with A 3 being the best EVB supplier, and the introduction of the GPHFOWG operator gives A 3 A 5 A 2 A 4 A 1 , with A 3 being the best alternative.
Evidently, the results obtained by the GPHFOWA and GPHFOWG operators are consistent with the desired result of the OEM.In a similar way, we assign the values 1/10, 1/5, 1/2, 1, 2, 5, 10 to λ and use the GPHFOWA operator and the GPHFOWG operator to determine the final ranking of alternatives.The results are given in Table 7 and Figs. 4 and 5.
(1) Evidently, the scores obtained by the GPHFOWA operator increase as λ increases, whereas the scores obtained by the GPHFOWG operator decrease as λ increases.In addition, the scores obtained by the GPH-FOWA operator are always greater than those obtained by the GPHFOWG operator.(2) When the GPHFOWA operator is applied, if λ = 0.1, 0.2, 0.5, or 1, the alternative A 3 is the best supplier, followed by A 2 and A 4 ; if λ = 5 or 10, A 4 becomes the optimum supplier, followed by A 2 .When the GPHFOWG operator is applied, the alternative A 3 remains the best supplier, followed by A 5 .Table 4 shows that the differences for A 2 and A 4 between the performances for each alternative are greater than for the counterparts of A 3 and A 5 .We can then reason that, compared with the GPHFWG operator, the GPHFWA operator has a higher level of compensation for the performance with respect to the given criteria, but the results obtained by the GPHFOWG operator are better, which is consistent with the preferences of the OEM.(3) For the GPHFOWA operator, along with the parameter λ increasing, the rankings of A 3 and A 5 decline, but the rankings of A 1 and A 4 increase.Given that the differences for A 1 and A 4 are greater than those for A 3 and A 5 , we infer that the compensation for the GPHFOWA operator increases along with λ. (4) For the GPHFOWG operator, the data in Table 7 cannot reveal the relationship between the compensation level of the GPHFOWG operator and the value of λ, mainly because different ACs are assigned to different alternatives, and the AC critically impacts the compensation level of the GPHFOWG operator, as illustrated below.To eliminate the impact of AC, we assign 0.6 to the AC for each alternative and apply the GPHFOWG operator with different values of λ.The results are presented in Table 8 and Figure 6.Obviously, as λ increases, the rankings of A 3 and A 5 with lower compensation levels rise, but the rankings slip for A 2 and A 4 , which have a higher level of compensation.Thus, we infer that the compensation level of the GPHFOWG operator decreases as λ increases.To further explore how AC affects the final ranking of alternatives, we maintain λ = 1 and change the AC.In this case, the decision team may have different ACs for each alternative.For convenience, we assign the same AC to all alternatives.The results obtained by the GPHFOWA and GPHFOWG operators are given in Table 9 and Figs.7 and  8.
(1) Table 9 shows clearly that the AC has a remarkable impact on the final ranking of alternatives.DMs should accurately output the AC according to their real feelings about alternatives.(2) Upon applying the GPHFOWA operator, the increase in AC increases the rankings of A 1 and A 4 and decreases the rankings of A 3 and A 5 .The results obtained by the GPHFOWG operator also lead to a similar conclusion.We therefore conclude that the compensation level of the GPHFOWA operator and of the GPHFOWG operator increases along with the AC.

C. COMPARISON BETWEEN PROPORTIONAL HESITANT FUZZY SETS AND HESITANT FUZZY SETS
In this section, we compare PHFSs with HFSs.[25] proposed a series of aggregation operators for HFS, including the GHFWA, GHFWG, GHFOWA, and GHFOWG operators.
To delve into the differences between PHFSs and HFSs, we make the following comparisons: GPHFWA vs GHFWA, GPHFWG vs GHFWG, GPHFOWA vs GHFOWA, and GPHFOWG vs GHFOWG.tion R = r ij m×n by using the GHFWA operator, , or the GHFWG operator, , where ϕ k is the weight of the kth DM, which is provided in advance, Step 1-3 : Determine the weights of criteria in accordance with Step 3 of Model 1 and represent the results as ω = (ω 1 , ω 2 , . . ., ω n ) T .
Step 1-4 : For each alternative, aggregate assessment information for all criteria into overall assessment information by using the GHFWA operator, , or the GHFWG operator, , where ω j is the weight of the kth criterion, n j=1 ω Step 1-5 : Rank the r i (i ∈ M ) in descending order by using Definitions 7 and and select the best alternative according to the ranking.
Next, we apply the Model 1 to the EVB-supplier-selection problem described above.During the process, the input information for Model 1 , including the original assessment information provided by DMs, the weights of DMs, and the incomplete preference information given by the decision team for the importance of criteria, remains the same with the input information of Model 1 to ensure comparability.The results obtained by using the GHFWA or GHFWG operators with λ = 1 are given in Table 10.The counterparts derived by using the GPHFWA or GPHFWG operators with λ = 1 are also given in Table 10.
• When the GHFWA operator is applied to the case study, A 4 becomes the best supplier, followed by A 5 .When the GPHFWA operator is used, A 4 remains the best supplier, followed by A 3 .The main difference is that the ranking of A 5 slips from 2 in the ranking obtained by the GHFWA operator to 5 in the ranking derived by the GPHFWA operator.Comparing the original assessment information of A 3 and A 5 in Table 4, we find that A 5 only outperforms A 3 on criteria C 1 and C 7 , but A 3 outperforms A 5 on criteria C 2 , C 3 , C 4 , C 5 , and C 6 .The final decision of selecting A 3 as the best supplier shows that the GHFWA operator is preferable to the GPHFWA operator.
• When the GHFWG operator is used, A 2 becomes the best supplier, followed by A 1 .However, the application of the GPHFWG operator leads to the result that A 3 is the best supplier, followed by A 4 .Remarkable differences exist between the results obtained by the GHFWG and GPHFWG operators.Comparing A 2 with A 3 with the help of the original assessment information in Table 4, we see that A 2 scores lower on criteria C 2 and C 5 , which are beyond the scope of tolerance, but A 3 has more balanced scores on all the criteria, which is the main reason why the OEM prefers A 3 to the other alternatives.Thus, we come to a similar conclusion that the GPHFWG operator outperforms the GHFWG operator in this case study.

2) GPHFOWA VS GHFOWA AND GPHFOWG VS GHFOWG
In this section, we construct another MCGDM model, called Model 2 and that is based on the GHFOWA or GHFOWG operator, by replacing the GPHFOWA operator or the GPHFOWG operator with the GHFOWA operator or the GHFOWG operator, respectively.In addition, note that, unlike the weights in the weighted means, which represent the relative importance of inputs, the weights in OWA functions are associated with ordered positions, meaning that the weight ω i reflects the importance of the ith ordered position.Thus, instead of the original weighting vector ϕ = (ϕ 1 , ϕ 2 , . . ., ϕ t ) T that indicates the relative degree of importance of DMs, we specify a new type of expert weight for the execution of OWA functions.Model 2 can be described as follows: Step 2-1 : Transform the original assessment information Step 2-2 : Determine the weights of DMs to facilitate the execution of the GHFOWA operator or the GHFOWG operator.We let the OEM give a value to measure their AC value, with AC ∈ [0, 1], for the performance of the decision team, and then introduce the AC into Method [M2] to determine the order weights (ϕ 1 , ϕ 2 , . . ., ϕ t ) for aggregating individual arguments on the expert level.
Step 2-3 : Fuse individual assessment information m×n |k ∈ T } into the collective assessment information R = r ij m×n by using the GHFOWA operator, or the GHFOWG operator, , where ϕ k is the weight of the kth ordered position in (r Step 2-4 : Determine the weights of criteria in accordance with Step 2-3 of Model 2 and represent the results as ω = (ω 1 , ω 2 , . . ., ω n ) T .
Next, we use Model 2 to address the EVB-supplierselection problem for this case.During the process, to compare the results from various angles, we assign 0.6 and 0.4 to the AC based on the performance of the decision team, and also assign the AC AC i (i ∈ M ) on each alternative 0.6 and 0.4.In addition, λ is set to unity.Four experiments are needed for the GHFOWA and GHFOWG operators.The results are given in Table 10.Note that, during the application of the GPHFOWA and GPHFOWG operators, the AC value AC i (i ∈ M ) for all alternatives should also be set to 0.6 and 0.4 to ensure comparability.The results obtained by using the GPHFOWA and GPHFOWG operators are presented in Table 10.
• For the GHFOWA operator, if AC i = 0.6 (i ∈ M ), A 4 is the best supplier; if AC i = 0.4 (i ∈ M ), A 5 becomes the best supplier.For the GPHFOWA operator, A 4 remains the best supplier independent of whether AC i = 0.6 or 0.4 (i ∈ M ).Thus, we infer that the results obtained by the GPHFOWA operator are more robust than those derived by the GHFOWA operator.
• When using the GHFOWG operator with AC i = 0.6 or 0.4 (i ∈ M ), A 5 remains the best supplier, followed by A 3 .When using the GPHFOWG operator with AC i = 0.6 or 0.4 (i ∈ M ), A 3 remains the best supplier, followed by A 5 .We cannot differentiate between the robustness of the two operators.However, as mentioned above, the results obtained by the GPHFOWG operator are preferable to those obtained by the GHFOWG operator.
Based on a comparative analysis, we find that the aggregation operators based on PHFSs, including GPHFWA, GPHFWG, GPHFOWA, and GPHFOWG, outperform the aggregation operators based on HFSs including GHFWA, GHFWG, GHFOWA, and GHFOWG, which indicates that PHFSs perform better than HFSs for uncertain MCGDM problems.The main reasons for this result can be summarized as follows: (1) Compared with HFSs, PHFSs introduce a new proportional dimension of information that is mined from the original assessment information to reduce the uncertainty associated with the original assessment information.Thus, MCGDM based on PHFSs can effectively improve the reliability of decision results.(2) Instead of aggregation operators, a different type of information processing is introduced during the aggregation process on the expert level, which is described in Step 1-2 of Model 1.This information fusion aims to generate proportional information while reserving original information as much as possible.Thus, it can mitigate the risk of information distortion.
In addition, it is easy to operate and has a high degree of interpretability.For example, the five DMs give the original assessment information for the performance of alternative A 1 with respect to quality (criterion C 2 ) in this case study, {0.8, 0.7}, {0.7}, {0.8}, {0.8, 0.7, 0.6}, {0.8, 0.7}.Evidently, the new type of information fusion on the expert level is preferable.(3) The aggregation operators based on HFSs and PHFSs necessitate all possible combinations of individual arguments.In the context of MCGDM, the aggregation operators based on PHFSs only need to be executed once, but the aggregation operators based on HFSs need to be executed twice, easily leading to information distortion and the combinatorial-explosion problem in combinatorics.Thus, the processing of the proposed MCGDM model based on PHFSs requires less computing time and is more efficient.

D. DISCUSSION
This section summarizes the conclusions obtained by the case study.
(1) The case study further verifies the theorems given in Sec.3; namely, that the score determined by the GPHFWA or GHPFOWA operator increases with increasing λ, the score obtained by the GPHFWG or GPHFOWG operator decreases with increasing λ, the scores obtained by the GPHFWA operator are always greater than those obtained by the GPHFWG operator, and the scores derived by the GPHFOWA operator are always greater than those obtained by the GPHFOWG operator.(2) Based on this case study, we find that the GPH-FWA and GPHFOWA operators have a higher level of compensation than the GPHFWG and GPHFOWG operators, respectively.But the results obtained by the GPHFWG and GPHFOWG operators are preferable.
In addition, the compensation level for the GPHFWA and GPHFOWA operators increases with increasing λ, whereas the compensation level for the GPHFWA and GPHFOWA operators decrease with increasing λ.Based on these conclusions, we give the following prerequisites for applying each PHFS-based aggregation operator: (1) When DMs output assessment information with no AC and have clear preferences on criteria regardless of whether the preference information is complete or incomplete, the GPHFWA or GPHFWG operator can be applied to fuse assessment information during the MCGDM process in the hesitant fuzzy context.For GPHFWA and GPHFWG operators, if limited compensation exists between criteria, or if DMs hope to mitigate the influence of outliers, the GPHFWG operator is more preferable.(2) When DMs output the hesitant fuzzy assessment information under optimistic or pessimistic conditions and have no preferences for criteria, or if DMs hope to flexibly choose aggregation operators ranging from the minimum to the maximum by specifying a parameter, the GPHFOWA or GPHFOWG operator can be used.Similarly, if limited compensation exists between criteria, or if DMs hope to mitigate the influence of outliers, the GPHFOWG operator is more preferable.

VI. CONCLUSION
By introducing a new proportional dimension, PHFSs offer outstanding advantages for modeling uncertainty.In this paper, we restrict our attention to expanding PHFS theory in terms of information fusion, constructing two MCGDM models involving PHFS-based aggregation techniques and exploring applications to which the two models may be applied.The three main contributions of this paper are summarized below.

FIGURE 1 .
FIGURE 1. Framework of the proposed MCGDM models.

FIGURE 2 .
FIGURE 2. Results obtained by the GPHFWA operator with different values of λ.

FIGURE 3 .
FIGURE 3. Results obtained by the GPHFWG operator with different values of λ.

FIGURE 4 .
FIGURE 4. Results obtained by the GPHFOWA operator with different values of λ.

FIGURE 5 .
FIGURE 5. Results obtained by the GPHFOWG operator with different values of λ.

1 )
GPHFWA VS GHFWA AND GPHFWG VS GHFWGIn accordance with Model 1, we develop a new MCGDM model, called Model 1 , by replacing the GPHFWA operator or the GPHFWG operator with the GHFWA operator or the GHFWG operator, respectively.Model 1 is based

FIGURE 8 .
FIGURE 8. Results obtained by the GPHFOWG operator with λ = 1 and different values of AC.

( 3 )
Attitudinal character, which can be characterized by the ORness value and used to guide the aggregation process, has a remarkable impact on the final ranking of alternatives.In fact, the main advantage of OWA functions over traditional averaging operators is that the former can flexibly provide an aggregation operator ranging between min and max by incorporating the AC of DMs.In addition, the compensation level of the GPHFOWA and GPHFOWG operators increases with increasing AC.Furthermore, OWA operators can flexibly model different degrees of compensation with the help of the ORness measure.When ORness takes the value of zero, OWA reduces to the max operator, which indicates full compensation between criteria.When ORness takes the value of unity, OWA reduces to the min operator, which indicates no compensation between criteria.A positive linear relationship exists between ORness measure and compensation level.(4) Based on a comparative analysis, we conclude that the introduction of PHFSs into the MCGDM process under a hesitant, uncertain context is preferable because the addition of a new information dimension reduces the uncertainty.In addition, instead of aggregation operators, we propose a method to fuse individual assessment information on the expert level and, simultaneously, transform from FSs or HFSs to PHFSs.The fusion of information by this method outperforms its counterpart involving aggregation operators.

TABLE 1 .
Criteria for selection of electric vehicle battery (EVB) supplier.

Table 4 ,
we find that the alternative A 4, which is a well-known overseas EVB supplier, performs well with respect to criteria including quality (C 2 ), service level (C 4 ), technical capability (C 5 ), and company profile (C 6 ).But the cost (C 1 ) and the geopolitical risk (C 7 )VOLUME 7, 2019

TABLE 2 .
Initial assessment information provided by DMs.

TABLE 5 .
Results obtained by the GPHFWA and GPHFWG operators.

TABLE 6 .
Weights of criteria in the case of different ACs for alternatives.

TABLE 7 .
Results obtained by GPHFOWA and GPHFOWG operators.

TABLE 10 .
Comparison of results obtained by aggregation operators based on HFSs and PHFSs.