A Single-Valued Extended Hesitant Fuzzy Score-Based Technique for Probabilistic Hesitant Fuzzy Multiple Criteria Decision-Making

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Introduction
Hesitant fuzzy set (HFS) as an extension of the fuzzy set [1] was introduced for reflecting the hesitancy of decision makers in providing their preferences over alternatives such that the membership degree of an element in HFS is represented by a set of values in [0, 1]. e concept of HFS is a field that still keeps attracting a significant amount of attention from researchers, and by owing to this concept, the other extensions of HFSs have been proposed in the literature [2][3][4][5][6] to overcome a number of corresponding challenges.
From diverse extensions of HFS, the concept of the extended hesitant fuzzy set (EHFS) is introduced first by Zhu and Xu [7] in terms of a function that returns a finite set of membership value-groups. en, Farhadinia and Herrera-Viedma [8] re-visited and revised the notion of EHFS as the Cartesian product of "n" HFSs in which each "n"-tupleformed element of EHFS is referred to as the opinion of some decision makers simultaneously.
Another interesting generalization of HFSs occurs when we are required to provide experts' evaluations based on two cases: whether experts have the same weight or whether each value in a hesitant fuzzy element (HFE) gets the same probability distribution? ese cases are covered by defining the concept of the probabilistic hesitant fuzzy set (PHFS) which was first developed by Zhu and Xu [9] to incorporate distribution information with the membership degrees included in hesitant fuzzy elements (HFEs). Furthermore, the PHFS concept has a great potential for handling multiple criteria decision-making (MCDM) processes in which both qualitative and quantitative criteria are to be considered [10][11][12][13][14].
Nowadays, among a large number of studies of PHFS notion, we may refer to the contribution of Zhang and Wu [15] in which two PHFS aggregation operators are developed by taking Archimedean t-norm and t-conorm into account. Following that work, Li and Wang [16] proposed the Hausdorff distance measure of PHFSs to extend a QUAL-Itative FLEXible multiple (QUALIFLEX) technique for evaluating green suppliers. Yue et al. [17] developed the application of probabilistic hesitant fuzzy elements (PHFEs) in MCDM problems by proposing a set of probabilistic hesitant fuzzy aggregation operators. Following that, Zeng et al. [18] introduced the uncertain probabilistic-ordered weighted averaging distance operator in order to unify the framework between the probability and the ordered weighted averaging operator. Ding et al. [19] dealt with the situation in which the weight information is incomplete, and then, they concentrated on the class of PHFE-based multiple attribute group decision-making.
In a completely updated study, Farhadinia [20] pointed out that there exist two kinds of normalization processes in dealing with PHFS decision-making problems, namely, the probabilistic normalization and cardinal normalization. We need to mention that, among the contributions considering different types of probabilistic-unification processes, the most eminent works are those of Zhang et al. [21], Farhadinia and Xu [22], Farhadinia and Herrera-Viedma [23], Li and Wang [24], Wu et al. [25], and Lin et al. [26]. Except Lin et al.'s [26] probabilistic-unification process, Farhadinia [20] demonstrated that the other probabilistic-unification processes considered in the later-mentioned contributions are not reasonable from a mathematical point of view. It can be seen that the probabilistic-unification processes of Lin et al. [26] and Farhadinia [20] give rise to the same result with this difference that the process of Lin et al. compromises the unification of probabilities and HFE parts simultaneously, and that of Farhadinia unifies firstly the probabilities part, and then, it does the corresponding HFE part.
Keeping the latter-mentioned applications of PHFS notion in mind, the subject of PHFS ranking technique has received significant attention in the recent years. Up to now, a variety of PHFE comparison techniques have been proposed as the combination of hesitancy degree and its corresponding probability. Taking these two notions into account, there have been considerable contributions done in the past on the PHFE comparison techniques which were developed by employing the score and deviation values of each PHFE [14,19,21]. For instance, Lin et al. [26] put forward two types of probabilistic hesitant fuzzy aggregation operators for specifying the ranking results of alternatives in decision-making problems. Jiang and Ma [27] proposed a PHFE comparison technique using the arithmetic-and geometric-mean scores. Song et al. [28] presented a possibility degree formula for ranking PHFEs in the case where different PHFEs have common or intersecting values. is comparison technique is able to realize the optimal sorting under the hesitant fuzzy environment, and of course, it can reduce effectively the complexity of computation. Krishankumar et al. [29] suggested a ranking technique which extends a well-known VIKOR approach to the PHFS context. Wu et al. [30] supplied an enhanced satisfaction degree function on the basis of probabilistic hesitant fuzzy cumulative residual entropy for ranking the alternatives involved in a MCDM. Last but not least, Farhadinia and Xu [31] developed a thorough review of PHFS comparison techniques in MCDM and introduced a kind of PHFE ranking technique which is based on the multiplying and exponential deformation formulas of each element of a PHFE. ey classified the PHFE measuring techniques in brief into the three classes which were called the elementbased processes for comparing PHFEs, the one step-based processes for comparing PHFEs, and the two step-based processes for comparing PHFEs.
However, the main objective of this study is to develop a class of score functions for capturing dependencies between PHFSs. Although the ranking of PHFSs has been discussed thoroughly before, the novelty presented here lies in the fact that the comparison is done inside the less dimensional space, referred here to as the single-valued EHFSs (SVEHFSs), and it has not yet been fully exploited. e notable characteristic of proposed SVEHFS-score functions is that not only they are projected from a highly dimensional space (i.e., the PHFS space) into a less dimensional space (i.e, the SVEHFS space) but also they offer a wide variety of interesting properties. Moreover, the proposed SVEHFS-based score functions proceed in less steps, and it relieves the laborious duty of using complex rules. Besides the latter advantages, we will demonstrate that the proposed SVEHFS-score functions can be more generalized to a wider class. e organization of this contribution is as follows. We firstly review the process of unification of PHFSs in Section 2. en, we demonstrate that how a unified PHFS is deduced to a SVEHFS in Section 3. Section 4 is devoted to introducing a new class of SVEHFS-based score functions for the unified PHFSs which provides the decision makers with more choices and flexibility. Subsequently, by re-encountering a number of MCDM problems, we indicate that the superiority of the proposed SVEHFS-score functions compare to the existing ones for PHFSs in Section 5. Section 6 concludes this contribution and provides some perspectives.

The Probabilistic-Unification Process of PHFSs
In the following part, we are going to review a number of basic notions which will be used frequently throughout this contribution. By taking the reference set of X into consideration, Torra [1] introduced the notion of hesitant fuzzy set (HFS) in terms of a function returning a finite subset of [0, 1] which is generally denoted by where h(x) ∈ [0, 1] is known as the hesitant fuzzy element (HFE) and denotes the possible membership degree of x ∈ X to the set H. ere is another way of representing HFS already described in the form of In order to emphasis on the probability occurrence of each possible value of HFE, Zhu [32] associated any element of HFE with its probability value as follows: 2 Complexity where ℘ h(x) stands for a probabilistic hesitant fuzzy element (PHFE). As can be observed, any PHFE ℘ h(x) is a pair of possible membership degree Z(x) and its probability distribution in the form of It is obvious that if all the values of ℘(x) are equal for any x ∈ X, then the PHFS ℘ H is reduced to a typical HFS.
Keeping the probabilistic-normalization and the cardinal-normalization procedures of PHFSs in mind, Farhadinia [20] represented a probabilistic-unification type of PHFSs. Anyway, it has been presented two PHFE probabilisticunification processes in earlier contributions, the one proposed by Farhadinia [20] and the other given by Lin et al. [26]. e main difference between these two processes is that Lin et al. [26] performed the unification process simultaneously for both the HFE part and its corresponding probability part, while Farhadinia [20] applied the unification process to probability part at the beginning and then partitioned the HFE part correspondingly. Regarding the same outcome of both Farhadinia's [20] and Lin et al.'s [26] procedures, we only consider the former one in the following.
By the use of Farhadinia's [20] algorithm which is seperated here as Algorithms 1 and 2, the initial partition of each PHFE probabilities is to be refined such that all the involved PHFEs have the same probability parts, while their corresponding HFE part remains unchanged. To explain Algorithm 1 and 2 briefly, we assume that ℘ h 1 � ∪ 〈Z 1 ,℘ 1 〉∈ ℘ h 1 We can summarise the outcome of the previous section as follows: the PHFE probabilistic-unification algorithm leads to the set of HFE and probability pairs whose second part is a fixed vector.
As mentioned before, the purpose of this contribution is to propose a class of score functions for PHFSs with less involved factors. is fact would help us greatly reduce the model construction effort without losing the generality for different PHFSs; meanwhile, their probability part is common. Such an effort will result in defining a fundamental concept, called here as the single-valued extended hesitant fuzzy set (SVEHFS).
In the sequel, we shall present some preliminaries which will be useful for the establishment of the desired results.

Complexity 3
Initial step: consider the input probability sets as We now let i � 1. Step Step 2: calculate the new probabilities: . . , l m . Else, go to the next step.
Initial step: we assume that ℘ * � ℘ 1 * , ℘ 2 * , . . . , ℘ l * * is to be the output of Phase 1 of Farhadinia's [20] algorithm Step 1: calculate the re-formatted probabilities as follows: Step 2: we re-arrange the HFE part of the first PHFE in the form of Step 3: in a similar manner as described above, we re-format the other PHFEs ℘ h 2 ,..., and ℘ h m to ℘ _ h 2 ,..., and ℘ _ h m .

Complexity
In a recent work, Zhu and Xu [7] introduced the notion of extended HFS (EHFS) in terms of a function which returns a finite set of membership value-groups. en, Farhadinia and Herrera-Viedma [8] indicated that each element of an EHFS, known as the extended hesitant fuzzy element (EHFE), is indeed a set of n-tuples which demonstrates the opinion of n number of decision makers. ey introduced an extended hesitant fuzzy set (EHFS) on the reference set X in the form of where which stands for an extended HFE (EHFE).
For instance, if we suppose that X � x 1 , x 2 is the reference set and h 1 { } are two EHFEs on X, then the EHFS H is characterized by

Complexity
Keeping the concept of EHFS in mind, we are now able to derive the concept of the single-valued extended hesitant fuzzy set (SVEHFS) as follows: )}〉| x ∈ X} be an extended hesitant fuzzy set (EHFS) on the reference set X. A single-valued extended hesitant fuzzy set (SVEHFS) is interpreted as the reduced form of H being characterized by where, for a fixed x ∈ X, which stands for a single-valued extended hesitant fuzzy element (SVEHFE).
To give a more specific example, let us consider again the above example of EHFS, but in the form of SVEHFS, suppose that X � x 1 , x 2 is the reference set and h 1 Now, we turn back to the beginning expression in this section where it was stated that all of the pairs involved in a unified PHFE correspond to a fixed vector as their probability part.
If we put aside the probability part of the unified PHFEs, then it gives rise to forming the corresponding SVEHFEs.
〉 are three arbitrary PHFEs. en, their unified forms can be derived as follows: If we put all the same probability part (℘ 1 * , ℘ 2 * , . . . , ℘ 6 * ) of the later unified PHFEs ℘ _ h 1 , ℘ _ h 2 , and ℘ _ h 3 aside, then the corresponding SVEHFEs are given as follows: Before ending this section, we are required to discuss about the issue of distance measures for SVEHFEs. Generally, an unified-PHFE distance measure is constructed using the different part of hesitancy and probability parts.
is is while the probability part of PHFEs is released in defining the concept of SVEHFEs. erefore, the probability difference part may not make sense in developing distance measures for SVEHFEs, and only the hesitancy difference part is kept instead. Now, if we assume that the weight of element x i ∈ X is to be denoted by w i , satisfying w i ∈ [0, 1] and N i�1 w i � 1, then a series of weighted distance measures for SVEHFSs _ x ∈ X} will be developed as the following: (1) e single-valued extended hesitant weighted distance measure: (2) e single-valued extended hesitant weighted Hausdorff distance measure: (3) e single-valued extended hesitant weighted hybrid distance measure: 6 Complexity (4) e generalized single-valued extended hesitant weighted hybrid distance measure: where 0 ≤ α, β ≤ 1, and α + β � 1.
x ∈ X} be two SVEHFSs. en, the weighted distance measures for SVEHFSs given by (21)-(24) satisfy the following properties: Proof. We only prove the above assertions for the distance measure d g (., .) given by (17), and the others can be deduced easily.
□ Axiom 1. Keeping equation (17) in mind, we easily deduce for any λ > 0. Now, by taking w i ∈ [0, 1] and N i�1 w i � 1, and moreover, 0 ≤ α, β ≤ 1 and α + β � 1, we result in is implies that in which both of them lead to |c 1 e inverse axiom will be easily proved in the same manner.

Axiom 3.
e proof is immediate from definition of distance measure d g (., .) given by (17).
for any j � (1, 2, . . . , m) and x i ∈ X. e latter inequalities give rise to |c 1 . . , m) and x i ∈ X, and therefore,

SVEHFS-Based Score Function for PHFSs
As will be shown later, the score function of SVEHFS is fundamentally defined in accordance with the score function of its SVEHFEs, and therefore, we only discuss the score functions for SVEHFEs. Now, we are in a position to introduce a class of SVEHFE-score functions by the help of SVEHFE distance measures given by (14)-(17).
As will be shown below, score function (22) satisfies the fundamental properties, known as monotonicity, boundary conditions, idempotency, and duality.
□ Definition 3. If Sc(.) stands for a score function of SVEHFEs, then which defines the dual form of Sc(.).
Property 4 (duality). e score function Sc(.) given by (22) Proof. Following from Definition 3, we get that 8 Complexity which defines a score function for SVEHFEs.
By the help of Property 5, we will be able to develop different formulas of score functions for SVEHFEs by taking into account different strictly monotone decreasing functions Θ: From this property, the following SVEHFE-score functions can be established: □

SVEHFS Score-Based Multiple Criteria Decision-Making
is section provides three practical case studies to demonstrate that the proposed concept of SVEHFS is effective enough in the field of score-based optimization methods.
Briefly speaking, the proposed SVEHFS score-based decision-making procedure is composed of the following three stages: the unification process of PHFSs, the reduction process of PHFSs to SVEHFSs, and the selection procedure. e first two stages have been served in Sections 2 and 3. e last stage given in Section 4 describes the process of ranking alternatives in accordance with their values of score function and selecting the best one with the greatest value. Now, in order for more systematically being understood, we give following Algorithm 3 (see Figure 4).

Case Study I.
In this portion, we adopt an optimization problem which was originally solved in [33] by the use of a probabilistic linguistic term set-(PLTS-) based algorithm. Here, in order to have a better understanding of how the proposed SVEHFS-(or PHFS-) based algorithm behaves over the later-mentioned multiple criteria decision-making problem, we transform PLTS information to PHFS (or SVEHFS) data. is is done by the help of eorem 1 in [34] in which the bijective transformation between PLTSs and PHFSs is explained.
A company needs to plan the development of large projects (strategy initiatives) for the next five years. To do this end, the company invites five experts to form the board of directors. Moreover, the company takes three possible projects A i (i � 1, 2, and 3) into consideration which should be evaluated based on their importance. ese projects should be ranked in accordance with four criteria of the benefit type which are suggested by the balanced scorecard methodology as follows: C 1 : financial perspective C 2 : the customer satisfaction C 3 : internal business process perspective C 4 : learning and growth perspective Now, by adopting Algorithm 3 and the assumption that five experts apply the linguistic term set S � s 0 � none, s 1 � verylow, s 2 � low, s 3 � medium, s 4 � high, s 5 � veryhigh, s 6 � perfect}, we are able to evaluate the projects A i (i � 1, 2, and 3) by means of PLTSs in Step 1. e corresponding data is presented in Table 1.
To save more space, we only present the transformation form of the probabilistic linguistic decision matrix into that of PHFSs as explained above. Consequently, the result will be that given in Table 2. Now, by the help of Step 2 of Algorithm 3, we are in a position to use the proposed unification process for the data of Table 2 and draw those results being summarized in Table 3.
In what follows, by the use of Step 3 of Algorithm 3, we will derive the corresponding SVEHFEs, as shown in Table 4.
Generally, the TOPSIS-based and aggregation-based techniques are chosen in accordance with the decision makers' need on one side, and on the other side, Pang et al.'s TOPSIS-and aggregation-based techniques [33] impose the extracondition of normalization by adding a number of artificial linguistic terms with "zero" probability. By imposing such artificial PLTS normalization process, the underlying optimization procedure will cause the computational process with more complexity. In contrast, the SVEHFS-score-based technique maintains the integrity and authenticity of decision information as far as possible, which results in much more reasonable decisions.

Case Study II.
In this part of contribution, we implement the proposed SVEHFS-score function for specifying the best Chinese hospital from a collection of considered hospitals. Such a problem was originally discussed by Song et al. [28], and then, it was more investigated by some other researchers Input: the probabilistic hesitant fuzzy decision matrix Output: the ranking of alternatives and the best one Step 1: build the probabilistic hesitant fuzzy decision matrix Step 2: extract the unified form of PHFSs from the decision matrix Step 3: reduced the probabilistic-unified PHFSs to so-called SVEHFSs Step 4: find the best alternative(s) in accordance with their SVEHFS score values ALGORITHM 3: Proposed SVEHFS score-based decision-making algorithm. Table 1: e probabilistic linguistic decision matrix.  Table 2: e probabilistic hesitant fuzzy decision matrix. Step 2 Unifying PHFSs

Reducing to SVEHFSs
Step 1 Step 4 Computing the Scores

Ranking of alternatives and the best one
The Best End Start Computation Process Outputs Section 3 Eq. (5) Section 4 Eq. (10) Section 2 Algorithm2.1 Step 3 Figure 4: Proposed SVEHFS score-based decision-making algorithm. Table  3: e unified probabilistic hesitant fuzzy decision matrix.  Table  4: e SVEHLFS decision matrix. including Zhang et al. [21] and Farhadinia and Herrera-Viedma [23]. e problem here is that we are seeking the best Chinese hospital with regards to the medical resource restriction and the old-age limitation of target population. In this regard, three criteria are mainly considered as C 1 : environment of health service, C 2 : treatment optimization, and C 3 : social resource allocation. e corresponding weight vector of criteria is supposed to be w � (0.2, 0.1, 0.7). For this optimization problem, we evaluate four candidate hospitals including A 1 : West China Hospital of Sichuan University, A 2 : Huashan Hospital of Fudan University, A 3 : Union Medical College Hospital, and A 4 : Chinese PLA General Hospital. Since one option is not able to describe the influence factor, a number of experts are asked to express their preferences related to the abovementioned hospitals based on available criteria in the form of PHFSs. Now, performing Step 1 of Algorithm 3 leads to constructing the following probabilistic hesitant fuzzy decision matrix (see Table 6).
Similar to what is discussed in Section 3 and applying Steps 2 and 3 of Algorithm 3, we will derive the corresponding unified PHFEs' and SVEHFEs' matrices, as shown in Tables 7 and 8 [28], and Farhadinia and Xu [22] techniques have been also presented in Tables 9 and 10.
As what can be observed from Tables 9 and 10, the most repeated alternative is A 2 which implies that the most appropriate hospital is the Huashan Hospital of Fudan University. is is exactly what we observe from the last three rows of Table 10 dedicated to the results of proposed SVEHFS-score functions. Now, let us conclude the part of this section with some discussions on the pros and cons of proposed SVEHFS-score functions. e techniques of Zhang et al. [21] and Song et al. [28] are restricted directly to the normalization process of PHFSs, and Farhadinia and Xu [22] techniques are related to the multiplying and exponential deformation formulas of each pair of possible membership degree and its associated probability.
is is while the proposed SVEHFS-score functions do not change the original form of PHFSs, and this can be seen as a pro. e other significant advantage of SVEHFS-based score functions over the other abovementioned techniques is its ease of use.

Case Study III.
Because of competition and limitation of research funding in universities of China, a few outstanding research topics are annually supported. In order to select the best research topic several aspects including practicality, innovativeness, and feasibility are taken into consideration.
In March 2018, the business school of university A in China asked three instructors to submit their research topics for evaluating which one is more suitable for granting the university research funding. In this project, three professors DM k (k � 1, 2, and 3) are invited for evaluating the quality of the three research topics A i (i � 1, 2, and 3) in accordance with three criteria: C j�1 : innovativeness, C j�2 : practicality, and C j�3 : feasibility. All the criteria are benefit types, and all the corresponding evaluations of three professors DM k (k � 1, 2, and 3) are represented in the form of PHFE-based decision matrices (see Tables 11-13).
By applying Steps 2 and 3 of Algorithm 3, the individual unified PHFEs are computed as the data given in Tables 14-16.
Following the process discussed by Li et al. [35], the decision makers' weights are obtained as   Table 6: e probabilistic hesitant fuzzy decision matrix.  Table  7: e unified probabilistic hesitant fuzzy decision matrix.   [22] first two step-based process multiplying deformation formula: Exponential deformation formula: Farhadinia and Xu's [22] second two step-based process multiplying deformation formula: Farhadinia and Xu's [22] third two step-based process multiplying deformation formula: Exponential deformation formula: 0.2075 0.0576 0.0827 0.0860

(36)
To save more space for convenient storage, we only list the subsequent results for ϖ d 1 .
Now, if we aggregate the individual SVEHFS matrices given in Tables 17-19 by the help of the following rule in which ϖ k stands for the weight of the decision makers DM k (k � 1, 2, and 3) and the notation (j) (for j � 1, . . . , m) denotes the jth element of collective SVEHFS, then the collective SVEHFS matrices can be derived in the form of Table 20. Table 21 shows the comparison outcomes of different techniques. e ranking results obtained by the techniques of Xu et al. [36,37] and Li et al. [35] are identical to those of proposed SVEHFS-score techniques. Such identical ranking results are possibly related to the same steps of processing which are performed using the latter-mentioned techniques. Briefly speaking, the common steps of these techniques are the integration of evaluation information given by the decision makers, the calculation of score value of the collective evaluation information, and the comparison of alternatives by the help of their score values.
e outcomes of such identical steps are seen in identical ranking results.
But, the result of classical ORESTE technique [38] is quite different from that of other abovementioned techniques. Such a different ranking result arises from the twostage integrating ranking process. e initial stage calculates utility values for determining the weak ranking of alternatives.
en, the subsequent stage will derive the preference, indifference, and incomparability relations with conflict analysis. Finally, the strong ranking of alternatives is extracted. However, due to such complicated two-stage procedure, the best alternative obtained from classical ORESTE technique does not appear convincing enough.
In summary, the comparison with other considered techniques suggests that the proposed SVEHFS-score techniques have superior performance and also less computational complexity.

Conclusion
Adopting a probability splitting algorithm for deriving an efficient probabilistic-unification process of PHFSs, we developed a class of score functions for SVEHFSs which are novel deformation of PHFSs. As we demonstrated here, the concept of SVEHFS belongs to a less dimensional space compared to that of PHFSs. Furthermore, we indicated that the proposed SVEHFS-based score functions satisfy a number of interesting properties. It may be of interest to mention that the proposed SVEHFS-based score functions are able to be more generalized to a wider class. Lastly, three case studies were prepared to illustrate the applicability and efficiency of proposed SVEHFS-based score functions compared to other existing PHFS-based techniques. In contrast to the other existing techniques for PHFSs, the SVEHFS-based score functions are associated with less complexity and computation requirements.
In future work, we will work towards opportunities to investigate the meaning and essence of SVEHFS-based score functions in the MCDM under probabilistic hesitant fuzzy setting.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.