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

Authors: Bahram Farhadinia; Atefeh Taghavi
Journal: Complexity
 (2021)

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## Abstract

The probabilistic hesitant fuzzy set (PHFS) is a worthwhile extension of the hesitant fuzzy set (HFS) which allows people to improve their quantitative assessment with the corresponding probability. Recently, in order to address the issue of difficulty in aggregating decision makers’ opinions, a probability splitting algorithm has been developed that drives an efficient probabilistic-unification process of PHFSs. Adopting such a unification process allows decision makers to disregard the probability part in developing fruitful theories of comparison of PHFSs. By keeping this feature in mind, we try to introduce a class of score functions for the notion of the single-valued extended hesitant fuzzy set (SVEHFS) as a novel deformation of PHFS. Interestingly, a SVEHFS not only belongs to a less dimensional space compared to that of PHFSs but also the proposed SVEHFS-based score functions satisfy a number of interesting properties. Eventually, some case studies of multiple criteria decision-making (MCDM) techniques under the PHFS environment are provided to demonstrate the effectiveness of proposed SVEHFS-based score functions.


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## Body

## 1. 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. The 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 [2–6] 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. Then, Farhadinia and Herrera-Viedma [8] re-visited and revised the notion of EHFS as the Cartesian product of “n” HFSs in which each “n”-tuple-formed 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? These 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 [10–14].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 QUALItative 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. This 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. They classified the PHFE measuring techniques in brief into the three classes which were called the element-based 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. The 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.The organization of this contribution is as follows. We firstly review the process of unification of PHFSs in Section2. Then, 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.


## 2. 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 ofX 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(1)H=x,hx:x∈X,where hx∈0,1 is known as the hesitant fuzzy element (HFE) and denotes the possible membership degree of x∈X to the set H.There is another way of representing HFS already described in the form of(2)H=x,∪ℏ∈hxℏ,x∈X.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:(3)H℘=x,℘hx:x∈X=x,∪ℏx,℘x∈℘hxℏx,℘xx∈Xwhere ℘hx stands for a probabilistic hesitant fuzzy element (PHFE).As can be observed, any PHFE℘hx is a pair of possible membership degree ℏx and its probability distribution in the form of ℘x∈0,1 such that ∑℘hx℘x=1 for any x∈X.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 probabilistic-unification processes in earlier contributions, the one proposed by Farhadinia [20] and the other given by Lin et al. [26]. The 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 Algorithms1 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 ℘h1=∪ℏ1,℘1∈℘h1ℏ1,℘1=ℏ11,℘11,…,ℏ1l1,℘1l1, ℘h2=∪ℏ2,℘2∈℘h2ℏ2,℘2=ℏ21,℘21,…,ℏ2l2,℘2l2, ..., and ℘hm=∪ℏm,℘m∈℘hmℏm,℘m=ℏm1,℘m1,…,ℏmlm,℘mlm indicate m arbitrary PHFEs whose probabilities can be separated in the forms of ℘11,℘12,…,℘1l1℘21,℘22,…,℘2l2,..., and ℘m1,℘m2,…,℘mlm, respectively. In this regard, the first phase of the PHFE probabilistic-unification process takes the following steps:Algorithm 1: Phase 1 of Farhadinia’s [20] algorithm.
Initial step: consider the input probability sets as℘11,℘12,…,℘1l1;℘21,℘22,…,℘2l2;⋮℘m1,℘m2,…,℘mlm.We now leti=1.Step 1: compute℘∗i=min℘1i,℘2i,…,℘mi.Step 2: calculate the new probabilities:℘1i≔max℘1i−℘∗i,0;℘2i≔max℘2i−℘∗i,0;⋮℘mi≔max℘mi−℘∗i,0.Now, if℘1i=℘2i=⋯=℘mi=0, then STOP, and return ℘∗=℘∗1,℘∗2,…,℘∗l∗ in which l∗≥maxl1,l2,…,lm. Else, go to the next step.Step 3: Define℘ji+1≔℘ji+1,if℘ji=0;℘ji,if℘ji=0;,forj=1,2,…,m.By settingi≔i+1, we return to Step 1.Algorithm 2: Phase 2 of Farhadinia’s [20] algorithm.
Initial step: we assume that℘∗=℘∗1,℘∗2,…,℘∗l∗ is to be the output of Phase 1 of Farhadinia’s [20] algorithmStep 1: calculate the re-formatted probabilities as follows:℘11=∑k=1k1℘∗k,℘12=∑k=k1+1k2℘∗k,⋮℘1l1=∑k=kl1+1l1℘∗k.Step 2: we re-arrange the HFE part of the first PHFE in the form ofℏ11,℘∗1,…,ℏ11,℘∗k1,ℏ12,℘∗k1+1,…,ℏ12,℘∗k2,⋮ℏ1l1,℘∗kl1+1,…,ℏ1l1,℘∗kl∗.In summary, the unified form of the PHFE℘h1 will be ℘h˙1=ℏ˙11,℘˙11,…,ℏ˙1k1,℘˙1k1,ℏ˙1k1+1,℘˙1k1+1,…,ℏ˙1k2,℘˙1k2,⋮ℏ˙1kl1+1,℘˙1kl1+1,…,ℏ˙1l∗,℘˙1l∗.Step 3: in a similar manner as described above, we re-format the other PHFEs℘h2,..., and ℘hm to ℘h˙2,..., and ℘h˙m.Farhadinia [20] indicated that the following results are the inherent advantages of the above-described algorithm.Lemma 1 (see [20]).
If∏∗:℘11,℘12,…,℘1l1,℘21,℘22,…,℘2l2,…,℘m1,℘m2,…,℘mlm⟶℘∗1,℘∗2,…,℘∗l∗ where l∗≥maxl1,l2,…,lm, then the aggregation operator ∏∗ is idempotent, commutative, and associative.Corollary 1 (see [20]).
Leth˙℘1=∪ℏ˙1,℘˙1∈h℘1ℏ˙1,℘˙1 and ℘h˙2=∪ℏ˙2,℘˙2∈℘h2ℏ˙2,℘˙2 be two unified PHFEs. Then, their corresponding probabilities sets are compatible (isomorphic).To gain a more clear understanding of Farhadinia’s [20] PHFE probabilistic-unification algorithm, we take the following three arbitrary PHFEs:(4)℘h1=0.3,0.2,0.6,0.5,0.8,0.3,℘h2=0.4,0.5,0.7,0.5,℘h3=0.2,0.1,0.5,0.7,0.9,0.2.It is apparent from illustrative Figures1–3 that the unified forms are obtained in the forms of(5)℘h˙1=0.3,0.1,0.3,0.1,0.6,0.3,0.6,0.2,0.8,0.1,0.8,0.2,℘h˙2=0.4,0.1,0.4,0.1,0.4,0.3,0.7,0.2,0.7,0.1,0.7,0.2,℘h˙3=0.2,0.1,0.5,0.1,0.5,0.3,0.5,0.2,0.5,0.1,0.9,0.2.By the use of the above illustrative example, we find that the PHFE probabilistic-unification algorithm enables us to gain a set of PHFEs whose probabilities are in the form of a fixed vector.Figure 1
Stage 1 of the unification process.Figure 2
Stage 2 of the unification process.Figure 3
Combining both Stages 1 and 2 of the unification process.


## 3. Reducing Unified PHFEs to SVEHFEs

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. This 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.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. Then, 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. They introduced an extended hesitant fuzzy set (EHFS) on the reference set X in the form of(6)H=x,hx|x∈X=x,∪γ1x,…,γmx∈hxγ1x,…,γmx|x∈X,where(7)h=∪γ1,…,γm∈hγ1,…,γm,which stands for an extended HFE (EHFE).For instance, if we suppose thatX=x1,x2 is the reference set and h1x=0.6,0.3,0.3,0.5,0.2,0.2 and h2x=0.3,0.2,0.1 are two EHFEs on X, then the EHFS H is characterized by(8)H=x1,h1x,x2,h2x=x1,0.6,0.3,0.3,0.5,0.2,0.2,x2,0.3,0.2,0.1.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:Definition 1.
LetH=x,∪γ1x,...,γmx∈hxγ1x,…,γmx|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(9)H˙=x,γ1x,…,γmx|x∈X,where, for a fixed x∈X,(10)hx=γ1x,…,γmx,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 thatX=x1,x2 is the reference set and h1x=0.6,0.3,0.3 and h2x=0.3,0.2,0.1 are two SVEHFEs on X. Then, the SVEHFS H˙ is characterized by(11)H˙=x1,h1x,x2,h2x=x1,0.6,0.3,0.3,x2,0.3,0.2,0.1.
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. For more explanation, we suppose that℘h1=ℏ11,℘11,ℏ12,℘12,ℏ13,℘13,℘h2=ℏ21,℘21,ℏ22,℘22, and ℘h3=ℏ31,℘31,ℏ32,℘32,ℏ33,℘33 are three arbitrary PHFEs. Then, their unified forms can be derived as follows:(12)℘h˙1=ℏ11,℘∗1,ℏ11,℘∗2,ℏ12,℘∗3,ℏ12,℘∗4,ℏ13,℘∗5,ℏ13,℘∗6,℘h˙2=ℏ21,℘∗1,ℏ21,℘∗2,ℏ21,℘∗3,ℏ22,℘∗4,ℏ22,℘∗5,ℏ22,℘∗6,℘h˙3=ℏ31,℘∗1,ℏ32,℘∗2,ℏ32,℘∗3,ℏ32,℘∗4,ℏ32,℘∗5,ℏ33,℘∗6.
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:(13)h˙1≔h1=ℏ11,ℏ11,ℏ12,ℏ12,ℏ13,ℏ13,h˙2≔h2=ℏ21,ℏ21,ℏ21,ℏ22,ℏ22,ℏ22,h˙3≔h3=ℏ31,ℏ32,ℏ32,ℏ32,ℏ32,ℏ33.
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. This is while the probability part of PHFEs is released in defining the concept of SVEHFEs. Therefore, 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 elementxi∈X is to be denoted by wi, satisfying wi∈0,1 and ∑i=1Nwi=1, then a series of weighted distance measures for SVEHFSs H˙1=x,h˙1x=γ11x,…,γm1x:x∈X and H˙2=x,h˙2x=γ12x,…,γm2x:x∈X will be developed as the following:(1)
The single-valued extended hesitant weighted distance measure:(14)d1H˙1,H˙2=∑i=1Nwi1m∑j=1mγj1xi−γj2xiλ1/λ,λ>0.(2)
The single-valued extended hesitant weighted Hausdorff distance measure:(15)d2H˙1,H˙2=∑i=1Nwimax1≤j≤mγj1xi−γj2xiλ1/λ,λ>0.(3)
The single-valued extended hesitant weighted hybrid distance measure:(16)d3H˙1,H˙2=∑i=1Nwi×121m∑j=1mγj1xi−γj2xiλ+max1≤j≤mγj1xi−γj2xiλ1/λ,λ>0.(4)
The generalized single-valued extended hesitant weighted hybrid distance measure:(17)dgH˙1,H˙2=∑i=1Nwi×α1m∑j=1mγj1xi−γj2xiλ+βmax1≤j≤mγj1xi−γj2xiλ1/λ,λ>0,where 0≤α,β≤1, and α+β=1.
Needless to say that all the abovementioned distance measuresd1.,., d2.,., and d3.,. can be derived from the generalized form of dg.,..Theorem 1.
LetH˙1=x,h˙1x=γ11x,…,γm1x:x∈X and H˙2=x,h˙2x=γ12x,…,γm2x:x∈X be two SVEHFSs. Then, the weighted distance measures for SVEHFSs given by (21)–(24) satisfy the following properties:(1)
0≤dH˙1,H˙2≤1(2)
dH˙1,H˙2=0 if and only if H˙1=H˙2(3)
dH˙1,H˙2=dH˙2,H˙1(4)
dH˙1,H˙2≤dH˙1,H˙3 if H˙1≤HH˙2≤HH˙3 implying that h˙1x≤hh˙2x≤hh˙3x for any x∈X, that is, γj1x≤γj2x≤γj3x for any j=1,2,…,mProof.
We only prove the above assertions for the distance measuredg.,. given by (17), and the others can be deduced easily.Axiom 1.
Keeping equation (17) in mind, we easily deduce that 0≤γj1xi−γj2xi≤1 for any 0≤γj1xi≤1 and 0≤γj2xi≤1 in which i=1,2,…,N and j=1,2,…,m.
These easily give rise to(18)0≤1m∑j=1mγj1xi−γj2xiλ≤1,and0≤max1≤j≤mγj1xi−γj2xiλ≤1,for any λ>0. Now, by taking wi∈0,1 and ∑i=1Nwi=1, and moreover, 0≤α,β≤1 and α+β=1, we result in 0≤dgH˙1,H˙2≤1.Axiom 2.
Let(19)dgH˙1,H˙2=∑i=1Nwi×α1m∑j=1mγj1xi−γj2xiλ+βmax1≤j≤mγj1xi−γj2xiλ1/λ=0.
This implies that(20)1m∑j=1mγj1xi−γj2xiλ=0,andmax1≤j≤mγj1xi−γj2xiλ=0,in which both of them lead to γj1xi−γj2xi=0, that is, γj1xi=γj2xi for any i=1,2,…,N and j=1,2,…,m. Thus, we conclude that H˙1=H˙2.
The inverse axiom will be easily proved in the same manner.Axiom 3.
The proof is immediate from definition of distance measuredg.,. given by (17).Axiom 4.
IfH˙1≤HH˙2≤HH˙3, then it implies that h˙1xi≤hh˙2xi≤hh˙3xi for any xi∈X, that is, γj1xi≤γj2xi≤γj3xi for any j=1,2,…,m and xi∈X. The latter inequalities give rise to γj1xi−γj2xi≤γj1xi−γj3xi for any k=1,2,…,m and xi∈X, and therefore,(21)dgH˙1,H˙2=∑i=1Nwi×α1m∑j=1mγj1xi−γj2xiλ+βmax1≤j≤mγj1xi−γj2xiλ1/λ≤∑i=1Nwi×α1m∑j=1mγj1xi−γj3xiλ+βmax1≤j≤mγj1xi−γj3xiλ1/λ=dgH˙1,H˙3.


## 4. 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).Definition 2.
Leth˙=γ1,…,γm be a SVEHFE. The score function Sc. is defined as(22)Sch˙≔Scγ1,…,γm=1−dγ1,…,γm,1,where d.,. is a distant measure for SVEHFE and 1 stands for the SVEHFE 1,…,1.
As will be shown below, score function (22) satisfies the fundamental properties, known as monotonicity, boundary conditions, idempotency, and duality.Property 1 (monotonicity).
Leth˙1=γ11,…,γm1 and h˙2=γ12,…,γm2 be two SVEHFEs such that h˙1≤hh˙2, that is, γj1≤γj2 for any j=1,2,…,m. Then, the score function Sc. given by (22) satisfies(23)Sch˙1≤Sch˙2.Proof.
From the fact thath˙1≤hh˙2≤h1 and the monotonicity property of any distance d.,., we find that dγ11,…,γm1,1≥dγ12,…,γm2,1 which easily implies that(24)Sch˙1=1−dγ11,…,γm1,1≤1−dγ12,…,γm2,1=Sch˙2.Property 2 (boundary conditions).
Let1=1,1,…,1 and 0=0,0,…,0 be One-SVEHFE and Zero-SVEHFE, respectively. Then, we conclude that the score function Sc. given by (22) satisfies(25)Sc1=1,andSc0=0.Proof.
By keeping the axiom0≤dh˙1,h˙2≤1 (for any SVEHFEs h˙1 and h˙2) in mind, the proof is evident.Property 3 (idempotency).
Leth˙=γ1,…,γm=γ,…,γ. If dγ,…,γ,1=1−γ, then the score function Sc. given by (22) satisfies(26)Sch˙=γ.Proof.
The proof is obvious.Definition 3.
IfSc. stands for a score function of SVEHFEs, then(27)DSch˙=DScγ1,…,γm≔1−Sc1−γ1,…,1−γm,which defines the dual form of Sc..Property 4 (duality).
The score functionSc. given by (22) satisfies(28)DDSch˙=Sch˙.Proof.
Following from Definition3, we get that(29)DDSch˙=D1−Sc1−γ1,…,1−γm≔1−1−Sc1−1−γ1,…,1−1−γm=Scγ1,…,γm.Property 5 (generalization).
LetΘ:0,1⟶0,1 be a strictly monotone decreasing real function and d.,. be a distance measure between SVEHFEs. Then,(30)ScΘh˙=Θdγ1,…,γm,1,which defines a score function for SVEHFEs.Proof.
We show thatScΘ. satisfies two fundamental properties, called above as monotonicity and boundary conditions.
Monotonicity property: from the fact thath˙1≤hh˙2≤h1 and the monotonicity property of any distance d.,., we find that dγ11,…,γm1,1≥dγ12,…,γm2,1. On the contrary, the latter inequality and the strictly monotone decreasing property of Θ give rise to(31)Θdγ11,…,γm1,1≤Θdγ12,…,γm2,1,which implies that(32)ScΘh˙1≤ScΘh˙2.
Boundary conditions’ property: consider the One-SVEHFE1=1,1,…,1 and the Zero-SVEHFE 0=0,0,…,0. Then, by keeping the axiom 0≤dh˙1,h˙2≤1 (for any SVEHFEs h˙1 and h˙2) in mind, we conclude easily that the score function ScΘ. given by (30) satisfies(33)ScΘ1=1,andScΘ0=0.
By the help of Property5, we will be able to develop different formulas of score functions for SVEHFEs by taking into account different strictly monotone decreasing functions Θ:0,1⟶0,1, for instance, (1) Θ1x=1−x; (2) Θ2x=1−x/1+x; (3) Θ3x=1−xex−1; (4) Θ4x=1−x2.
From this property, the following SVEHFE-score functions can be established:(34)ScΘ1h˙=1−dγ1,…,γm,1;ScΘ2h˙=1−dγ1,…,γm,11+dγ1,…,γm,1;ScΘ3h˙=1−dγ1,…,γm,1edγ1,…,γm,1−1;ScΘ4h˙=1−d2γ1,…,γm,1.


## 5. SVEHFS Score-Based Multiple Criteria Decision-Making

This 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. The first two stages have been served in Sections2 and 3. The 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 Algorithm3 (see Figure 4).Algorithm 3: Proposed SVEHFS score-based decision-making algorithm.
Input: the probabilistic hesitant fuzzy decision matrixOutput: the ranking of alternatives and the best oneStep 1: build the probabilistic hesitant fuzzy decision matrixStep 2: extract the unified form of PHFSs from the decision matrixStep 3: reduced the probabilistic-unified PHFSs to so-called SVEHFSsStep 4: find the best alternative(s) in accordance with their SVEHFS score valuesFigure 4
Proposed SVEHFS score-based decision-making algorithm.


### 5.1. 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. This is done by the help of Theorem 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 projectsAii=1,2,and3 into consideration which should be evaluated based on their importance. These projects should be ranked in accordance with four criteria of the benefit type which are suggested by the balanced scorecard methodology as follows:C1: financial perspectiveC2: the customer satisfactionC3: internal business process perspectiveC4: learning and growth perspectiveNow, by adopting Algorithm3 and the assumption that five experts apply the linguistic term set S=s0=none,s1=verylow,s2=low,s3=medium,s4=high,s5=veryhigh,s6=perfect, we are able to evaluate the projects Aii=1,2,and3 by means of PLTSs in Step 1. The corresponding data is presented in Table 1.Table 1
The probabilistic linguistic decision matrix.
C1C2C3C4A1s30.4,s40.6s20.2,s40.8s30.2,s40.8s30.4,s50.6A2s30.8,s50.2s20.3,s30.4,s40.3s10.3,s20.4,s30.3s30.8,s40.2A3s30.6,s40.4s30.6,s40.2,s50.2s30.4,s40.2,s50.4s40.7,s60.3To 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 Table2.Table 2
The probabilistic hesitant fuzzy decision matrix.
C1C2C3C4A10.50.4,0.670.60.330.2,0.670.80.50.2,0.670.80.50.4,0.830.6A20.50.8,0.830.20.330.3,0.50.4,0.670.30.170.3,0.330.4,0.50.30.50.8,0.670.2A30.50.6,0.670.40.50.6,0.670.2,0.830.20.50.4,0.670.2,0.830.40.670.7,10.3Now, by the help of Step 2 of Algorithm3, 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.Table 3
The unified probabilistic hesitant fuzzy decision matrix.
C1C2C3C4A10.50.2,0.50.1,0.50.1,0.670.2,0.670.1,0.670.1,0.670.20.330.2,0.670.1,0.670.1,0.670.2,0.670.1,0.670.1,0.670.20.50.2,0.670.1,0.670.1,0.670.2,0.670.1,0.670.1,0.670.20.50.2,0.50.1,0.50.1,0.830.2,0.830.1,0.830.1,0.830.2A20.50.2,0.50.1,0.50.1,0.50.2,0.50.1,0.50.1,0.830.20.330.2,0.330.1,0.50.1,0.50.2,0.50.1,0.670.1,0.670.20.170.2,0.170.1,0.330.1,0.330.2,0.330.1,0.50.1,0.50.20.50.2,0.50.1,0.50.1,0.50.2,0.50.1,0.50.1,0.670.2A30.50.2,0.50.1,0.50.1,0.50.2,0.670.1,0.670.1,0.670.20.50.2,0.50.1,0.50.1,0.50.2,0.670.1,0.670.1,0.830.20.50.2,0.50.1,0.50.1,0.670.2,0.830.1,0.830.1,0.830.20.670.2,0.670.1,0.670.1,0.670.2,0.670.1,10.1,10.2In what follows, by the use of Step 3 of Algorithm3, we will derive the corresponding SVEHFEs, as shown in Table 4.Table 4
The SVEHLFS decision matrix.
C1C2C3C4A10.5,0.5,0.5,0.67,0.67,0.67,0.670.33,0.67,0.67,0.67,0.67,0.67,0.670.5,0.67,0.67,0.67,0.67,0.67,0.670.5,0.5,0.5,0.83,0.83,0.83,0.83A20.5,0.5,0.5,0.5,0.5,0.5,0.830.33,0.33,0.5,0.5,0.5,0.67,0.670.17,0.17,0.33,0.33,0.33,0.5,0.50.5,0.5,0.5,0.5,0.5,0.5,0.67A30.5,0.5,0.5,0.5,0.67,0.67,0.670.5,0.5,0.5,0.5,0.67,0.67,0.830.5,0.5,0.5,0.67,0.83,0.83,0.830.67,0.67,0.67,0.67,0.67,1,1If we now consider the weight vector of criteriaCii=1,2,3,4 in the form of w=0.138,0.304,0.416,0.142 together with λ=1 for distance measures d1.,., d2.,., and d3.,. given, respectively, by (14)–(16); then, following Step 4 of Algorithm 3, the proposed SVEHFS-score function Sc. gives rise to the priorities of projects listed in Table 5. In addition to these results, the output of Pang et al.’s TOPSIS-based and aggregation-based techniques [33] has been presented in Table 5.Table 5
Ranking results of the different techniques.
Ranking orderOptimal alternativePang et al.’s TOPSIS-based technique [33]A1>A3>A2A1Pang et al.’s aggregation-based technique [33]A1>A3>A2A1The proposed SVEHFS-score functionScd1A3>A1>A2A3The proposed SVEHFS-score functionScd2A3>A1>A2A3The proposed SVEHFS-score functionScd3A3>A1>A2A3Generally, 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.


### 5.2. 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 including Zhang et al. [21] and Farhadinia and Herrera-Viedma [23].The 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 asC1: environment of health service, C2: treatment optimization, and C3: social resource allocation. The 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 A1: West China Hospital of Sichuan University, A2: Huashan Hospital of Fudan University, A3: Union Medical College Hospital, and A4: 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 Algorithm3 leads to constructing the following probabilistic hesitant fuzzy decision matrix (see Table 6).Table 6
The probabilistic hesitant fuzzy decision matrix.
Environment of health serviceC1Treatment optimizationC2Social resource allocationC3West China HospitalA10.50.4,0.70.60.910.30.2,0.50.8Huashan HospitalA20.80.3,0.90.70.510.80.4,0.90.6Union Medical College HospitalA30.510.70.5,0.90.50.80.6,0.90.4PLA General HospitalA40.80.5,0.90.50.30.5,0.60.50.71Similar to what is discussed in Section3 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, respectively.Table 7
The unified probabilistic hesitant fuzzy decision matrix.
C1C2C3A10.50.2,0.50.1,0.50.1,0.70.1,0.70.1,0.70.40.90.2,0.90.1,0.90.1,0.90.1,0.90.1,0.90.40.30.2,0.50.1,0.50.1,0.50.1,0.50.1,0.50.4A20.80.2,0.80.1,0.90.1,0.90.1,0.90.1,0.90.40.50.2,0.50.1,0.50.1,0.50.1,0.50.1,0.50.40.80.2,0.80.1,0.80.1,0.90.1,0.90.1,0.90.4A30.50.2,0.50.1,0.50.1,0.50.1,0.50.1,0.50.40.70.2,0.70.1,0.70.1,0.70.1,0.90.1,0.90.40.80.2,0.80.1,0.80.1,0.80.1,0.80.1,0.90.4A40.80.2,0.80.1,0.80.1,0.80.1,0.90.1,0.90.40.30.2,0.30.1,0.30.1,0.30.1,0.70.1,0.70.40.70.2,0.70.1,0.70.1,0.70.1,0.70.1,0.70.4Table 8
The SVEHFS decision matrix.
C1C2C3A10.5,0.5,0.5,0.7,0.7,0.70.9,0.9,0.9,0.9,0.9,0.90.3,0.5,0.5,0.5,0.5,0.5A20.8,0.8,0.8,0.9,0.9,0.90.5,0.5,0.5,0.5,0.5,0.50.8,0.8,0.8,0.9,0.9,0.9A30.5,0.5,0.5,0.5,0.5,0.50.7,0.7,0.7,0.7,0.9,0.90.8,0.8,0.8,0.8,0.8,0.9A40.8,0.8,0.8,0.8,0.9,0.90.3,0.3,0.3,0.3,0.7,0.70.7,0.7,0.7,0.7,0.7,0.7Following Step 4 of Algorithm3, if we now consider the weight vector of criteria Cii=1,2,3 in the form of w=0.2,0.1,0.7 together with λ=1 for distance measures d1.,., d2.,., and d3.,. given, respectively, by (14)–(16), then the proposed SVEHFS-score function Sc. gives rise to the priority of Chinese hospitals listed in Tables 9 and 10. In addition to these results, the output of Zhang et al.’s [21], Song et al.’s [28], and Farhadinia and Xu [22] techniques have been also presented in Tables 9 and 10.Table 9
Ranking results of Chinese hospitals.
Score functionScore of hospitalsRanking orderOptimal alternativeS¯d10.8555 0.7884 0.8002 0.8186A2>A3>A4>A1A2S¯d10.1268 0.0410 0.0546 0.0771A2>A3>A4>A1A2S¯d20.8618 0.7951 0.8007 0.8186A2>A3>A4>A1A2S¯d20.1406 0.0432 0.0595 0.0774A2>A3>A4>A1A2S¯d30.9631 0.9063 0.8310 0.8255A4>A3>A2>A1A4S¯d30.2075 0.0576 0.0827 0.0860A2>A3>A4>A1A2S¯d40.9093 0.8473 0.8156 0.8220A3>A4>A2>A1A3S¯d40.1671 0.0493 0.0686 0.0815A2>A3>A4>A1A2S¯AM0.1445 0.2116 0.1998 0.1814A2>A3>A4>A1A2S¯AM0.8732 0.9590 0.9454 0.9229A2>A3>A4>A1A2S¯GM0.1054 0.1871 0.1977 0.1814A3>A2>A4>A1A3S¯GM0.8711 0.9589 0.9451 0.9229A2>A3>A4>A1A2S¯Min0.0369 0.0937 0.1690 0.1745A4>A3>A2>A1A4S¯Min0.7925 0.9424 0.9173 0.9140A2>A3>A4>A1A2S¯Max0.2957 0.3704 0.2328 0.1880A2>A1>A3>A4A2S¯Max0.9399 0.9742 0.9707 0.9312A2>A3>A1>A4A2S¯P0.0001 0.0012 0.0015 0.0011A3>A2>A4>A1A3S¯P0.5758 0.8455 0.7979 0.7255A2>A3>A1>A4A2S¯BS0.5779 0.8464 0.7992 0.7257A2>A3>A1>A4A2S¯BS1 1 1 1A1>A2>A3>A4A1S¯F0.0007 0.0112 0.0112 0.0069A3>A2>A4>A1A3S¯F0.9993 1.0000 1.0000 0.9998A2>A3>A1>A4A2Table 10
Continues from Table9.
TechniqueRanking orderOptimal alternativeZhang et al.’s [21]A2>A3>A4>A1A2Song et al.’s [28]A2>0.838A3>0.819A4>1A1A2Farhadinia and Xu’s [22] first two step-based process multiplying deformation formula: S¯1A1=0.144, S¯1A2=0.2116, S¯1A3=0.199, and S¯1A4=0.1814A2>A3>A4>A1A2Exponential deformation formula:S¯1A1=0.873, S¯1A2=0.9590, S¯1A3=0.945, and S¯1A4=0.9229A2>A3>A4>A1A2Farhadinia and Xu’s [22] second two step-based process multiplying deformation formula: S¯2A1=0.1445, S¯2A2=0.2116, S¯2A3=0.1998, and S¯2A4=0.1814A2>A3>A4>A1A2Exponential deformation formula:S¯2A1=0.8732, S¯2A2=0.9590, S¯2A3=0.9454, and S¯2A4=0.9229A2>A3>A4>A1A2Farhadinia and Xu’s [22] third two step-based process multiplying deformation formula: R¯3A1=0.1445,0.1376, R¯3A2=0.2116,0.1171, R¯3A3=0.1998,0.0428, R¯3A4=0.1814,0.1824R¯3A2≥lexR¯3A3≥lexR¯3A4≥lexR¯3A1A2>A3>A4>A1A2Exponential deformation formula:R¯3A1=0.8732,0.1499, R¯3A2=0.9590,0.1114, R¯3A3=0.9454,0.0475, and R¯3A4=0.9229,0.0430R¯3A2≥lexR¯3A3≥lexR¯3A4≥lexR¯3A1A2>A3>A4>A1A2The proposed SVEHFS-score functionScd1: Scd1A1=0.5471, Scd1A2=0.8735, Scd1A3=0.7483, and Scd1A4=0.7000A2>A3>A4>A1A2The proposed SVEHFS-score functionScd2Scd2A1=0.4310, Scd2A2=0.7970, Scd2A3=0.7300, and Scd2A4=0.6800A2>A3>A4>A1A2The proposed SVEHFS-score functionScd2Scd3A1=0.4890, Scd3A2=0.8352, Scd3A3=0.7391, and Scd3A4=0.6900A2>A3>A4>A1A2As what can be observed from Tables9 and 10, the most repeated alternative is A2 which implies that the most appropriate hospital is the Huashan Hospital of Fudan University. This 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. The 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. This is while the proposed SVEHFS-score functions do not change the original form of PHFSs, and this can be seen as a pro. The other significant advantage of SVEHFS-based score functions over the other abovementioned techniques is its ease of use.


### 5.3. 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 professorsDMkk=1,2,and3 are invited for evaluating the quality of the three research topics Aii=1,2,and3 in accordance with three criteria: Cj=1: innovativeness, Cj=2: practicality, and Cj=3: feasibility. All the criteria are benefit types, and all the corresponding evaluations of three professors DMkk=1,2,and3 are represented in the form of PHFE-based decision matrices (see Tables 11–13).Table 11
Evaluation information provided byDM1.
C1C2C3A10.30.3,0.40.4,0.50.30.40.3,0.50.4,0.60.30.21A20.710.30.5,0.40.50.80.5,0.90.5A30.60.5,0.80.50.70.5,0.90.50.30.5,0.40.5Table 12
Evaluation information provided byDM2.
C1C2C3A10.40.5,0.50.50.610.50.3,0.70.4,0.80.3A20.30.5,0.40.50.40.3,0.50.4,0.60.30.60.5,0.70.5A30.50.5,0.60.50.80.5,0.90.50.61Table 13
Evaluation information provided byDM3.
C1C2C3A10.10.5,0.30.50.30.3,0.40.4,0.50.30.60.5,0.70.5A20.70.3,0.80.4,0.90.30.50.3,0.60.4,0.80.30.31A30.40.5,0.50.50.910.70.5,0.80.5By applying Steps 2 and 3 of Algorithm3, the individual unified PHFEs are computed as the data given in Tables 14–16.Table 14
The unified probabilistic hesitant fuzzy decision matrix forDM1.
C1C2C3A10.30.3,0.40.2,0.40.2,0.50.30.40.3,0.50.2,0.50.2,0.60.30.20.3,0.20.2,0.20.2,0.20.3A20.70.3,0.70.2,0.70.2,0.70.30.30.3,0.30.2,0.40.2,0.40.30.80.3,0.80.2,0.90.2,0.90.3A30.60.3,0.60.2,0.80.2,0.80.30.70.3,0.70.2,0.90.2,0.90.30.30.3,0.30.2,0.40.3,0.40.3Table 15
The unified probabilistic hesitant fuzzy decision matrix forDM2.
C1C2C3A10.40.3,0.40.2,0.50.2,0.50.30.60.3,0.60.2,0.60.2,0.60.30.50.3,0.70.2,0.70.2,0.80.3A20.30.3,0.30.2,0.40.2,0.40.30.40.3,0.50.2,0.50.2,0.60.30.60.3,0.60.2,0.70.2,0.70.3A30.50.3,0.50.2,0.60.2,0.60.30.80.3,0.80.2,0.90.2,0.90.30.60.3,0.60.2,0.60.2,0.60.3Table 16
The unified probabilistic hesitant fuzzy decision matrix forDM3.
C1C2C3A10.10.3,0.10.2,0.30.2,0.30.30.30.3,0.40.2,0.40.2,0.50.30.60.3,0.60.2,0.70.2,0.70.3A20.70.3,0.80.2,0.80.2,0.90.30.50.3,0.60.2,0.60.2,0.80.30.30.3,0.30.2,0.30.2,0.30.3A30.40.3,0.40.2,0.50.2,0.50.30.90.3,0.90.2,0.90.2,0.90.30.70.3,0.70.2,0.80.2,0.80.3Following the process discussed by Li et al. [35], the decision makers’ weights are obtained as(35)ϖk=∑j=1n∑i=1m−1∑g=i+1mdγijk,γgjk∑k=1z∑j=1n∑i=1m−1∑g=i+1mdγijk,γgjk,k=1,2,and3,in which d.,. is a distance measure.If we implement distance measuresd1.,., d2.,., and d3.,. provided, respectively, by (14)–(16) with λ=1, then the weight vectors will be(36)ϖd1=ϖ1,ϖ2,ϖ3=0.3545,0.2421,0.4034,ϖd2=ϖ1,ϖ2,ϖ3=0.3864,0.1932,0.4204,ϖd3=ϖ1,ϖ2,ϖ3=0.3927,0.1859,0.4215.To save more space for convenient storage, we only list the subsequent results forϖd1.Now, if we aggregate the individual SVEHFS matrices given in Tables17–19 by the help of the following rule(37)h˙=⊕k=13ϖk×h˙k=1−∏k=131−ℏ˙kϖk1,…,1−∏k=131−ℏ˙kϖkm,in which ϖk stands for the weight of the decision makers DMkk=1,2,and3 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 17
The SVEHFS decision matrix forDM1.
C1C2C3A1blue0.3,0.4,0.4,0.50.4,0.5,0.5,0.60.2,0.2,0.2,0.2A20.7,0.7,0.7,0.70.3,0.3,0.4,0.40.8,0.8,0.9,0.9A30.6,0.6,0.8,0.80.7,0.7,0.9,0.90.3,0.3,0.4,0.4Table 18
The SVEHFS decision matrix forDM2.
C1C2C3A10.4,0.4,0.5,0.50.6,0.6,0.6,0.60.5,0.7,0.7,0.8A20.3,0.3,0.4,0.40.4,0.5,0.5,0.60.6,0.6,0.7,0.7A30.5,0.5,0.6,0.60.8,0.8,0.9,0.90.6,0.6,0.6,0.6Table 19
The SVEHFS decision matrix forDM3.
C1C2C3A10.1,0.1,0.3,0.30.3,0.4,0.4,0.50.6,0.6,0.7,0.7A20.7,0.8,0.8,0.90.5,0.6,0.6,0.80.3,0.3,0.3,0.3A30.4,0.4,0.5,0.50.9,0.9,0.9,0.90.7,0.7,0.8,0.8Table 20
The collective SVEHFS decision matrix.
C1C2C3A10.2537,0.2934,0.3891,0.42730.4212,0.4901,0.4901,0.56230.4602,0.5230,0.5753,0.6150A20.6317,0.6873,0.6987,0.77220.4112,0.4851,0.5125,0.65080.6079,0.6079,0.7140,0.7140A30.5028,0.5028,0.6577,0.65770.8254,0.8254,0.9000,0.90000.5657,0.5657,0.6508,0.6508Table21 shows the comparison outcomes of different techniques. The 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. The outcomes of such identical steps are seen in identical ranking results.Table 21
Ranking results of the different techniques.
TechniqueScore of alternativesRanking orderOptimal alternativePHFWA-based technique of Xu et al. [36]0.45840.59900.7245A1<A2<A3A3PHFWG-based technique of Xu et al. [36]0.41170.52840.6909A1<A2<A3A3Xu et al. [37]0.28040.36530.7974A1<A2<A3A3Classical ORESTE [38]4.81727.13066.7856A2<A3<A1A1Li et al. [35]0.25630.19330.1212A1<A2<A3A3The proposed SVEHFS-score functionScd10.45830.62440.6837A1<A2<A3A3The proposed SVEHFS-score functionScd20.37840.55030.6313A1<A2<A3A3The proposed SVEHFS-score functionScd30.41830.58730.6575A1<A2<A3A3But, the result of classical ORESTE technique [38] is quite different from that of other abovementioned techniques. Such a different ranking result arises from the two-stage integrating ranking process. The initial stage calculates utility values for determining the weak ranking of alternatives. Then, 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.


## 5.1. 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. This is done by the help of Theorem 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 projectsAii=1,2,and3 into consideration which should be evaluated based on their importance. These projects should be ranked in accordance with four criteria of the benefit type which are suggested by the balanced scorecard methodology as follows:C1: financial perspectiveC2: the customer satisfactionC3: internal business process perspectiveC4: learning and growth perspectiveNow, by adopting Algorithm3 and the assumption that five experts apply the linguistic term set S=s0=none,s1=verylow,s2=low,s3=medium,s4=high,s5=veryhigh,s6=perfect, we are able to evaluate the projects Aii=1,2,and3 by means of PLTSs in Step 1. The corresponding data is presented in Table 1.Table 1
The probabilistic linguistic decision matrix.
C1C2C3C4A1s30.4,s40.6s20.2,s40.8s30.2,s40.8s30.4,s50.6A2s30.8,s50.2s20.3,s30.4,s40.3s10.3,s20.4,s30.3s30.8,s40.2A3s30.6,s40.4s30.6,s40.2,s50.2s30.4,s40.2,s50.4s40.7,s60.3To 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 Table2.Table 2
The probabilistic hesitant fuzzy decision matrix.
C1C2C3C4A10.50.4,0.670.60.330.2,0.670.80.50.2,0.670.80.50.4,0.830.6A20.50.8,0.830.20.330.3,0.50.4,0.670.30.170.3,0.330.4,0.50.30.50.8,0.670.2A30.50.6,0.670.40.50.6,0.670.2,0.830.20.50.4,0.670.2,0.830.40.670.7,10.3Now, by the help of Step 2 of Algorithm3, 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.Table 3
The unified probabilistic hesitant fuzzy decision matrix.
C1C2C3C4A10.50.2,0.50.1,0.50.1,0.670.2,0.670.1,0.670.1,0.670.20.330.2,0.670.1,0.670.1,0.670.2,0.670.1,0.670.1,0.670.20.50.2,0.670.1,0.670.1,0.670.2,0.670.1,0.670.1,0.670.20.50.2,0.50.1,0.50.1,0.830.2,0.830.1,0.830.1,0.830.2A20.50.2,0.50.1,0.50.1,0.50.2,0.50.1,0.50.1,0.830.20.330.2,0.330.1,0.50.1,0.50.2,0.50.1,0.670.1,0.670.20.170.2,0.170.1,0.330.1,0.330.2,0.330.1,0.50.1,0.50.20.50.2,0.50.1,0.50.1,0.50.2,0.50.1,0.50.1,0.670.2A30.50.2,0.50.1,0.50.1,0.50.2,0.670.1,0.670.1,0.670.20.50.2,0.50.1,0.50.1,0.50.2,0.670.1,0.670.1,0.830.20.50.2,0.50.1,0.50.1,0.670.2,0.830.1,0.830.1,0.830.20.670.2,0.670.1,0.670.1,0.670.2,0.670.1,10.1,10.2In what follows, by the use of Step 3 of Algorithm3, we will derive the corresponding SVEHFEs, as shown in Table 4.Table 4
The SVEHLFS decision matrix.
C1C2C3C4A10.5,0.5,0.5,0.67,0.67,0.67,0.670.33,0.67,0.67,0.67,0.67,0.67,0.670.5,0.67,0.67,0.67,0.67,0.67,0.670.5,0.5,0.5,0.83,0.83,0.83,0.83A20.5,0.5,0.5,0.5,0.5,0.5,0.830.33,0.33,0.5,0.5,0.5,0.67,0.670.17,0.17,0.33,0.33,0.33,0.5,0.50.5,0.5,0.5,0.5,0.5,0.5,0.67A30.5,0.5,0.5,0.5,0.67,0.67,0.670.5,0.5,0.5,0.5,0.67,0.67,0.830.5,0.5,0.5,0.67,0.83,0.83,0.830.67,0.67,0.67,0.67,0.67,1,1If we now consider the weight vector of criteriaCii=1,2,3,4 in the form of w=0.138,0.304,0.416,0.142 together with λ=1 for distance measures d1.,., d2.,., and d3.,. given, respectively, by (14)–(16); then, following Step 4 of Algorithm 3, the proposed SVEHFS-score function Sc. gives rise to the priorities of projects listed in Table 5. In addition to these results, the output of Pang et al.’s TOPSIS-based and aggregation-based techniques [33] has been presented in Table 5.Table 5
Ranking results of the different techniques.
Ranking orderOptimal alternativePang et al.’s TOPSIS-based technique [33]A1>A3>A2A1Pang et al.’s aggregation-based technique [33]A1>A3>A2A1The proposed SVEHFS-score functionScd1A3>A1>A2A3The proposed SVEHFS-score functionScd2A3>A1>A2A3The proposed SVEHFS-score functionScd3A3>A1>A2A3Generally, 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.


## 5.2. 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 including Zhang et al. [21] and Farhadinia and Herrera-Viedma [23].The 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 asC1: environment of health service, C2: treatment optimization, and C3: social resource allocation. The 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 A1: West China Hospital of Sichuan University, A2: Huashan Hospital of Fudan University, A3: Union Medical College Hospital, and A4: 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 Algorithm3 leads to constructing the following probabilistic hesitant fuzzy decision matrix (see Table 6).Table 6
The probabilistic hesitant fuzzy decision matrix.
Environment of health serviceC1Treatment optimizationC2Social resource allocationC3West China HospitalA10.50.4,0.70.60.910.30.2,0.50.8Huashan HospitalA20.80.3,0.90.70.510.80.4,0.90.6Union Medical College HospitalA30.510.70.5,0.90.50.80.6,0.90.4PLA General HospitalA40.80.5,0.90.50.30.5,0.60.50.71Similar to what is discussed in Section3 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, respectively.Table 7
The unified probabilistic hesitant fuzzy decision matrix.
C1C2C3A10.50.2,0.50.1,0.50.1,0.70.1,0.70.1,0.70.40.90.2,0.90.1,0.90.1,0.90.1,0.90.1,0.90.40.30.2,0.50.1,0.50.1,0.50.1,0.50.1,0.50.4A20.80.2,0.80.1,0.90.1,0.90.1,0.90.1,0.90.40.50.2,0.50.1,0.50.1,0.50.1,0.50.1,0.50.40.80.2,0.80.1,0.80.1,0.90.1,0.90.1,0.90.4A30.50.2,0.50.1,0.50.1,0.50.1,0.50.1,0.50.40.70.2,0.70.1,0.70.1,0.70.1,0.90.1,0.90.40.80.2,0.80.1,0.80.1,0.80.1,0.80.1,0.90.4A40.80.2,0.80.1,0.80.1,0.80.1,0.90.1,0.90.40.30.2,0.30.1,0.30.1,0.30.1,0.70.1,0.70.40.70.2,0.70.1,0.70.1,0.70.1,0.70.1,0.70.4Table 8
The SVEHFS decision matrix.
C1C2C3A10.5,0.5,0.5,0.7,0.7,0.70.9,0.9,0.9,0.9,0.9,0.90.3,0.5,0.5,0.5,0.5,0.5A20.8,0.8,0.8,0.9,0.9,0.90.5,0.5,0.5,0.5,0.5,0.50.8,0.8,0.8,0.9,0.9,0.9A30.5,0.5,0.5,0.5,0.5,0.50.7,0.7,0.7,0.7,0.9,0.90.8,0.8,0.8,0.8,0.8,0.9A40.8,0.8,0.8,0.8,0.9,0.90.3,0.3,0.3,0.3,0.7,0.70.7,0.7,0.7,0.7,0.7,0.7Following Step 4 of Algorithm3, if we now consider the weight vector of criteria Cii=1,2,3 in the form of w=0.2,0.1,0.7 together with λ=1 for distance measures d1.,., d2.,., and d3.,. given, respectively, by (14)–(16), then the proposed SVEHFS-score function Sc. gives rise to the priority of Chinese hospitals listed in Tables 9 and 10. In addition to these results, the output of Zhang et al.’s [21], Song et al.’s [28], and Farhadinia and Xu [22] techniques have been also presented in Tables 9 and 10.Table 9
Ranking results of Chinese hospitals.
Score functionScore of hospitalsRanking orderOptimal alternativeS¯d10.8555 0.7884 0.8002 0.8186A2>A3>A4>A1A2S¯d10.1268 0.0410 0.0546 0.0771A2>A3>A4>A1A2S¯d20.8618 0.7951 0.8007 0.8186A2>A3>A4>A1A2S¯d20.1406 0.0432 0.0595 0.0774A2>A3>A4>A1A2S¯d30.9631 0.9063 0.8310 0.8255A4>A3>A2>A1A4S¯d30.2075 0.0576 0.0827 0.0860A2>A3>A4>A1A2S¯d40.9093 0.8473 0.8156 0.8220A3>A4>A2>A1A3S¯d40.1671 0.0493 0.0686 0.0815A2>A3>A4>A1A2S¯AM0.1445 0.2116 0.1998 0.1814A2>A3>A4>A1A2S¯AM0.8732 0.9590 0.9454 0.9229A2>A3>A4>A1A2S¯GM0.1054 0.1871 0.1977 0.1814A3>A2>A4>A1A3S¯GM0.8711 0.9589 0.9451 0.9229A2>A3>A4>A1A2S¯Min0.0369 0.0937 0.1690 0.1745A4>A3>A2>A1A4S¯Min0.7925 0.9424 0.9173 0.9140A2>A3>A4>A1A2S¯Max0.2957 0.3704 0.2328 0.1880A2>A1>A3>A4A2S¯Max0.9399 0.9742 0.9707 0.9312A2>A3>A1>A4A2S¯P0.0001 0.0012 0.0015 0.0011A3>A2>A4>A1A3S¯P0.5758 0.8455 0.7979 0.7255A2>A3>A1>A4A2S¯BS0.5779 0.8464 0.7992 0.7257A2>A3>A1>A4A2S¯BS1 1 1 1A1>A2>A3>A4A1S¯F0.0007 0.0112 0.0112 0.0069A3>A2>A4>A1A3S¯F0.9993 1.0000 1.0000 0.9998A2>A3>A1>A4A2Table 10
Continues from Table9.
TechniqueRanking orderOptimal alternativeZhang et al.’s [21]A2>A3>A4>A1A2Song et al.’s [28]A2>0.838A3>0.819A4>1A1A2Farhadinia and Xu’s [22] first two step-based process multiplying deformation formula: S¯1A1=0.144, S¯1A2=0.2116, S¯1A3=0.199, and S¯1A4=0.1814A2>A3>A4>A1A2Exponential deformation formula:S¯1A1=0.873, S¯1A2=0.9590, S¯1A3=0.945, and S¯1A4=0.9229A2>A3>A4>A1A2Farhadinia and Xu’s [22] second two step-based process multiplying deformation formula: S¯2A1=0.1445, S¯2A2=0.2116, S¯2A3=0.1998, and S¯2A4=0.1814A2>A3>A4>A1A2Exponential deformation formula:S¯2A1=0.8732, S¯2A2=0.9590, S¯2A3=0.9454, and S¯2A4=0.9229A2>A3>A4>A1A2Farhadinia and Xu’s [22] third two step-based process multiplying deformation formula: R¯3A1=0.1445,0.1376, R¯3A2=0.2116,0.1171, R¯3A3=0.1998,0.0428, R¯3A4=0.1814,0.1824R¯3A2≥lexR¯3A3≥lexR¯3A4≥lexR¯3A1A2>A3>A4>A1A2Exponential deformation formula:R¯3A1=0.8732,0.1499, R¯3A2=0.9590,0.1114, R¯3A3=0.9454,0.0475, and R¯3A4=0.9229,0.0430R¯3A2≥lexR¯3A3≥lexR¯3A4≥lexR¯3A1A2>A3>A4>A1A2The proposed SVEHFS-score functionScd1: Scd1A1=0.5471, Scd1A2=0.8735, Scd1A3=0.7483, and Scd1A4=0.7000A2>A3>A4>A1A2The proposed SVEHFS-score functionScd2Scd2A1=0.4310, Scd2A2=0.7970, Scd2A3=0.7300, and Scd2A4=0.6800A2>A3>A4>A1A2The proposed SVEHFS-score functionScd2Scd3A1=0.4890, Scd3A2=0.8352, Scd3A3=0.7391, and Scd3A4=0.6900A2>A3>A4>A1A2As what can be observed from Tables9 and 10, the most repeated alternative is A2 which implies that the most appropriate hospital is the Huashan Hospital of Fudan University. This 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. The 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. This is while the proposed SVEHFS-score functions do not change the original form of PHFSs, and this can be seen as a pro. The other significant advantage of SVEHFS-based score functions over the other abovementioned techniques is its ease of use.


## 5.3. 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 professorsDMkk=1,2,and3 are invited for evaluating the quality of the three research topics Aii=1,2,and3 in accordance with three criteria: Cj=1: innovativeness, Cj=2: practicality, and Cj=3: feasibility. All the criteria are benefit types, and all the corresponding evaluations of three professors DMkk=1,2,and3 are represented in the form of PHFE-based decision matrices (see Tables 11–13).Table 11
Evaluation information provided byDM1.
C1C2C3A10.30.3,0.40.4,0.50.30.40.3,0.50.4,0.60.30.21A20.710.30.5,0.40.50.80.5,0.90.5A30.60.5,0.80.50.70.5,0.90.50.30.5,0.40.5Table 12
Evaluation information provided byDM2.
C1C2C3A10.40.5,0.50.50.610.50.3,0.70.4,0.80.3A20.30.5,0.40.50.40.3,0.50.4,0.60.30.60.5,0.70.5A30.50.5,0.60.50.80.5,0.90.50.61Table 13
Evaluation information provided byDM3.
C1C2C3A10.10.5,0.30.50.30.3,0.40.4,0.50.30.60.5,0.70.5A20.70.3,0.80.4,0.90.30.50.3,0.60.4,0.80.30.31A30.40.5,0.50.50.910.70.5,0.80.5By applying Steps 2 and 3 of Algorithm3, the individual unified PHFEs are computed as the data given in Tables 14–16.Table 14
The unified probabilistic hesitant fuzzy decision matrix forDM1.
C1C2C3A10.30.3,0.40.2,0.40.2,0.50.30.40.3,0.50.2,0.50.2,0.60.30.20.3,0.20.2,0.20.2,0.20.3A20.70.3,0.70.2,0.70.2,0.70.30.30.3,0.30.2,0.40.2,0.40.30.80.3,0.80.2,0.90.2,0.90.3A30.60.3,0.60.2,0.80.2,0.80.30.70.3,0.70.2,0.90.2,0.90.30.30.3,0.30.2,0.40.3,0.40.3Table 15
The unified probabilistic hesitant fuzzy decision matrix forDM2.
C1C2C3A10.40.3,0.40.2,0.50.2,0.50.30.60.3,0.60.2,0.60.2,0.60.30.50.3,0.70.2,0.70.2,0.80.3A20.30.3,0.30.2,0.40.2,0.40.30.40.3,0.50.2,0.50.2,0.60.30.60.3,0.60.2,0.70.2,0.70.3A30.50.3,0.50.2,0.60.2,0.60.30.80.3,0.80.2,0.90.2,0.90.30.60.3,0.60.2,0.60.2,0.60.3Table 16
The unified probabilistic hesitant fuzzy decision matrix forDM3.
C1C2C3A10.10.3,0.10.2,0.30.2,0.30.30.30.3,0.40.2,0.40.2,0.50.30.60.3,0.60.2,0.70.2,0.70.3A20.70.3,0.80.2,0.80.2,0.90.30.50.3,0.60.2,0.60.2,0.80.30.30.3,0.30.2,0.30.2,0.30.3A30.40.3,0.40.2,0.50.2,0.50.30.90.3,0.90.2,0.90.2,0.90.30.70.3,0.70.2,0.80.2,0.80.3Following the process discussed by Li et al. [35], the decision makers’ weights are obtained as(35)ϖk=∑j=1n∑i=1m−1∑g=i+1mdγijk,γgjk∑k=1z∑j=1n∑i=1m−1∑g=i+1mdγijk,γgjk,k=1,2,and3,in which d.,. is a distance measure.If we implement distance measuresd1.,., d2.,., and d3.,. provided, respectively, by (14)–(16) with λ=1, then the weight vectors will be(36)ϖd1=ϖ1,ϖ2,ϖ3=0.3545,0.2421,0.4034,ϖd2=ϖ1,ϖ2,ϖ3=0.3864,0.1932,0.4204,ϖd3=ϖ1,ϖ2,ϖ3=0.3927,0.1859,0.4215.To save more space for convenient storage, we only list the subsequent results forϖd1.Now, if we aggregate the individual SVEHFS matrices given in Tables17–19 by the help of the following rule(37)h˙=⊕k=13ϖk×h˙k=1−∏k=131−ℏ˙kϖk1,…,1−∏k=131−ℏ˙kϖkm,in which ϖk stands for the weight of the decision makers DMkk=1,2,and3 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 17
The SVEHFS decision matrix forDM1.
C1C2C3A1blue0.3,0.4,0.4,0.50.4,0.5,0.5,0.60.2,0.2,0.2,0.2A20.7,0.7,0.7,0.70.3,0.3,0.4,0.40.8,0.8,0.9,0.9A30.6,0.6,0.8,0.80.7,0.7,0.9,0.90.3,0.3,0.4,0.4Table 18
The SVEHFS decision matrix forDM2.
C1C2C3A10.4,0.4,0.5,0.50.6,0.6,0.6,0.60.5,0.7,0.7,0.8A20.3,0.3,0.4,0.40.4,0.5,0.5,0.60.6,0.6,0.7,0.7A30.5,0.5,0.6,0.60.8,0.8,0.9,0.90.6,0.6,0.6,0.6Table 19
The SVEHFS decision matrix forDM3.
C1C2C3A10.1,0.1,0.3,0.30.3,0.4,0.4,0.50.6,0.6,0.7,0.7A20.7,0.8,0.8,0.90.5,0.6,0.6,0.80.3,0.3,0.3,0.3A30.4,0.4,0.5,0.50.9,0.9,0.9,0.90.7,0.7,0.8,0.8Table 20
The collective SVEHFS decision matrix.
C1C2C3A10.2537,0.2934,0.3891,0.42730.4212,0.4901,0.4901,0.56230.4602,0.5230,0.5753,0.6150A20.6317,0.6873,0.6987,0.77220.4112,0.4851,0.5125,0.65080.6079,0.6079,0.7140,0.7140A30.5028,0.5028,0.6577,0.65770.8254,0.8254,0.9000,0.90000.5657,0.5657,0.6508,0.6508Table21 shows the comparison outcomes of different techniques. The 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. The outcomes of such identical steps are seen in identical ranking results.Table 21
Ranking results of the different techniques.
TechniqueScore of alternativesRanking orderOptimal alternativePHFWA-based technique of Xu et al. [36]0.45840.59900.7245A1<A2<A3A3PHFWG-based technique of Xu et al. [36]0.41170.52840.6909A1<A2<A3A3Xu et al. [37]0.28040.36530.7974A1<A2<A3A3Classical ORESTE [38]4.81727.13066.7856A2<A3<A1A1Li et al. [35]0.25630.19330.1212A1<A2<A3A3The proposed SVEHFS-score functionScd10.45830.62440.6837A1<A2<A3A3The proposed SVEHFS-score functionScd20.37840.55030.6313A1<A2<A3A3The proposed SVEHFS-score functionScd30.41830.58730.6575A1<A2<A3A3But, the result of classical ORESTE technique [38] is quite different from that of other abovementioned techniques. Such a different ranking result arises from the two-stage integrating ranking process. The initial stage calculates utility values for determining the weak ranking of alternatives. Then, 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.


## 6. 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.



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*Source: 2454738-2021-09-20.xml*