Three-way decision based on canonical soft sets of hesitant fuzzy sets

: The theory of three-way decision is built on the philosophy of thinking in threes. The essence of three-way decision is trisecting the whole and taking di ﬀ erent strategies for di ﬀ erent parts accordingly. The theory of three-way decision has been successfully implemented to diverse ﬁelds since it provides an elegant and e ﬃ cient solution for solving complicated problems. In this paper, a useful representation for hesitant fuzzy sets is obtained by means of canonical soft sets. We also deﬁne unit interval parameterized soft sets and their derived hesitant fuzzy sets. Mutual representations and inner connections between hesitant fuzzy sets and soft sets are examined. With the help of soft rough sets, a generalized rough model based on hesitant fuzzy sets is established. A novel three-way decision method is presented for solving decision-making problems by means of hesitant fuzzy sets and canonical soft sets. Finally, a numerical example regarding peer review of research articles is given to illustrate the validity and e ﬃ cacy of the proposed method.


Introduction
Inspired by Pawlak's rough sets [37] and probabilistic rough sets [9,48], Yao [49][50][51] proposed the theory of three-way decision (3WD) which provides reasonable semantic interpretation to the positive, negative and boundary regions of rough sets.The core of 3WD is trisecting the whole and taking different strategies for different parts.We can simplify complex problems by dividing the whole into three parts.3WD provide us with a framework to solve the generalized uncertainty problem.It might be considered as part of a more complete decision-making strategy which in situations of e.g., large computational burden, should only refine the analysis of the alternatives belonging to the positive (and possibly the boundary) regions.The philosophy of thinking in threes includes not only splitting the set of alternatives in three parts, it can also evoke thinking in three views, three components, three categories, three levels or three dimensions [53].Different interpretations of ternary options make 3WD become a useful and universal theoretical method to solve practical problems.For instance, in a social choice context, Alcantud and Laruelle [2] characterized a pragmatical model of voting with three actions.Recently, Laruelle [25] has argued that citizens in democratic countries tend to abstain because they cannot voice their dissatisfaction in their ballots.She reports the results of a field experiment that took place during the 2017 French presidential elections: Real voters were entirely satisfied with ternary voting [2], and understood well its aim and purpose.Many scholars have studied and expanded the three decision theories and applied them to many disciplines in the past decade.Other researchers have represented several classifications of methods based on the 3WD theory and favorably practiced in many fields.For example, Yu et al. [55] offered an effective automatic process by applying the decision-theoretic rough set model to clustering.Jia et al. [24] developed a mathematical model for optimal representation of the decision-theoretic rough set.Liang and Liu [28] proposed a risk decision model based on hesitant fuzzy rough sets.Liang et al. [29] presented a 3WD model with linguistic features for decision-making problems.Yao [52] established the TAO ("trisecting-acting-outcome") framework as a fundamental model for 3WD.Hu et al. [23] presented two kinds of 3WD approaches and examined their properties.Li et al. [26] introduced 3WD models based on subset-evaluation which generalize the original models.Liu et al. [30] proposed a new 3WD model with intuitionistic fuzzy numbers.Li et al. [27] generalized Pawlak's classical model and introduced rough set models on two universes.Yang and Yao [47] presented two semantics for interpreting soft sets and constructed two soft set-based 3WD models.Yao [54] investigated set-theoretic models of 3WD and illustrated the essential role of the TAO framework in 3WD.
In addition to rough set theory which we have mentioned above, there are other models of uncertain knowledge whose advantages are used in our study.They incorporate features related to indiscernibility or granularity (rough set theory), partial membership (fuzzy set theory and its extension under hesitancy), and parameterized descriptions (soft set theory).
In 1965, Zadeh [56] acquainted fuzzy set (FS) theory to manage the impreciseness in the data.After its presence, many scholars have spread them to L-fuzzy sets [21], IVFSs ("interval-valued fuzzy sets") [42] and IFSs ("Intuitionistic fuzzy sets") [5].In practical application, when determining the membership degree of an element to a fuzzy set, people often hesitate between two or more possible values and find it difficult to choose.In order to solve this problem, Torra [41] proposed the concept of a hesitant fuzzy set (HFS).Compared with classical sets and FSs, HFSs contain all the possible membership values corresponding to each alternative, which if properly handled, can reduce the information loss in the decision-making process.Based on 3WD theory, Wang et al. [43] developed a novel approach to MADM ("multiple attribute decision making") under a hesitant fuzzy setting.
To build a more comprehensive theoretical structure for representing and administering uncertainty, the concept of a soft set (SS) was born from a different narrative that concerns parameterization [36].
As a new type of nonstandard sets, the SS involves not only the domain of objects, but also a parameter space related to the object domain.In recent years, the research on soft set theory has developed quickly.Maji et al. [34] applied the theory of SSs to solve a decision-making problem using rough mathematics.Maji et al. [33] proposed several operations of SSs.They also considered the fuzzy extension of SSs in [31].
Later on, Maji et al. [32] joined SSs with the features of IFSs and hence stated the notion of an IFSS ("intuitionistic fuzzy soft set").Peng et al. [38] combined SSs with PFSs ("Pythagorean fuzzy sets") to put forward the concept of Pythagorean fuzzy soft sets.Athira et al. [6] proposed some new entropy measures for Pythagorean fuzzy soft sets.By virtue of Archimedean t-norm and t-conorm, Garg and Arora [20] generalized the MSM ("Maclaurin symmetric mean") aggregation operators to IFSSs.Chen et al. [8] developed a parameterization reduction method for SSs.Ali et al. [4] revisited and improved some basic operations such as "intersection", "union", "difference", between the pairs of SSs.Garg and Arora [19] developed a correlation coefficient-based TOPSIS method to address decision-making problems with intuitionistic fuzzy soft information.A concept of generalized fuzzy soft sets was stated and investigated by Majumdar and Samanta [35].Feng et al. [17] extended the preference ranking organization method for enrichment evaluation with IFSSs.Agarwal et al. [1] defined the notion of GIFSSs ("generalized intuitionistic fuzzy soft sets") and considered practical applications of GIFSSs.Feng et al. [13] improved the GIFSS and simplified this notion as a combination of an IFSS over a universal set and an IFS in a parameter set.Garg and Arora [18] put forth group-based GIFSSs and discussed their applications to group decision-making problems.
Returning to the motivational topic of rough set (RS) theory, we recall that it was introduced by Pawlak in 1982, who attempted to analyze the granularity-based uncertainty during the decision process [37].Driven by the features of FSs and RSs, Dubois and Prade [10] defined the notion of rough fuzzy sets and fuzzy rough sets.Raszikowaka and Kerre [39] further extended the concepts presented by Dubois and Prade to (I, T )-fuzzy rough sets.Using the intuitionistic fuzzy overlap function and its residual implication, Wen et al. [44] presented a new type of intuitionistic fuzzy rough sets.Given the important role of FSs and RSs, Feng et al. [14] combined SSs with them.Further, Feng et al. [16] put forth the concept of soft rough sets and demonstrated that Pawlak's RS model is a special case of the presented concept when the SS in a soft approximation space is a partition SS.By combining fuzzy soft sets with fuzzy rough sets, Sun and Ma [40] put forward soft fuzzy rough sets and considered their application to decision making problems.Xie and Gong [46] presented a notion called hesitant soft fuzzy rough set based on SSs and HFSs.They ascertained the relationship between hesitant fuzzy rough sets and hesitant soft fuzzy rough sets as well.
The main goal of our paper is to take advantage of a novel construction of SSs derived from HFSs, in order to justify a 3WD strategy that acts on hesitant fuzzy data.In passing, we study some theoretical facts that expand our knowledge about the new SS produced from an HFS, which we term as the canonical soft set.The rest of this article is organized as follows.Section 2 gives a review of the rudiments on FSs, HFSs, SSs, RSs and 3WD.In Section 3, the notions of canonical soft sets, unit interval parameterized SSs, and derived HFSs are introduced.The relationships between canonical soft sets and derived HFSs are discussed.In Section 4, a novel 3WD algorithm is proposed to address group decision-making problems by using canonical soft sets of HFSs.In addition, an example regarding paper review is given to illustrate the validity of the proposed algorithm in Section 5. A summary of this study is given in Section 6.

Preliminaries
Some basic concepts related to FSs, SSs, RSs and 3WD are briefly reviewed in this section.Through the paper, denotes the universal set and I = [0, 1].For any nonempty set K, we use the following notations: • P(K) represents the "power set" of K; • P * (K) represents the family of all nonempty subsets of K; • F * (K) represents the family of all nonempty finite subsets of K; • F n (K) represents the family of all finite subsets E of K whose cardinality satisfies |E| = n (n ≥ 1).

Fuzzy sets and HFSs
Zadeh [56] suggested that a certain type of uncertainty of information can be quantified by partial membership values.Consequently, a fuzzy set M in is defined by a mapping M : → I which represents the membership function of M.
This definition extends the usual notion of a characteristic function, which in set theory, is equivalent to a standard subset.Now when u ∈ , the value M(u) is the degree to which u belongs to M. Let F S( ) denote the class of all FSs on .A number of relevant operations of fuzzy sets can be introduced.
With the "min-max" system, Zadeh defined the "intersection", "union", and "complement" operations of FSs as follows: where M, N ∈ F S( ) and τ ∈ .When M(τ) ≤ N(τ) for all τ ∈ , then M ⊆ N. It is obvious that M = N if and only if M ⊆ N and N ⊆ M.
When an object has various viable membership values, it is often tough to ascertain its precise membership grade.For this, a concept of HFSs ("hesitant fuzzy sets") was proposed by Torra [41].In an HFS, the membership grades of objects are represented by sets containing several (maybe infinitely many) values in the unit interval I.In the following, we recall some rudiments regarding the theory of HFSs.
According to [45], an HFE ("hesitant fuzzy element") means a subset of I.That is, the power set P(I) of I consists of all HFEs.Given an HFS h and τ ∈ , the HFE h(τ) contains all the possible membership grades of τ to the HFS h.Henceforth, HF S( ) denotes the class of all HFSs on .Definition 2.2.An HFS h : → P * (I) is said to be a regular HFS (RHFS) on .
In practical applications, it is justifiable to assume that the employed HFSs are regular.This means that at least one value in I must be assigned for each element of as its possible membership grade.Henceforth, the class of all RHFSs on is denoted by RHF S( ).

Definition 2.3. [7] A typical hesitant fuzzy set (THFS) on is a mapping h : → F * (I).
A nonempty finite subset of I is called a typical hesitant fuzzy element (THFE).It is clear that the family of all THFEs coincides with F * (I).Conventionally, the possible membership grades in a THFE E are listed in ascending order.In other words, E is written in the form Particularly, the THFE E = {1} is said to be full and E = {0} is referred to as the empty HFE.Accordingly, { τ, {1} : τ ∈ } is called the ideal (or full) HFS on , and { τ, {0} : τ ∈ } is called the anti-ideal (or empty) HFS on In the following, we denote the class of all THFSs on by T HF S( ).

Definition 2.4. [3]
A THFS h on is n-typical if there exists a positive integer n such that |h(τ)| = n for all τ ∈ .
To put it in another way, an n-typical hesitant fuzzy set is a mapping h : → F n (I) for some n ≥ 1.For an HFS, there are multiple values in HFEs since there is more than one criterion for determining membership grades.This can be explained by a common application scenario.When multiple experts determine the membership grades of an object, there might be different values for the same object because different experts have distinguishing opinions.

Soft sets
In order to establish a theory framework to describe and deal with uncertainty, Molodtsov [36] proposed the important concept of SSs from the perspective of parameterization.Let E be a set containing all the relevant parameters of objects in .Conventionally, the set E is called the parameter space.For convenience, we can simply denote E by E. The pair ( , E) is named as a soft universe.In practical applications, the parameter space E includes all attributes and characteristics of the object in .The mathematical definition of SSs is as follows: Definition 2.5.[36] Assume that ( , E) is a soft universe.An ordered pair T = (G, B) is called a soft set over .The subset B ⊆ E is the parameter set of T and the approximate function of T is the mapping G : B → P( ).
is a family of non-empty subsets of such that for every τ ∈ , there exists an α ∈ Λ with τ ∈ C α .A covering C of is said to be a partition of if Proposition 2.12.Suppose that T = (G, B) is a soft set over .Then the following are equivalent: (1) T is an injective partition soft set; (2) T is a disjoint covering soft set.
Proof.Suppose that T = (G, B) is an injective partition soft set over .By Definition 2.9, {G(a) | a ∈ B} forms a partition of .Thus T is a covering soft set such that

Rough sets and soft rough sets
Let Γ represent an equivalence relation on .The pair ( , Γ) is said to be a Pawlak approximation space [37].For any τ 1 , τ 2 ∈ , if (τ 1 , τ 2 ) ∈ Γ we said that τ 1 and τ 2 are Γ-indiscernible.The equivalence class of τ 1 ∈ is denoted by [τ 1 ] Γ .The quotient set of with respect to Γ is denoted by /Γ.Assume that Γ is an equivalence relation on and Υ is a subset of .Then Υ can be characterized by means of the following two approximations: The sets Γ * Υ and Γ * Υ are called the lower and upper approximations of Υ with respect to the approximation space ( , Γ).We say that the set Υ is definable if Γ * Υ = Γ * Υ; otherwise, Υ is said to be rough or inexact.
Furthermore, we can define the positive region, boundary region and negative region of Υ respectively as follows: Assume that C = Γ * Υ and D = Γ * Υ where Υ ⊆ .Sometimes, a rough set can also be represented by the pair (C, D) ∈ P( ) × P( ).If Υ is defined by a predicate H and τ ∈ , we have the following: Feng et al. [16] considered the generalization of Pawlak's rough sets based on soft set theory, and put forward the concept of soft rough sets.Different from rough sets based on Pawlak approximation space, soft rough sets are based on the granular structure given by a soft approximation space (SAS).An SAS is an ordered pair ( , T ), where T is a soft set over .More specifically, the soft lower and upper approximations are defined in the following way: Definition 2.13.[16] Assume that T = (G, B) is a soft set over , H ⊆ and P = ( , T ) is a soft approximation space.The soft lower approximation apr P (H) and the soft upper approximation apr P (H) of H with respect to the SAS P = ( , T ) are defined as: When apr P (H) = apr P (H), we say that the set H is soft P-definable; otherwise, H is said to be a soft P-rough set.
Below the concepts given in Definition 2.13 are reformulated: Proposition 2.14.[16] Suppose that T = (G, B) is a soft set over and P = ( , T ) is an SAS.Then apr where H is a subset of .

Three-way decision
3WD is an uncertain framework based on theory of probability rough sets and decision rough sets.The basic philosophy of 3WD is thinking in threes which means to understand and process a whole through three different and related parts.Based on the trisecting-acting-outcome framework, Yao [52] has proposed a generalized 3WD model.Hereinafter, the framework trisecting-acting-outcome is simply named TAO.The TAO model of 3WD concentrates on the following three works: (1) To divide the universe into three parts; (2) To take actions on the three parts separately; (3) To optimize the results and obtain a desirable outcome.
It is clear that the TAO model of 3WD is an interpretative case of thinking in threes.The fundamental notion of 3WD is trisecting the whole.Three parts of trisection can be seen as three different aspects, perspectives or components of the whole and their integration represents and covers the whole.Hence the three parts are both independent and interrelated.
Yao [51] introduced a general model for building a trisection with a single evaluation function.In this model, a linearly ordered set (L, ≤) is used as the codomain of an evaluation function, which specifies the scale of the evaluation.Given two evaluation values n, m ∈ L, n < m means n ≤ m and n m.Moreover, we also write n < m as m > n.
Definition 2.17.[51] Let E : −→ (L, ≤) be an evaluation function on .Given two thresholds λ, µ ∈ L with λ < µ, a trisection of can be derived as follows: Intuitively, we refer to elements in the disjoint sets Pos [µ,.) (E), Bnd (λ,µ) (E) and Neg (.,λ] (E) as positive, neutral and negative objects, respectively.The above model plays a fundamental role in constructing set-theoretic models of 3WDs [51].It should be noted that how to select easy-to-understand and meaningful scales is an issue of great importance.In many practical applications, it is sufficient to consider subsets of real numbers as evaluation scales.More specifically, as a special case of Definition 2.17, we have the following notion.
Definition 2.18.[54] Suppose that E : −→ [b, a] is an evaluation on where b < a are two real numbers.Given two thresholds γ, δ with b ≤ γ < δ ≤ a, a trisection of can be derived as follows: Elements in the disjoint sets H [δ,a] (E), M (γ,δ) (E) and L [b,γ] (E) are called objects with high, medium and low values, respectively.
To facilitate the reading of this article, the abbreviations of some frequently used terminologies are listed in Table 1.

Mutual representations between HFSs and SSs
In this section, we define some useful concepts including canonical soft sets, unit interval parameterized SSs and derived HFSs.The relationships among these notions are examined in detail.Definition 3.1.Suppose that h : → P(I) is an HFS.Then is called the value space of the HFS h.
In other words, the value space of an HFS is a set containing all possible membership values of objects in .Definition 3.2.Let h : → P(I) be an HFS with its value space V h .Then Note that Co T (ν) contains all the parameters of the alternative ν, according to the information carried by T .Definition 3.5.Let T = (G, B) be a UIPSS over and ν ∈ .Then we can define an HFS h T : → P(I) on by which is said to be the derived hesitant fuzzy set (DHFS) of the UIPSS T .
Proof.Let T = (G, B) be a serial UIPSS over and h T : → P(I) be the DHFS of T .By Definition 3.5, the DHFS h T of T is given by h Firstly, we prove V h T = B.By Definition 3.1, we have Conversely, assume that T = (G, B) is a serial UIPSS.By Definition 2.6, we have G( ) ∅ for any ∈ B. This means that for any ∈ B there exist a ν 0 ∈ such that ν 0 ∈ G( ).Then, by definition, we have ∈ Co T (ν 0 ) and then ∈ h T (ν 0 ).It is clear that ∈ V h T for any ∈ B. Thus we have B ⊆ V h T .We have proved V h T = B.
It should be noted that C h T = T might not hold when the UIPSS T is not serial.To demonstrate this, let us consider the following example.
Example 3.9.Let = {τ 1 , τ 2 , τ 3 , τ 4 } and T = (G, B) be an UIPSS over .The parameter set of T is B = {0.1,0.5, 0.7, 0.9, 1} and the approximate function of T is given by The tabular representation of T is shown in Table 3.By Definition 3.5, the DHFS h T of T is an HFS with its HFEs given by: h T (τ 1 ) = {0.1,0.5, 0.7}, h T (τ 2 ) = {0.5, 0.7}, h T (τ 3 ) = {0.5, 0.9} and h T (τ 4 ) = {0.1,0.7, 0.9}.According to Definition 3.6, we can obtain the CSS C h T = ( Ĉh T , V h T ) of the DHFS h T , which is shown in Table 4.As we know, the parameter set V h T of C h T is the value space of the DHFS h T .It is clear that V h T = {0.1,0.5, 0.7, 0.9} {0.1, 0.5, 0.7, 0.9, 1} = B.This shows that C h T T if T is not a serial UIPSS.Table 3.The UIPSS T = (G, B) over .0.1 0.5 0.7 0.9 1 0.1 0.5 0.7 0.9 Theorem 3.10.Suppose that h is an HFS on and Proof.Let h : → P(I) be an HFS on and C h = ( Ĉh , V h ) be the CSS of h.By Definition 3.6, C h is an UIPSS and Ĉh (a) = h −1 (a) for all a ∈ V h .Let h C h be the DHFS of C h and be given by Assume that a ∈ h(ν 0 ) for any ν 0 ∈ .By Definition 3.2, it is clear that ν 0 ∈ h −1 (a).Then we have ν 0 ∈ Ĉh (a).From the definition of DHFS, it is easy to see a ∈ h C h (ν 0 ).Thus we have h(ν 0 ) ⊆ h C h (ν 0 ) for any ν 0 ∈ .Conversely, assume that a ∈ h C h (ν 0 ).By definition of DHFS, we have ν 0 ∈ Ĉh (a) which also show that ν 0 ∈ h −1 (a).From the definition of value cosets, it is easy to see that a ∈ h(ν 0 ).Thus we have h C h (ν 0 ) ⊆ h(ν 0 ) for any ν 0 ∈ .We have proved that h C h (ν 0 ) = h(ν 0 ) for any ν 0 ∈ .
The following result presents some characterizations of RHFSs in terms of their CSSs.Theorem 3.11.Assume that h is an HFS on with its CSS C h = ( Ĉh , V h ).Then the following are equivalent: (1) h is an RHFS; (2) C h is a full soft set; (3) C h is a covering soft set.
where Υ is a subset of .Furthermore, the positive, boundary and the negative regions of Υ can be defined from h as follows: Example 4.1.Suppose that a transport company intends to deliver a batch of relief supplies to the disaster area.There are four available transport modes = {τ 1 , τ 2 , τ 3 , τ 4 }.The meaning of the objects in is given as below: • τ 1 : highway transportation; • τ 2 : air transportation; • τ 3 : railway transportation; • τ 4 : sea transportation.
The transportation company invites several experts to evaluate the four methods of transportation and take into account factors such as time, cost and weather.Based on the evaluation results, an HFS h 2 is constructed.Assume that the h 2 is given by h is the value space of h 2 .By Definition 3.6, we can establish the CSS C h 2 = (h −1 2 , V 2 ) of HFS h 2 .The obtained results are shown in Table 5.
and the soft upper approximation of Υ is

Moreover, we can obtain the positive region Pos
These results indicate that the air transportation is the best choice for this task while the sea transportation should not be considered.
The next result simplifies the computation of the soft upper approximations of singletons.Notice that their soft lower approximations are quite trivial: Proposition 4.2.Let h be an HFS on and C h = ( Ĉh , V h ) be the CSS of h .For every τ i ∈ , the soft upper approximation of {τ i } can be calculated by Proof.According to Eq (4.2), we immediately have By Definition 3.2 and Definition 3.6, we can deduce that

It follows that {τ
, which completes the proof.

A 3WD method with CSSs of HFSs
Suppose there is a director who is in charge of a group of specialists.All the specialists are requested to evaluate each alternative in a universe of discourse = {τ 1 , τ 2 , • • • , τ n }.For the sake of simplicity, we assume that the evaluation scores given by the experts are chosen from the unit interval [0,1], and a higher score means a more positive evaluation on the concerned alternative.Based on these evaluation results, an HFS h can be constructed on , which stores the decision information collected from the group of specialists.After checking the evaluation from the expert group, the director select several alternatives in to form a pre-decision set R. In this 3WD process, the main purpose is to derive a trisection of as a general consensus among all the specialists together with the director.In the following, we present a novel approach for solving 3WD problems by virtue of HFSs and their CSSs.
Firstly, we can transform the HFS h into its CSS C h = ( Ĉh , V h ).Using the CSS C h , we construct an SAS P = ( , C h ).Based on the SAS P, we can calculate the soft upper approximations 3).It should be noted that {τ i } h can be interpreted as the set of alternatives in which have at least one evaluation score in common with the alternative τ i .
Secondly, we should select an appropriate evaluation function E : −→ [0, 1] which plays a core role in 3WD process.As pointed out by Hu [22], a cogent evaluation function should satisfy the monotonicity axiom.In other words, an alternative with higher value of evaluation function is considered to be more positive than an alternative with lower value.Actually, this can also be seen from the trisection as given in Definition 2.17.Furthermore, it is more reasonable and flexible for us to consider the following two cases when addressing decision-making problems in real-world scenarios: • The director would like to select favorable alternatives in to form a positive pre-decision set R; • The director would like to select rejective alternatives in to form a negative pre-decision set R.
In light of the above-mentioned issues, we define the evaluation function as follows: where τ i ∈ (i = 1, 2, • • • , n), "\" denotes the set difference operation and R is the pre-decision set.
Finally, given two thresholds γ, δ with 0 ≤ γ < δ ≤ 1, we can obtain the trisection of all alternatives in as follows: The alternatives in high value region H [δ,1] (E) should be accepted.The alternatives in medium value region M (γ,δ) (E) need further consideration.The alternatives in low value region L [0,γ] (E) should be rejected.The complete process of the proposed method is summarized as Algorithm 1 in Table 6.

An illustrative example
To confirm the reliability of the proposed 3WD method with CSSs of HFSs, we consider an example regarding peer review of research papers given as follows.
In the peer-review process of academic papers, each manuscript is reviewed by a group of referees who are professional in the field of their expertise.The reviewers are usually invited by an Associate Editor of an academic journal.Based on the comments from all the reviewers and the Associate Editor, the Editor-in-Chief can make a final decision to accept, revise or reject the manuscript.
In the following, we demonstrate how to help the Editor-in-Chief to achieve a general consensus among all the reviewers and the Associate Editor so as to make final decisions on seven papers by using Algorithm 1.
Step 1.Based on Definition 3.6, we first transform the 3-THFS h into its CSS C h = ( Ĉh , V h ) as shown in Table 7.It is easy to see that By Definition 3.2, we can compute the value cosets of all evaluation scores in V h .The obtained results are as follows: τ 4 , τ 5 , τ 6 }, h −1 (0.8) = {τ 7 } and h −1 (0.9) = {τ 6 , τ 7 }.It should be noted that the approximation function of the CSS C h is specified by these value cosets.
Step 3. Now, let us consider the evaluation function E : −→ [0, 1] in this particular case.It is clear that the pre-decision set R = {τ 1 , τ 3 , τ 4 } is negative since it contains the papers that the Associate Editor intends to reject.According to Eq (4.4), we calculate the evaluation values E(τ i ) of each paper In this context, the evaluation value E(τ i ) can be interpreted as the proportion of papers in {τ i } h which are not rejected by the Associate Editor.For instance, the evaluation value E(τ 1 ) = 0.4 shows that 40% of the papers which have at least one evaluation score in common with τ 1 are not rejected by the Associate Editor.Based on the evaluation values, we can rank τ i and τ j as follows: • τ i τ j if and only if E(τ i ) > E(τ j ); • τ i ≺ τ j if and only if E(τ i ) < E(τ j ); • τ i ≈ τ j if and only if E(τ i ) = E(τ j ).
Step 4. Next, we need to select a pair of thresholds γ, δ from the unit interval [0,1] with γ < δ.These thresholds are used to attain the trisection of the universe of discourse.In this problem, we choose γ = 0.4 and δ = 0.7 as the low and high thresholds, respectively.
Step 6.It can be seen that the above results are obtained by taking into account both the evaluation scores given by three referees and the pre-decision provided by the Associate Editor.According to the obtained trisection, the Editor-in-Chief is advised to make the final decision regarding papers in as follows: (1) The papers τ 6 , τ 7 in high value region H [0.7,1] (E) should be accepted; (2) The papers τ 2 , τ 4 , τ 5 in medium value region M (0.4,0.7) (E) should be revised; (3) The papers τ 1 , τ 3 in low value region L [0,0.4] (E) should be rejected.

Conclusions
We defined the unit interval parameterized SS, CSS and derived HFS in this study.With the proposed notions, mutual representations between HFSs and SSs have been investigated.By transforming an HFS into its CSS, we developed a soft rough model based on HFSs.By virtue of this soft rough model, a novel 3WD method has been proposed for solving decision-making problems with hesitant fuzzy information.A numerical example was presented to illustrate the validity and feasibility of our 3WD method.It has shown that the proposed approach achieved a general consensus between the expert group and the director.In the future, we will further explore other potential applications of CSSs and examine more properties of HFSs from the perspective of SSs.In addition, it would be interesting to discover inner connections between probabilistic HFSs [57] and probabilistic SSs [11] by using similar techniques.These links might prompt new 3WD techniques in the context of group decision-making with partial memberships, hesitation, and probabilistic information.
This shows that T is a disjoint covering soft set over .Conversely, assume that T is a disjoint covering soft set over .According to Definition 2.8, T is both full and serial.Thus we have b∈B G(b) = and G(b) ∅ for all b ∈ B. Since T is disjoint, we can deduce that T is an injective soft set such that G(a) G(b) implies G(a) ∩ G(b) = ∅ for all a, b ∈ B. Thus {G(a) | a ∈ B} forms a partition of .This shows that T is an injective partition soft set over .