Attribute reduction algorithms for hesitant fuzzy decision systems based on hypergraphs

Abstract

Rough set theory is a powerful tool for handling uncertainty and vagueness in various fields. The hesitant fuzzy rough set, as a generalization of rough sets, can solve more complex problems. However, existing hesitant fuzzy rough sets do not satisfy the inclusive property. To address this issue, a novel hesitant fuzzy rough set model based on dual score functions is proposed. Four generalized hesitant fuzzy rough sets and their discernibility matrices are also presented. Additionally, the lower approximation distribution reductions can be obtained by the discernibility matrix. Meanwhile, hypergraphs provide an accurate description of relationships between multiple objects and offer a concise operational approach. Then it is discovered that finding the lower approximation distribution reductions of a hesitant fuzzy decision system is equivalent to finding the minimal transversals of its hypergraph. Moreover, an improved algorithm for hesitant fuzzy decision systems based on hypergraphs is presented to accelerate the reduction process. Finally, the proposed algorithm is applied to the hybrid data of Hepatitis C Virus from UCI to demonstrate its feasibility.

Keywords

Attribute reduction hesitant fuzzy rough set hypergraph hesitant fuzzy decision system approximation operator

1 Introduction

The contemporary research on information systems and knowledge discovery has proposed further promotion of classical set theory, e.g., fuzzy set theory (short for FST) and rough set theory (short for RST). The notion of FST is introduced by Zadeh [1] as a mathematical way to provide a wide variety of methods for analyzing imprecise data. Several extensions from FST have been developed, e.g., intuitionistic fuzzy sets [2], interval-valued fuzzy sets [3], type-2 fuzzy sets [4], type-n fuzzy sets [4], Pythagorean fuzzy sets [5] and hesitant fuzzy sets [6]. While the RST is initially proposed by Pawlak [7] as an important way to research information systems characterized by incomplete and insufficient information. This theory uses an indiscernibility relation to classify objects into lower and upper approximations. Both FST and RST can solve the problem of information granulation: the FST is centred on the fuzzy information granulation [8 –10], while the RST is focused on the crisp information granulation. In real life scenarios, FST and RST have many restrictions. Therefore, it evokes a natural question concerning possible connections between FST and RST. Rough fuzzy sets [11], fuzzy rough sets [12], intuitionistic fuzzy rough sets [13, 14], multi-fuzzy rough sets [15, 16] and hesitant fuzzy rough sets [17, 18] are proposed successively. Specially, the hesitant fuzzy rough sets is presented by Yang et al. [17]. However, there is a problem that the order of two hesitant fuzzy sets representing the inclusion relation on the hesitant fuzzy power set is not necessarily antisymmetric. Then, a novel framework to research hesitant fuzzy rough sets is proposed by Zhang et al. [19] which overcomes the problem of the previous one. But the new one is also imperfect, since there is no inclusion relation between lower and upper approximation.

In pattern recognition, the attribute reduction [20 –22] has been a popular area of research for nearly two decades. Within the framework of RST, the attribute reduction is a method to remove irrelevant and redundant attributes to reduce the data dimension without affecting the classification capability of the information system. Until now, the discernibility matrix-based attribute reduction approach is still a common channel to obtain all reductions. The first reduction algorithm based on a discernibility matrix with equivalence relations is proposed by Skowron and Rauszer [23]. An ordered attribute reduction algorithm based on a discernibility matrix is presented by Wang et al. [24]. However, a problem about the reduction algorithm based on a discernibility matrix is presented in [23, 25], i.e., the procedures for transforming discernibility functions from their conjunctive normal forms (short for CNFs) to the disjunctive normal forms (short for DNFs) are excessively time consuming. To address this problem, an accelerated algorithm that converts the discernibility function from its CNFs to its DNFs is proposed by Borowik and Luba [26], and an attribute reduction accelerator based on a fuzzy discernibility matrix is proposed to accelerate the procedures in [27].

Hypergraphs, which can accurately describe relationships between multiple objects, have been proposed as an alternative to traditional graphs. Recently, it was discovered that both attribute reductions based on discernibility matrix and vertex cover problems of hypergraphs can be achieved through Boolean logical operations. This led to the establishment of a connection between minimal vertex covers of hypergraphs and attribute reductions in fuzzy rough sets. Building upon this finding, Chen et al. [28] proposed a novel graph-based attribute reduction algorithm that can speed up feature selection in the fuzzy rough set framework by finding all minimal transversals of a hypergraph.

Based on the above introductions, the research about attribute reduction and its accelerated algorithms for a hesitant fuzzy decision information system (short for HFDS) is relatively lacking. To overcome the inadequate problems of the hesitant fuzzy rough sets proposed by Zhang et al. [19], a novel hesitant fuzzy rough set constructed by a pair of dual score functions is proposed. Further, a reduction approach based on the proposed hesitant fuzzy rough set is constructed. Meanwhile, the process for transforming discernibility functions is simplified when using hypergraphs to find reductions. Therefore, an attribute reduction algorithm for HFDS based on hypergraphs is presented by drawing the idea of Chen et al. [28] to accelerate the reduction process.

In this paper, a decision system with binary relations is converted into a HFDS using a hesitant fuzzy relation (short for HFR). By using a pair of dual score functions of hesitant fuzzy sets, a novel hesitant fuzzy rough approximation operators (short for HF-RAOps) based on dual score functions is proposed. Since there are different extension methods, four HF-RAOps are constructed, i.e., score function based optimism-optimism HF-RAOps, score function based optimism-pessimism HF-RAOps, score function based pessimism-optimism HF-RAOps and score function based pessimism-pessimism HF-RAOps. For an inconsistent HFDS, lower approximation distribution consistent sets and lower approximation distribution reductions are obtained by using the four newly arrived HF-RAOps. A fuzzy granule of hesitant fuzzy sets is also proposed in order to construct an attribute reduction approach based on the score function based hesitant fuzzy rough sets for obtaining the lower approximation distribution reductions. Then different discernibility matrices and all the reductions of an inconsistent HFDS are obtained. Next it is proved that searching the lower approximation distribution reductions of a HFDS is equivalent to searching the minimal transversals of its hypergraph. On the basis of the algorithm proposed by Chen et al. [28], an improved one for this approach to find all the lower approximation distribution reductions is presented. Finally, using Hepatitis C Virus (HCV) from UCI, which is hybrid data for data analysis, the feasibility of the proposed algorithm is shown.

The main theoretical contributions are summarized as follows:

A novel hesitant fuzzy rough approximation operator based on dual score functions is proposed which overcomes the problem of the previous one, i.e., there is no inclusion relation between lower and upper approximations;

Since decision makers have different attitudes, four hesitant fuzzy rough approximation operators are constructed to be applied in different situations;

The inducing method for transforming a hesitant fuzzy decision information system into a hypergraph is given. Subsequently, the relevant theoretical properties of the induced hypergraph are discussed;

It is proved that searching the lower approximation distribution reductions of a hesitant fuzzy decision system is equivalent to searching the minimal transversals of its hypergraph, and then an improved algorithm to find all the lower approximation distribution reductions is proposed.

2 Preliminaries

2.1 Hesitant fuzzy sets and hesitant fuzzy relations

In this subsection, some related notions of hesitant fuzzy sets and HFRs are briefly reviewed.

Definition 1. [6] Let U be an arbitrary nonempty set. A hesitant fuzzy set α on U is represented by a function h_α (u) that returns a subset of [0, 1] when applied to U, which is expressed as the following notation $α = {< u, h_{α} (u) > | u \in U},$ (1) where h_α (u) is a finite set of some different values in [0, 1], representing the possible membership degrees of the element u ∈ U in α. Furthermore, h = h_α (u) is known as a hesitant fuzzy element (short for HFE). The set of all hesitant fuzzy sets on U, named the hesitant fuzzy power set of U, is recorded by HF (U).

Obviously, a hesitant fuzzy set α can degenerate into a fuzzy set if and only if there is only one element in h_α (u). Note that the number of values in different HFEs may be different. Assume that L (h_α (u)) denotes the number of values in h_α (u). To facilitate the operation, the following assumptions are formulated by Xu and Xia [29, 30]:

All the elements in h_α (u) are listed in ascending order, then $h_{α}^{σ (ξ)} (u)$ is known as the ξ-th largest value in h_α (u).

For any two HFEs h_α (u) and h_β (u), if L (h_α (u)) ≠ L (h_β (u)), then L = max{ L (h_α (u)) , L (h_β (u)) }. If the elements in h_α (u) are less than the elements in h_β (u), an extension of h_α (x) can be deemed optimistically by repeating its maximum element until it has the same number with h_β (u). We generalize it further in this paper, assuming that there may be 0 in HFEs, then an extension of h_α (u) can also be deemed pessimistically by repeating its minimum element until it has the same number with h_β (u).

For any two HFEs h_α (u) and h_β (u), h_α (u) ⪯ h_β (u) if and only if $h_{α}^{σ (ξ)} (u) \leq h_{β}^{σ (ξ)} (u), ξ = 1, 2, . . ., L$ , L = max{ L (h_α (u)) , L (h_β (u)) }.

Example 1. If U ={ u₁, u₂, u₃ } is the universe of discourse, h_α (u₁) ={ 0.3, 0.5 }, h_α (u₂) ={ 0.3, 0.4, 0.5 }, h_α (u₃) ={ 0, 0.9 } are the HFEs of u_i (i = 1, 2, 3) to the set α, respectively. Then α can be deemed as a hesitant fuzzy set, that is,

$\begin{matrix} α = {< u_{1}, {0.3, 0.5} >, < u_{2}, {0.3, 0.4, 0.5} >, \\ < u_{3}, {0, 0.9} >} . \end{matrix}$

To compare h_α (u₁) and h_α (u₂), let L = max { 2, 3 } = 3. There are fewer elements in h_α (u₁) than in h_α (u₂), it can be optimistically repeated its maximum element, i.e. h_α (u₁) ={ 0.3, 0.5, 0.5 }, then h_α (u₂) ⪯ h_α (u₁). Similarly, it can also be pessimistically repeated its minimum element, i.e. h_α (u₁) ={ 0.3, 0.3, 0.5 }, then h_α (u₁) ⪯ h_α (u₂).

Definition 2. ([31]) Let U be a finite nonempty universe. Suppose that α and β are two hesitant fuzzy sets, then

the complement of α is recorded by α^c, namely that ∀u ∈ U, $\begin{matrix} h_{α^{c}} (u) = \sim h_{α} (u) \\ = {1 - h_{α}^{σ (ξ)} (u) | ξ = 1, 2, . . ., L}; \end{matrix}$

the union of α and β is recorded by α ∪ β, namely that ∀u ∈ U, $\begin{matrix} h_{α \cup β} (u) = h_{α} (u) ⊻ h_{β} (u) \\ = {h_{α}^{σ (ξ)} (u) \lor h_{β}^{σ (ξ)} (u) | ξ = 1, 2, . . ., L}; \end{matrix}$

the intersection of α and β is recorded by α ∩ β, namely that ∀u ∈ U, $\begin{matrix} h_{α \cap β} (u) = h_{α} (u) ⊽ h_{β} (u) \\ = {h_{α}^{σ (ξ)} (u) \land h_{β}^{σ (ξ)} (u) | ξ = 1, 2, . . ., L}, \end{matrix}$

where

h_{α}^{σ (ξ)} (u)

and

h_{β}^{σ (ξ)} (u)

are the ξ-th largest value in h_α (u) and h_β (u), L = max{ L (h_α (u)) , L (h_β (u)) }.

Example 2. (Continue to Example 1) Let $β = {< u_{1}, {0.3, 0.4, 0.9} >, < u_{2}, {0.2, 0.6, 0.7} >, < u_{3}, {0.1, 0.9} >},$ then we have $α^{c} = {< u_{1}, {0.5, 0.7} >, < u_{2}, {0.5, 0.6, 0.7} >, < u_{3}, {0.1, 1} >};$ $β^{c} = {< u_{1}, {0.1, 0.6, 0.7} >, < u_{2}, {0.3, 0.4, 0.8} >, < u_{3}, {0.1, 0.9} >} .$

To calculate α ∪ β and α ∩ β, let α and β be optimistically repeated its maximum element, $α \cup β = {< u_{1}, {0.3, 0.5, 0.9} >, < u_{2}, {0.3, 0.6, 0.7} >, < u_{3}, {0.1, 0.9, 0.9} >};$ $α \cap β = {< u_{1}, {0.3, 0.4, 0.5} >, < u_{2}, {0.2, 0.4, 0.5} >, < u_{3}, {0, 0.9, 0.9} >} .$

Similarly, α and β can also be pessimistically repeated its minimum element, $α \cup β = {< u_{1}, {0.3, 0.4, 0.9} >, < u_{2}, {0.3, 0.6, 0.7} >, < u_{3}, {0.1, 0.1, 0.9} >};$ $α \cap β = {< u_{1}, {0.3, 0.3, 0.5} >, < u_{2}, {0.2, 0.4, 0.5} >, < u_{3}, {0, 0, 0.9} >} .$

Definition 3. ([17]) Let U be a finite nonempty universe. A HFR hr on U is a hesitant fuzzy set such that hr ∈ HF (U × U), namely, hr is given by $hr = {< (u_{1}, u_{2}), h_{hr} (u_{1}, u_{2}) > | (u_{1}, u_{2}) \in U \times U},$ (2) where h_hr (u₁, u₂) is a finite set of some different values in [0, 1], representing the possible membership degrees of the relationship between u₁ and u₂.

Definition 4. ([17]) Let U be a finite nonempty universe, hr be a HFR on U. Then

hr is regarded as serial if and only if ∀u₁ ∈ U, there exists u₂ ∈ U such that h_hr (u₁, u₂) ={ 1 };

hr is regarded as reflexive if and only if ∀u ∈ U, h_hr (u, u) ={ 1 };

hr is regarded as symmetric if and only if ∀u₁, u₂ ∈ U, h_hr (u₁, u₂) = h_hr (u₂, u₁);

hr is regarded as transitive if and only if ∀u₁, u₂, u₃ ∈ U, h_hr (u₁, u₂) ⊽h_hr (u₂, u₃) ⪯ h_hr (u₁, u₃).

2.2 The score function of hesitant fuzzy sets

In this subsection, for any HFE $h_{α} (u) = {h_{α}^{σ (1)} (u), h_{α}^{σ (2)} (u), . . ., h_{α}^{σ (n)} (u)},$ the score function of h_α (u) will be discussed, denoted by S : [0, 1] ⁿ → [0, 1], the following properties hold [32]:

Monotone non-decreasing property: for another HFE $h_{β} (u) = {h_{β}^{σ (1)} (u), h_{β}^{σ (2)} (u), . . ., h_{β}^{σ (n)} (u)},$ if $h_{α}^{σ (1)} (u) \leq h_{β}^{σ (1)} (u), h_{α}^{σ (2)} (u) \leq h_{β}^{σ (2)} (u), . . ., h_{α}^{σ (n)} (u) \leq h_{β}^{σ (n)} (u)$ , then S (h_α (u)) ≤ S (h_β (u)), where $h_{α}^{σ (ξ)} (u)$ and $h_{β}^{σ (ξ)} (u)$ are the ξ-th largest value in h_α (u) and h_β (u).

Boundary conditions property:

$S ({0, 0, . . ., 0}) = 0, S ({1, 1, . . ., 1}) = 1,$ h_α (u) ={ 0 } and h_α (u) ={ 1 } are known as the empty HFE and the full HFE, respectively.

Now, two score functions of HFEs satisfying the above properties will be introduced.

Definition 5. Let U be a finite nonempty universe, for any HFE $h_{α} (u) = {h_{α}^{σ (1)} (u), h_{α}^{σ (2)} (u), . . ., h_{α}^{σ (n)} (u)},$ the following function is obtained: $s (h_{α} (u)) = 1 - \prod_{ξ = 1}^{L} (1 - h_{α}^{σ (ξ)} (u))^{\frac{1}{L}},$ (3) where L stands for the number of values in h_α (u), $h_{α}^{σ (ξ)} (u)$ is the ξ-th largest value in h_α (u), then s (h_α (u)) is a score function of HFEs, denoted by s.

To facilitate subsequent discussions, for any HFE h_α (u), let s^d (h_α (u)) =1 - s (∼ h_α (u)). Since s is monotonic, it is clear that s^d (h_α (u)) is also monotonic with respect to h_α (u); s^d ({ 0, 0, . . . , 0 }) =1 - s ({ 1, 1, . . . , 1 }) =1 - 1 =0, s^d ({ 1, 1, . . . , 1 }) =1 - s ({ 0, 0, . . . , 0 }) =1 - 0 =1, so s^d (h_α (u)) is bounded. Therefore, s^d (h_α (u)) is also a score function of h_α (u), then s^d (h_α (u)) is called a dual score function of s (h_α (u)), denoted by s^d.

Theorem 1. Suppose that U is a finite nonempty universe. For any HFE $h_{α} (u) = {h_{α}^{σ (1)} (u), h_{α}^{σ (2)} (u), . . ., h_{α}^{σ (n)} (u)}$ , s and s^d satisfy:

If ${h_{α}^{δ (1)} (u), h_{α}^{δ (2)} (u), . . ., h_{α}^{δ (n)} (u)}$ is a permutation of h_α (u), then $s ({h_{α}^{δ (1)} (u), h_{α}^{δ (2)} (u), . . ., h_{α}^{δ (n)} (u)}) = s (h_{α} (u)),$ $s^{d} ({h_{α}^{δ (1)} (u), h_{α}^{δ (2)} (u), . . ., h_{α}^{δ (n)} (u)}) = s^{d} (h_{α} (u)) .$

If all the values of h_α (u) are the same, i.e., $p = h_{α}^{σ (ξ)} (u), ξ = 1, 2, . . ., n$ , then $s (h_{α} (u)) = s ({p, p, . . ., p}) = p,$ $s^{d} (h_{α} (u)) = s^{d} ({p, p, . . ., p}) = p .$

If (s^d) ^d (h_α (u)) =1 - s^d (∼ h_α (u)), then (s^d) ^d (h_α (u)) = s (h_α (u)) .

s^d (h_α (u)) ≤ s (h_α (u)) .

Proof. (1), (2) are obvious, so we only prove (3) and (4).

(3) (s^d) ^d (h_α (u)) =1 - s^d (∼ h_α (u)) =1 - [1 - s (h_α (u))] = s (h_α (u)) .

(4) In order to prove s^d (h_α (u)) ≤ s (h_α (u)), we firstly need to prove $\prod_{i = 1}^{n} h_{i}^{λ_{i}} \leq \sum_{i = 1}^{n} λ_{i} h_{i}$ , where $\sum_{i = 1}^{n} λ_{i} = 1$ . This inequality follows the strict convexity of the function f (u) = e^u, u ∈ [- ∞ , + ∞], i.e., $f ({tu}_{1} + (1 - t) u_{2}) \leq tf (u_{1}) + (1 - t) f (u_{2}), t \in [0, 1] .$

Then we have $\begin{matrix} \prod_{i = 1}^{n} h_{i}^{λ_{i}} = e^{ln \prod_{i = 1}^{n} h_{i}^{λ_{i}}} = e^{\sum_{i = 1}^{n} λ_{i} ln h_{i}} \leq \sum_{i = 1}^{n} λ_{i} e^{ln h_{i}} = \sum_{i = 1}^{n} λ_{i} h_{i}, \end{matrix}$ and subsequently,

$\begin{matrix} s^{d} (h_{α} (u)) & = 1 - s (\sim h_{α} (u)) = \prod_{ξ = 1}^{L} (h_{α}^{σ (ξ)} (u))^{\frac{1}{L}} \\ \leq \sum_{ξ = 1}^{L} \frac{1}{L} (h_{α}^{σ (ξ)} (u)) = 1 - \sum_{ξ = 1}^{L} \frac{1}{L} (1 - h_{α}^{σ (ξ)} (u)) \\ \leq 1 - \prod_{ξ = 1}^{L} (1 - h_{α}^{σ (ξ)} (u))^{\frac{1}{L}} = s (h_{α} (u)), \end{matrix}$ where L stands for the number of values in h_α (u), $h_{α}^{σ (ξ)} (u)$ is the ξ-th largest value in h_α (u).

In order to facilitate subsequent discussions, for any HFR hr and hesitant fuzzy set α, u, u₁, u₂ ∈ U, both h_hr (u₁, u₂) and h_α (u) are HFEs, let hr (u₁, u₂) and α (u) be briefed as h_hr (u₁, u₂) and h_α (u), respectively.

2.3 Hypergraphs

In this subsection, some basic notions related to hypergraphs are introduced.

Definition 6. ([33]) A hypergraph can be defined as $H = (N, A)$ , where N is a finite nonempty set of vertices and A is a family of subsets of N. The elements of A are named hyperedges.

In contrast to edges of the traditional graph, hyperedges are arbitrary sets of vertices, which can collect an arbitrary number of vertices. Therefore, it can be considered as an extension of traditional graphs.

Definition 7. ([33]) Let $H = (N, A)$ be a hypergraph. A transversal of $H$ is a subset Q ⊆ N that has a nonempty intersection with each hyperedge of $H$ . That is, Q is a set of vertices that covers all the hyperedges. Furthermore, Q is a minimal transversal if and only if any proper subset of Q is not a transversal. The set of all minimal transversals is recorded by $T (H)$ .

In order to facilitate the subsequent search for the minimal transversal, for any vertex w, the number of hyperedges containing w is the degree of w of $H$ , denoted by $d_{H} (w)$ .

Example 3. Suppose that $H = (N, A)$ is a hypergraph, where N ={ w₁, w₂, w₃, w₄ } is the set of vertices and A ={{ w₁w₂ } , { w₂w₃w₄ } , { w₄ }} is the set of hyperedges. Then $d_{H} (w_{1}) = 1$ , $d_{H} (w_{2}) = 2$ , $d_{H} (w_{3}) = 1$ , $d_{H} (w_{4}) = 2$ .

Suppose that $H = (N, A)$ is a hypergraph. A Boolean function $f_{H}$ for $H$ can be defined as follows, which is composed of n Boolean variables $w_{1}^{*}, w_{2}^{*}, . . ., w_{n}^{*}$ corresponding to the vertices w₁, w₂, . . . , w_n, respectively. $f_{H} (w_{1}^{*}, w_{2}^{*}, . . ., w_{n}^{*}) = \land {\lor e_{i} | e_{i} \in A},$ (4) where ∨e_i is the disjunction of all variables w^∗ such that w^∗ ∈ e_i.

Theorem 2. ([34]) Suppose that $H = (N, A)$ is a hypergraph. Q ⊆ N is a minimal transversal of $H$ if and only if $\land_{w_{i} \in Q} w_{i}^{*}$ is a prime implicant of the Boolean function $f_{H}$ .

The above theorem demonstrates that if $f_{H} (w_{1}^{*}, w_{2}^{*}, . . ., w_{n}^{*}) = \land {\lor e_{i} | e_{i} \in A} = \lor_{i = 1}^{t} (\land_{j = 1}^{l_{i}} w_{j}^{*}),$ where $\land_{j = 1}^{l_{i}} w_{j}^{*} (i \leq t)$ are all the prime implicants of the Boolean function $f_{H}$ , let $Q_{i} = {w_{j}^{*} | j \leq l_{i}}$ , then $T (H) = {Q_{1}, Q_{2}, . . ., Q_{t}}$ is the set of all minimal transversals of $H$ .

3 A novel attribute reduction approach for HFDSs

In this section, four score function based hesitant fuzzy rough sets are constructed by using the score functions proposed in Section 2.2. And then a discernibility matrix based on the HF-RAOps and all the reductions of a HFDS can be obtained.

3.1 A construction method of hesitant fuzzy rough sets

Due to the specificity of hesitant fuzzy sets, it is difficult for hesitant fuzzy sets to satisfy the traditional inclusion relation. Therefore, the inclusion relation of hesitant fuzzy sets is proposed as follows.

Definition 8. Let U be a finite nonempty universe, ∀α, β ∈ HF (U), β ⊆ α if and only if ∀u ∈ U, s^d (h_β (u)) ≤ s^d (h_α (u)).

Definition 9. Let U be a finite nonempty universe, hr be a reflexive HFR over U, then the pair (U, hr) is known as a reflexive hesitant fuzzy approximation space. ∀α ∈ HF (U), the lower and upper approximations of α with respect to (U, hr) are recorded by $\underline{hr} (α)$ and $\bar{hr} (α)$ , respectively, ∀u₁, u₂ ∈ U, $h_{\underline{hr} (α)} (u_{1}) = \land_{u_{2} \in U} [s^{d} (\sim hr (u_{1}, u_{2})) \lor s^{d} (α (u_{2}))],$ (5) $h_{\bar{hr} (α)} (u_{1}) = \lor_{u_{2} \in U} [s (hr (u_{1}, u_{2})) \land s (α (u_{2}))] .$ (6)

The pair $(\underline{hr} (α), \bar{hr} (α))$ is called a score function based hesitant fuzzy rough set of α with respect to (U, hr). $\underline{hr} (α), \bar{hr} (α) : HF (U) ⟶ HF (U)$ are known as the lower and upper score function based HF-RAOps, respectively.

Obviously, the upper and lower score function based HF-RAOps are single-valued for any element u ∈ U. For the convenience of the subsequent expression, let $h_{\underline{hr} (α)} (u)$ and $h_{\bar{hr} (α)} (u)$ be abbreviated as $\underline{hr} (α) (u)$ and $\bar{hr} (α) (u)$ .

Example 4. Suppose that (U, hr) is a reflexive hesitant fuzzy approximation space. U ={ u₁, u₂, u₃ }, hr is a reflexive HFR on U defined by the following matrix: $hr = (\begin{matrix} {1} & {0.2, 0.6} & {0.4, 0.6, 0.7} \\ {0.2, 0.6} & {1} & {0.1, 0.7} \\ {0.4, 0.6, 0.7} & {0.1, 0.7} & {1} \end{matrix}) .$

Obviously, the HFR hr is reflexive and symmetric. Given a hesitant fuzzy set $α = {< u_{1}, {0.3, 0.4, 0.6} >, < u_{2}, {0.4, 0.8} >, < u_{3}, {0.4, 0.7} >},$ by Definition 9, we have $\begin{matrix} \underline{hr} (α) (u_{1}) & = \land_{u \in U} [s^{d} (\sim hr (u_{1}, u)) \lor s^{d} (α (u))] \\ = [s^{d} (\sim hr (u_{1}, u_{1})) \lor s^{d} (α (u_{1}))] \land [s^{d} (\sim hr (u_{1}, u_{2})) \lor s^{d} (α (u_{2}))] \\ \land [s^{d} (\sim hr (u_{1}, u_{3})) \lor s^{d} (α (u_{3}))] \\ = 0.416 \land 0.635 \land 0.5809 \\ = 0.416 . \end{matrix}$

Similarly, we have $\begin{matrix} \underline{hr} (α) (u_{2}) & = 0.504, \underline{hr} (α) (u_{3}) = 0.416, \\ \bar{hr} (α) (u_{1}) & = 0.584, \bar{hr} (α) (u_{2}) = 0.7116, \bar{hr} (α) (u_{3}) = 0.622 . \end{matrix}$

Theorem 3. Suppose that (U, hr) is a reflexive hesitant fuzzy approximation space. For ∀α, β ∈ HF (U), the lower and upper score function based HF-RAOps induced from (U, hr), the following properties hold:

$\underline{hr} (\emptyset) = \bar{hr} (\emptyset) = \emptyset;$

$\underline{hr} (U) = \bar{hr} (U) = U;$

$\underline{hr} (α) \subseteq α \subseteq \bar{hr} (α);$

$\underline{hr} (α^{c}) = (\bar{hr} (α))^{c}, \bar{hr} (α^{c}) = (\underline{hr} (α))^{c};$

$α \subseteq β \Rightarrow \underline{hr} (α) \subseteq \underline{hr} (β), \bar{hr} (α) \subseteq \bar{hr} (β) .$

Proof.

For ∀u ∈ U, h_∅ (u) ={ 0 }. Then $\begin{matrix} \underline{hr} (\emptyset) (u) & = \land_{u^{'} \in U} [s^{d} (\sim hr (u, u^{'})) \lor s^{d} (\emptyset (u^{'}))] = \land_{u^{'} \in U} [s^{d} (\sim hr (u, u^{'})) \lor s^{d} ({0})] \\ = \land_{u^{'} \in U} [s^{d} (\sim hr (u, u^{'}))] = [s^{d} (\sim hr (u, u))] \land [\land_{u^{'} \neq u} s^{d} (\sim hr (u, u^{'}))] \\ = 0 \land [\land_{u^{'} \neq u} s^{d} (\sim hr (u, u^{'}))] = 0 . \end{matrix}$ By the above discussion, $\underline{hr} (\emptyset) = \emptyset$ is obtained.

Similarly, $\bar{hr} (\emptyset) = \emptyset$ is obtained.

For ∀u ∈ U, h_U (u) ={ 1 }. Then $\begin{matrix} \underline{hr} (U) (u) & = \land_{u^{'} \in U} [s^{d} (\sim hr (u, u^{'})) \lor s^{d} (U (u^{'}))] \\ = \land_{u^{'} \in U} [s^{d} (\sim hr (u, u^{'})) \lor s^{d} ({1})] \\ = \land_{u^{'} \in U} [s^{d} ({1})] = 1 . \end{matrix}$ By the above discussion, $\underline{hr} (U) = U$ is obtained.

Similarly, $\bar{hr} (U) = U$ is obtained.

For ∀u ∈ U, $\begin{matrix} \underline{hr} (α) (u) & = \land_{u^{'} \in U} [s^{d} (\sim hr (u, u^{'})) \lor s^{d} (α (u^{'}))] \\ = [s^{d} (\sim hr (u, u)) \lor s^{d} (α (u))] \land [\land_{u^{'} \neq u} [s^{d} (\sim hr (u, u^{'})) \lor s^{d} (α (u^{'}))]] \\ = s^{d} (α (u)) \land [\land_{u^{'} \neq u} [s^{d} (\sim hr (u, u^{'})) \lor s^{d} (α (u^{'}))]] \\ \leq s^{d} (α (u)), \end{matrix}$ then $\underline{hr} (α) \subseteq α$ is obtained.

Similarly, for ∀u ∈ U, $\begin{matrix} \bar{hr} (α) (u) & = \lor_{u^{'} \in U} [s (hr (u, u^{'})) \land s (α (u^{'}))] \\ = [s (hr (u, u)) \land s (α (u))] \lor [\lor_{u^{'} \neq u} [s (hr (u, u^{'})) \land s (α (u^{'}))]] \\ = s (α (u)) \lor [\lor_{u^{'} \neq u} [s (hr (u, u^{'})) \land s (α (u^{'}))]] \\ \geq s (α (u)) \geq s^{d} (α (u)), \end{matrix}$ then $α \subseteq \bar{hr} (α)$ is obtained.

For ∀u ∈ U, $\begin{matrix} \underline{hr} (α^{c}) (u) & = \land_{u^{'} \in U} [s^{d} (\sim hr (u, u^{'})) \lor s^{d} (\sim α (u^{'}))] \\ = \land_{u^{'} \in U} [(1 - s (hr (u, u^{'}))) \lor (1 - s (α (u^{'})))] \\ = \land_{u^{'} \in U} [1 - (s (hr (u, u^{'})) \land s (α (u^{'})))] \\ = 1 - \lor_{u^{'} \in U} [s (hr (u, u^{'})) \land s (α (u^{'}))] \\ = 1 - \bar{hr} (α) (u), \end{matrix}$ then $\underline{hr} (α^{c}) = (\bar{hr} (α))^{c}$ is obtained.

Similarly, it is not hard to prove that $\bar{hr} (α^{c}) = (\underline{hr} (α))^{c} .$

Since α ⊆ β, by Definition 8 we have s^d (α (u)) ≤ s^d (β (u)) for each u ∈ U, then it can be concluded that s^d (∼ hr (u, u′)) ∨ s^d (α (u)) ≤ s^d (∼ hr (u, u′)) ∨ s^d (β (u)) for each u′ ∈ U, i.e., $\underline{hr} (α) (u) \leq \underline{hr} (β) (u)$ . Finally, $\underline{hr} (α) \subseteq \underline{hr} (β)$ is obtained.

Similarly, it is not hard to prove that $\bar{hr} (α) \subseteq \bar{hr} (β)$ .

3.2 Lower approximation distribution reductions and its judgment theorems

In this subsection, a notion of lower approximation distribution reductions in an inconsistent HFDS and its judgment theorems are proposed.

Definition 10. Let (U, C ∪ D) be a decision system. $HR = {{hr}_{1}, {hr}_{2}, . . ., {hr}_{m}}$ is a family of HFRs from the condition attribute set C, R_D is a HFR from the decision attribute set D. Then $S = (U, HR \cup R_{D})$ is called a hesitant fuzzy decision system (short for HFDS).

By Definition 10, a decision system can be translated into a HFDS. Denote $Sim (R) = ⊓ {{hr}_{i} | {hr}_{i} \in R},$ (7) where $Sim (HR) (u, u^{'}) = {hr (u, u^{'}), hr \in HR | s (hr (u, u^{'})) = min_{{hr}_{i} \in HR} s ({hr}_{i} (u, u^{'}))}$ , ∀u, u′ ∈ U. It means that $Sim (HR) (u, u^{'})$ is the HFR with minimum value under the score function s. Obviously, $Sim (HR)$ is also a HFR.

Combined with Section 2.1, for any two HFEs h_α (u) and h_β (u), if L (h_α (u)) ≠ L (h_β (u)) and h_α (u) has the fewer elements, an extension of h_α (u) can be deemed optimistically by repeating its maximum element or pessimistically by repeating its minimum element. ∀u, u′ ∈ U, let $Sim (HR)^{l} (u, u^{'})$ and $R_{D}^{l} (u, u^{'})$ be extensions of $Sim (HR) (u, u^{'})$ and R_D (u, u′) using the optimistic extensions, $Sim (HR)^{b} (u, u^{'})$ and $R_{D}^{b} (u, u^{'})$ be extensions of $Sim (HR) (u, u^{'})$ and R_D (u, u′) using the pessimistic extensions. By Definition 9, ∀u, u′, u″ ∈ U, the following four situations can be obtained:

An optimistic attitude can be used to deal with the HFR under the condition attribute set and decision attribute set. Next, score function based optimism-optimism HF-RAOps can be constructed, i.e., ${\underline{Sim (HR)}}^{l} ([u^{″}]_{R_{D}}^{l}) (u) = \land_{u^{'} \in U} [s^{d} (\sim Sim (HR)^{l} (u, u^{'})) \lor s^{d} (R_{D}^{l} (u^{″}, u^{'}))],$ ${\bar{Sim (HR)}}^{l} ([u^{″}]_{R_{D}}^{l}) (u) = \lor_{u^{'} \in U} [s (Sim (HR)^{l} (u, u^{'})) \land s (R_{D}^{l} (u^{″}, u^{'}))],$ where $Sim (HR)^{l} (u, u^{'})$ and $R_{D}^{l} (u^{″}, u^{'})$ are obtained by using the optimistic extensions.

An optimistic attitude can be used to deal with the HFR under the condition attribute set. Meanwhile, a pessimistic attitude can be used to deal with the HFR under the decision attribute set. Then, score function based optimism-pessimism HF-RAOps can be constructed, i.e., ${\underline{Sim (HR)}}^{l} ([u^{″}]_{R_{D}}^{b}) (u) = \land_{u^{'} \in U} [s^{d} (\sim Sim (HR)^{l} (u, u^{'})) \lor s^{d} (R_{D}^{b} (u^{″}, u^{'}))],$ ${\bar{Sim (HR)}}^{l} ([u^{″}]_{R_{D}}^{b}) (u) = \lor_{u^{'} \in U} [s (Sim (HR)^{l} (u, u^{'})) \land s (R_{D}^{b} (u^{″}, u^{'}))],$ where $Sim (HR)^{l} (u, u^{'})$ is obtained by using the optimistic extension, $R_{D}^{b} (u^{″}, u^{'})$ is obtained by using the pessimistic extension.

A pessimistic attitude can be used to deal with the HFR under the condition attribute set. Meanwhile, an optimistic attitude can be used to deal with the HFR under the decision attribute set. Then, score function based pessimism-optimism HF-RAOps can be constructed, i.e., ${\underline{Sim (HR)}}^{b} ([u^{″}]_{R_{D}}^{l}) (u) = \land_{u^{'} \in U} [s^{d} (\sim Sim (HR)^{b} (u, u^{'})) \lor s^{d} (R_{D}^{l} (u^{″}, u^{'}))],$ ${\bar{Sim (HR)}}^{b} ([u^{″}]_{R_{D}}^{l}) (u) = \lor_{u^{'} \in U} [s (Sim (HR)^{b} (u, u^{'})) \land s (R_{D}^{l} (u^{″}, u^{'}))],$ where $Sim (HR)^{b} (u, u^{'})$ is obtained by using the pessimistic extension, $R_{D}^{l} (u^{″}, u^{'})$ is obtained by using the optimistic extension.

A pessimistic attitude can be used to deal with the HFR under the condition attribute set and decision attribute set. Next, score function based pessimism-pessimism HF-RAOps can be constructed, i.e., ${\underline{Sim (HR)}}^{b} ([u^{″}]_{R_{D}}^{b}) (u) = \land_{u^{'} \in U} [s^{d} (\sim Sim (HR)^{b} (u, u^{'})) \lor s^{d} (R_{D}^{b} (u^{″}, u^{'}))],$ ${\bar{Sim (HR)}}^{b} ([u^{″}]_{R_{D}}^{b}) (u) = \lor_{u^{'} \in U} [s (Sim (HR)^{b} (u, u^{'})) \land s (R_{D}^{b} (u^{″}, u^{'}))],$ where $Sim (HR)^{b} (u, u^{'})$ and $R_{D}^{b} (u^{″}, u^{'})$ are obtained by using the pessimistic extensions.

Example 5. Suppose that $S = (U, HR \cup R_{D})$ is a HFDS, where $HR = {{hr}_{1}, {hr}_{2}, {hr}_{3}, {hr}_{4}}$ is a family of HFRs on U defined by the following matrices: ${hr}_{1} = (\begin{matrix} {1} & {0, 0.6} & {0.5, 0.6, 0.7} \\ {0, 0.6} & {1} & {0.1, 0.9} \\ {0.5, 0.6, 0.7} & {0.1, 0.9} & {1} \end{matrix}),$ ${hr}_{2} = (\begin{matrix} {1} & {0.5, 0.7} & {0, 0.5} \\ {0.5, 0.7} & {1} & {0.3, 0.7, 0.9} \\ {0, 0.5} & {0.3, 0.7, 0.9} & {1} \end{matrix}),$ ${hr}_{3} = (\begin{matrix} {1} & {0.2, 0.7} & {0, 0.4, 0.6} \\ {0.2, 0.7} & {1} & {0.1, 0.3, 0.8} \\ {0, 0.4, 0.6} & {0.1, 0.3, 0.8} & {1} \end{matrix}),$ ${hr}_{4} = (\begin{matrix} {1} & {0.4, 0.5, 0.8} & {0, 0.6} \\ {0.4, 0.5, 0.8} & {1} & {0.2, 0.8} \\ {0, 0.6} & {0.2, 0.8} & {1} \end{matrix}) .$

The HFRs can be processed by using the optimistic extensions as follows: ${hr}_{1}^{l} = (\begin{matrix} {1, 1, 1} & {0, 0.6, 0.6} & {0.5, 0.6, 0.7} \\ {0, 0.6, 0.6} & {1, 1, 1} & {0.1, 0.9, 0.9} \\ {0.5, 0.6, 0.7} & {0.1, 0.9, 0.9} & {1, 1, 1} \end{matrix}),$ ${hr}_{2}^{l} = (\begin{matrix} {1, 1, 1} & {0.5, 0.7, 0.7} & {0, 0.5, 0.5} \\ {0.5, 0.7, 0.7} & {1, 1, 1} & {0.3, 0.7, 0.9} \\ {0, 0.5, 0.5} & {0.3, 0.7, 0.9} & {1, 1, 1} \end{matrix}),$ ${hr}_{3}^{l} = (\begin{matrix} {1, 1, 1} & {0.2, 0.7, 0.7} & {0, 0.4, 0.6} \\ {0.2, 0.7, 0.7} & {1, 1, 1} & {0.1, 0.3, 0.8} \\ {0, 0.4, 0.6} & {0.1, 0.3, 0.8} & {1, 1, 1} \end{matrix}),$ ${hr}_{4}^{l} = (\begin{matrix} {1, 1, 1} & {0.4, 0.5, 0.8} & {0, 0.6, 0.6} \\ {0.4, 0.5, 0.8} & {1, 1, 1} & {0.2, 0.8, 0.8} \\ {0, 0.6, 0.6} & {0.2, 0.8, 0.8} & {1, 1, 1} \end{matrix}) .$

The HFRs can also be processed by using the pessimistic extensions as follows: ${hr}_{1}^{b} = (\begin{matrix} {1, 1, 1} & {0, 0, 0.6} & {0.5, 0.6, 0.7} \\ {0, 0, 0.6} & {1, 1, 1} & {0.1, 0.1, 0.9} \\ {0.5, 0.6, 0.7} & {0.1, 0.1, 0.9} & {1, 1, 1} \end{matrix}),$ ${hr}_{2}^{b} = (\begin{matrix} {1, 1, 1} & {0.5, 0.5, 0.7} & {0, 0, 0.5} \\ {0.5, 0.5, 0.7} & {1, 1, 1} & {0.3, 0.7, 0.9} \\ {0, 0, 0.5} & {0.3, 0.7, 0.9} & {1, 1, 1} \end{matrix}),$ ${hr}_{3}^{b} = (\begin{matrix} {1, 1, 1} & {0.2, 0.2, 0.7} & {0, 0.4, 0.6} \\ {0.2, 0.2, 0.7} & {1, 1, 1} & {0.1, 0.3, 0.8} \\ {0, 0.4, 0.6} & {0.1, 0.3, 0.8} & {1, 1, 1} \end{matrix}),$ ${hr}_{4}^{b} = (\begin{matrix} {1, 1, 1} & {0.4, 0.5, 0.8} & {0, 0, 0.6} \\ {0.4, 0.5, 0.8} & {1, 1, 1} & {0.2, 0.2, 0.8} \\ {0, 0, 0.6} & {0.2, 0.2, 0.8} & {1, 1, 1} \end{matrix}) .$

Then, we have $Sim (HR)^{l} = (\begin{matrix} {1, 1, 1} & {0, 0.6, 0.6} & {0, 0.5, 0.5} \\ {0, 0.6, 0.6} & {1, 1, 1} & {0.1, 0.3, 0.8} \\ {0, 0.5, 0.5} & {0.1, 0.3, 0.8} & {1, 1, 1} \end{matrix}),$ $Sim (HR)^{b} = (\begin{matrix} {1, 1, 1} & {0, 0, 0.6} & {0, 0, 0.5} \\ {0, 0, 0.6} & {1, 1, 1} & {0.2, 0.2, 0.8} \\ {0, 0, 0.5} & {0.2, 0.2, 0.8} & {1, 1, 1} \end{matrix}) .$

Reductions and discernibility matrices based on the above four situations will be obtained as follows.

Definition 11. Let $S = (U, HR \cup R_{D})$ be a HFDS. $Sim (HR)$ is a HFR from the condition attribute set, R_D is a HFR from the decision attribute set. $Sim (HR)^{•}$ and $R_{D}^{⊙}$ are obtained using the optimistic extensions or pessimistic extensions (• , ⊙ ∈ { l, b }). If $Sim (HR)^{•} (u, u^{'}) \subseteq R_{D}^{⊙} (u, u^{'})$ , then S is known as a • -⊙ consistent HFDS. If ∃u, u′ ∈ U, $Sim (HR)^{•} (u, u^{'}) ⊈ R_{D}^{⊙} (u, u^{'})$ , then S is known as a • -⊙ inconsistent HFDS.

The attribute reductions of a • -⊙ inconsistent HFDS are discussed as follows.

Theorem 4. Suppose that $S = (U, HR \cup R_{D})$ is a • -⊙ inconsistent HFDS (• , ⊙ ∈ { l, b }), where $HR = {{hr}_{1}, {hr}_{2}, . . ., {hr}_{m}}$ is a family of HFRs. For any hesitant fuzzy set α, $\forall HP \subseteq HR$ , then $\underline{Sim (HP)^{•}} (α^{⊙}) \subseteq \underline{Sim (HR)^{•}} (α^{⊙})$ .

Proof. Since $HP \subseteq HR$ , it is got that $s (Sim (HP)^{•} (u, u^{'})) \geq s (Sim (HR)^{•} (u, u^{'}))$ , ∀u, u′ ∈ U. According to the definition of s^d, $1 - s^{d} (\sim Sim (HP)^{•} (u, u^{'})) \geq 1 - s^{d} (\sim Sim (HR)^{•} (u, u^{'}))$ . Further, $s^{d} (\sim Sim (HP)^{•} (u, u^{'})) \leq s^{d} (\sim Sim (HR)^{•} (u, u^{'}))$ can be obtained, from which it is concluded that $s^{d} (\sim Sim (HP)^{•} (u, u^{'})) \lor s^{d} (α^{⊙} (u^{'})) \leq s^{d} (\sim Sim (HR)^{•} (u, u^{'})) \lor s^{d} (α^{⊙} (u^{'})) .$ (8)

It follows that $h_{\underline{Sim (HP)^{•}} (α^{⊙})} (u) \leq h_{\underline{Sim (HR)^{•}} (α^{⊙})} (u)$ . Thus, $\underline{Sim (HP)^{•}} (α^{⊙}) \subseteq \underline{Sim (HR)^{•}} (α^{⊙})$ is obtained.

Next, the fuzzy granule of hesitant fuzzy sets is proposed by drawing on the idea of Chen [28] in order to construct an attribute reduction method based on the score function based hesitant fuzzy rough sets.

Definition 12. Let $S = (U, HR \cup R_{D})$ be a • -⊙ inconsistent HFDS (• , ⊙ ∈ { l, b }). For any HFR hr, given a real number λ ∈ (0, 1], u ∈ U, a hesitant fuzzy granule for ∀u′ ∈ U is defined as $[u_{λ}]_{hr} (u^{'}) = {\begin{matrix} 0, s^{d} (\sim hr (u, u^{'})) \geq λ, \\ λ, s^{d} (\sim hr (u, u^{'})) < λ . \end{matrix}$

Theorem 5. Suppose that $S = (U, HR \cup R_{D})$ is a • -⊙ inconsistent HFDS (• , ⊙ ∈ { l, b }), where $HR = {{hr}_{1}, {hr}_{2}, . . ., {hr}_{m}}$ is a family of HFRs. For any hesitant fuzzy set α, $\forall HP \subseteq HR$ , then

$[u_{λ}]_{Sim (HR)^{•}} \subseteq [u_{λ}]_{Sim (HP)^{•}}$ , ∀u ∈ U, λ ∈ (0, 1];

$\underline{Sim (HP)^{•}} (α^{⊙}) (u) = \sup {λ | [u_{λ}]_{Sim (HP)^{•}} \subseteq α^{⊙}, λ \in (0, 1]}$ , ∀u ∈ U.

Proof.

By Theorem 4, it can be easily proved.

Given an object u ∈ U, let $\underline{Sim (HP)^{•}} (α^{⊙}) (u) = λ^{*}$ .

If λ^∗ = 0, $[u_{λ^{*}}]_{Sim (HP)^{•}} \subseteq α^{⊙}$ is obvious.

If λ^∗ ≠ 0, it is also got that $[u_{λ^{*}}]_{Sim (HP)^{•}} \subseteq α^{⊙}$ . In fact, if $[u_{λ^{*}}]_{Sim (HP)^{•}} ⊈ α^{⊙}$ , there exists u₀ ∈ U such that $s^{d} (α^{⊙} (u_{0})) < [u_{λ^{*}}]_{Sim (HP)^{•}} (u_{0}) = λ^{*}$ , which yields $\begin{matrix} λ^{*} & = \underline{Sim (HP)^{•}} (α^{⊙}) (u) \\ = \land_{y \in U} [s^{d} (\sim Sim ({HP}^{•}) (u, u^{'})) \lor s^{d} (α^{⊙} (u^{'}))] \\ \leq s^{d} (\sim Sim (HP)^{•} (u, u_{0})) \lor s^{d} (α^{⊙} (u_{0})) \\ = s^{d} (\sim Sim (HP)^{•} (u, u_{0})) . \end{matrix}$ By Definition 12, $[u_{λ^{*}}]_{Sim (HP)^{•}} (u_{0}) = 0$ is obtained, i.e., λ^∗ = 0. It is in contradiction with $\underline{Sim (HP)^{•}} (α^{⊙}) (u) = λ^{*} \neq 0$ . Therefore, $[u_{λ^{*}}]_{Sim (HP)^{•}} \subseteq α^{⊙}$ .

Next, it is proved that λ^∗ is the maximum value that satisfies $[u_{λ^{*}}]_{Sim (HP)^{•}} \subseteq α^{⊙}$ .

If there exists λ′ > λ^∗ such that $[u_{λ^{'}}]_{Sim (HP)^{•}} \subseteq α^{⊙}$ , then $[u_{λ^{'}}]_{Sim (HP)^{•}} (u^{'}) \leq s^{d} (α^{⊙} (u^{'}))$ for each u′ ∈ U, which yields $\begin{matrix} λ^{*} & = \underline{Sim (HP)^{•}} (α^{⊙}) (u) \\ = \land_{u^{'} \in U} [s^{d} (\sim Sim (HP)^{•} (u, u^{'})) \lor s^{d} (α^{⊙} (u^{'}))] \\ \geq \land_{u^{'} \in U} [s^{d} (\sim Sim (HP)^{•} (u, u^{'})) \lor [u_{λ^{'}}]_{Sim (HP)^{•}} (u^{'})] \\ = λ^{'}, \end{matrix}$ a contradiction with $\underline{Sim (HP)^{•}} (α^{⊙}) (u) = λ^{*}$ .

From the above conclusions, it is obtained that $\underline{Sim (HP)^{•}} (α^{⊙}) (u) = \sup {λ | [u_{λ}]_{Sim (HP)^{•}} \subseteq α^{⊙}, λ \in (0, 1]}$ for ∀u ∈ U.

Definition 13. Let $S = (U, HR \cup R_{D})$ be a • -⊙ inconsistent HFDS (• , ⊙ ∈ { l, b }), where $HR = {{hr}_{1}, {hr}_{2}, . . ., {hr}_{m}}$ is a family of HFRs. For $\forall HP \subseteq HR, u \in U$ , $μ_{Sim (HP)^{•}} (u) = (\underline{Sim (HP)^{•}} ([u_{1}]_{R_{D}^{⊙}}) (u), \underline{Sim (HP)^{•}} ([u_{2}]_{R_{D}^{⊙}}) (u), . . ., \underline{Sim (HP)^{•}} ([u_{n}]_{R_{D}^{⊙}}) (u)),$ $μ_{Sim (HP)^{•}} (u)$ is known as a lower approximation distribution function of u with respect to $HP$ . If $μ_{Sim (HP)^{•}} (u) = μ_{Sim (HR)^{•}} (u)$ for ∀u ∈ U, then $HP$ is known as a lower approximation distribution consistent set; if $HP$ is a lower approximation distribution consistent set, and any proper subset of $HP$ is not a lower approximation distribution consistent set, then $HP$ is known as a lower approximation distribution reduction of S.

A lower approximation distribution reduction $HP$ is a minimal subset of $HR$ to keep the lower approximation distribution function of every object with respect to $HP$ the same as that with respect to $HR$ . The set of all lower approximation distribution reductions of S is denoted by red (S).

Subsequently, some judgment theorems of lower approximation distribution reductions are presented in detail.

Theorem 6. Suppose that $S = (U, HR \cup R_{D})$ is a • -⊙ inconsistent HFDS (• , ⊙ ∈ { l, b }), where $HR = {{hr}_{1}, {hr}_{2}, . . ., {hr}_{m}}$ is a family of HFRs. Then $HP \subseteq HR$ is a lower approximation distribution consistent set of S if and only if $HP$ satisfies $[u_{λ}^{'}]_{Sim (HP)^{•}} \subseteq [u]_{R_{D}^{⊙}}$ for $\forall u, u^{'} \in U, λ = \underline{Sim (HR)^{•}} ([u]_{R_{D}^{⊙}}) (u^{'})$ .

Proof. ⇒ If $HP$ is a lower approximation distribution consistent set of S, then $μ_{Sim (HP)^{•}} (u) = μ_{Sim (HR)^{•}} (u)$ for ∀u ∈ U. By Theorem 5, it is obtained that $\begin{matrix} λ & = \underline{Sim (HR)^{•}} ([u]_{R_{D}^{⊙}}) (u^{'}) \\ = \underline{Sim (HP)^{•}} ([u]_{R_{D}^{⊙}}) (u^{'}) \\ = \sup {λ | [u_{λ}^{'}]_{Sim (HP)^{•}} \subseteq [u]_{R_{D}^{⊙}}, λ \in (0, 1]} . \end{matrix}$

This implies that $[u_{λ}^{'}]_{Sim (HP)^{•}} \subseteq [u]_{R_{D}^{⊙}}$ .

⟸ If $HP$ satisfies $[u_{λ}^{'}]_{Sim (HP)^{•}} \subseteq [u]_{R_{D}^{⊙}}$ for $\forall u, u^{'} \in U, λ = \underline{Sim (HR)^{•}} ([u]_{R_{D}^{⊙}}) (u^{'})$ . Let $λ^{*} = \underline{Sim (HP)^{•}} ([u]_{R_{D}^{⊙}}) (u^{'})$ . Clearly, $\underline{Sim (HR)^{•}} ([u]_{R_{D}^{⊙}}) (u^{'}) \geq \underline{Sim (HP)^{•}} ([u]_{R_{D}^{⊙}}) (u^{'})$ , which is λ^∗ ≤ λ. By Theorem 5, it is obtained that $\begin{matrix} λ^{*} & = \underline{Sim (HP)^{•}} ([u]_{R_{D}^{⊙}}) (u^{'}) \\ = \sup {λ | [u_{λ}^{'}]_{Sim (HP)^{•}} \subseteq [u]_{R_{D}^{⊙}}, λ \in (0, 1]} \\ \geq λ . \end{matrix}$

Therefore, λ^∗ = λ, i.e., $\underline{Sim (HR)^{•}} ([u]_{R_{D}^{⊙}}) (u^{'}) = \underline{Sim (HP)^{•}} ([u]_{R_{D}^{⊙}}) (u^{'})$ for ∀u, u′ ∈ U. This implies that $HP$ is a lower approximation distribution consistent set.

By the previous discussions, a method will be obtained to calculate the lower approximation distribution reductions.

Definition 14. Let $S = (U, HR \cup R_{D})$ be a • -⊙ inconsistent HFDS (• , ⊙ ∈ { l, b }), where U ={ u₁, u₂, . . . , u_n }, $HR = {{hr}_{1}, {hr}_{2}, . . ., {hr}_{m}}$ is a family of HFRs inducing from the condition attribute set and R_D is a HFR inducing from the decision attribute set. The discernibility matrix of $S = (U, HR \cup R_{D})$ with respect to $HR$ is denoted as $M_{• ⊙} = (m_{ij}^{• ⊙})_{n \times n}$ , and $m_{ij}^{• ⊙}$ is calculated by: $\begin{matrix} m_{ij}^{• ⊙} = {\begin{matrix} {hr \in HR | [{u_{j}}_{λ_{ij}}]_{{hr}^{•}} \subseteq [u_{i}]_{R_{D}^{⊙}}}, & u_{i} \neq u_{j}, \\ \emptyset, & otherwise, \end{matrix} \end{matrix}$ (9) where $λ_{ij} = \underline{Sim (HR)^{•}} ([u_{i}]_{R_{D}^{⊙}}) (u_{j})$ .

Based on the different extension methods, we have four discernibility matrices:

The discernibility matrix $M_{ll} = (m_{ij}^{ll})_{n \times n}$ based on the score function based optimism-optimism HF-RAOps;

The discernibility matrix $M_{lb} = (m_{ij}^{lb})_{n \times n}$ based on the score function based optimism-pessimism HF-RAOps;

The discernibility matrix $M_{bl} = (m_{ij}^{bl})_{n \times n}$ based on the score function based pessimism-optimism HF-RAOps;

The discernibility matrix $M_{bb} = (m_{ij}^{bb})_{n \times n}$ based on the score function based pessimism-pessimism HF-RAOps.

According to Theorem 6, the equivalence condition about lower approximation distribution consistent sets is further promoted as follows.

Theorem 7. Suppose that $S = (U, HR \cup R_{D})$ is a • -⊙ inconsistent HFDS (• , ⊙ ∈ { l, b }). M . (M . ∈ { M_ll, M_lb, M_bl, M_bb }) is the discernibility matrix of S, $m_{ij}^{• ⊙} \in M .$ , $HP \subseteq HR$ is a lower approximation distribution consistent set of S if and only if $HP \cap m_{ij}^{• ⊙} \neq \emptyset$ for every $m_{ij}^{• ⊙} \neq \emptyset$ .

Proof. The proof is obvious according to Theorem 6 and the definition of $m_{ij}^{• ⊙}$ .

Example 6. (Continue to Example 5) Let R_D be a HFR on U defined by the following matrix: $R_{D} = (\begin{matrix} {1} & {0.5, 0.8} & {0.5, 0.6, 0.9} \\ {0.5, 0.8} & {1} & {0.4, 0.7} \\ {0.5, 0.6, 0.9} & {0.4, 0.7} & {1} \end{matrix}) .$

Similarly, the HFR R_D can be processed using the optimistic extension or pessimistic extension as follows: $\begin{matrix} [u_{1}]_{R_{D}^{l}} & = ({1, 1, 1}, {0.5, 0.8, 0.8}, {0.5, 0.6, 0.9}); \\ [u_{2}]_{R_{D}^{l}} & = ({0.5, 0.8, 0.8}, {1, 1, 1}, {0.4, 0.7, 0.7}); \\ [u_{3}]_{R_{D}^{l}} & = ({0.5, 0.6, 0.9}, {0.4, 0.7, 0.7}, {1, 1, 1}); \\ [u_{1}]_{R_{D}^{b}} & = ({1, 1, 1}, {0.5, 0.5, 0.8}, {0.5, 0.6, 0.9}); \\ [u_{2}]_{R_{D}^{b}} & = ({0.5, 0.5, 0.8}, {1, 1, 1}, {0.4, 0.4, 0.7}); \\ [u_{3}]_{R_{D}^{b}} & = ({0.5, 0.6, 0.9}, {0.4, 0.4, 0.7}, {1, 1, 1}) . \end{matrix}$

By Definition 9, four lower score function based HF-RAOps of R_D with respect to $Sim (HR)$ are obtained as follows: $\begin{matrix} \underline{Sim (HR)^{l}} ([u_{1}]_{R_{D}^{l}}) & = {0.6463, 0.6463, 0.6463}, \\ \underline{Sim (HR)^{l}} ([u_{2}]_{R_{D}^{l}}) & = {0.6300, 0.5809, 0.5809}, \\ \underline{Sim (HR)^{l}} ([u_{3}]_{R_{D}^{l}}) & = {0.5809, 0.5809, 0.5809}, \\ \underline{Sim (HR)^{l}} ([u_{1}]_{R_{D}^{b}}) & = {0.5848, 0.5848, 0.5848}, \\ \underline{Sim (HR)^{l}} ([u_{2}]_{R_{D}^{b}}) & = {0.5848, 0.5013, 0.4820}, \\ \underline{Sim (HR)^{l}} ([u_{3}]_{R_{D}^{b}}) & = {0.5429, 0.4820, 0.5013}, \\ \underline{Sim (HR)^{b}} ([u_{1}]_{R_{D}^{l}}) & = {0.7368, 0.6463, 0.6463}, \\ \underline{Sim (HR)^{b}} ([u_{2}]_{R_{D}^{l}}) & = {0.6840, 0.5809, 0.5809}, \\ \underline{Sim (HR)^{b}} ([u_{3}]_{R_{D}^{l}}) & = {0.6463, 0.5809, 0.5809}, \\ \underline{Sim (HR)^{b}} ([u_{1}]_{R_{D}^{b}}) & = {0.7368, 0.5848, 0.5848}, \\ \underline{Sim (HR)^{b}} ([u_{2}]_{R_{D}^{b}}) & = {0.5848, 0.5040, 0.4820}, \\ \underline{Sim (HR)^{b}} ([u_{3}]_{R_{D}^{b}}) & = {0.6463, 0.4820, 0.5040} . \end{matrix}$

Then the hesitant fuzzy granule $[u_{λ}]_{Sim (HR)^{•}}$ can be calculated. Moreover, four discernibility matrices are obtained as follows:

$M_{ll} = (\begin{matrix} \emptyset & {{hr}_{4}} & \emptyset \\ {{hr}_{2}, {hr}_{4}} & \emptyset & \emptyset \\ {{hr}_{1}, {hr}_{2}, {hr}_{3}} & \emptyset & \emptyset \end{matrix}), M_{bl} = (\begin{matrix} \emptyset & {{hr}_{4}} & \emptyset \\ \emptyset & \emptyset & \emptyset \\ \emptyset & \emptyset & \emptyset \end{matrix}),$ $M_{lb} = (\begin{matrix} \emptyset & \emptyset & {{hr}_{4}} \\ \emptyset & \emptyset & \emptyset \\ {{hr}_{1}, {hr}_{2}, {hr}_{3}} & \emptyset & \emptyset \end{matrix}), M_{bb} = (\begin{matrix} \emptyset & \emptyset & {{hr}_{4}} \\ \emptyset & \emptyset & \emptyset \\ \emptyset & \emptyset & \emptyset \end{matrix}) .$

Given a • -⊙ inconsistent HFDS $S = (U, HR \cup R_{D})$ , $HR = {{hr}_{1}, {hr}_{2}, . . ., {hr}_{m}}$ is a family of HFRs. A discernibility function f_S for S can be defined as follows, which is composed of m Boolean variables ${hr}_{1}^{*}, {hr}_{2}^{*}, . . ., {hr}_{m}^{*}$ corresponding to the HFRs hr₁, hr₂, . . . , hr_m, respectively. $f_{S} ({hr}_{1}^{*}, {hr}_{2}^{*}, . . ., {hr}_{m}^{*}) = \land {\lor m_{ij}^{• ⊙} | m_{ij}^{• ⊙} \in M ., m_{ij}^{• ⊙} \neq \emptyset},$ (10) where M .∈ { M_ll, M_lb, M_bl, M_bb }, $\lor m_{ij}^{• ⊙}$ is the disjunction of all variables hr^∗ such that ${hr}^{*} \in m_{ij}^{• ⊙}$ .

Theorem 8. Suppose that $S = (U, HR \cup R_{D})$ is a • -⊙ inconsistent HFDS (• , ⊙ ∈ { l, b }). $HP \subseteq HR$ is a lower approximation distribution reduction of S if and only if $\land_{{hr}_{i} \in HP} {hr}_{i}^{*}$ is a prime implicant of the discernibility function f_S.

Proof. ⇒ If $HP \subseteq HR$ is a lower approximation distribution reduction of S, then the intersection of $HP$ and the non-empty elements in M . (M .∈ { M_ll, M_lb, M_bl, M_bb }) are non-empty, and any proper subset of $HP$ is not a lower approximation distribution consistent set. $\forall hr \in HP$ , $\exists m_{ij}^{• ⊙} \in M .$ , then $(HP - {hr}) \cap m_{ij}^{• ⊙} = \emptyset$ . It shows that $HP$ is the smallest set which intersects with the non-empty elements in M . are non-empty. Thus, $\land_{{hr}_{i} \in HP} {hr}_{i}^{*}$ is a prime implicant of the discernibility function f_S.

⟸ If $\land_{{hr}_{i} \in HP} {hr}_{i}^{*}$ is a prime implicant of the discernibility function f_S, then $\forall m_{ij}^{• ⊙} \in M ., HP \cap m_{ij}^{• ⊙} \neq \emptyset$ . It shows that $HP$ is a lower approximation distribution consistent set of S. $\forall hr \in HP$ , $\exists m_{ij}^{• ⊙} \in M .$ , then $(HP - {hr}) \cap m_{ij}^{• ⊙} = \emptyset$ . Thus, $(HP - {hr})$ is not a lower approximation distribution consistent set. To summarize, $HP \subseteq HR$ is a lower approximation distribution reduction of S.

Theorem 8 demonstrates that if $\begin{matrix} f_{S} ({hr}_{1}^{*}, {hr}_{2}^{*}, . . ., {hr}_{m}^{*}) = \\ \land {\lor m_{ij}^{• ⊙} | m_{ij}^{• ⊙} \in M ., m_{ij}^{• ⊙} \neq \emptyset} = \lor_{i = 1}^{t} {\land_{j = 1}^{l_{i}} {hr}_{j}^{*}}, \end{matrix}$ where M .∈ { M_ll, M_lb, M_bl, M_bb }, $\land_{j = 1}^{l_{i}} {hr}_{j}^{*} (i \leq t)$ are all the prime implicants of the discernibility function f_S, let ${HP}_{i} = {{hr}_{j}^{*} | j \leq l_{i}}$ , then $red (S) = {{HP}_{1}, {HP}_{2}, . . ., {HP}_{t}}$ is the set of all lower approximation distribution reductions of S.

Example 7. (Continue to Example 6) By Theorem 8 and the discernibility matrix M . ∈ { M_ll, M_lb, M_bl, M_bb }, the lower approximation distribution reductions can be calculated as follows,

M^ll:

$\begin{matrix} f_{S} ({hr}_{1}^{*}, {hr}_{2}^{*}, {hr}_{3}^{*}, {hr}_{4}^{*}) & = ({hr}_{1} \lor {hr}_{2} \lor {hr}_{3}) \land ({hr}_{2} \lor {hr}_{4}) \land ({hr}_{4}) \\ = ({hr}_{1} \land {hr}_{4}) \lor ({hr}_{2} \land {hr}_{4}) \lor ({hr}_{3} \land {hr}_{4}); \end{matrix}$ M^lb: $\begin{matrix} f_{S} ({hr}_{1}^{*}, {hr}_{2}^{*}, {hr}_{3}^{*}, {hr}_{4}^{*}) & = ({hr}_{1} \lor {hr}_{2} \lor {hr}_{3}) \land ({hr}_{4}) \\ = ({hr}_{1} \land {hr}_{4}) \lor ({hr}_{2} \land {hr}_{4}) \lor ({hr}_{3} \land {hr}_{4}); \end{matrix}$ M^bl and M^bb: $\begin{matrix} f_{S} ({hr}_{1}^{*}, {hr}_{2}^{*}, {hr}_{3}^{*}, {hr}_{4}^{*}) & = {hr}_{4} . \end{matrix}$

Thus, the reductions of S is red (S) ={{ hr₁, hr₄ } , { hr₂, hr₄ } , { hr₃, hr₄ }} when the discernibility matrix is based on the score function based optimism-optimism or optimism-pessimism HF-RAOps; the reductions of S is red (S) ={{ hr₄ }} when the discernibility matrix is based on the score function based pessimism-optimism or pessimism-pessimism HF-RAOps.

4 An attribute reduction algorithm based on hypergraphs

In this section, a hypergraph induced from a HFDS can be constructed. Then the relationship between the lower approximation distribution reductions of a HFDS and the minimal transversala of its hypergraph will be discussed. Finally, a novel attribute reduction algorithm based on hypergraphs is obtained.

Definition 15. Let $S = (U, HR \cup R_{D})$ be a • -⊙ inconsistent HFDS (• , ⊙ ∈ { l, b }). M . (M . ∈ { M_ll, M_lb, M_bl, M_bb }) is the discernibility matrix of S and $Δ = {m_{ij}^{• ⊙} \in M . | m_{ij}^{• ⊙} \neq \emptyset}$ . Then a hypergraph $H = (HR, Δ)$ can be induced from S.

It is noteworthy that the hyperedges of $H = (HR, Δ)$ are nonempty terms of the discernibility matrix M . (M . ∈ { M_ll, M_lb, M_bl, M_bb }). If some nonempty terms of the discernibility matrix are identical, then it means that the induced hypergraph has parallel hyperedges.

Combined with Theorem 2 and Theorem 8, the following theorem hold.

Theorem 9. Suppose that $H = (HR, Δ)$ is an induced hypergraph from a • -⊙ inconsistent HFDS $S = (U, HR \cup R_{D})$ , then $T (H) = red (S)$ .

Proof. To prove this, it just need to prove that the discernibility function f_S of S is equal to the Boolean function $f_{H}$ of $H$ . In fact, by Definition 15, it shows that the hyperedges of a hypergraph are nonempty term of the discernibility matrix. Therefore, by the definition of f_S and $f_{H}$ , the equation clearly holds.

The above theorem shows that finding the lower approximation distribution reductions of a HFDS is equivalent to finding the minimal transversals of its hypergraph.

Definition 16. Let $S = (U, HR \cup R_{D})$ be a • -⊙ inconsistent HFDS (• , ⊙ ∈ { l, b }), where U ={ u₁, u₂, . . . , u_n }, $HR = {{hr}_{1}, {hr}_{2}, . . ., {hr}_{m}}$ . $\forall hr \in HR$ , a crisp relation $(B)_{hr} = (b_{ij}^{• ⊙})_{n \times n}$ with respect to hr can be defined as follows, $b_{ij}^{• ⊙} = {\begin{matrix} 1, [{u_{j}}_{λ_{ij}}]_{{hr}^{•}} \subseteq [u_{i}]_{R_{D}^{⊙}} and u_{i} \neq u_{j}, \\ 0, otherwise, \end{matrix}$ (11) where $λ_{ij} = \underline{Sim (HR)^{•}} ([u_{i}]_{R_{D}^{⊙}}) (u_{j})$ .

Theorem 10. Suppose that $H = (HR, Δ)$ is an induced hypergraph from a • -⊙ inconsistent HFDS $S = (U, HR \cup R_{D})$ . $\forall hr \in HR$ , the degree of the vertex hr is $d_{H} (hr) = \sum_{b_{ij}^{• ⊙} \in (B)_{hr}} b_{ij}^{• ⊙}$ .

Proof. The proof is straightforward by Definitions 15, 16 and the definition of $m_{ij}^{• ⊙}$ .

Next, an example will be used to illustrate the above definition and theorem.

Example 8. (Continue to Example 7) Considering the discernibility matrix M_ll based on the optimism-optimism score function based HF-RAOps. According to Definition 15, the hypergraph $H = (HR, Δ)$ induced from S is shown as follows,

$\begin{matrix} HR = {{hr}_{1}, {hr}_{2}, {hr}_{3}, {hr}_{4}}, \\ Δ = {{{hr}_{4}}, {{hr}_{1}, {hr}_{2}, {hr}_{3}}, {{hr}_{2}, {hr}_{4}}} (Fig . 1) . \end{matrix}$

By Definition 16 and Theorem 10, it is easy to get $\begin{matrix} (B)_{{hr}_{1}} = (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}), (B)_{{hr}_{2}} = (\begin{matrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}), \\ (B)_{{hr}_{3}} = (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}), (B)_{{hr}_{4}} = (\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}), \end{matrix}$ then we can calculate $d_{H} ({hr}_{1}) = 1$ , $d_{H} ({hr}_{2}) = 2$ , $d_{H} ({hr}_{3}) = 1$ , $d_{H} ({hr}_{4}) = 2$ . From Fig. 1, it is known that the degrees of the vertices are consistent with the calculation.

Fig. 1

The hypergraph of $H = (HR, Δ)$ .

Suppose that $H = (HR, Δ)$ is an induced hypergraph from a • -⊙ inconsistent HFDS $S = (U, HR \cup R_{D})$ . $\forall {hr}^{*} \in HR$ , if all the edges incident to hr^∗ of $H$ are deleted, then an edge-deleted sub-graph is obtained which can be defined as $H_{hr} = (HR, Δ - {m_{ij}^{• ⊙} \in M . | {hr}^{*} \in m_{ij}^{• ⊙}})$ , M .∈ { M_ll, M_lb, M_bl, M_bb }. In order to calculate the degree of the vertices of $H_{{hr}^{*}}$ , a τ-cuts matrix is introduced. Let C = (c_ij) _n×n be a n × n matrix, τ ∈ [- ∞ , + ∞], then a τ-cuts matrix of C is denoted by $C_{τ} = (c_{ij}^{τ})_{n \times n}$ , where $c_{ij}^{τ} = {\begin{matrix} 1, c_{ij} \geq τ, \\ 0, c_{ij} < τ . \end{matrix}$

Theorem 11. Suppose that $H = (HR, Δ)$ is an induced hypergraph from a • -⊙ inconsistent HFDS $S = (U, HR \cup R_{D})$ . $\forall {hr}^{*} \in HR$ , $H_{{hr}^{*}} = (HR, Δ - {m_{ij} \in M . | {hr}^{*} \in m_{ij}})$ is an edge-deleted sub-graph of hypergraph $H$ . $\forall hr \in HR$ , denote C^{hr-hr^∗} = (B) _hr - (B) _{hr
^∗}, then $d_{H_{{hr}^{*}}} (hr) = \sum_{c_{ij}^{τ} \in C_{0}^{hr - {hr}^{*}}} c_{ij}^{τ}$ .

Proof. Obviously, C^{hr-hr^∗} is also a matrix. Let $C_{0}^{hr - {hr}^{*}}$ be a 0-cuts matrix of C^{hr-hr^∗}. By the definition of $H_{{hr}^{*}}$ , it implies that all hyperedges incident to the vertex hr^∗ of $H$ have been deleted. Therefore, $\forall hr \in HR$ , $\begin{matrix} d_{H_{{hr}^{*}}} (hr) & = d_{H} (hr) - | {m_{ij}^{• ⊙} | hr, {hr}^{*} \in m_{ij}^{• ⊙}} | \\ = \sum_{b_{ij}^{• ⊙} \in (B)_{hr}} b_{ij}^{• ⊙} - \sum_{{(b_{1})}_{ij}^{• ⊙} \in (B)_{hr}, {(b_{2})}_{ij}^{• ⊙} \in (B)_{{hr}^{*}}} \\ \min {{(b_{1})}_{ij}^{• ⊙}, {(b_{2})}_{ij}^{• ⊙}} . \end{matrix}$

By the definition of $C_{0}^{hr - {hr}^{*}}$ , $d_{H_{{hr}^{*}}} (hr) = \sum_{c_{ij}^{τ} \in C_{0}^{hr - {hr}^{*}}} c_{ij}^{τ}$ is obtained.

Theorem 11 implies that the degree of the vertex hr of $H_{{hr}^{*}}$ is the sum of nonempty items in $C_{0}^{hr - {hr}^{*}}$ .

A feature selection approach for fuzzy decision systems based on the graph theory is proposed by Chen et al. [28]. Suppose that $H = (HR, Δ)$ is an induced hypergraph from a fuzzy decision system $S = (U, HR \cup R_{D})$ , a forward search can be applied to greedily search the vertex hr with the largest degree of $H$ . The vertex hr will be added to the reduction set, then all hyperedges containing the vertex can be deleted. Finally, it is calculated that the degree of each vertex and repeated the above process until the degree of all vertices is 0. This approach is also applicable to find the lower approximation distribution reductions in this paper.

In the following, the above process is explained by an example.

Example 9. (Continue to Example 8) Considering the discernibility matrix M_ll based on the optimism-optimism score function based HF-RAOps. It is found that hr₂ and hr₄ have the largest degrees. If hr₂ is chosen, all the edges incident to hr₂ of $H$ are deleted by Theorem 11, then $H_{{hr}_{2}} = (HR, Δ - {m_{ij}^{ll} \in M_{ll} | {hr}_{2} \in m_{ij}^{ll}})$ can be obtained (Fig. 2).

Fig. 2

The sub-graph of hypergraph $H = (HR, Δ)$ .

Let C^hr₁-hr₂ = (B) _{hr
₁} - (B) _{hr
₂}, C^hr₃-hr₂ = (B) _{hr
₃} - (B) _{hr
₂}, C^hr₄-hr₂ = (B) _{hr
₄} - (B) _{hr
₂}. It can be obtained that $\begin{matrix} \begin{matrix} C^{{hr}_{1} - {hr}_{2}} = (\begin{matrix} 0 & 0 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}), C^{{hr}_{3} - {hr}_{2}} = (\begin{matrix} 0 & 0 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}), \\ C^{{hr}_{4} - {hr}_{2}} = (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ - 1 & 0 & 0 \end{matrix}), \end{matrix} \end{matrix}$

thus $d_{H_{{hr}_{2}}} (R_{1}) = \sum_{c_{ij}^{0} \in C_{0}^{{hr}_{1} - {hr}_{2}}} c_{ij}^{0} = 0, d_{H_{{hr}_{2}}} (R_{3}) = \sum_{c_{ij}^{0} \in C_{0}^{{hr}_{3} - {hr}_{2}}} c_{ij}^{0} = 0, d_{H_{{hr}_{2}}} ({hr}_{4}) = \sum_{c_{ij}^{0} \in C_{0}^{{hr}_{4} - {hr}_{2}}} c_{ij}^{0} = 1$ can be obtained. Now, hr₄ have the largest degree. Repeat the above process. Finally, the result {hr₂, hr₄} is got. Similarly, if hr₄ is chosen, the result {hr₁, hr₄} , { hr₂, hr₄ } , { hr₃, hr₄ } could be got. In summary, the reduction red (S) ={{ hr₁, hr₄ } , { hr₂, hr₄ } , { hr₃, hr₄ }} can be obtained when the discernibility matrix is based on the optimism-optimism score function based HF-RAOps. It has the same result as Example 7.

All the reductions can be obtained by using the discernibility matrix-based attribute reduction algorithm and the reduction algorithm proposed by Chen [28] can only obtain one reduction by greedily search. Therefore, Algorithm 1, an improved algorithm, can be proposed to find all the lower approximation distribution reductions.

Algorithm 1

A score function based hesitant fuzzy rough attribute reduction algorithm based on hypergraphs(HFARH)

Input: A HFDS $S = (U, HR \cup R_{D})$ ;

Output: All the lower approximation distribution reductions red.

1: Let red =∅ , s = 0, dh = ∅ , k = 0 ;

2: Calculate $Sim (HR)$ ;

3: for each u_i, u_j ∈ U

4: Calculate $λ = \underline{Sim (HR)^{•}} ([u_{i}]_{R_{D}^{⊙}}) (u_{j})$ ; (•, ⊙ ∈ { l, b })

5: end for

6: for each ${hr}_{i} \in HR$ do

7: Calculate $(B)_{{hr}_{i}}, d_{H} ({hr}_{i}) = \sum_{b_{ij}^{• ⊙} \in (B)_{{hr}_{i}}} b_{ij}^{• ⊙}$ ;

8: $s = s + d_{H} ({hr}_{i})$ ;

9: $dh (i) = d_{H} ({hr}_{i})$ ;

10: end for

11: Let $DR = {hr \in HR | d_{H} (hr) = max_{{hr}_{i} \in HR} d_{H} ({hr}_{i})}$ and DH ={ dh₁, . . . , dh_|DR| }, where dh₁ = ⋯ = dh_|DR| = dh; (i.e., let DH be a family of dhs which is the vector of degrees for each vertex and DR be a family of vertices with the largest degree, if multiple vertices have the largest degree at the same time, they are all recorded in DR. DR corresponds to DH one by one.)

12: while DR≠ ∅ do

13: k = k + 1;

14: r (k) = DR (k);

15: while s ≠ 0

16: $hrmax = {hr \in HR | d_{H_{r (k)}} (hr) = max_{{hr}_{i} \in HR} d_{H_{r (k)}} ({hr}_{i})}$ ;

17: $H = H_{r (k)}$ ;

18: if |hrmax|>1 then

19: for j = 1 : (|hrmax|-1) do

20: dr = r (k) ∪ hrmax (j + 1);

21: DR ={ DR, dr };

22: for each ${hr}_{i} \in HR$ do

23: Calculate $(B)_{{hr}_{i}}, d_{H_{hrmax (j + 1)}} ({hr}_{i})$ ;

24: $dh (i) = d_{H_{hrmax (j + 1)}} ({hr}_{i})$ ;

25: end for

26: DH ={ DH, dh };

27: end for

28: end if

29: r (k) = r (k) ∪ hrmax (1) , s = 0; (The wait-to-be-reduced set r (k) and the largest degree of vertex take the union.)

30: for each ${hr}_{i} \in HR$ do

31: Calculate $(B)_{{hr}_{i}}, d_{H_{hrmax (j + 1)}} ({hr}_{i})$ ;

32: $s = s + d_{H_{hrmax (j + 1)}} ({hr}_{i})$ ;

33: end for

34: end while

35: red ={ red, r (k) } , DR (k) = ∅;

36: end while

37: return red

In order to understand how the algorithm finds all the lower approximation distribution reductions, an example will be used to illustrate the whole loop portion of Algorithm 1. (All data in this example are artificially given. In real problems, it should be calculated according to the corresponding formulas.)

Example 10. Let $S = (U, HR \cup R_{D})$ be a • -⊙ inconsistent HFDS, where $HR = {{hr}_{1}, {hr}_{2}, {hr}_{3}, {hr}_{4}, {hr}_{5}, {hr}_{6}, {hr}_{7}}$ is a family of HFRs. DR is a family of the wait-to-be-reduced set, dhs is a vector of degrees for each vertex in different states and DH is a family of dh, DR corresponds to DH one by one. Suppose hr₂, hr₃, hr₅ have the largest degree in the initial state, then the process of reductions can be obtained as follows:

According to the given conditions, DR ={{ hr₂ } , { hr₃ } , { hr₅ }} and DH ={ dh₁, dh₂, dh₃ } are obtained. Obviously, dh₁, dh₂ and dh₃ are the same vector. {hr₂} is chosen at first, then $H = H_{{hr}_{2}}$ . By dh₁, assume that hr₁ and hr₆ have the largest degree in $H_{{hr}_{2}}$ . hr₁ is chosen to continue the process, while {hr₂, hr₆} is recorded in DR. Let dh₄ be the vector of degrees for each vertex at this time, then DR ={{ hr₂ } , { hr₃ } , { hr₅ } , { hr₂, hr₆ }} and DH ={ dh₁, dh₂, dh₃, dh₄ } are obtained. Continue to delete hr₁ in this hypergraph. Assume that hr₄ has the largest degree. Since the degree of all vertices is not 0, the process is continuing. Continue to delete hr₄ in this hypergraph. Assume that hr₆ has the largest degree. The degree of all vertices are still not 0, the process is continuing. Continue to delete hr₆ in this hypergraph. Assume that the degree of all vertices is 0 at this point. The first reduction {hr₁, hr₂, hr₄, hr₆} is obtained, then $red = {{{hr}_{1}, {hr}_{2}, {hr}_{4}, {hr}_{6}}} .$ {hr₂} is replaced with an empty set in DR, i.e., DR ={ ∅ , { hr₃ } , { hr₅ } , { hr₂, hr₆ }}. End of the first loop (Fig. 3(a)).

{hr₃} is chosen in order to continue the process. By dh₂, assume that hr₁ and hr₆ have the largest degree in $H_{{hr}_{3}}$ . hr₁ is chosen to continue the process, while {hr₃, hr₆} is recorded in DR. Let dh₅ be the vector of degrees for each vertex at this time, then dh₅ is recorded in DH. Continue to delete hr₁ in this hypergraph. Assume that the degree of all vertices is 0 at this point. The second reduction {hr₁, hr₃} is obtained, then $red = {{{hr}_{1}, {hr}_{2}, {hr}_{4}, {hr}_{6}}, {{hr}_{1}, {hr}_{3}}} .$ {hr₃} is replaced with an empty set in DR, i.e., DR ={ ∅ , ∅ , { hr₅ } , { hr₂, hr₆ } , { hr₃, hr₆ }}. End of the second loop (Fig. 3(b)).

{hr₅} is chosen in order to continue the process. By dh₃, assume that hr₄ has the largest degree in $H_{{hr}_{5}}$ . Continue to delete hr₄ in this hypergraph. Assume that the degree of all vertices is 0 at this point. The third reduction {hr₄, hr₅} is obtained, then $red = {{{hr}_{1}, {hr}_{2}, {hr}_{4}, {hr}_{6}}, {{hr}_{1}, {hr}_{3}}, {{hr}_{4}, {hr}_{5}}} .$ {hr₅} is replaced with an empty set in DR, i.e., DR ={ ∅ , ∅ , ∅ , { hr₂, hr₆ } , { hr₃, hr₆ }}. End of the third loop (Fig. 3(c)).

{hr₂, hr₆} is chosen in order to continue the process. hr₂ and hr₆ are deleted successively in the hypergraph. By dh₄, assume that hr₃ has the largest degree in $H_{{hr}_{6}}$ . Continue to delete hr₃ in this hypergraph. Assume that the degree of all vertices is 0 at this point. The fourth reduction {hr₂, hr₃, hr₆} is obtained, then $\begin{matrix} red = {{{hr}_{1}, {hr}_{2}, {hr}_{4}, {hr}_{6}}, {{hr}_{1}, {hr}_{3}}, \\ {{hr}_{4}, {hr}_{5}}, {{hr}_{2}, {hr}_{3}, {hr}_{6}}} . \end{matrix}$ {hr₂, hr₆} is replaced with an empty set in DR, i.e., DR ={ ∅ , ∅ , ∅ , ∅ , { hr₃, hr₆ }}. End of the fourth loop (Fig. 3(d)).

{hr₃, hr₆} is chosen in order to continue the process. hr₃ and hr₆ are deleted successively in the hypergraph. By dh₅, assume that hr₂ has the largest degree in $H_{{hr}_{6}}$ . Continue to delete hr₂ in this hypergraph. Assume that the degree of all vertices is 0 at this point. The fifth reduction {hr₂, hr₃, hr₆} is obtained, it is the same as the fourth reduction. Thus, $\begin{matrix} red = {{{hr}_{1}, {hr}_{2}, {hr}_{4}, {hr}_{6}}, {{hr}_{1}, {hr}_{3}}, \\ {{hr}_{4}, {hr}_{5}}, {{hr}_{2}, {hr}_{3}, {hr}_{6}}} . \end{matrix}$ {hr₃, hr₆} is replaced with an empty set in DR, i.e., DR ={ ∅ , ∅ , ∅ , ∅ , ∅ }. End of the fifth loop (Fig. 3(e)).

At this point, there are only empty sets in DR, so the program jumps out of the loop and outputs red.

Fig. 3

The process of reductions(a - e).

5 Numerical experiments

In this section, the hybrid data of Hepatitis C Virus (HCV) from UCI is used for data analysis to reflect the superiority of the proposed model and algorithm (https://archive-beta.ics.uci.edu/ml/datasets/ hepatitis+c+virus+hcv+for+egyptian+patients). The experiment is performed on a personal computer with Intel(R) Core(TM) i5 - 8265U CPU@ 3.40GHz. All algorithms are implemented in MatlabR2016a.

5.1 Depictions of the HCV and data preprocessing

In order to better treat HCV patients and save public resources, it is usually necessary to differentiate the disease severity of HCV patients. Due to different areas of expertise, an expert group consisting of multiple experts is usually required for diagnosis. Meantime, due to the limited number of authoritative medical experts, it is necessary to select a suitable general medical expert group to replace the authoritative medical expert group without delaying the medical diagnosis of HCV patients.

To evaluate the performance of HFARH, the following experimental procedure will be set.

The publicly available data set will be used in our experiment. The number of samples selected is controlled between 5 and 9. The samples have 21 attributes which are Body Mass Index, Fever, Vomiting, Headache, Diarrhea, Fatigue, Bone ache, Jaundice, Epigastria pain, White Blood Cells, Red Blood Cells, Hemoglobin, Platelet, Aspartate transaminase ratio, RNA Base, RNA 4, RNA 12, RNA end-of-treatment, RNA elongation factor, Baseline Histological Grading, Baseline Histological.

Suppose that an authoritative medical expert group and several general medical expert groups are given. Then, according to the experts in the relevant fields, the HFRs between samples can be given by expert groups in Table 1. According to Table 1, it is found that ${HR}_{2}$ contains four HFRs, ${HR}_{1}$ , ${HR}_{4}$ and ${HR}_{7}$ contain five HFRs, ${HR}_{3}$ , ${HR}_{5}$ and ${HR}_{6}$ contain six HFRs. The general medical expert group ${HR}_{1}$ is taken as an example. The HFR ${hr}_{1}^{1}$ about Body Mass Index, Fever, Headache and Bone ache are given by the same expert, the HFR ${hr}_{2}^{1}$ about Vomiting, Diarrhea, Aspartate transaminase ratio, RNA Base, RNA 4 and Baseline Histological are given by another expert, etc.

Since the data is hybrid, the binary relation appropriately need to be properly selected. The construction of HFR hr about attributes c₁, c₂, c₃ by general medical expert group ${HR}_{i}$ is taken as an example. If c₁ is a nominal attribute, the value of the fuzzy similarity degree r_{c
₁} (u_i, u_j) between the samples u_i and u_j could be measured as: $r_{c_{1}} (u_{i}, u_{j}) = {\begin{matrix} 1, c_{1} (u_{i}) = c_{1} (u_{j}), \\ 0, otherwise . \end{matrix}$ If c₂ is a numerical attribute, the value of the fuzzy similarity degree r_{c
₂} (u_i, u_j) between the samples u_i and u_j could be measured as: $r_{c_{2}} (u_{i}, u_{j}) = \frac{2 (c_{2} (u_{i}) \land c_{2} (u_{j}))}{c_{2} (u_{i} + c_{2} (u_{j}))},$ or $\begin{matrix} r_{c_{2}} (u_{i}, u_{j}) = {\begin{matrix} c_{2} (u_{i}) \land c_{2} (u_{j}), & c_{2} (u_{i}) \neq c_{2} (u_{j}), \\ 1, & c_{2} (u_{i}) = c_{2} (u_{j}), \end{matrix} \end{matrix}$ where c₂ (u_i) ∧ c₂ (u_j) takes the smaller of c₂ (u_i) and c₂ (u_j). If c₃ is a ordinal attribute, let df be a function to encode ordinal attribute values, then ordinal attribute can be transformed into numerical attribute to calculate the value of the fuzzy similarity degree r_{c
₃} (u_i, u_j) between u_i and u_j. If C is the condition attribute set, let hr (u_i, u_j) ={ r_c (u_i, u_j) |c ∈ C } for ∀u_i, u_j, then we can obtain the HFR hr.

According to Table 1, seven HFDSs are generated, i.e., $S = (U, {HR}_{i} \cup R_{D}) (i = 1, 2, . . ., 7)$ , where ${HR}_{i} = {{hr}_{i}^{1}, {hr}_{i}^{2}, \dots}$ . Then seven experiments are conducted to better illustrate the superiority of this model. Each experiment will get four types of the lower approximation distribution reductions based on different score function based HF-RAOps (i.e., the optimism-optimism score function based HF-RAOps, the optimism-pessimism score function based HF-RAOps, the pessimism-optimism score function based HF-RAOps and the pessimism-pessimism score function based HF-RAOps).

Table 1
The HFRs given by different expert groups

Attributes The expert groups

${HR}_{1}$ ${HR}_{2}$ ${HR}_{3}$ ${HR}_{4}$ ${HR}_{5}$ ${HR}_{6}$ ${HR}_{7}$ R _D

Body Mass Index ${hr}_{1}^{1}$ ${hr}_{1}^{2}$ ${hr}_{2}^{3}$ ${hr}_{3}^{4}$ ${hr}_{1}^{5}$ ${hr}_{5}^{6}$ ${hr}_{3}^{7}$ R _D

Fever ${hr}_{1}^{1}$ ${hr}_{3}^{2}$ ${hr}_{6}^{3}$ ${hr}_{1}^{4}$ ${hr}_{1}^{5}$ ${hr}_{5}^{6}$ ${hr}_{3}^{7}$ R _D

Vomiting ${hr}_{2}^{1}$ ${hr}_{3}^{2}$ ${hr}_{6}^{3}$ ${hr}_{3}^{4}$ ${hr}_{3}^{5}$ ${hr}_{1}^{6}$ ${hr}_{3}^{7}$ R _D

Headache ${hr}_{1}^{1}$ ${hr}_{1}^{2}$ ${hr}_{3}^{3}$ ${hr}_{1}^{4}$ ${hr}_{2}^{5}$ ${hr}_{5}^{6}$ ${hr}_{2}^{7}$ R _D

Diarrhea ${hr}_{2}^{1}$ ${hr}_{2}^{2}$ ${hr}_{4}^{3}$ ${hr}_{5}^{4}$ ${hr}_{2}^{5}$ ${hr}_{3}^{6}$ ${hr}_{1}^{7}$ R _D

Fatigue ${hr}_{4}^{1}$ ${hr}_{1}^{2}$ ${hr}_{2}^{3}$ ${hr}_{1}^{4}$ ${hr}_{3}^{5}$ ${hr}_{6}^{6}$ ${hr}_{3}^{7}$ R _D

Bone ache ${hr}_{1}^{1}$ ${hr}_{4}^{2}$ ${hr}_{4}^{3}$ ${hr}_{3}^{4}$ ${hr}_{5}^{5}$ ${hr}_{6}^{6}$ ${hr}_{4}^{7}$ R _D

Jaundice ${hr}_{3}^{1}$ ${hr}_{1}^{2}$ ${hr}_{2}^{3}$ ${hr}_{3}^{4}$ ${hr}_{1}^{5}$ ${hr}_{5}^{6}$ ${hr}_{3}^{7}$ R _D

Epigastria pain ${hr}_{3}^{1}$ ${hr}_{1}^{2}$ ${hr}_{2}^{3}$ ${hr}_{3}^{4}$ ${hr}_{1}^{5}$ ${hr}_{5}^{6}$ ${hr}_{3}^{7}$ R _D

White Blood Cells ${hr}_{5}^{1}$ ${hr}_{3}^{2}$ ${hr}_{4}^{3}$ ${hr}_{1}^{4}$ ${hr}_{3}^{5}$ ${hr}_{6}^{6}$ ${hr}_{1}^{7}$ R _D

Red Blood Cells ${hr}_{1}^{4}$ ${hr}_{4}^{2}$ ${hr}_{2}^{3}$ ${hr}_{5}^{4}$ ${hr}_{6}^{5}$ ${hr}_{6}^{6}$ ${hr}_{5}^{7}$ R _D

Hemoglobin ${hr}_{5}^{1}$ ${hr}_{1}^{2}$ ${hr}_{1}^{3}$ ${hr}_{3}^{4}$ ${hr}_{1}^{5}$ ${hr}_{5}^{6}$ ${hr}_{2}^{7}$ R _D

Platelet ${hr}_{4}^{1}$ ${hr}_{2}^{2}$ ${hr}_{3}^{3}$ ${hr}_{1}^{4}$ ${hr}_{3}^{5}$ ${hr}_{4}^{6}$ ${hr}_{6}^{7}$ R _D

Aspartate transaminase ratio ${hr}_{2}^{1}$ ${hr}_{3}^{2}$ ${hr}_{1}^{3}$ ${hr}_{3}^{4}$ ${hr}_{2}^{5}$ ${hr}_{4}^{6}$ ${hr}_{3}^{7}$ R _D

RNA Base ${hr}_{2}^{1}$ ${hr}_{1}^{2}$ ${hr}_{4}^{3}$ ${hr}_{3}^{4}$ ${hr}_{2}^{5}$ ${hr}_{5}^{6}$ ${hr}_{3}^{7}$ R _D

RNA 4 ${hr}_{2}^{1}$ ${hr}_{2}^{2}$ ${hr}_{1}^{3}$ ${hr}_{4}^{4}$ ${hr}_{3}^{5}$ ${hr}_{1}^{6}$ ${hr}_{3}^{7}$ R _D

RNA 12 ${hr}_{4}^{1}$ ${hr}_{4}^{2}$ ${hr}_{2}^{3}$ ${hr}_{5}^{4}$ ${hr}_{5}^{5}$ ${hr}_{5}^{6}$ ${hr}_{1}^{7}$ R _D

RNA end-of-treatment ${hr}_{3}^{1}$ ${hr}_{3}^{2}$ ${hr}_{1}^{3}$ ${hr}_{2}^{4}$ ${hr}_{6}^{5}$ ${hr}_{6}^{6}$ ${hr}_{2}^{7}$ R _D

RNA elongation factor ${hr}_{5}^{1}$ ${hr}_{2}^{2}$ ${hr}_{5}^{3}$ ${hr}_{5}^{4}$ ${hr}_{1}^{5}$ ${hr}_{3}^{6}$ ${hr}_{4}^{7}$ R _D

Baseline Histological Grading ${hr}_{5}^{1}$ ${hr}_{1}^{2}$ ${hr}_{6}^{3}$ ${hr}_{3}^{4}$ ${hr}_{1}^{5}$ ${hr}_{5}^{6}$ ${hr}_{3}^{7}$ R _D

Baseline Histological ${hr}_{2}^{1}$ ${hr}_{2}^{2}$ ${hr}_{3}^{3}$ ${hr}_{1}^{4}$ ${hr}_{3}^{5}$ ${hr}_{2}^{6}$ ${hr}_{5}^{7}$ R _D

Attributes	The expert groups
Body Mass Index	${hr}_{1}^{1}$	${hr}_{1}^{2}$	${hr}_{2}^{3}$	${hr}_{3}^{4}$	${hr}_{1}^{5}$	${hr}_{5}^{6}$	${hr}_{3}^{7}$	R _D
Fever	${hr}_{1}^{1}$	${hr}_{3}^{2}$	${hr}_{6}^{3}$	${hr}_{1}^{4}$	${hr}_{1}^{5}$	${hr}_{5}^{6}$	${hr}_{3}^{7}$	R _D
Vomiting	${hr}_{2}^{1}$	${hr}_{3}^{2}$	${hr}_{6}^{3}$	${hr}_{3}^{4}$	${hr}_{3}^{5}$	${hr}_{1}^{6}$	${hr}_{3}^{7}$	R _D
Headache	${hr}_{1}^{1}$	${hr}_{1}^{2}$	${hr}_{3}^{3}$	${hr}_{1}^{4}$	${hr}_{2}^{5}$	${hr}_{5}^{6}$	${hr}_{2}^{7}$	R _D
Diarrhea	${hr}_{2}^{1}$	${hr}_{2}^{2}$	${hr}_{4}^{3}$	${hr}_{5}^{4}$	${hr}_{2}^{5}$	${hr}_{3}^{6}$	${hr}_{1}^{7}$	R _D
Fatigue	${hr}_{4}^{1}$	${hr}_{1}^{2}$	${hr}_{2}^{3}$	${hr}_{1}^{4}$	${hr}_{3}^{5}$	${hr}_{6}^{6}$	${hr}_{3}^{7}$	R _D
Bone ache	${hr}_{1}^{1}$	${hr}_{4}^{2}$	${hr}_{4}^{3}$	${hr}_{3}^{4}$	${hr}_{5}^{5}$	${hr}_{6}^{6}$	${hr}_{4}^{7}$	R _D
Jaundice	${hr}_{3}^{1}$	${hr}_{1}^{2}$	${hr}_{2}^{3}$	${hr}_{3}^{4}$	${hr}_{1}^{5}$	${hr}_{5}^{6}$	${hr}_{3}^{7}$	R _D
Epigastria pain	${hr}_{3}^{1}$	${hr}_{1}^{2}$	${hr}_{2}^{3}$	${hr}_{3}^{4}$	${hr}_{1}^{5}$	${hr}_{5}^{6}$	${hr}_{3}^{7}$	R _D
White Blood Cells	${hr}_{5}^{1}$	${hr}_{3}^{2}$	${hr}_{4}^{3}$	${hr}_{1}^{4}$	${hr}_{3}^{5}$	${hr}_{6}^{6}$	${hr}_{1}^{7}$	R _D
Red Blood Cells	${hr}_{1}^{4}$	${hr}_{4}^{2}$	${hr}_{2}^{3}$	${hr}_{5}^{4}$	${hr}_{6}^{5}$	${hr}_{6}^{6}$	${hr}_{5}^{7}$	R _D
Hemoglobin	${hr}_{5}^{1}$	${hr}_{1}^{2}$	${hr}_{1}^{3}$	${hr}_{3}^{4}$	${hr}_{1}^{5}$	${hr}_{5}^{6}$	${hr}_{2}^{7}$	R _D
Platelet	${hr}_{4}^{1}$	${hr}_{2}^{2}$	${hr}_{3}^{3}$	${hr}_{1}^{4}$	${hr}_{3}^{5}$	${hr}_{4}^{6}$	${hr}_{6}^{7}$	R _D
Aspartate transaminase ratio	${hr}_{2}^{1}$	${hr}_{3}^{2}$	${hr}_{1}^{3}$	${hr}_{3}^{4}$	${hr}_{2}^{5}$	${hr}_{4}^{6}$	${hr}_{3}^{7}$	R _D
RNA Base	${hr}_{2}^{1}$	${hr}_{1}^{2}$	${hr}_{4}^{3}$	${hr}_{3}^{4}$	${hr}_{2}^{5}$	${hr}_{5}^{6}$	${hr}_{3}^{7}$	R _D
RNA 4	${hr}_{2}^{1}$	${hr}_{2}^{2}$	${hr}_{1}^{3}$	${hr}_{4}^{4}$	${hr}_{3}^{5}$	${hr}_{1}^{6}$	${hr}_{3}^{7}$	R _D
RNA 12	${hr}_{4}^{1}$	${hr}_{4}^{2}$	${hr}_{2}^{3}$	${hr}_{5}^{4}$	${hr}_{5}^{5}$	${hr}_{5}^{6}$	${hr}_{1}^{7}$	R _D
RNA end-of-treatment	${hr}_{3}^{1}$	${hr}_{3}^{2}$	${hr}_{1}^{3}$	${hr}_{2}^{4}$	${hr}_{6}^{5}$	${hr}_{6}^{6}$	${hr}_{2}^{7}$	R _D
RNA elongation factor	${hr}_{5}^{1}$	${hr}_{2}^{2}$	${hr}_{5}^{3}$	${hr}_{5}^{4}$	${hr}_{1}^{5}$	${hr}_{3}^{6}$	${hr}_{4}^{7}$	R _D
Baseline Histological Grading	${hr}_{5}^{1}$	${hr}_{1}^{2}$	${hr}_{6}^{3}$	${hr}_{3}^{4}$	${hr}_{1}^{5}$	${hr}_{5}^{6}$	${hr}_{3}^{7}$	R _D
Baseline Histological	${hr}_{2}^{1}$	${hr}_{2}^{2}$	${hr}_{3}^{3}$	${hr}_{1}^{4}$	${hr}_{3}^{5}$	${hr}_{2}^{6}$	${hr}_{5}^{7}$	R _D

5.2 Experimental results and analyses

In this subsection, the number of nonempty items in discernibility matrices, the running time, the number of the lower approximation distribution reductions red and the average number of HFRs in reductions are studied in the experiments.

Tables 2 , 3 and 4 record the number of nonempty items in different discernibility matrices for different HFDSs with five samples, seven samples and nine samples, where |Δ| represents the number of nonempty items in Δ. Remove extreme data, from these tables we can find that the number of nonempty items in M_ll is approximately equal to the number of nonempty items in M_lb, and the number of nonempty items in M_bl is approximately equal to the number of nonempty items in M_bb. The number of nonempty items in M_ll or M_lb is obviously more than the number of nonempty items in M_bl or M_bb. Combined with the relevant definitions, how to deal with HFRs inducing from the condition attribute set is more important than how to deal with HFRs inducing from the decision attribute set in this model. Meanwhile, the number of nonempty items in discernibility matrix using the optimistic condition is more than using the pessimistic condition.

Table 2
Related data in the reduction process with 5 samples

Systems |Δ| = | { m_ij ∈ M . |m_ij ≠ ∅ } |

M . = M_ll M . = M_lb M . = M_bl M . = M_bb

$(U, {HR}_{1} \cup R_{D})$ 7 6 7 5

$(U, {HR}_{2} \cup R_{D})$ 8 8 5 7

$(U, {HR}_{3} \cup R_{D})$ 8 10 6 7

$(U, {HR}_{4} \cup R_{D})$ 8 9 7 7

$(U, {HR}_{5} \cup R_{D})$ 11 9 9 3

$(U, {HR}_{6} \cup R_{D})$ 7 8 6 7

$(U, {HR}_{7} \cup R_{D})$ 7 9 6 8

Systems	\|Δ\| = \| { m_ij ∈ M . \|m_ij ≠ ∅ } \|
$(U, {HR}_{1} \cup R_{D})$	7	6	7	5
$(U, {HR}_{2} \cup R_{D})$	8	8	5	7
$(U, {HR}_{3} \cup R_{D})$	8	10	6	7
$(U, {HR}_{4} \cup R_{D})$	8	9	7	7
$(U, {HR}_{5} \cup R_{D})$	11	9	9	3
$(U, {HR}_{6} \cup R_{D})$	7	8	6	7
$(U, {HR}_{7} \cup R_{D})$	7	9	6	8

Table 3

Related data in the reduction process with 7 samples

Systems	\|Δ\| = \| { m_ij ∈ M . \|m_ij ≠ ∅ } \|
	M . = M_ll	M . = M_lb	M . = M_bl	M . = M_bb
$(U, {HR}_{1} \cup R_{D})$	20	19	10	11
$(U, {HR}_{2} \cup R_{D})$	19	17	15	14
$(U, {HR}_{3} \cup R_{D})$	19	20	8	13
$(U, {HR}_{4} \cup R_{D})$	20	21	9	12
$(U, {HR}_{5} \cup R_{D})$	21	23	14	13
$(U, {HR}_{6} \cup R_{D})$	17	19	16	14
$(U, {HR}_{7} \cup R_{D})$	18	19	13	13

Table 4

Related data in the reduction process with 9 samples

Systems	\|Δ\| = \| { m_ij ∈ M . \|m_ij ≠ ∅ } \|
	M . = M_ll	M . = M_lb	M . = M_bl	M . = M_bb
$(U, {HR}_{1} \cup R_{D})$	25	27	17	16
$(U, {HR}_{2} \cup R_{D})$	29	27	20	17
$(U, {HR}_{3} \cup R_{D})$	24	28	16	17
$(U, {HR}_{4} \cup R_{D})$	19	24	11	12
$(U, {HR}_{5} \cup R_{D})$	23	22	19	13
$(U, {HR}_{6} \cup R_{D})$	25	27	16	16
$(U, {HR}_{7} \cup R_{D})$	20	24	14	11

Figure 4 records the running time in different HFDSs with five samples, seven samples and nine samples. According to Table 1, it is found that ${HR}_{2}$ contains four HFRs, ${HR}_{1}$ , ${HR}_{4}$ and ${HR}_{7}$ contain five HFRs, ${HR}_{3}$ , ${HR}_{5}$ and ${HR}_{6}$ contain six HFRs. When the number of samples is small, the running time of the program is acceptable. However, the order of the discernibility matrix increases as the number of samples increases, then the running time increases significantly. By comparing different HFDSs, it is found that the number of HFRs have less effect in the running time of the program.

Fig. 4

The running time of different HFDSs under five samples, seven samples and nine samples.

Tables 5 , 6 and 7 record the number of reductions for HFDSs with five samples, seven samples and nine samples, where |red| represents the number of sets in red. Since the optimistic extension and pessimistic extension are different, the condition of the optimism-pessimism score function based HF-RAOps is stronger than that of the optimism-optimism score function based HF-RAOps. Similarly, the condition of the pessimism-pessimism score function based HF-RAOps is stronger than that of the pessimism-optimism score function based HF-RAOps. However, the condition of the optimism-optimism score function based HF-RAOps is stronger than that of the pessimism-optimism score function based HF-RAOps. Moreover, the condition of the optimism-pessimism score function based HF-RAOps is stronger than that of the pessimism-pessimism score function based HF-RAOps. The above analysis is confirmed in Tables 8 , 9 and 10.

Table 5

The reductions with 5 samples

Systems	\|red\|
	M . = M_ll	M . = M_lb	M . = M_bl	M . = M_bb
$(U, {HR}_{1} \cup R_{D})$	1	1	1	1
$(U, {HR}_{2} \cup R_{D})$	2	2	1	1
$(U, {HR}_{3} \cup R_{D})$	4	4	2	2
$(U, {HR}_{4} \cup R_{D})$	1	2	1	1
$(U, {HR}_{5} \cup R_{D})$	2	4	2	2
$(U, {HR}_{6} \cup R_{D})$	2	2	2	2
$(U, {HR}_{7} \cup R_{D})$	1	1	1	1

Table 6

The reductions with 7 samples

Systems	\|red\|
	M . = M_ll	M . = M_lb	M . = M_bl	M . = M_bb
$(U, {HR}_{1} \cup R_{D})$	2	4	1	1
$(U, {HR}_{2} \cup R_{D})$	2	2	1	1
$(U, {HR}_{3} \cup R_{D})$	4	4	1	1
$(U, {HR}_{4} \cup R_{D})$	2	2	2	2
$(U, {HR}_{5} \cup R_{D})$	2	2	1	1
$(U, {HR}_{6} \cup R_{D})$	1	1	1	1
$(U, {HR}_{7} \cup R_{D})$	2	2	2	2

Table 7

The reductions with 9 samples

Systems	\|red\|
	M . = M_ll	M . = M_lb	M . = M_bl	M . = M_bb
$(U, {HR}_{1} \cup R_{D})$	1	1	1	1
$(U, {HR}_{2} \cup R_{D})$	1	1	1	1
$(U, {HR}_{3} \cup R_{D})$	2	2	1	2
$(U, {HR}_{4} \cup R_{D})$	2	2	1	1
$(U, {HR}_{5} \cup R_{D})$	2	2	1	1
$(U, {HR}_{6} \cup R_{D})$	2	2	1	1
$(U, {HR}_{7} \cup R_{D})$	1	1	1	1

Table 8

The average number of HFRs with 5 samples in reductions

Systems	$\| \bar{r} \|$
	M . = M_ll	M . = M_lb	M . = M_bl	M . = M_bb
$(U, {HR}_{1} \cup R_{D})$	5	5	5	5
$(U, {HR}_{2} \cup R_{D})$	3.5	3.5	3	3
$(U, {HR}_{3} \cup R_{D})$	5	5.25	4.5	4.5
$(U, {HR}_{4} \cup R_{D})$	4	4.5	4	4
$(U, {HR}_{5} \cup R_{D})$	4.5	5	4.5	4.5
$(U, {HR}_{6} \cup R_{D})$	5.5	5.5	5.5	5.5
$(U, {HR}_{7} \cup R_{D})$	5	5	5	5

Table 9

The average number of HFRs with 7 samples in reductions

Systems	$\| \bar{r} \|$
	M . = M_ll	M . = M_lb	M . = M_bl	M . = M_bb
$(U, {HR}_{1} \cup R_{D})$	4	4.25	3	4
$(U, {HR}_{2} \cup R_{D})$	3.5	3.5	3	3
$(U, {HR}_{3} \cup R_{D})$	5	5	5	5
$(U, {HR}_{4} \cup R_{D})$	4.5	4.5	4.5	4.5
$(U, {HR}_{5} \cup R_{D})$	5.5	5.5	5	5
$(U, {HR}_{6} \cup R_{D})$	6	6	6	6
$(U, {HR}_{7} \cup R_{D})$	4.5	4.5	4.5	4.5

Table 10

The average number of HFRs with 9 samples in reductions

Systems	$\| \bar{r} \|$
	M . = M_ll	M . = M_lb	M . = M_bl	M . = M_bb
$(U, {HR}_{1} \cup R_{D})$	5	5	5	5
$(U, {HR}_{2} \cup R_{D})$	4	4	4	4
$(U, {HR}_{3} \cup R_{D})$	5.5	5.5	5	5.5
$(U, {HR}_{4} \cup R_{D})$	4	4.5	4	4
$(U, {HR}_{5} \cup R_{D})$	5.5	5.5	5	5
$(U, {HR}_{6} \cup R_{D})$	5	5.5	5	5
$(U, {HR}_{7} \cup R_{D})$	5	5	5	5

Tables 8 , 9 and 10 record the average number of HFRs for HFDSs with five samples, seven samples and nine samples, where $| \bar{r} |$ represents the average number of HFRs in reductions. From these tables, it is found that the average number of HFRs under the optimism-pessimism score function based HF-RAOps is essentially the largest, and the average number of HFRs under the pessimism-optimism score function based HF-RAOps is essentially the smallest. There is essentially no significant difference in the average number of HFRs under the optimism-optimism score function based HF-RAOps and the optimism-pessimism score function based HF-RAOps. Similarly, the average number of individuals under the pessimism-pessimism score function based HF-RAOps and the pessimism-optimism score function based HF-RAOps is essentially the same. Therefore, the relationship between different HF-RAOps is further verified by discussing the average number of HFRs in reductions.

6 Conclusion

Inspired by Chen [28], a novel HF-RAOps based on dual score functions and an improved algorithm for this approach were proposed to find all the lower approximation distribution reductions. Firstly, since there are different extension methods, four generalized HF-RAOps about the proposed HF-RAOps were constructed. Next, the lower approximation distribution consistent sets and lower approximation distribution reductions of an inconsistent HFDS were obtained by using the four newly arrived HF-RAOps, and some equivalence conditions about the approximate distribution reductions were proved successively. Therefore, four discernibility matrices and all the reductions were obtained under an inconsistent HFDS. Meanwhile, through the theory related to hypergraphs, it was proved that finding the lower approximation distribution reductions of a HFDS is equivalent to finding the minimal transversals of its hypergraph. Based on these discussions, an improved attribute reduction algorithm for HFDS based on hypergraphs was proposed. Finally, using Hepatitis C Virus (HCV) from UCI, which is hybrid data for data analysis, the superiority of the proposed model and algorithm was shown. Through experiments, it was found that when decision makers holding optimistic attitudes deal with pessimistic data, they get results under loose conditions, but the results are risky. When decision makers holding pessimistic attitudes deal with optimistic data, they get results under strict conditions, but the results are lower risky. Meantime, the running time of the proposed algorithm is highly sensitive to the number of samples and less sensitive to the number of HFRs.

For further research, the following are possible directions: building on the concepts presented in this paper, the topic of incremental attribute reductions in the context of dynamic HFDSs will be discussed; and the conflict analysis based on three-way decisions under HFDSs will be investigated using the proposed hesitant fuzzy rough set.

Footnotes

Acknowledgements

Funding: This study was funded by the National Natural Science Foundation of China (No. 62076088), by the Natural Science Foundation of Hebei Province (No. A2020208004), by the Graduate Innovation Funding Project of Shijiazhuang Tiedao University (No. YC2022063).

References

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets andSystems 20(1) (1986), 87–96.

Gorzalczany

M.B.

, A method of inference in approximate reasoningbased on interval-valued fuzzy sets, Fuzzy Sets and Systems 21(1) (1987), 1–17.

Dubois

and Prade

, Fuzzy sets and systems: theory andapplications, The Journal of Symbolic Logic 47(3) (1982), 198–198.

Yager

R.R.

, Pythagorean Membership Grades in Multicriteria DecisionMaking, IEEE Trans Fuzzy Systems 22(4) (2014), 958–965.

Torra

and Narukawa

, On hesitant fuzzy sets and decision, The 18th IEEE international conference on fuzzy systems (2009), 1378–1382.

Pawlak

, Rough sets, International Journal of Computer & Information Sciences 11(5) (1982), 341–356.

Akram

, Nawaz

H.S.

and Deveci

, Attribute reduction and information granulation in Pythagorean fuzzy formal contexts, Expert Systems With Applications 222 (2023), 119794.

Z.P.

, Guo

H.Y.

and Wang

L.D.

, A hybrid method of time seriesforecasting based on information granulation and dynamic selectionstrategy, Journal of Intelligent & Fuzzy Systems 44(2) (2023), 2367–2379.

10.

Chen

L.J.

, Luo

D.M.

, Wang

, Li

Z.W.

and Xie

N.X.

, Measures of uncertainty for a fuzzy probabilistic approximation space, Journal of Intelligent & Fuzzy Systems 42(4) (2022), 3615–3638.

11.

Dubois

and Prade

, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems 17(2) (1990), 191–209.

12.

Morsi

N.N.

and Yakout

M.M.

, Axiomatics for fuzzy rough sets, Fuzzy Sets & Systems 100(1-3) (1998), 327–342.

13.

Cornelis

, Martine

D.C.

and Kerre

E.E.

, Intuitionistic fuzzy roughsets: at the crossroads of imperfect knowledge, Expert Systems 20(5) (2003), 260–270.

14.

Yun

S.M.

and Lee

S.J.

, New approach to intuitionistic fuzzy roughsets, International Journal of Fuzzy Logic and Intelligent Systems 20(2) (2020), 129–137.

15.

Varma

and John

S.J.

, Rough approximations of multi-fuzzy sets, International Journal of Granular Computing Rough Sets & Intelligent Systems 3(4) (2014), 327–344.

16.

Gayathri

and Sunil

J.J.

, Multi-fuzzy Rough Sets based on Implicators and Continuous t-norms, Fuzzy Information and Engineering 9(4) (2017), 515–537.

17.

Yang

, Song

, Qi

, et al., Constructive and axiomaticapproaches to hesitant fuzzy rough set, Soft Computing 18(6) (2014), 1067–1077.

18.

Jin

C.X.

, Mi

J.S.

, Li

F.C.

and Liang

M.S.

, A novel probabilistic hesitant fuzzy rough set based multi-criteria decision-making method, Information Sciences 608(2022), 489–516.

19.

Zhang

H.D.

, Lan

, et al, On novel hesitant fuzzy rough sets, Soft Computing 23(22) (2019), 11357–11371.

20.

Sun

J.Q.

, Wang

P.X.

, Yu

H.L.

and Yang

X.B.

, A constraint scoreguided meta-heuristic searching to attribute reduction, Journalof Intelligent & Fuzzy Systems 44(3) (2023), 4779–4800.

21.

Tan

G.X.

, Attribute reduction for multiset-valued data based on FRIC-model, Journal of Intelligent & Fuzzy Systems 44(1) (2023), 75–90.

22.

Zhang

X.Y.

, Fan

Y.R.

, Yao

Y.S.

and Yang

J.L.

, Class-specific attribute reducts based on neighborhood rough sets, Journal of Intelligent & Fuzzy Systems 43(6) (2022), 7891–7910.

23.

Skowron

and Rauszer

, The Discernibility Matrices and Functions in Information Systems, Intelligent Decision Support (1992), 331–362.

24.

Wang

and Wang

, Reduction algorithms based on discernibility matrix: The ordered attributes method, Journal of Computer Science and Technology 16(6) (2001), 489–504.

25.

Wong

and Ziarko

, On optimal decision rules in decision tables, Bulletin of the Polish Academy of Sciences Mathematics 33(11-12) (1985), 693–696.

26.

Borowik

and Luba

, Fast Algorithm of Attribute Reduction Based on the Complementation of Boolean Function, Springer International Publishing (2014), 25–41.

27.

Pandu

and Sai Prasad

P.S.V.S.

, Map Reduce based parallel fuzzy-rough attribute reduction using discernibility matrix, Applied Intelligence 1(52) (2021), 1–20.

28.

Chen

, Mi

J.S.

and Lin

, A graph approach for fuzzy-rough feature selection, Fuzzy Sets and Systems 391(2019), 96–116.

29.

Xia

M.M.

and Xu

Z.S.

, Hesitant fuzzy information aggregation indecision making, International Journal of Approximate Reasoning 52(3) (2010), 395–407.

30.

Z.S.

and Xia

M.M.

, Distance and similarity measures for hesitant fuzzy sets, Information Sciences 181(11) (2011), 2128–2138.

31.

Liao

H.C.

and Xu

Z.S.

, A VIKOR-based method for hesitant fuzzy multi-criteria decision making, Fuzzy Optimization and Decision Making: A Journal of Modeling and Computation Under Uncertainty 12(4) (2013), 373–392.

32.

Farhadinia

, A series of score functions for hesitant fuzzy sets, Information Sciences 277(2014), 102–110.

33.

Demange

and Paschos

V.T.

, Graphs and hypergraphs, Mathématiques Informatique Et Sciences Humaines 135(1996), 51–66.

34.

Bondy

J.A.

and Murty

, Graph Theory with Applications, Journal of the Operational Research Society 28(419) (1977), 237–238.

Attributes	The expert groups
	${HR}_{1}$	${HR}_{2}$	${HR}_{3}$	${HR}_{4}$	${HR}_{5}$	${HR}_{6}$	${HR}_{7}$	R _D
Body Mass Index	${hr}_{1}^{1}$	${hr}_{1}^{2}$	${hr}_{2}^{3}$	${hr}_{3}^{4}$	${hr}_{1}^{5}$	${hr}_{5}^{6}$	${hr}_{3}^{7}$	R _D
Fever	${hr}_{1}^{1}$	${hr}_{3}^{2}$	${hr}_{6}^{3}$	${hr}_{1}^{4}$	${hr}_{1}^{5}$	${hr}_{5}^{6}$	${hr}_{3}^{7}$	R _D
Vomiting	${hr}_{2}^{1}$	${hr}_{3}^{2}$	${hr}_{6}^{3}$	${hr}_{3}^{4}$	${hr}_{3}^{5}$	${hr}_{1}^{6}$	${hr}_{3}^{7}$	R _D
Headache	${hr}_{1}^{1}$	${hr}_{1}^{2}$	${hr}_{3}^{3}$	${hr}_{1}^{4}$	${hr}_{2}^{5}$	${hr}_{5}^{6}$	${hr}_{2}^{7}$	R _D
Diarrhea	${hr}_{2}^{1}$	${hr}_{2}^{2}$	${hr}_{4}^{3}$	${hr}_{5}^{4}$	${hr}_{2}^{5}$	${hr}_{3}^{6}$	${hr}_{1}^{7}$	R _D
Fatigue	${hr}_{4}^{1}$	${hr}_{1}^{2}$	${hr}_{2}^{3}$	${hr}_{1}^{4}$	${hr}_{3}^{5}$	${hr}_{6}^{6}$	${hr}_{3}^{7}$	R _D
Bone ache	${hr}_{1}^{1}$	${hr}_{4}^{2}$	${hr}_{4}^{3}$	${hr}_{3}^{4}$	${hr}_{5}^{5}$	${hr}_{6}^{6}$	${hr}_{4}^{7}$	R _D
Jaundice	${hr}_{3}^{1}$	${hr}_{1}^{2}$	${hr}_{2}^{3}$	${hr}_{3}^{4}$	${hr}_{1}^{5}$	${hr}_{5}^{6}$	${hr}_{3}^{7}$	R _D
Epigastria pain	${hr}_{3}^{1}$	${hr}_{1}^{2}$	${hr}_{2}^{3}$	${hr}_{3}^{4}$	${hr}_{1}^{5}$	${hr}_{5}^{6}$	${hr}_{3}^{7}$	R _D
White Blood Cells	${hr}_{5}^{1}$	${hr}_{3}^{2}$	${hr}_{4}^{3}$	${hr}_{1}^{4}$	${hr}_{3}^{5}$	${hr}_{6}^{6}$	${hr}_{1}^{7}$	R _D
Red Blood Cells	${hr}_{1}^{4}$	${hr}_{4}^{2}$	${hr}_{2}^{3}$	${hr}_{5}^{4}$	${hr}_{6}^{5}$	${hr}_{6}^{6}$	${hr}_{5}^{7}$	R _D
Hemoglobin	${hr}_{5}^{1}$	${hr}_{1}^{2}$	${hr}_{1}^{3}$	${hr}_{3}^{4}$	${hr}_{1}^{5}$	${hr}_{5}^{6}$	${hr}_{2}^{7}$	R _D
Platelet	${hr}_{4}^{1}$	${hr}_{2}^{2}$	${hr}_{3}^{3}$	${hr}_{1}^{4}$	${hr}_{3}^{5}$	${hr}_{4}^{6}$	${hr}_{6}^{7}$	R _D
Aspartate transaminase ratio	${hr}_{2}^{1}$	${hr}_{3}^{2}$	${hr}_{1}^{3}$	${hr}_{3}^{4}$	${hr}_{2}^{5}$	${hr}_{4}^{6}$	${hr}_{3}^{7}$	R _D
RNA Base	${hr}_{2}^{1}$	${hr}_{1}^{2}$	${hr}_{4}^{3}$	${hr}_{3}^{4}$	${hr}_{2}^{5}$	${hr}_{5}^{6}$	${hr}_{3}^{7}$	R _D
RNA 4	${hr}_{2}^{1}$	${hr}_{2}^{2}$	${hr}_{1}^{3}$	${hr}_{4}^{4}$	${hr}_{3}^{5}$	${hr}_{1}^{6}$	${hr}_{3}^{7}$	R _D
RNA 12	${hr}_{4}^{1}$	${hr}_{4}^{2}$	${hr}_{2}^{3}$	${hr}_{5}^{4}$	${hr}_{5}^{5}$	${hr}_{5}^{6}$	${hr}_{1}^{7}$	R _D
RNA end-of-treatment	${hr}_{3}^{1}$	${hr}_{3}^{2}$	${hr}_{1}^{3}$	${hr}_{2}^{4}$	${hr}_{6}^{5}$	${hr}_{6}^{6}$	${hr}_{2}^{7}$	R _D
RNA elongation factor	${hr}_{5}^{1}$	${hr}_{2}^{2}$	${hr}_{5}^{3}$	${hr}_{5}^{4}$	${hr}_{1}^{5}$	${hr}_{3}^{6}$	${hr}_{4}^{7}$	R _D
Baseline Histological Grading	${hr}_{5}^{1}$	${hr}_{1}^{2}$	${hr}_{6}^{3}$	${hr}_{3}^{4}$	${hr}_{1}^{5}$	${hr}_{5}^{6}$	${hr}_{3}^{7}$	R _D
Baseline Histological	${hr}_{2}^{1}$	${hr}_{2}^{2}$	${hr}_{3}^{3}$	${hr}_{1}^{4}$	${hr}_{3}^{5}$	${hr}_{2}^{6}$	${hr}_{5}^{7}$	R _D