Hesitant fuzzy graphs,hesitant fuzzy hypergraphs and fuzzy graph decisions 1

1 Introduction

Since the contemporary concern about knowledge representation and information systems has put forward useful extensions of classical set theory such as fuzzy set theory and fuzzy graph theory. The concept of fuzzy set was originally proposed by Zadeh [28, 29] in 1965, offers a wide variety of techniques for analyzing imprecise date. While the fuzzy graph theory, introduced by Rosenfeld [17] in 1975. It soon fuzzy graph theory and fuzzy set are combined to deal with some common uncertain problems. On the one hand, graph theory as a classical mathematical tool has numerous applications to problems in computer science, system analysis, decision making system, and so on. Recently, many scholars have made a lot of generalization of fuzzy graph. Mordeson et al. [14] proposed the cartesian product, synthesis and union of fuzzy graph in 1994. In order to analyze fuzzy graph, in 2003, Bhutani et al. [8] defined a concept and some properties of M-strong fuzzy graphs. Considering that interval fuzzy sets can express information imperfection efficiently, the concept of interval fuzzy graph is introduced by Akram [1–3]. With the development of the fuzzy graph theory, R.H.Goetschel [11] first explored the concept of fuzzy hypergraph in 1995, and gave a screenshot of fuzzy hypergraph. Subsequently, R.H.Goetschel et al. [12] studied the operation of fuzzy hypergraphs. In order to use fuzzy hypergraph to process fuzzy information, in 2017, Gong and Wang [21] discussed the equivalence relationship between fuzzy hypergraphs and fuzzy formal concept analysis and fuzzy information systems. On the other hand, considering the complexity of the decision makers, the differences of cognitive and thinking, for the same question often have different opinions. For the sake of attempting to better capture the possible subjectivity, uncertainty of the evaluations, et cetera, Zadeh’s introduction of fuzzy set was subsequently extended to different types for various applications. In recent years, Torra [18, 19] introduced the concept of hesitant fuzzy set as an extension of the fuzzy set in which the membership degree of a given element, and allowed the membership degree of element is a set of possible values. The problem of the hesitant fuzzy multi-criteria decision making has attracted researchers’ attention. Xia et al. [23] studied the operator of hesitant fuzzy information integration and applies it to decision problems. Then Xu et al. [24, 25] defined the similarity measure on this basis. Wei et al. [22] proposed several classes of interval hesitant fuzzy aggregation operators and applies them to multi-criteria decision making problems. Picture fuzzy sets can also be viewed as a gener alization of fuzzy sets, each of their nembers is described by four values in [0, 1], corresponding to for kinds of attitudes of decision makers: yes, neutrality, against and abstained. Neutrosophic set is a powerful general formal framework that generalizes the concept of fuzzy set and intuitionistic fuzzy set, in neutrosophic set, indeterminacy is quantified explicitly, and truth membership, indeterminacy membership, and falsity membership are independent. So the hesitant fuzzy set can deal with the uncertain problem better.

Recently, multi-attribute decision making widely exists in the social and economic life and is an important branch of the modern decision theory [26]. Many scholars have studied this kind of problem. In picture fuzzy environment, The new order is given by Si, A. et al. [4] in 2019, different group decision making methods are presented by Shahzaib Ashraf et al. [5] Then, the picture fuzzy aggregation operator for multi-attribute decision-making problems are presented by Qiyas et al. [16] At the same time, multi-attribute decision making methods in different environments have also been given, which can be referred to in the literature. [6, 15]. The social network group decision making (SNGDM) consists of a group of decision makers who express their opinions on a set of alternatives with the aim of choosing the best alternative, Dong et al. [10] proposed a minimum adjustment consensus framework for the social network group decision-making with incomplete linguistic preference relations. When consensus among decision makers is reached, a selection process is applied usually by fusing the preferences of individual decision makers into a collective preference from which the final ordering of the considered alternatives is derived. However, under the background of indecisiveness, decision-makers have different views on the same problem due to its attributes. In this case, it is more intuitive to solve such problems with hesitant fuzzy hypergraphs. Since the hesitant fuzzy set and its extensions indeed describe the thoughts of experts better because of a better tolerance, we introduce the definition of hesitant fuzzy hypergraph based on hesitant fuzzy sets and fuzzy hypergraph to solve these problems. As an important research direction in the field of artificial intelligence, granular computing [13, 27] is mainly used to describe uncertain, fuzzy and incomplete information, provided a solution method based on the relationship between grain and grain. In this paper, the multi-attribute decision making problems are transformed into a hesitant fuzzy hyperhraph model, granular computing is used to make the optimal decision.

The rest of this paper is organized as follows. Section 2 presents some basic results of the hesitant fuzzy sets and fuzzy graph. The definition of the hesitant fuzzy graph and its operations are proposed and investigated in the Section 3. In Section 4 and Section 5, we give the concept of the hesitant fuzzy hypergraph, characterize the level cut, and discuss the graph operations of the hesitant fuzzy hypergraph. The equivalent representation of hesitant fuzzy hypergraph, hesitant fuzzy information table and hesitant fuzzy formal contexts are proved and studied in the Section 7. Finally, as an application, the hesitant fuzzy hypergraph could be applied in the granular computing of information system which shows in the Section 8 and 9.

2 Preliminaries

In this section, we first give the concept of hesitation fuzzy set, and then give the basic method to judge the quality of hesitant fuzzy elements, at last we introduced the definition of fuzzy graph.

Definition 2.1. [19] Let V be a nonempty finite domain. A hesitant fuzzy set on V can be expressed as a function h, which is a function that every element on V maps to a subset on the interval [0, 1], and its mathematical is expressed as A = { < x , h A ( x ) > | x ∈ V } , where h _A  (x) is a set of some values in [0, 1], representing some possible membersip degree of element x on the set A, and h = h _A  (x) is called the hesitation fuzzy element. In addition, we call all of the hesitantly fuzzy sets on V is HF (V). In the classical collections, the hierarchical method is reflected by the inclusion relations between the collections, while in the classical fuzzy set, the hierarchical method is realized by comparing the size of membership function. As an extension of the classical fuzzy set, we give the following sorting method to compare the hesitation fuzzy elements.

Definition 2.2. [19] Let h be a hesitation fuzzy element, we call S ( h A ( x ) ) = 1 l ( h A ( x ) ) ∑ r ∈ h A ( x ) r is the score function of h, where l (h _A  (x)) is the number of elements in h _A  (x). For two hesitation fuzzy elements h₁ and h₂, if S (h₁) > S (h₂), h₁ is beter than h₂, denoting as h₁ > h₂. If S (h₁) = S (h₂), it is said that h₁ is the same as h₂, denoting as h₁ ∼ h₂. However, according to the above definition, the score function of hesitation fuzzy elements cannot be distinguished efficiently in some cases.

Example 2.3. Let h _{A 1}  (x) = {0.1, 0.3, 0.5}, h _{A 2}  (x) = {0.2, 0.4, 0.3}, h _{A 3}  (x) = {0.3, 0.2, 0.4}. Therefore, S ( h A 1 ( x ) ) = 0.1 + 0.3 + 0.5 3 = 0.3 , S ( h A 2 ( x ) ) = 0.2 + 0.4 + 0.3 3 = 0.3 , S ( h A 3 ( x ) ) = 0.3 + 0.2 + 0.4 3 = 0.3 .

Because of S (h _{A 1}  (x))=S (h _{A 2}  (x))=S (h _{A 3}  (x)), it is impossible to judge whether h₁, h₂, h₃ is best by the score function. In order to further distinguish, we give the concept of hesitation fuzzy elements variance var ( h A ( x ) ) = 1 l ( h A ( x ) ) ∑ r ∈ h A ( x ) ( r - S ( h A ( x ) ) ) 2 . According to the above definition, a general method for the comparisn of hesitation fuzzy elements is given. Let h _{A 1}  (x) and h _{A 2}  (x) be two hesitation fuzzy elements. 1 ) IfS ( h A 1 ( x ) ) > S ( h A 2 ( x ) ) , then h A 1 ( x ) > h A 2 ( x ) . 2 ) IfS ( h A 1 ( x ) ) < S ( h A 2 ( x ) ) , then h A 1 ( x ) < h A 2 ( x ) . 3 ) IfS ( h A 1 ( x ) ) = S ( h A 2 ( x ) ) , var ( h A 1 ( x ) ) > var ( h A 2 ( x ) ) , then h A 1 ( x ) < h A 2 ( x ) , ifS ( h A 1 ( x ) ) = S ( h A 2 ( x ) ) , var ( h A 1 ( x ) ) < var ( h A 2 ( x ) ) , then h A 1 ( x ) > h A 2 ( x ) , ifS ( h A 1 ( x ) ) = S ( h A 2 ( x ) ) , var ( h A 1 ( x ) ) = var ( h A 2 ( x ) ) , then h A 1 ( x ) = h A 2 ( x ) .

Definition 2.4. [17] Fuzzy graph G is defined as a sequence pair from G = (σ, μ), where σ is a fuzzy subset of the V, μ is called a fuzzy symmetrical relation on σ. That is to say: σ : V → [0, 1], μ : V × V → [0, 1], such that for all u ∈ V, v ∈ V,there is μ (uv) ≤ σ (u) ∧ σ (v). Where uv is the edge between vertex u and v,σ is the fuzzy vertex set on V, and μ is the fuzzy edge set on E.

3 Hesitant fuzzy graph

In order to study the graph theory in the context of hesitation fuzzy, we give the definition of the hesitant fuzzy graph.

Definition 3.1. Let V be a nonempty finite domain, A and B be two hesitation fuzzy sets on V and V². Satisfying h A : V → P ( [ 0 , 1 ] ) , h B : E ⊆ V × V → P ( [ 0 , 1 ] ) . Where P ([0, 1]) is the power set on [0, 1], which makes h _B  (xy) ≤ min {h _A  (x) , h _A  (y)} valid for any x ∈ V, y ∈ V. Then, the sequence pair G = (A, B) is called the hesitation fuzzy graph on G^* = (V, E). A is called the hesitation fuzzy vertex set on V, and B is called the hesitation fuzzy edge set on E. And B is the symmetric hesitation fuzzy relation on A.

Example 3.1 Let G = (A, B) be a hesitation fuzzy graph defined on the fuzzy graph G^* = (V, E). If V = {x₁, x₂, x₃, x₄}, E = {x₁x₂, x₂x₃, x₃x₄, x₄x₁}. A is a hesitant fuzzy sets on V, can be expressed by A = {< x₁, {0.6, 0.8} > , < x₂, {0.3, 0.7, 0.9} > , < x₃, {0.5, 0.7} > , < x₄, {0.5, 0.6, 0.7} >}. Then B is a hesitant fuzzy sets on E ⊆ V × V, can be expressed as B = {< x₁x₂, {0.1, 0.2} > , < x₂x₃, {0.2} > , < x₃x₄, {0.2, 0.3, 0.4} > , < x₄x₁, {0.3, 0.5} >}. As a result, the hesitant fuzzy graph can be expressed as Fig. 1.

OPEN IN VIEWER

Fig. 1

A hesitant fuzzy graph

Remark For a hesitant fuzzy graph G = (A, B). If h _B  (xy) = min(h _A  (x) , h _A  (y)), for xy ∈ E. Then G is called a strong hasitant fuzzy graph.

Definition 3.2. Let G₁ = (A₁, B₁), G₂ = (A₂, B₂) be the hesitant fuzzy graph of fuzzy graph G 1 * = ( V 1 , E 1 ) , G 2 * = ( V 2 , E 2 ) respectively. Then

1) The cartesian product of G₁ and G₂, G₁ × G₂ = (A₁ × A₂, B₁ × B₂) is expressed as h_A1×A2 (x₁, x₂) = min {h _{A 1}  (x₁) , h _{A 2}  (x₂)}, ∀ (x₁, x₂) ∈ V₁ × V₂, h_B1×B2 ((x, x₂) , (x, y₂)) = min {h _{A 1}  (x) , h _{B 2}  (x₂y₂)}, ∀x ∈ V₁, x₂y₂ ∈ E₂, h_B1×B2 ((x₁, z) , (y₁, z)) = min {h _{B 1}  (x₁y₁) , h _{B 2}  (z)}, ∀z ∈ V₂, x₁y₁ ∈ E₁. 2) The union of G₁ and G₂, G₁ ∪ G₂ = (A₁ ∪ A₂, B₁ ∪ B₂) is expressed as h_A1∪A2 (x) = h _{A 1}  (x), ∀x ∈ V₁, x ∉ V₂, h_A1∪A2 (x) = h _{A 2}  (x), ∀x ∉ V₁, x ∈ V₂, h_A1∪A2 (x)=max{h _{A 1}  (x) , h _{A 2}  (x)}, ∀x ∈ V₁ ∩ V₂, h_B1∪B2 (xy) = h _{B 1}  (xy), ∀xy ∈ E₁, xy ∉ E₂, h_B1∪B2 (xy) = h _{B 2}  (xy), ∀xy ∉ E₁, xy ∈ E₂, h_B1∪B2 (xy)=max{h _{B 1}  (xy) , h _{B 2}  (xy)}, ∀xy ∈ E₁ ∩ E₂. 3) The intersection of G₁ and G₂, G₁ ∩ G₂ = (A₁ ∩ A₂, B₁ ∩ B₂) is expressed as h_A1∩A2 (x) = Φ, ∀x ∈ V₁, x ∉ V₂, h_A1∩A2 (x) = Φ, ∀x ∉ V₁, x ∈ V₂, h_A1∩A2 (x) = min {h _{A 1}  (x) , h _{A 2}  (x)}, ∀x ∈ V₁ ∩ V₂, h_B1∩B2 (xy) = Φ, ∀xy ∈ E₁, xy ∉ E₂, h_B1∩B2 (xy) = Φ, ∀xy ∉ E₁, xy ∈ E₂, h_B1∩B2 (xy) = min {h _{B 1}  (xy) , h _{B 2}  (xy)}, ∀xy ∈ E₁ ∩ E₂. 4) The join-graph of G₁ and G₂, G₁ + G₂ = (A₁ + A₂, B₁ + B₂) is expressed as: h_A1+A2 (x) = h_A1∪A2 (x), ∀x ∈ V₁ ∪ V₂, h_B1+B2 (xy) = h_B1∪B2 (xy), ∀xy ∈ E₁ ∪ E₂, h_B1+B2 (xy) = min {h _{A 1}  (x) , h _{A 2}  (y)}, ∀_xy ∈ E′, where E′ prime refers to the set of all connecting points V₁ and V₂.

Definition 3.3. Let G = (A, B) be a hesitant fuzzy graph based on G^* = (V, E). For λ ∈ [0, 1], we call G λ ′ = ( A λ , B λ ) as the λ-cut of G = (A, B), is defined by A _λ  = {x ∈ V|S (h _A  (x)) ≥ λ}, B _λ  = {xy ∈ E|S (h _B  (xy)) ≥ λ}.

The characteristic of graph theory is to transform a problem into an image of binary relation graph, by studying a series of structural properties to reveal the nature of the problem, so as to solve the problem. As an extension of the simple graph, hypergraphs and fuzzy hypergraphs because an edge can contain multiple vertices. Therefore, it provides an effective analysis method for solving complex problems. Let’s first recall the definition of fuzzy hypergraph.

Definition 4.1. [11] Let V = {x₁, x₂ ⋯ x _n } be a finite set, ɛ  = {e₁, e₂ ⋯ e _m } be a nontrivial fuzzy subset on V, that is ɛ  ∈ 2 ^V . We call H = ( V , ɛ ) is a fuzzy hypergraph on V, if and only if ∀i ∈ {1, 2, ⋯ , m}, e _i  ≠ Φ and ∪ i = 1 m { supp e _i |e _i  ∈  ɛ } = V. Based on the definition of fuzzy hypergraph given above, in order to solve the problem of the graph in the hesitant fuzzy environment under the support of the hesitant fuzzy set. We give the definition of the hesitant fuzzy hypergraph below, and discuss its correlation graph operation and further analyze the graph.

Definition 4.2. Let V = {< x₁, h _V  (x₁) > , < x₂, h _V  (x₂) > , ⋯ , < x _n , h _V  (x _n ) >} be a finite hesitant fuzzy set, ɛ  = {< e₁, h _ɛ  (e₁) > , < e₂, h _ɛ  (e₂) > , ⋯ , < e _m , h _ɛ  (e _m ) >} be a nontrivial hesitant fuzzy subset on V, for ∀i ∈ 1, 2, ⋯ , m, if it satisfies h _ɛ  (e _i ) ≠ Φ, ∪ i = 1 m { supp e _i |e _i  ∈  ɛ } = V, and h _ɛ  (e _i ) ≤ ∧ _{x∈e i} h _V  (x), where x is a subset on e _i . We call H = ( V , ɛ ) is a hesitant fuzzy hypergraph on V.

Example 4.3. Let H = ( V , ɛ ) be a hesitant fuzzy hypergraph on V. If V = { < x 1 , { 0.3 , 0.6 } > , < x 2 , { 0.5 , 0.7 , 0.8 } > , < x 3 , { 0.2 , 0.4 , 0.5 } > , < x 4 , { 0.3 , 0.7 } > , < x 5 , { 0.2 } > , < x 6 , { 0.4 , 0.7 } > , < x 7 , { 0.2 , 0.3 } > } , ɛ = { e 1 , e 2 , e 3 , e 4 } , and e₁ = {x₁, x₂, x₃}, e₂ = {x₃, x₄},e₃ = {x₄, x₅, x₆}, e₄ = {x₂, x₇}.

Because of h _ɛ  (xy) ≤ min {h _V  (x) , h _V  (y)}, we have h _ɛ  (e₁) = {0.1, 0.2}, h _ɛ  (e₂) = {0.1},h _ɛ  (e₃) = {0.1, 0.2}, h _ɛ  (e₄) = {0.1, 0.2}. The one of the hesitant fuzzy hypergraphs is shown in Fig. 2.

OPEN IN VIEWER

Fig. 2

A hesitant fuzzy hypergraph of seven vertices and four edges.

Based on the definition 4.2, we give the λ-cut of the hesitant fuzzy hypergraph.

Definition 4.4. Let H = ( V , ɛ ) be a hesitant fuzzy hypergraph on V, where h _V  (x) → [0, 1], h _ɛ  (e) → [0, 1]. Suppose 0 < λ ≤ 1, the λ-cut hypergraph of the hesitant fuzzy hypergraph H is defined as V _λ  = {x ∈ V|S (h _V  (x)) ≥ λ}, ɛ _λ  = {e ∈  ɛ |S (h _ɛ  (e)) ≥ λ}. With the help of the cut of hesitating fuzzy hypergraph, we can segment the hypergraph to obtain the subgraph that meets our requirements, and then conduct further research on the problem.

According to the natural generalization of fuzzy graph, the wertex set and the edge set of hypergraph are represented by hesitant fuzzy set σ and μ respectively. So another definition of hesitant fuzzy graph is given.

Theorem 4.5. Hesitant fuzzy hypergraph can be represented by a ternary array H = ( V , σ , μ ) , sometimes V can be omitted, where σ and μ respectively are hesitant fuzzy vertex set and hesitant fuzzy hyperedge set of V and ɛ , and satisfies: (1) suppσ = V. (2) h _ɛ  (μ) ≤ min h _V  (σ), for V ∈  ɛ . And V =∪ suppμ.

Proof. Let σ ⊂ V, μ ⊂  ɛ . By definition 3, H = ( V , ɛ ) is a hesitant fuzzy hypergraph, and satisfes h _ɛ  (e _i ) ≠ Φ, ∪ i = 1 m { supp e _i |e _i  ∈  ɛ } = V, and h _ɛ  (e _i ) ≤ ∧ _{x∈e i} h _V  (x). However, for σ ⊂ V, supp e _i  ⊂ σ ⊂ V, then we have suppσ⊂ suppe _i  = V. For e _i  ⊂ μ, then ∪ i = 1 m { supp e _i |e _i ∈  ɛ } = ∪ suppe _i  = V.

Definition 5.1. Let H 1 = ( σ 1 , μ 1 ) and H 2 = ( σ 2 , μ 2 ) be two hesitant fuzzy hypergraphs, where σ₁ and σ₂ respectively are the hesitant fuzzy subset of vertex set V₁ and V₂. μ₁ and μ₂ respectively are the hesitant fuzzy subset of hyperedge set ɛ ₁ and ɛ ₂. Then the cartesian product of H 1 and H 2 , H = H 1 □ H 2 is defined as ∀ (x₁, x₂) ∈ V₁ × V₂, h_σ1×σ2 (x₁, x₂) = min {h _{σ 1}  (x₁) , σ₂ (x₂)}. ∀x₁ ∈ V₁, ∀e₂ ∈  ɛ ₂, h_μ1×μ2 ({x₁} × e₂) = min {h _{σ 1}  (x₁) , h _{μ 2}  (e₂)}. ∀e₁ ∈  ɛ ₁, ∀x₂ ∈ V₂, h_μ1×μ2 (e₁ × {x₂}) = min {h _{μ 1}  (e₁) , h _{σ 2}  (x₂)}.

Theorem 5.2. If H 1 and H 2 be two hesitant fuzzy hypergraph, then the cartesian product of H 1 and H 2 , H = H 1 □ H 2 is a hesitant fuzzy hypergraph.

Proof. Let v₁ ∈ V₁, e₁ ∈  ɛ ₁, and e₁ contain p vertices, where 1 ≤ p ≤ m. Let v₂ ∈ V₂, e₂ ∈  ɛ ₂, e contain q vertices, where 1 ≤ q ≤ n. Then

( h μ 1 × μ 2 ( { v 1 } × e 2 ) ) = min [ h σ 1 ( v 1 ) , h μ 2 ( e 2 ) ] ≤ min [ h σ 1 ( v 1 ) , min v 2 ∈ e 2 h σ 2 ( v 2 ) ] = min { h σ 1 ( v 1 ) , min [ h σ 2 ( v 21 ) , h σ 2 ( v 22 ) , ⋯ , h σ 2 ( v 2 q ) ] } = min { min [ h σ 1 ( v 1 ) , h σ 2 ( v 21 ) ] , min [ h σ 1 ( v 1 ) , h σ 2 ( v 22 ) ] , ⋯ , min [ h σ 1 ( v 1 ) , h σ 2 ( v 2 q ) ] } = min [ h σ 1 × σ 2 ( v 1 , v 21 ) , h σ 1 × σ 2 ( v 1 , v 22 ) , ⋯ , h σ 1 × σ 2 ( v 1 , v 2 q ) ] = min v 1 ∈ e 1 , v 2 ∈ e 2 h σ 1 × σ 2 ( v 1 , v 2 )

Similarly, h μ 1 × μ 2 ( { v 2 } × e 1 ) = min v 1 ∈ e 1 , v 2 ∈ e 2 h σ 1 × σ 2 ( v 1 , v 2 )

Definition 5.3. Let H 1 = ( V 1 , ɛ 1 ) and H 2 = ( V 2 , ɛ 2 ) be two hesitant fuzzy hypergraphs. Let σ _i and μ _i respectively are the hesitant fuzzy subset of V _i and ɛ _i , i = 1, 2. Then the union of H 1 and H 2 can be defined as H 1 ∪ H 2 = ( V 1 ∪ V 2 , ɛ 1 ∪ ɛ 2 ) , where the hesitant fuzzy subsets of V₁ ∪ V₂ and ɛ ₁ ∪  ɛ ₂ are σ₁ ∪ σ₂ and μ₁ ∪ μ₂ expressed as If x ∈ V₁, x ∉ V₂, then h_σ1∪σ2 (x) = h _{σ 1}  (x). If x ∈ V₂, x ∉ V₁, then h_σ1∪σ2 (x) = h _{σ 2}  (x). If x ∈ V₁ ∩ V₂, then h_σ1∪σ2 (x) = max {h _{σ 1}  (x) , h _{σ 2}  (x)}. If e ∈  ɛ ₁, e ∉  ɛ ₂, then h_μ1∪μ2 (e) = h _{μ 1}  (e). If e ∈  ɛ ₂, e ∉  ɛ ₁, then h_μ1∪μ2 (e) = h _{μ 2}  (e). If e ∈  ɛ ₁ ∩  ɛ ₂, then h_μ1∪μ2 (e) = max {h _{μ 1}  (e) , h _{μ 2}  (e)}.

Theorem 5.4. If H 1 and H 2 are two hesitant fuzzy hypergraphs, then H 1 ∪ H 2 is also hesitant fuzzy hypergraph.

Proof. According to definition 5.3, we can prove it in the following. (i). If e ∈  ɛ ₁, and e ∉  ɛ ₂, then h μ 1 ∪ μ 2 ( e ) = h μ 1 ( e 1 ) ≤ min v 1 ∈ e 1 h σ 1 ( V 1 ) = min { h σ 1 ( v 11 ) , ⋯ , h σ 1 ( v 1 p ) } . If v₁₁, v₁₂, ⋯ , v_1p ∈ V₁ but v₁₁, v₁₂, ⋯ , v_1p ∉ V₂. Then min { h σ ( v 11 ) , h σ ( v 12 ) , ⋯ , h σ ( v 1 p ) } = min { h σ 1 ∪ σ 2 ( v 11 ) , h σ 1 ∪ σ 2 ( v 12 ) , ⋯ , h σ 1 ∪ σ 2 ( v 1 p ) } . (ii). If e ∈  ɛ ₂, and e ∉  ɛ ₁, then h μ 1 ∪ μ 2 ( e ) = h μ 2 ( e 2 ) ≤ min v 2 ∈ e 2 h σ 2 ( V 2 ) = min { h σ 2 ( v 21 ) , ⋯ , h σ 2 ( v 2 p ) } . If v₂₁, v₂₂, ⋯ , v_2p ∈ V₂ but v₂₁, v₂₂, ⋯ , v_2p ∉ V₁, then min { h σ ( v 21 ) , h σ ( v 22 ) , ⋯ , h σ ( v 2 p ) } = min { h σ 1 ∪ σ 2 ( v 21 ) , h σ 1 ∪ σ 2 ( v 22 ) , ⋯ , h σ 1 ∪ σ 2 ( v 2 p ) } . (iii). If e ∈  ɛ ₁ ∩  ɛ ₂. Then ( h μ 1 ∪ μ 2 ( e ) ) = max [ h μ 1 ( e ) , h μ 2 ( e ) ] ≤ max { min [ h σ 1 ( v 11 ) , h σ 1 ( v 12 ) , ⋯ , h σ 1 ( v 1 p ) , h σ 2 ( v 21 ) , h σ 2 ( v 22 ) , ⋯ , h σ 2 ( v 2 p ) ] } ≤ min { max [ σ 1 ( v 11 ) , σ 2 ( v 21 ) ] , max [ σ 1 ( v 12 ) , σ 2 ( v 22 ) ] , ⋯ , max [ σ 1 ( v 1 p ) , σ 2 ( v 2 p ) ] } = min { h σ 1 ∪ σ 2 ( v i 1 ) , h σ 1 ∪ σ 2 ( v i 2 ) , ⋯ , h σ 1 ∪ σ 2 ( v ip ) } . ( i = 1 , 2 )

Example 5.5. Let H 1 = ( V 1 , ɛ 1 ) and H 2 = ( V 2 , ɛ 2 ) be two hesitant fuzzy hypergraphs. V₁ = {a, b, c, d, e}, V₂ = {a, c, d, e, f}, ɛ ₁ = {e₁₁, e₁₂}, ɛ ₂ = {e₂₁, e₂₂}. Where e₁₁ = {a, b, c}, e₁₂ = {c, d, e}, e₂₁ = {c, d, e}, e₂₂ = {e, f, a}. As shown in the Fig.3.

OPEN IN VIEWER

Fig. 3

The hesitant fuzzy hypergraph of H = H 1 ∪ H 2

The vertex sets V₁, V₂, the superedge sets ɛ ₁, ɛ ₂ respectively defined as V₁ = {< a, {0.5, 0.7} > , < b, {0.4} > , < c, {0.6, 0.8} > , < d, {0.2} > , < e, {0.5, 0.7} >}, V₂ = {< a, {0.3, 0.5} > , < c, {0.6} > , < d, {0.3, 0.5} > , < e, {0.5} > , < f, {0.6} >}. ɛ ₁ = {< e₁₁, {0.3, 0.4} > , < e₁₂, {0.2} >}, ɛ ₂ = {< e₂₁, {0.2, 0.3} > , < e₂₂, {0.1, 0.2} >}. By definition 4.2 and 5.3. We can get the vertex set of H = H 1 ∪ H 2 are V = {a, b, c, d, e, f}, the super edge of H = H 1 ∪ H 2 are ɛ  = {e₁, e₂, e₃}. Where e₁ = {a, b, c}, e₂ = {c, d, e}, e₃ = {a, e, f}. And V = {< a, {0.5, 0.7} > , < b, {0.4} > , < c, {0.6, 0.8} > , < d, {0.3, 0.5} > , < e, {0.5, 0.7} > , < f, {0.6} >}, ɛ  = {< e₁, {0.3, 0.4} > , < e₂, {0.2, 0.3} > , < e₃, {0.1, 0.2} >}. Then the operational graph is expressed as

Definition 5.6. The join-graph of H 1 = ( V 1 , ɛ 1 ) and H 2 = ( V 2 , ɛ 2 ) can be defined as H = H 1 + H 2 = ( V 1 + V 2 , ɛ 1 + ɛ 2 + ɛ ′ ) . Let σ _i and μ _i be hesitant fuzzy subsets of V _i and ɛ _i , i = 1, 2. The hesitant fuzzy subset of the join-graph is h_σ1+σ2 (x) = h_σ1∪σ2 (x). When e ∈  ɛ ₁ ∪  ɛ ₂, there is h_μ1+μ2 (e) = h_μ1∪μ2 (e). When e = {x₁₁, x₁₂, ⋯ , x_1s, x₂₁, x₂₂, ⋯ , x_2t} ∈  ɛ ′, h_μ1+μ2 (e) = min {h _{σ 1}  (x₁₁) , ⋯ , h _{σ 1}  (1s) , h _{σ 2}  (21) , h _{σ 2}  (22) , ⋯ , h _{σ 2}  (2t)}, where x₁₁, x₁₂, ⋯ , x_1s represents a point in V₁, x₂₁, x₂₂, ⋯ , x_2trepresents a point in V₂.

The study of the formal concept analysis was initially carried out under the classical formal background, but the traditional formal concept aalysis can not solve the problem of more and more uncertain and fuzzy information. So we discuss and analyze it in the hesitant fuzzy enviroment. Since in the formal concept analysis, date is presented in formal context, we give corresponding definition.

Definition 6.1. A hesitant fuzzy formal context is a triplet K = ( G , M , R ˜ = φ ( G × M ) ) , where G is a set of objects, M is a set of attributes. R ˜ is a hesitant fuzzy set on domain G × M. Each pair (g, m) ∈ K has a membership h R ˜ ( g , m ) , satisfies 0 ≤ h R ˜ ( g , m ) ≤ 1 . The set R ˜ = φ ( G × M ) = { < ( g , m ) , h R ˜ ( g , m ) > | ∀ g ∈ G , m ∈ M } is a hesitant fuzzy relation on G × M.

Definition 6.2. [13] Let H be a hesitant fuzzy hypregraph on a set V = {v₁, v₂, ⋯ , v _m } with hesitant fuzzy hyperedges ɛ ₁, ɛ ₂, ⋯, ɛ _n . We associate to H a hesitant fuzzy information system as follows, the attribute set is V, and the object set is { ɛ ₁, ɛ ₂, ⋯, ɛ _n }. The value set is [0, 1], and the information map is defined naturally as F H = { h ɛ j ( v i ) , if v i ∈ ɛ j , i = 1 , 2 , ⋯ , m . j = 1 , 2 , ⋯ , n . 0 , otherwise .

where h _{ɛ j}  (v _i ) ∈ [0, 1] and h _{ɛ j}  (v _i ) represents the membership degree of v _i  ∈  ɛ _j . If and only if h _{ɛ j}  (v _i ) ≥ λ, we consider object ɛ _j has attribute v _i .

Example 6.3. Let H be a hesitant fuzzy hypergraph with vertex set V = {v₁, v₂, v₃, v₄} and hesitant fuzzy hyperedges ɛ  = { ɛ ₁, ɛ ₂, ɛ ₃, ɛ ₄, ɛ ₅}, if λ ≥ 0.5. Where ɛ 1 = Φ , h ɛ 2 ( v 1 ) = { 0.8 , 0.6 } , h ɛ 3 ( v 2 , v 3 ) = { { 0.6 , 0.5 } , { 0.5 } } ,

h ɛ 4 ( v 1 , v 2 , v 3 ) = { { 0.5 } , { 0.6 , 0.7 } , { 0.6 } } , h ɛ 5 ( v 1 , v 2 , v 3 ) = { { 0.5 , 0.6 } , { 0.6 , 0.7 } , { 0.6 } } . Then the hesitant fuzzy information table is shown in Table 6.1.

Definition 6.4. Let I ˜ = < U , A , [ 0 , 1 ] , F > be a hesitant fuzzy information table, K = ( G , M , R ˜ = φ ( G × M ) ) is a hesitant fuzzy formal context. We can set U = G, A = M, if g R ˜ m ,F (g, m) = h _g  (m), otherwise, F (g, m) =0. We can equalize the hesitant fuzzy formal context and hesitant fuzzy information table. For the scope of the present paper, we only considered the numberical attributes of hesitant fuzzy information table, let us show it by an example.

Example 6.5. Let us consider the hesitant fuzzy information table based on the universe U = {x₁, x₂, x₃, x₄, x₅}, attribute set is A = {a₁, a₂, a₃, a₄, a₅, a₆} by Table 6.2.

This hesitant fuzzy information table can be represented as a hesitant fuzzy formal context, if the hesitant fuzzy formal context is K = ( G , M , R ˜ = φ ( G × M ) ) , the attribute set is M = ∪ _a∈AM _a , and M _a  = {f (u, a) |u ∈ U}. So, the binary relation R ˜ is defined according to the following definition. Let u ∈ U, a ∈ A. If ∃a ∈ A, s.t m = f (u, a). Then ( u , m ) ∈ R ˜ . If F (u, a) = f (u, a) = h _u  (a) >0. Then h R ˜ ( g , m ) = h u ( a ) . So, we obtain the hesitant fuzzy formal context in Table 6.3.

As already shown in 6.3, the hesitant fuzzy information table and the hesitant fuzzy formal context are two equivalent ways to represent data. So, a hesitant fuzzy hypergraph and hesitant fuzzy formal context are also equivalent. Therefore, a hesitant fuzzy formal context K = ( G , M , R ˜ ) can be associated to a hesitant fuzzy hypergraph H = ( V , ɛ ) . Let G =  ɛ  = { ɛ ₁, ɛ ₂, ⋯ , ɛ _m }. M = V = {x₁, x₂, ⋯ , x _n }, and ɛ j R ˜ v i ⇔ h ɛ j ( v i ) . That is to say its membership degree could be represented as v _i  ∈  ɛ _j  .

Granular computing is mainly used to deal with uncertain, fuzzy and incomplete information, and through the relations between the grain and grain to provide a solution for the problem. So the grain structure based on the graph or the hypergraph model is good for grain structure. In a graph or hypergraph, we use vertices for grains and deges or hyperedges for relations between particles. Through the graph granularity level division, open to our multi-decision problem to find the best choice.

A hesitant fuzzy subset of theset of all vertices on the graph is called a grain g, so clearly,the total set V of vertices is the largest grain, the empty set Φ is the smallest grain, the power set 2 ^V of the vertex set Vcontains all the grains on the graph.

Definition 7.1. All vertices on graph G are classified as granules according to the given granulation standaed, that is, the graph has completed a granulation. The set of all particles obtained after one granulation is called a level of the graph G. For the two levels l₁ and l₂ of graph G, for any grain g₁ in l₁, there is a grain g₁ in l₂, s.t g₁ ∈ g₂, then said the level of l₁ is finer than l₂, the level of l₂ is coarser than l₁.

We use one vertex of the hesitant fuzzy hypergraph to represent an object, and use a hesitant fuzzy hypergraph corresponds to the objects which have relation r _i . And use h _g  (x _i ) represent the membership degree of x _i beionging to the grain g, where h _g  (x _i ) ∈ [0, 1], then we can get a hesitant fuzzy hypergraph model of granular computing. The λ-cut of this hesitant fuzzy hypergraph is H λ = ( V λ , ɛ λ ) , we use the λ-cut of hesitant fuzzy hyperedge ɛ _λ to denote a granule, then the problem can be hierarchica.

Example 7.2. Let U = {x₁, x₂, x₃, x₄, x₅}, R = {r₁, r₂, r₃, r₄, r₅}. The sets of objects which have relation r _i  (1 ≤ i ≤ 5) is shown in a, and corresponding to hypergraph H = ( V , ɛ ) , we give a membership degree of x _i beionging to hesitant fuzzy hypergraph. Let X = {x₁, x₂, x₃, x₄, x₅}, then we can get a hesitant fuzzy relation matrix. If λ = 0.5, then r₁ = {(x₃) , (x₅)}, r₂ = {(x₁, x₂)}, r₃, r₄, r₅ = Φ. As shown in Fig.4.(b).

OPEN IN VIEWER

Fig. 4

A partition of granules within a level.

In the process of solving the problem, we realize the multi-level understanding of the problem through different grains of a hierarchical structure, and realize the multi-perspective view of the problem through several hierarchical structures. So, in multi-attribute decision analysis,we can formalize the problem, convert it into a hesitant fuzzy hypergraph to get the best solution which we need. We present an algorithm for hesitant fuzzy multi-attribute decision making:

Algorithm A decision-making method based on granular computing and hesitant fuzzy multi-attribute decision making.

Input Hesitant fuzzy multi-attribute decision system S =< U, A, f >, where U = {u₁, u₂, ⋯ , u _n }, A = {a₁, a₂, ⋯ , a _m }.

Output The ranking results of each alternative step 1. Union the vertices which have a relationship together to a unit O, this of this unit O as a hesitant fuzzy hyperedge, then we can construct the m-level of the model.

step 2. Calculate the score function for each attribute, that is S ( h a i ( u j ) ) = 1 l ( h a i ( u j ) ) ∑ i = 1 , 2 , ⋯ , m . j = 1 , 2 , ⋯ , n h a i ( u j )

step 3. Let λ _i  = S (h _{a i}  (u),i = 1, 2, ⋯ , m, where 0 < λ < 1. Use the λ _i which we calculated to cut the m-level of hesitant fuzzy hypergraph, then get the λ-cut of hypergraph. Let’s integrate the u that satisfies this condition, then get a grain g _i satisfies the attribute a _i .

step 4. By integrating the elements in each grain g _i , can get the attributes that each object satisfies. Then sort the number of attributes that each solution satisfies.

step 5. According to step 4, give the best plan and alternative plan.

According to the construction of the hesitant fuzzy hypergraph and the application of granular computing, the above algorithm is obtained.The algorithm can not only sort the alternatives of multi-attribute decision problems, but also clearly see the membership degree of each attribute to the object, and the decision maker can make a satisfactory choice.

Taking an enterprise planning a future development plan as an example. The enterprise leader chooses three technicians according to the financial situation x₁, customers satisfaction x₂,future market x₃, talent introductionx₄ were evaluated. The best one was selected from the five development plans u₁, u₂, u₃, u₄, u₅. Because 3 technicians come from different departments, the evaluation of various aspects may be different, that is to say the value of hesitant fuzzy evaluation is generated. We may not know enough about a certain aspect, fail to make judgment, and do not give the hesitation fuzzy judgment value. The evaluation information table is as Table 8.1.

step 1. According to the information table, the above objects and attributes are formed into hesitant fuzzy hypergraph, and the hesitant fuzzy hyperedges are x₁, x₂, x₃, x₄. Then x₁ = {u₁, u₂, u₃, u₄, u₅}, x₂ = {u₁, u₂, u₃, u₄, u₅}, x₃ = {u₁, u₂, u₃, u₄, u₅}, x₄ = {u₁, u₂, u₃, u₄, u₅}.

step 2. According to the information table, calculate the score function of each x _i , S ( h x i ( u j ) ) = 1 l ( h x i ( u j ) ) ∑ j = 1 , 2 , ⋯ , 5 h x i ( u j ) , i = 1, 2, 3, 4.

S ( h x 1 ( u ) ) = 0.4 + 0.6 + 0.5 + 0.3 + 0.7 + 0.6 + 0.6 7 = 0.528 , S ( h x 2 ( u ) ) = 0.5 + 0.6 + 0.7 + 0.4 + 0.9 + 0.5 + 0.6 + 0.7 + 0.6 + 0.8 + 0.7 11 = 0.636 , S ( h x 3 ( u ) ) = 0.8 + 0.5 + 0.7 + 0.6 + 0.7 + 0.9 + 0.65 + 0.75 8 = 0.7 , S ( h x 4 ( u ) ) = 0.4 + 0.6 + 0.7 + 0.3 + 0.3 + 0.6 + 0.6 + 0.7 + 0.8 + 0.5 + 0.6 11 = 0.557 .

step 3. Let λ₁ = 0.528,λ₂ = 0.636,λ₃ = 0.7,λ₄ = 0.557. We use each λ _i to cut the hyperedges x _i , so we have grain g _i , see Fig.5.

OPEN IN VIEWER

Fig. 5

A partition of granules within a level

step 4. We can get g₁ = {u₄, u₅}, g₂ = {u₂, u₄, u₅}, g₃ = {u₁, u₄, u₅}, g₄ = {u₂, u₄} through the step 3. Therefore, we can calculate the results as follows the scheme that reaches the expected attribute x₁ are u₄, u₅, the scheme that reaches the expected attribute x₂ are u₂, u₄, u₅, the scheme that reaches the expected attribute x₃ are u₁, u₄, u₅, and the scheme that reaches the expected attribute x₄ are u₂, u₄.

step 5. The sequence of the best scheme is obtained according to the number of satisfying attributes:u₄ > u₅ > u₂ > u₁ > u₃. Therefore, the best plan is u₄, and the alternative plan is u₅.

The following is the calculation of this example based on the merit and demerit of fuzzy elements with score function in [19]. According to the above table, it can be obtained. S ( h u 1 ( x ) ) = 0.4 + 0.6 + 0.5 + 0.6 + 0.7 + 0.8 + 0.4 + 0.6 8 = 0.575 , S ( h u 2 ( x ) ) = 0.5 + 0.4 + 0.9 + 0.5 + 0.7 + 0.7 6 = 0.617 , S ( h u 3 ( x ) ) = 0.3 + 0.7 + 0.5 + 0.6 + 0.7 + 0.6 + 0.3 + 0.3 + 0.6 9 = 0.511 , S ( h u 4 ( x ) ) = 0.6 + 0.6 + 0.8 + 0.7 + 0.9 + 0.6 + 0.7 + 0.8 8 = 0.7125 , S ( h u 5 ( x ) ) = 0.6 + 0.7 + 0.65 + 0.75 + 0.5 + 0.6 6 = 0.63 . Then there is S (h _{u 4}  (x)) > S (h _{u 5}  (x)) > S (h _{u 2}  (x)) > S (h _{u 1}  (x)) > S (h _{u 3}  (x)), the sequence of schemes is: u₄ > u₅ > u₂ > u₁ > u₃. According to the resuile, it can be known that, the multi-attribute decision making algorithm based on granular computing of this paper is the same as that calculated by score function. It shows that the algorithm is effective and feasible.Compared with the method of calculating score function, this algorithm has the following characteristics:

(1) The ranking obtained by calculating the score function can only determine the merits of each scheme, but not the satisfaction degree of each attribute. This algorithm can not only sort the decision problems, but also visually see the properties of the objects, which can be more clear when making decisions to make a choice.

(2)The algorithm proposed in this paper is based on the granular computing. For multi-attribute problems,hesitant fuzzy hypergraphs can be constructed, by λ-cut the hesitant fuzzy hypergraph, where we can find out what we want in different grains. It is of better significance to explain the decision problem in choosing different attributes.

As a generalization of classical fuzzy sets, hesitant fuzzy sets are more accurate in processing fuzzy information because their membership degree is not unique. In this paper, the definition of hesitant fuzzy hypergraph is proposed and a new algorithm based on granular computing is proposed by combining a hesitant fuzzy graph with multi-attribute problem. Using the hesitant fuzzy hypergraph model,by calculating the score function of each attribute, each subgraph of the hesitant fuzzy hypergraph is obtained. The algorithm can intuitively show the attributes satisfied by each object and has a good explanatory ability for decision making, compared with conventional algorithms, the optimal decision can be made to the satisfaction of every decision maker.

The further study on multi-attribute decision making based on hesitant fuzzy hypergraph is still needed to overcome two shortcomings of the present work. First of all, it is difficult to get the hesitant fuzzy relational data, which influenced the description and analysis of each attributes. Second, for the attributes with different weights, how to use the hesitant fuzzy hypergraph to make decisions is the main direction of future research.