Directed hypergraphs under m -polar fuzzy environment

1 Introduction

In mathematics, graph theory is the knowledge of graphs used to ideal the pairwise relationship among objects. Graph theory is an efficacious structure used in different fields of life, including socially to scrutinize the diffusion mechanism, in biology node exemplify the regions of certain species and the edges symbolize the migration path between the regions. In mathematics, computer science and engineering, graphs are used for the designing and the result of combinatorial optimization. Graphs are only practicable for forming of the pairwise intercommunication. But in many problems, data deals with more than two objects. To handle such type of synergy, we use a hyperedge, as it contains more than two objects. Hypergraph is a elongated form of ordinary graphs as it contain a definable collection of objects. Hypergraphs have been broadly and profoundly studied in Berge [9]. In real-world problems, hypergraph techniques are more beneficial in many regions including declustering problems to enhance the enforcement of parallel databases [15]. In various fields of discrete mathematics, hypergraphs are used to model ideas and systems. The rewriting systems, problem solving, databases and logic programming can be described using hypergraphs [8]. In computer science, undirected hypergraphs [13] are suitable. But in deductive databases and in model checking, directed hypergraphs are suited. To model different combinatorial structures, directed hypergraphs are used.

In science and technology, there are many problematic phenomena, in which given data is not precise. To overcome such problems, we adopt such mathematical models that involve elements of ambiguity planted on fuzzy set theory. The notion of fuzzy set was given by Zadeh [25] in 1965 whose membership value lie in [0, 1]. In different fields of life including chemistry, economics, computer science, engineering, medical and decision-making problems, fuzzy sets are trifling a durable performance. In 1986, Atanassov [7] popularized the idea of intuitionistic fuzzy sets. There are many problems which based on two-sided knowledge, one is positive and other is negative. To deal them, in 1994, Zhang [26] introduced the bipolar fuzzy set as the continuation of fuzzy set whose membership function ranges over [-1, 1]. By using bipolar fuzzy models, a variety of decision-making problems based on two-sided data can be solved. Recently, the concept of bipolar fuzzy set was elongated to m-polar fuzzy set by Chen et al. [11] whose membership function lies in [0, 1] ^m , where m is an ordinal number representing the m-tuples aspects of an object. To represent the interrelationship between different individuals, m-polar fuzzy sets can be used. In various real world problems, information consists n parameters (n ≥ 2), that is, multiple information exist which cannot be handled by fuzzy sets and bipolar fuzzy sets because these are single and double information models, respectively. They do not give any information about multiple information, therefore, we need an m-polar fuzzy set.

The initial perception about fuzzy graph was given by Kaufmann [14] based on Zadeh’s fuzzy relation in 1973. The conception of fuzzy graphs structure was expressed by Rosenfeld [19] to deal with relations involving uncertainty. After that, Bhattacharya [10] gave some comments on fuzzy graphs. Mordeson and Chang-Shyh [17] considerded operations on fuzzy graphs. Akram and Sarwar [6] introduced novel applications of m-polar fuzzy competition graphs in decision support system. The idea of fuzzy hypergraphs was studied by Kaufmann [14]. The notion of interval-valued fuzzy hypergraphs was imported by Chen [12]. Lee-Kwang and keon-Myung [16] redefined the fuzzy hypergraph. The view of transversals of m-polar fuzzy hypergraph with applications was explained by Akram and Sarwar [5]. In 2013, Parvathi and Thilagavathi defined the intuitionistic fuzzy directed hypergraphs [20]. Myithili et al. [18] examined the certain types of intuitionistic fuzzy directed hypergraphs. Akram and Luqman [3] formalized the many concepts of bipolar fuzzy directed hypergraphs. Akram and Luqman [4] also illustrated a new decision-making method based on bipolar neutrosophic directed hypergraphs. In this research study, we introduce the notion of m-polar fuzzy directed hypergraphs, and present some properties. We depict certain operations on m-polar fuzzy directed hypergraphs. We also consider an application of m-polar fuzzy directed hypergraphs in businees strategy company. For other notations and applications, readers are referred to [1, 21–24].

2 Directed hypergraphs under m-polar fuzzy environment

Definition 2.1. [13] A directed hypergraph(DH) is a hypergraph with directed hyperedges. A directed hyperedge or hyperarc is an ordered pair E = (X, Y) of (possibly empty) disjoint subsets of vertices. X is the tail of E, while Y is its head.

Definition 2.2. [13] A sequence of crisp hypergraphs H _i = (V _i , E _i ), 1 ≤ i ≤ n, is said to ordered if H₁ ⊂ H₂ ⊂ …, H _n . The sequence {H _i  | 1 ≤ i ≤ n} is said to be simply ordered if it is ordered, and if whenever E ⊂ E_i+1 ∖ E _i , then E ⊈ V _i .

Definition 2.3. [11] An m-polar fuzzy set (or a [0, 1] ^m -set) on V is a mapping A : V → [0, 1] ^m . The degree of an element x ∈ V is denoted by A (x) = (P₁ ∘ A (x), P₂ ∘ A (x) , ⋯, P _m ∘ A (x)) , where P _i ∘ A : [0, 1] ^m → [0, 1] is the i- th projection mapping. The set of all m-polar fuzzy sets on V is denoted by m (V).

We note that [0, 1] ^m (m-power of [0, 1]) is considered a poset with the point-wise order ≤, where m is an arbitrary ordinal number (we make an appointment that m = {n|n < m} when m > 0), ≤ is defined by x ≤ y ⇔ p _i (x) ≤ p _i (y) for each i ∈ m (x, y ∈ [0, 1] ^m ), and p _i : [0, 1] ^m → [0, 1] is the i-th projection mapping (i ∈ m). 0 = (0, 0, …, 0) is the smallest element in [0, 1] ^m and 1 = (1, 1, …, 1) is the largest element in [0, 1] ^m .

Definition 2.4. An m-polar fuzzy directed hypergraph (mFDH, for short) with underlying set V is an ordered pair H = (σ, ɛ) where, σ is non-empty set of vertices and ɛ is a family of m-polar fuzzy (mF, for short) directed hyperarcs (or hyperedges).

An mF directed hyperarc (or hyperedge) e _i ∈ ɛ is an ordered pair (t (e _i ), h (e _i )), such that, t (e _i )≠ ∅, is called its tail and h (e _i ) ≠ t (e _i ) is its head, such that P k o ɛ i ( { v 1 , v 2 , . . . , v s } ) ≤ inf { P k o σ i ( v 1 ) , P k o σ i ( v 2 ) , . . . , P k o σ i ( v s ) } for all v₁, v₂, …, v _s ∈ V,  1 ≤ k ≤ m.

Definition 2.5. Let H = (σ, ɛ) be an mFDH. The order of H, denoted by O (H), is defined as O (H) = ∑_x∈V ∧ σ _i (x). The size of H, denoted by S (H), is defined by S (H) = ∑_{e k ⊂V}ɛ (e _k ).

In an mFDH, the vertices v _i and v _j are adjacent vertices if they both belong to the same mF directed hyperedge. Two mF directed hyperedges e _i and e _j are called adjacent if they have non-empty intersection. That is, supp (e _i )∩ supp (e _j ) ≠ ∅, i ≠ j.

Definition 2.6. An mFDH, H = (σ, ɛ) is simple if it contains no repeated directed hyperedges, i.e., if e _j , e _k ∈ ɛ and e _j ⊆ e _k then e _j = e _k . An mFDH, H = (σ, ɛ) is called support simple if e _j , e _k ∈ ɛ and supp (e _j ) = supp (e _k ) and e _j ⊆ e _k , then e _j = e _k . An mFDH, H = (σ, ɛ) is called strongly support simple if e _j , e _k ∈ ɛ and supp (e _j ) = supp (e _k ), then e _j = e _k .

Example 2.1. Consider a 3-polar fuzzy directed hypergraph H = (σ, ɛ), such that σ = {σ₁, σ₂, σ₃, σ₄, σ₅} is the family of 3-polar fuzzy subsets on V = {v₁, v₂, v₃, v₄, v₅, v₆}, as shown in Fig. 1, such that σ 1 = { ( v 1 , 0.1 , 0.2 , 0.3 ) , ( v 2 , 0.3 . , 0.4 , 0.4 ) , ( v 3 , 0.1 , 0.3 , 0.4 ) } , σ 2 = { ( v 5 , 0.4 , 0.3 , 0.3 ) , ( v 6 , 0.2 , 0.2 , 0.3 ) , ( v 7 , 0.1 , 0.1 , 0.4 ) } , σ 3 = { ( v 3 , 0.1 , 0.3 , 0.4 ) , ( v 4 , 0.4 , 0.3 , 0.2 ) , ( v 7 , 0.1 , 0.1 , 0.4 ) } .

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Fig.1

3-polar fuzzy directed hypergraph.

The 3-polar fuzzy relation ɛ is defined as ɛ (v₁, v₂, v₇) = (0.1, 0.1, 0.3), ɛ (v₅, v₆, v₇) = (0.1, 0.1, 0.3), ɛ (v₃, v₄, v₇) = (0.1, 0.1, 0.2). Clearly, H is simple, strongly support simple and support simple, that is, it contains no repeated directed hyperedges and if whenever e _j , e _k ∈ ɛ and supp (e _j ) = supp (e _k ), then e _j = e _k . Further, O (H) = (1.6, 1.8, 2.3) and S (H) = (0.3, 0.3, 0.8).

Definition 2.7. Let ɛ = (ɛ^-, ɛ⁺) be a directed mF hyperedge in an mFDH. Then the vertex set ɛ^- is called the mF in-set and the vertex set ɛ⁺ is called the mF out-set of the directed hyperedge ɛ. It is not necessary that the sets ɛ^-, ɛ⁺ will be disjoint. The hyperedge ɛ is called the join of the vertices of ɛ^- and ɛ⁺.

Definition 2.8. The in-degree D H - ( v ) of a vertex v in an mFDH is defined as the sum of membership degrees of all those directed hyperedges such that v is contained in their out-set, that is, D H - ( v ) = ∑ v ∈ h ( e i ) ɛ ( e i ) , 1 ≤ k ≤ m .

The out-degree D H + ( v ) of a vertex v in an mFDH is defined as the sum of membership degrees of all those directed hyperedges such that v is contained in their in-set, that is, D H + ( v ) = ∑ v ∈ t ( e i ) ɛ ( e i ) , 1 ≤ k ≤ m .

Definition 2.9. An mFDH, H = (σ, ɛ) is said to be k-regular if in-degrees and out-degrees of all vertices in H are same.

Example 2.2. Consider a 3-polar fuzzy directed hypergraph H = (σ, ɛ) as shown in Fig. 2, where σ = {σ₁, σ₂, σ₃, σ₄} is the family of 3-polar fuzzy subsets on V = {v₁, v₂, v₃, v₄, v₅, v₆} and σ 1 = { ( v 1 , 0.2 , 0.3 , 0.5 ) , ( v 2 , 0.2 , 0.3 , 0.5 ) , ( v 4 , 0.2 , 0.3 , 0.5 ) } , σ 2 = { ( v 4 , 0.2 , 0.3 , 0.5 ) , ( v 5 , 0.2 , 0.3 , 0.5 ) , ( v 6 , 0.2 , 0.3 , 0.5 ) } , σ 3 = { ( v 3 , 0.2 , 0.3 , 0.5 ) , ( v 5 , 0.2 , 0.3 , 0.5 ) , ( v 6 , 0.2 , 0.3 , 0.5 ) } , σ 4 = { ( v 1 , 0.2 , 0.3 , 0.5 ) , ( v 2 , 0.2 , 0.3 , 0.5 ) , ( v 3 , 0.2 , 0.3 , 0.5 ) } .

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Fig.2

Regular 3-polar fuzzy hypergraph.

By routine calculations, we see that the 3-polar fuzzy directed hypergraph is regular. Note that, D H - ( v 1 ) = ( 0.2 , 0.3 , 0.5 ) = D H + ( v 1 ) and D H - ( v 2 ) = ( 0.2 , 0.3 , 0.5 ) = D H + ( v 2 ) . Similarly, D H - ( v 3 ) = D H + ( v 3 ) , D H - ( v 4 ) = D H + ( v 4 ) , D H - ( v 5 ) = D H + ( v 5 ) . Hence H is regular 3-polar fuzzy directed hypergraph.

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Fig.3

Directed hyperpath (denoted by a thick line).

Definition 2.10. An mF directed hyperpath of length k in an mFDH is defined as a sequence v₁, e₁, v₂, e₂, …, e _k , v_k+1 of distinct vertices and directed hyperedges such that 1.

ɛ (e _i ) >0, i = 1, 2, …, k,

The consecutive pairs (v _i , v_i+1) are called the directed arcs of the directed hyperpath. The path is shown by a thick line in Fig. 3.

Definition 2.11. The incidence matrix of an mFDH, H = (σ, ɛ) is characterized by an n × m matrix [a _ij ] as follows: a ij = { P k o ɛ j ( v i ) , if v i ∈ ɛ j , 0 , otherwise .

Definition 2.12. An mFDH is called elementary if P _k oɛ _ij : V ⟶ [0, 1] ^m are constant functions, P _k oɛ _ij is taken as the membership degree of vertex i to hyperedge j.

Proposition 2.1. In an mFDH, when m-polar fuzzy vertices have constant membership degrees, then mF directed hyperedges are elementary.

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Fig.4

Elementary 3-polar fuzzy hypergraph.

Example 2.3. Consider a 3-polar fuzzy directed hypergraph H = (σ, ɛ), where σ = {σ₁, σ₂, σ₃} be the family of 3-polar fuzzy subsets on V = {v₁, v₂, v₃, v₄, v₅}. The corresponding incidence matrix is given in Table 1. The corresponding elementary 3-polar fuzzy directed hypergraph is shown in Fig. 4.

Definition 2.13. Let H = (σ, ɛ) be an mFDH. Suppose μ = (μ₁, μ₂, …, μ _m ) ∈ [0, 1] ^m . The μ -level is defined as ɛ _μ = {v ∈ σ | P _k oσ (v) ≥ μ _k }. The crisp directed hypergraph H _μ = (σ _μ , ɛ _μ ), such that:

ɛ _μ = {v ∈ σ | P _k oσ (v) ≥ μ _k },  1 ≤ k ≤ m.

Definition 2.14. Let H = (σ, ɛ) be an mFDH and H _μt = (σ _μt , ɛ_μtnbi> ) be the μ_t -level DHs of H. The sequence { μ₁ , μ₂ , μ₃ , …, μ_nitii } of m-tuples, where μ₁ > μ₂ > … μ_nitii > 0 and μ_nitii = h (H)(height of mFDH), such that the following properties: 1.

If μ_i+1 < α ≤ μ_t , then ɛ _α = ɛ _μt ,

are satisfied, is called a fundamental sequence (FS) of H. The sequence is denoted by FS (H). The μ_t -level hypergraphs {H _μ1 , H _μ2 , …, H _μnitii } are called the core hypergraphs of H. This is also called core set of H and is denoted by c (H).

Definition 2.15. Let H = (σ, ɛ) be an mFDH and FS (H) = { μ₁, μ₂, μ₃, …, μ_nitii }. If for each e ∈ ɛ and each μ_t ∈ FS (H), e _μ = ɛ _{μ t} , for all μ ∈ (μ₊₁, μ_tnbi>] , then H is called sectionally elementary.

Definition 2.16. Let H = (σ, ɛ) be an mFDH and c (H) = {H _μ1 , H _μ2 , …, H _μnitii }. H is said to be ordered if c (H) is ordered. That is, H _μ1 ⊂ H _μ2 ⊂ …⊂ H _μnitii . The mFDH is called simply ordered if the sequence {H _μ1 , H _μ2 , …, H _μnitii } is simply ordered.

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Fig.5

3-polar fuzzy directed hypergraph.

Example 2.4. Consider a 3-polar fuzzy directed hypergraph H = (σ, ɛ) as shown in Fig. 5 and given by incidence matrix in Table 2. By computing the μ _i -level 3-polar fuzzy directed hypergraphs of H, we have ɛ_{(0.8,0.6,0.5)} = {v₂, v₃}, ɛ_{(0.6,0.4,0.3)} = {v₂, v₃} and ɛ_{(0.5,0.3,0.2)} = {v₂, v₃, v₅, v₆}. Note that H_{(0.8,0.6,0.5)} = H_{(0.6,0.4,0.3)} and H_{(0.8,0.6,0.5)} ⊆ H_{(0.5,0.3,0.2)}. The fundamental sequence is FS (H) = {(0.8, 0.6, 0.5), (0.5, 0.3, 0.2)}.

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Fig.6

H induced fundamental sequence.

Furthermore, H_{(0.8,0.6,0.5)} ≠H_{(0.6,0.4,0.3)}. H is not sectionally elementary since ɛ_2(μ) ≠ ɛ_{2(0.8,0.6,0.5)} for μ = (0.6, 0.4, 0.3). The 3-polar fuzzy directed hypergraph is ordered, and the set of core hypergraphs is c (H) = {H₁ = H_{(0.8,0.6,0.5)}, H₂ = H_{(0.5,0.3,0.2)}}. The induced fundamental sequence of H is given in Fig. 6.

Proposition 2.2. Let H = (σ, ɛ) be an mFDH, the following conditions hold: \begin description\item [(a)] If H = (σ, ɛ) is an elementary mFDH, then H is ordered. \item [(b)] If H is an ordered mFDH with c (H) = {H _μ1 , H _μ2 , …, H _μnitii } and if H _μnitii is simple, then H is elementary. \end description

Definition 2.17. Let H = (σ, ɛ) be an mFDH. The index matrix of H is defined by

Now we present certain operations on mFDHs.

Definition 2.18. Let H₁ = (σ₁, ɛ₁) and H₂ = (σ₂, ɛ₂) be two mFDHs. The addition of two mFDHs over a fixed set V is denoted by H₁boxplusH₂ = (σ₁ ∪ σ₂, ɛ₁ ∪ ɛ₂) and defined as, P k o ( σ 1 ∪ σ 2 ) ( v r ) = { P k o σ 1 ( v r ) if v r ∈ σ 1 \ σ 2 , P k o σ 2 ( v r ) if v r ∈ σ 2 \ σ 1 , max { P k o σ 1 ( v r ) , P k o σ 2 ( v r ) } if v r ∈ σ 1 ∩ σ 2 , 0 otherwise .(1) P k o ( ɛ 1 ∪ ɛ 2 ) ( e rs ) = { P k o ɛ 1 ( e ij ) if v r = v i ∈ σ 1 and v s = v j ∈ σ 1 \ σ 2 , P k o ɛ 2 ( e p q ) if v r = v p ∈ σ 2 and v s = v q ∈ σ 2 \ σ 1 , max { P k o ɛ 1 ( e ij ) , if v r = v i = v p ∈ σ 1 ∩ σ 2 and P k o ɛ 2 ( e pq ) } v s = v j = v q ∈ σ 1 ∩ σ 2 , 0 otherwise .(2)

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Fig.7

H₁.

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Fig.8

H₂.

Example 2.5. Let H₁ = (σ₁, ɛ₁) and H₂ = (σ₂, ɛ₂) be two 3-polar fuzzy directed hypergraphs, where σ₁={v₁, v₂, …, v₅}, ɛ₁ = {({v₁, v₂}, v₃), ({v₁, v₄}, v₅), {{v₂}, v₅} and σ₂ = {v₁, v₂, …, v₆}, ɛ₂ = {({v₁, v₂}, v₅), ({v₄, v₆}, v₃), {{v₁, v₄}, v₆} as shown in Figs. 7 and 8, respectively. The index matrix of H₁ is given in Table 4, where σ₁ = {v₁, v₂, …, v₅}. The index matrix of H₂ is given in Table 5, where σ₂ = {v₁, v₂, …, v₆}. The index matrix of H₁boxplusH₂ is given in Table 6, where σ₁ ∪ σ₂ = {v₁, v₂, …, v₆}. The membership functions for vertices and edges are calculated by utilizing Equations 1 and 2, respectively. The graph of H₁boxplusH₂ is shown in the Fig. 9.Definition 2.19. Let H₁ = (σ₁, ɛ₁) and H₂ = (σ₂, ɛ₂) be two mFDHs. The vertex-wise multiplication of two mFDHs over a fixed set V is denoted by H₁ ⊗ H₂=(σ₁ ⊗ σ₂, ɛ₁ ⊗ ɛ₂) and defined as, P k o ( σ 1 ⊗ σ 2 ) = min { P k o σ 1 ( v r ) , P k o σ 2 ( v r ) } if v r ∈ σ 1 ∩ σ 2 ,(3) P k o ( ɛ 1 ⊗ ɛ 2 ) ( e rs ) = min { P k o ɛ 1 ( e ij ) , P k o σ 2 ( e pq ) } if v r = v i = v p ∈ σ 1 ∩ σ 2 , v s = v j = v q ∈ σ 1 ∩ σ 2 .(4)

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Fig.9

H₁boxplusH₂.

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Fig.10

H₁ ⊗ H₂.

Example 2.6. Let H₁ = (σ₁, ɛ₁) and H₂ = (σ₂, ɛ₂) be two 3-polar fuzzy directed hypergraphs as shown in Figs. 7 and 8 respectively. The index matrix of H₁ ⊗ H₂ is shown in Table 7, where σ₁ ∩ σ₂ = {v₁, v₂, …, v₅}. The membership functions for vertices and edges are calculated by applying Equations 3 and 4, respectively. The graph of H₁ ⊗ H₂ is shown in the Fig. 10.

Definition 2.20. Let H₁ = (σ₁, ɛ₁) and H₂ = (σ₂, ɛ₂) be two mFDHs. The structural subtraction of two mFDHs over a fixed set V is denoted by H₁boxminusH₂=(σ₂ - σ₁, ɛ₂ - ɛ₁) and defined as,

P k o ( σ 2 - σ 1 ) ( v r ) = { P k o σ 1 ( v r ) if v r ∈ σ 1 , P k o σ 2 ( v r ) if v r ∈ σ 2 , 0 otherwise .(5)

P k o ( ɛ 2 - ɛ 1 ) ( e rs ) = P k o ɛ 1 ( e ij ) if v r = v i ∈ σ 2 - σ 1 and v s = v j ∈ σ 2 - σ 1 .(6) The graph H₁boxminusH₂ is empty when σ₂ - σ₁ =∅.

Example 2.7. Let H₁ = (σ₁, ɛ₁) and H₂ = (σ₂, ɛ₂) be two 3-polar fuzzy directed hypergraphs as shown in Figs. 7 and 8 respectively. The index matrix of H₁boxminusH₂ is shown in Table 8, where σ₂ - σ₁ = {v₆}. The membership functions for vertices and edges are calculated by applying Equations 5 and 6, respectively.

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Fig.11

H₁boxminusH₂.

The graph H₁boxminusH₂ is shown in following Fig. 11.Definition 2.21. Let H₁ = (σ₁, ɛ₁) and H₂ = (σ₂, ɛ₂) be two mFDHs. The multiplication of two mFDHs H₁ and H₂, denoted by H₁ ⊙ H₂ = (σ₁ ⊙ σ₂, ɛ₁ ⊙ ɛ₂), is defined as P _k o (σ₁ ⊙ σ₂) (v _r ) = { P k o σ 1 ( v r ) if v r ∈ σ 1 , P k o σ 2 ( v r ) if v r ∈ σ 2 , min { P k o σ 1 ( v r ) , P k o σ 2 ( v r ) } if v r ∈ σ 1 ∩ σ 2 .(7) P k o ( ɛ 1 ⊙ ɛ 2 ) ( e rs ) = { P k o ɛ 1 ( e ij ) if v r = v i ∈ σ 1 , and v s = v j ∈ σ 1 \ σ 2 P k o ɛ 2 ( e pq ) if v r = v p ∈ σ 2 and v s = v q ∈ σ 2 \ σ 1 , max i , q { min j , p { P k o ɛ 1 ( e ij ) , if v r = v i ∈ σ 1 ∩ σ 2 P k o ɛ 2 ( e pq ) } } and v s = v q ∈ σ 1 ∩ σ 2(8)

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Fig.12

H₁ ⊙ H₂.

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Fig.13

3-polar fuzzy directed hypergraph model between investors and business strategy companies.

Example 2.8. The index matrix of graph H₁ ⊙ H₂ is shown in Table 9, where σ₂ ∪ (σ₁ - σ₂) = {v₁, v₂, v₃, …, v₆} is given in Table 9. The membership values of vertices and edges are calculated by applying formulae 7 and 8. The graph of H₁ ⊙ H₂ is shown in Fig. 12.

Decision-making is regarded as the intellectual process resulting in the selection of a belief or a course of action among several alternative possibilities. Every decision-making process produces a final choice, which may or may not prompt action. Decision-making is the process of identifying and choosing alternatives based on the values, preferences and beliefs of the decision-maker. Problems in almost every credible discipline, including decision-making can be handled using graphical models.

Business strategy company: A business strategy is a registered plan on how an organization is setting out to fulfill their ambitions. A business strategy have a variety of successful key of principles that sketch how a company will go about achieving their dreams in business. It deals with competitors, look at their needs and expectations of customers and will examine the long term growth and sustainability of their organization.

In this fast running world where every investor is searching out a best business strategy company so that they invest their money on the company to promote the business and to compete their competitors. Then to select a good marketing business company which will achieve its goals, meet the expectations and sustain a competitive advantage in the marketplace, we develop a 3-polar fuzzy directed hypergraphical model that how an investor can choice the greatest salubrious company to promote the business by following a step by step procedure. A 3-polar fuzzy directed hypergraph demonstrating a group of investors as members of different business strategy companies is shown in Fig. 13.

If an investor wants to adopt the most suitable and powerful business company to which he works and get the progress in business, the following procedure can help the investors. Firstly, one should think about the cooperative contribution of investors towards the company, which can be find out by means of membership values of 3-polar fuzzy directed hypergraphs. The membership values given in Table 9 show the collective interest of investors towards the company.

The first membership value showing the how much investors invest money on company, second showing the sharp-minded quality of investors to run the business and third showing how can strongly they make production by working with company. It can be noticed that the company C has strong collective interest of investors which is maximum among all others companies. Secondly, one should do his research on the powerful impacts of all under consideration companies on their investors. The membership degrees of all company nodes show their effects on their investors as given in Table 10.

The membership values showing three different positive effects of company on investor, first one shows how much a company is financially strong already, second showing its business growth in the market and third one showing the strong competitive position of company. Note that, company C has most benefits for investors. Thirdly, an investor can observe the influence of a company by calculating its in-degrees and out-degrees. In-degrees show the percentage of investors joining the company and out-degrees show the percentage of investors leaving that company. The in-degrees and out-degrees of all business strategy companies are given in Tables 11 and 12.

Hence, a best business strategy company has maximum in-degrees and minimum out-degrees. However, in case when two companies have same minimum out-degrees, then we compare their in-degrees. Similarly, when in-degrees same, we compare out-degrees. From all above discussion, we conclude that company C is the most appropriate company to fulfill the requirements of the investors because it is more financially strong, best in competitive position and business growth of this company is more suitable to run the business and compete the competitors. The method of searching out the constructive and profitable business strategy company is explained in the following algorithm.

Algorithm 1. Input the membership values of all nodes (investors) v₁, v₂, …, v _n . 2. Determine the augmentation of investors towards companies by calculating the membership values of all directed hyperedges as: P k o ɛ r ≤ inf { P k ov 1 , P k ov 2 , . . . , P k ov n } , 1 ≤ k ≤ m . 3. Obtain the most suitable company as: max P k o ɛ r . 4. Find the company having strong and more benefits for investors as: max P k ov r . where all v _r here are vertices represent the different business strategy company.\\ 5. Find the profitable influence of companies v _r on the investors by calculating the in-degrees D^- (v _r ) as: ∑ v r ∈ h ( ɛ r ) P k o ɛ r . 5. Find the profitless impact of companies v _k on the investors by calculating the out-degrees D⁺ (v _r ) as: ∑ v r ∈ t ( ɛ r ) P k o ɛ r . 6. Obtain the most advantageous business strategy company as: ( max D - ( v r ) , min D + ( v r ) ) .

The algorithm runs linearly and its net time complexity is O(n) where, n is the number of membership values of all nodes(investors).

A directed hypergraph is a generalization of directed graphs in which each directed hyperedge has one or more source (tail) nodes and has one or more destination (head) nodes. Directed hypergraphs have fructiferous appliances in different fields of life including medical, social analysis, network modeling and many others fields. In real world problems, data comes from m factors, that is, multiple information arises which can not be disentangled by fuzzy system and any other system, so we need m-ploar fuzzy set to handle the multiple uncertainty and vagueness in data. m-polar fuzzy set are helpful to deal m-tuples properties of an object. mFDH are playing a useful model in handling many complications of life. One of the most effective use of mFDH is that it is helpful to handle problems of business. In this research paper, we have studied the certain concepts and operations of mFDHs and use of mFDH in decision-making problem to solve our real world problem. We will extend our research work on hybrid models of mF rough sets such as, (1) soft mF rough hypergraphs, (2) rough soft mF hypergraphs.