A New Method to Decision-Making with Fuzzy Competition Hypergraphs

: Hypergraph theory is the most developed tool for demonstrating various practical problems in different domains of science and technology. Sometimes, information in a network model is uncertain and vague in nature. In this paper, our main focus is to apply the powerful methodology of fuzziness to generalize the notion of competition hypergraphs and fuzzy competition graphs. We introduce various new concepts, including fuzzy column hypergraphs, fuzzy row hypergraphs, fuzzy competition hypergraphs, fuzzy k -competition hypergraphs and fuzzy neighbourhood hypergraphs, strong hyperedges, k th strength of competition and symmetric properties. We design certain algorithms for constructing different types of fuzzy competition hypergraphs. We also present applications of fuzzy competition hypergraphs in decision support systems, including predator–prey relations in ecological niche, social networks and business marketing.


Introduction
In mathematical modeling, competition graphs are sufficient to specify well defined behaviors of objects and specifically predator-prey relations.In 1968, while studying applications of graph theory in ecology, Cohen introduced the notion of a competition graph.Competition graphs have been applied to various fields of biological sciences and technology.After the strong motivation of energy and food competition in food webs between species, competition graphs were a part of active research in recent years.In 2004, Sonntag and Teichert [1] introduced the notion of competition hypergraphs.These representations are crisp hypergraphs that do not describe all the competitions of real-world problems.These models contain uncertainty and fuzzy in nature for problems that are more relevant to everyday life, including critical writing style of a writer, predator-prey relationship, trading relationship among different communities, honesty leadership quality of a politician and, signal strength of wireless devices.Motivating from this idea, we have applied the notion of fuzzy sets to competition hypergraphs to study the problems having nonlinear uncertainties.
In 1965, Zadeh [2] introduced the strong mathematical notion of fuzzy set in order to discuss the phenomena of vagueness and uncertainty in various real-life problems.Using the concept of fuzzy relations introduced by Zadeh [3], the idea of fuzzy graph was given by Kaufmann [4].The fuzzy relations in fuzzy sets were studied by Rosenfeld [5] and he introduced the structure of fuzzy graphs, obtaining analysis of various graph theoretical concepts.Lee-kwang and Lee [6]  extended the notion of fuzzy hypergraphs whose idea was first discussed by Kaufmann [4].Later, the idea of fuzzy hypergraph was studied by Goetschel in [7,8].The concept of interval-valued fuzzy hypergraphs was initiated by Chen [9] and Parvathi et al. [10] generalized the idea of hypergraphs to intuitionistic fuzzy hypergraphs.Moreover, Akram and Dudek [11], Akram and Luqman [12][13][14], and Akram and Shahzadi [15] have discussed certain extensions of fuzzy hypergraphs with applications.Samanta and Pal [16] studied fuzzy k-competition graphs and p-competition graphs.Later, Samanta et al. [17] introduced the concept of m-step fuzzy competition graphs.Applying the idea of bipolar fuzzy sets to competition graphs, Alshehri and Akram [18] introduced the notion of bipolar fuzzy competition graphs and applied this idea to economic systems.Furthermore, the study of bipolar fuzzy competition graphs was discussed by Sarwar and Akram in [19].Certain competition graphs based on neutrosophic environment were described in [20,21].In this research paper, we introduce the concept of fuzzy competition hypergraphs as a generalized case of fuzzy competition graphs.We study various new concepts, including fuzzy column hypergraphs, fuzzy row hypergraphs, fuzzy competition hypergraphs, fuzzy k-competition hypergraphs and fuzzy neighbourhood hypergraphs and investigate some of their interesting properties.We design certain algorithms for the construction of different types of fuzzy competition hypergraphs.We also present applications of fuzzy competition hypergraphs in decision support systems, including food webs, social networks and business marketing.

Fuzzy Competition Hypergraphs
In this section, we discuss various types of fuzzy competition hypergraphs with certain properties and algorithms.Definition 2. Let A = [x ij ] n×n be the adjacency matrix of a fuzzy digraph #» G = (µ, #» λ ) on a non-empty set X.The fuzzy row hypergraph of #» G, denoted by R • H( #» G) = (µ, λ r ), having the same set of vertices as #» G and the set of hyperedges is defined as {x 1 , x 2 , . . ., x r }|A(x ij ) > 0, r ≥ 2, for each 1 ≤ i ≤ r, x i ∈ X, for some 1 ≤ j ≤ n .
The degree of membership of hyperedges is defined as #» G and the set of hyperedges is defined as {x 1 , x 2 , . . ., x s }|A(x ji ) > 0, s ≥ 2, for each 1 ≤ i ≤ s, x i ∈ X, for some 1 ≤ j ≤ n .
The degree of membership of hyperedges is defined as The methods for computing fuzzy row hypergraph and fuzzy column hypergraph are given in Algorithms A1 and A2, respectively.Example 1.Consider the universe X = {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 }, µ a fuzzy set on X and #» λ a fuzzy relation in X as defined in Tables 1 and 2, respectively.The fuzzy digraph #» G = (µ, #» λ ) is shown in Figure 1.The adjacency matrix of #» G is given in Table 3.Using Algorithm A1 and Table 3, there are three hyperedges E 2 = {x 1 , x 5 , x 6 }, E 3 = {x 2 , x 5 } and E 4 = {x 3 , x 5 }, corresponding to the columns x 2 , x 3 and x 4 of adjacency matrix, in fuzzy row hypergraph of #» G.The membership degree of the hyperedges is calculated as The fuzzy row hypergraph is shown in Figure 2. Using Algorithm A2 and Table 3, the hyperedges in fuzzy column hypergraph of x 5 }, corresponding to the rows x 2 , x 5 and x 6 of the adjacency matrix.The membership degree of the hyperedges is calculated as The fuzzy column hypergraph is given in Figure 3.The degree of membership of hyperedges is defined as,

having the same set of vertices as
#» G and the set of hyperedges is defined as, The degree of membership of hyperedges is defined as, The methods for computing fuzzy row hypergraph and fuzzy column hypergraph are given in Algorithm 4.1 and 4.2, respectively.
Example 2.1.consider the universe X = {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 }, µ a fuzzy set on X and #» λ a fuzzy relation in X as defined in

Table 3: Adjacency matrix
and Table 3, there are three hyperedges E 2 = {x 1 , x 5 , x 6 }, E 3 = {x 2 , x 5 } and E 4 = {x 3 , x 5 }, corresponding to the columns x 2 , x 3 and x 4 of adjacency matrix, in fuzzy row hypergraph of #» G.The membership degree of the hyperedges is calculated as, The fuzzy row hypergraph is shown in Fig. 2.2.Using Algorithm 4.2 and Table 3, the hyperedges in fuzzy column hypergraph of } and E 6 = {x 2 , x 5 }, corresponding to the rows x 2 , x 5 and x 6 of the adjacency matrix.The membership degree of the hyperedges is calculated as, The fuzzy column hypergraph is given in Fig. 2  rresponding to the rows x 2 , x 5 and x 6 of the adjacency matrix.The membership degree of the peredges is calculated as, Definition 5. [16] A fuzzy out neighbourhood of a vertex x of a fuzzy digraph Definition 6. [16] The fuzzy in neighbourhood of vertex x of a fuzzy digraph is a fuzzy set ) be a fuzzy digraph on a non-empty set X.The fuzzy competition hypergraph CH( #» G) = (µ, λ c ) on X having the same vertex set as #» G and there is a hyperedge consisting of vertices x 1 , x 2 , . . . ,x s if N + (x 1 ) ∩ N + (x 2 ) ∩ . . .∩ N + (x s ) = ∅.The degree of membership of hyperedge E = {x 1 , x 2 , . . ., x s } is defined as The method for constructing fuzzy competition hypergraph of a fuzzy digraph is given in Algorithm A3.

Lemma 1. The fuzzy competition hypergraph of a fuzzy digraph
#» G is a fuzzy row hypergraph of #» G.
Example 2. Consider the fuzzy digraph given in Figure 1.The fuzzy out neighbourhood and fuzzy in neighbourhood of all the vertices are given in Table 4.

For hyperedge
x 1 x 2 x 3 x 4 x 5 x 6 x 1 x 2 x 3 x 4 x 5 x 6  G and there is hyperedge consisting of vertices The method for the construction of fuzzy double competition hypergraph is given in Algorithm 4.4.
Lemma 2.2.The fuzzy double competition hypergraph is the intersection of fuzzy row hypergraph and fuzzy column hypergraph.
The method for the construction of fuzzy double competition hypergraph is given in Algorithm A4.

Proof. Let
#» G = (µ, #» λ ) be a fuzzy digraph; then, for any hyperedge It follows that the fuzzy double competition hypergraph is the intersection of a fuzzy row hypergraph and fuzzy column hypergraph.show that the fuzzy double competition hypergraph is the intersection of fuzzy row hypergraph and fuzzy column hypergraph.

Definition 9. Let
G and there is hyperedge consisting of vertices x 1 , x 2 , . . ., The fuzzy niche hypergraph is the union of fuzzy row hypergraph and fuzzy column hypergraph.
Example 4. The fuzzy niche hypergraph of Figure 1 is shown in Figure 7, which is the union of Figures 2 and 3.
Proof.On contrary assume that for some y ∈ X either supp(N Then by definition of fuzzy niche hypergraph, z is adjacent to all vertices X ′ in H ∪ I t .A contradiction to the fact that z ∈ X t . Lemma 2.5.Let H be a fuzzy hypergraph with n(H) = t < ∞ and H ∪ I t is a fuzzy niche hypergraph of an acyclic fuzzy digraph #» G then for all z ∈ X t , N + (z) = ∅ and N − (z) = ∅.
Proof.On contrary assume that X + z = {y 1 , y 2 , . . ., y s } and Proof.On the contrary, assume that, for some y ∈ X, either supp(N Then, by definition of a fuzzy niche hypergraph, z is adjacent to all vertices X in H ∪ I t -a contradiction to the fact that z ∈ X t . Lemma 5. Let H be a fuzzy hypergraph with n(H) = t < ∞ and H ∪ I t is a fuzzy niche hypergraph of an acyclic fuzzy digraph #» G then for all z ∈ X t , N + (z) = ∅ and N − (z) = ∅.

Definition 12.
Let k be a non-negative real number number; then, the fuzzy k-competition hypergraph of a fuzzy digraph , which has the same fuzzy vertex set as in #» G and there is a hyperedge The membership degree of the hyperedge E is defined as , Example 5.The fuzzy 0.2−competition hypergraph of Figure 1 is given in Figure 8.
Proof.Assume that 2 by Definition 2.10.Definition 2.11.Let k be a non-negative real number number then the fuzzy k−competition hypergraph of a fuzzy digraph which has the same fuzzy vertex set as in #» G and there is a hyperedge The membership degree of the hyperedge E is defined as, where,  Remark 1.For k = 0, a fuzzy k-competition hypergraph is simply a fuzzy competition hypergraph.

Fuzzy Neighbourhood Hypergraphs
The concepts of fuzzy open neighbourhood and fuzzy closed neighbourhood are given in Definition 13.Definition 13. [16] The fuzzy open neighbourhood of a vertex y in a fuzzy graph G = (µ, λ) is a fuzzy set N (y) = (X y , µ y ), where X y = {w|λ(yw) > 0} and µ y : X y → [0, 1] a membership function defined by µ y (w) = λ(yw).
The fuzzy closed neighbourhood hypergraph is defined on the same lines in the following definition.
Example 6.Consider the fuzzy graph G = (µ, λ) on set Y = {y 1 , y 2 , y 3 , y 4 } as shown in Figure 9.The fuzzy open neighbourhoods are given in Table 5.
Define a relation f : X → X by f (y i ) = y j if y j ∈ supp(N (y i )) as shown in Figure 10.
The degree of membership of each hyperedge can be computed using Definition 15 as follows.

Definition 17.
Let k be a non-negative real number; then, the fuzzy (k)-competition hypergraph of a fuzzy graph G = (µ, λ) is a fuzzy hypergraph N k (G) = (µ, λ kc ) having the same fuzzy set of vertices as G and there is a hyperedge where Definition 19.[16] Let #» G = (µ, #» λ ) be a fuzzy digraph.The underlying fuzzy graph of The relations between fuzzy neighbourhood hypergraphs and fuzzy competition hypergraphs are given in the following theorems.

Theorem 3. Let
#» G = (µ, #» λ ) be a symmetric fuzzy digraph without any loops; then, C k ( Clearly, the fuzzy k-competition hypergraph C k ( #» G) and the underlying fuzzy graph have the same fuzzy set of vertices as #» G. Hence, N k (U ( #» G)) has the same vertex set as #» G.It remains only to show that λ kc (xw) = λ kc (xw) for every x, w ∈ X.Thus, there are two cases.
Case 1: If, for each Case 2: If, for some Hence, λ kc (E) = λ kc (E).Since x 1 , x 2 , . . ., x r were taken to be arbitrary, the result holds for all hyperedges E of C k ( #» G).
Proof.Let U ( #» G) = (µ, λ) be an underlying fuzzy graph corresponding to fuzzy digraph as well as the underlying fuzzy graph have the same vertex set as ] has the same fuzzy vertex set as #» G.It remains only to show that λ kc ({x 1 , x 2 , . . ., x r }) = λ kc ({x 1 , x 2 , . . ., x r }) for every x 1 , x 2 , . . . ,x r ∈ X.As the fuzzy digraph has a loop at every vertex, the fuzzy out neighbourhood contains the vertex itself.There are two cases.

Applications of Fuzzy Competition Hypergraphs
In this section, we present several applications of fuzzy competition hypergraphs in food webs, business marketing and social networks.

Identifying Predator-Prey Relations in Ecosystems
We now present application of fuzzy competition hypergraphs in order to describe the interconnection of food chains between species, flow of energy and predator-prey relationship in ecosystems.The strength of competition between species represents the competition for food and common preys of species.We will discuss a method to give a description of species relationship, danger to the population growth rate of certain species, powerful animals in ecological niches and lack of food for weak animals.
Competition graphs arose in connection with an application in food webs.However, in some cases, competition hypergraphs provide a detailed description of predator-prey relations than competition graphs.In a competition hypergraph, it is assumed that vertices are defined clearly but in real-world problems, vertices are not defined precisely.As an example, species may be of different type like vegetarian, non-vegetarian, weak or strong.
Fuzzy food webs can be used to describe the combination of food chains that are interconnected by a fuzzy network of food relationship.There are many interesting variations of the notion of fuzzy competition hypergraph in ecological interpretation.For instance, two species may have a common prey (fuzzy competition hypergraph), a common enemy (fuzzy common enemy hypergraph), both common prey and common enemy (fuzzy competition common enemy hypergraph), and either a common prey or a common enemy (fuzzy niche hypergraph).We now discuss a type of fuzzy competition hypergraph in which species have common enemies known as fuzzy common enemy hypergraph.
Let #» G = (µ, #» λ ) be a fuzzy food web.The fuzzy common enemy hypergraph CH( G and there is a hyperedge consisting of vertices x 1 , x 2 , . . ., The strength of common enemies between species can be calculated using Algorithm A3.Consider the example of a fuzzy food web of 13 species giraffe, lion, vulture, rhinoceros, African skunk, fiscal shrike, grasshopper, baboon, leopard, snake, caracal, mouse and impala.The degree of membership of each species represents the species' ability of resource defence.The degree of membership of each directed edge represents the strength to which the prey is harmful for the predator.The fuzzy food web is shown in Figure 12.The directed edge between the giraffe and the lion shows that the giraffe is eaten by the lion and similarly.
The degree of membership of the lion is 0.9, which shows that the lion has 90% ability of resource defence, i.e., it can defend itself against other animals as well as survive many days if the lion does not find any food.The directed edge between giraffe and lion has degree of membership 0.25, which represents that the giraffe is 25% harmful for the lion because a giraffe can kill a lion with its long legs.This is an acyclic fuzzy digraph.The fuzzy out neighbourhoods are given in Table 7.
The fuzzy common-enemy hypergraph is shown in Figure 13.The hyperedges in Figure 13 show that there are common enemies between giraffe and rhinoceros, rhinoceros, African skunk and leopard, grasshopper and snake, mouse and impala, and baboon and impala.The membership value of each hyperedge represents the degree of common enemies among the species.
The hyperedge {impala, baboon} has a maximum degree of membership, which shows that the impala and the baboon have the largest number of common enemies, whereas the mouse and the impala have the least number of common enemies.{x 1 , x 2 , . . ., x s } is defined as, The strength of common enemies between species can be calculated using Algorithm 4.3.Consider the example of a fuzzy food web of 13 species giraffe, lion, vulture, rhinoceros, African skunk, fiscal shrike, grasshopper, baboon, leopard, snake, caracal, mouse and impala.The degree of membership of each specie represents the specie's ability of resource defence.The degree of membership of each directed edge represents the strength to which the prey is harmful for predator.The fuzzy food web is shown in Fig. 3.1.The directed edge between giraffe and lion shows that giraffe is eaten by lion and similarly.The degree of membership of lion is 0.9, which shows that lion has 90% ability of resource defence, i.e, it can defend itself against other animals as well as can survive many days if the lion does'nt find any food.The directed edge between giraffe and lion has degree of membership 0.25 which represents that giraffe is 25% harmful for lion because a giraffe can kill a lion with its long legs.This is an acyclic fuzzy digraph.The fuzzy out neighbourhoods are given in Table 7.
The fuzzy common-enemy hypergraph is shown in Fig. 3.2.The hyperedges in Fig. 3.2 show that there are common enemies between giraffe and rhinoceros, rhinoceros, African skunk and leopard, grasshopper and snake, mouse and impala, baboon and impala.The membership value of each hyperedge represents the degree of common enemies among the species.
The hyperedge {impala, baboon} has maximum degree of membership which shows that impala and baboon has largest number of common enemies whereas mouse and impala has least number of common enemies.

Identifying Competitors in Business Market
Fuzzy competition hypergraphs are a key approach to study the competition, profit and loss, market power and rivalry among buyers and sellers using fuzziness in hypergraphical structures.We now discuss a method to study the business competition for power and profit, success and business failure and, demanding products in market.
In business market, there are competitive rivalries among companies which are endeavoring to increase the demand and profit of their product.More than one companies in market sell the identical products.Since various companies regularly market identical of products, every

Identifying Competitors in the Business Market
Fuzzy competition hypergraphs are a key approach to studying the competition, profit and loss, market power and rivalry among buyers and sellers using fuzziness in hypergraphical structures.We now discuss a method to study the business competition for power and profit, success and business failure, and demanding products in market.
In the business market, there are competitive rivalries among companies that are endeavoring to increase the demand and profit of their product.More than one company in the market sells identical products.Since various companies regularly market identical products, every company wants to attract a consumer's attention to its product.There is always a competitive situation in the business market.Hypergraph theory is a key approach to studying the competitive behavior of buyers and sellers using structures of hypergraphs.In some cases, these structures do not study the level of competition, profit and loss between the companies.As an example, companies may have different reputations in the market according to market power and rivalry.These are fuzzy concepts and motivates the necessity of fuzzy competition hypergraphs.The competition among companies can be studied using a fuzzy competition hypergraph known as fuzzy enmity hypergraph.
We present a method for calculating the strength of competition of companies in the following Algorithm 1.

1.
Input the adjacency matrix [x ij ] n×n of bipolar fuzzy digraph Construct the table of fuzzy out neighbourhoods of all the companies.
do i from 1 → n 5.
Calculate the degree of each vertex as, S(x i ) = ∑ x i ∈E λ c (E) where E is a hyperedge in fuzzy enmity hypergraph.6. end do 7.
S(x i ) denotes the strength of competition of each company x i , 1 ≤ i ≤ n.
Consider the example of a marketing competition between seven companies DEL, CB, HW, AK, LR, RP, SONY, RA, LR, three retailers, one retailer outlet and one multinational brand as shown in Figure 14.The vertices represent companies, retailers, outlets and brands.The degree of membership of each vertex represents the strength of rivalry (aggression) of each company in the market.The degree of membership of each directed edge # » xy represents the degree of rejectability of company's x product by company y.The strength of competition of each company can be discussed using fuzzy competition hypergraph known as fuzzy enmity hypergraph.The fuzzy out neighbouhoods are calculated in Table 8.The fuzzy enmity hypergraph of Figure 14 is shown in Figure 15.The degree of membership of each hyperedge shows the strength of rivalry between the companies.The strength of rivalry of each company is calculated in Table 9, which shows its enmity value within the business market.Table 9 shows that SONY is the biggest rival company among other companies.

Finding Influential Communities in a Social Network
Fuzzy competition hypergraphs have a wide range of applications in decision-making problems and decision support systems based on social networking.To elaborate on the necessity of the idea discussed in this paper, we apply the notion of fuzzy competition hypergraphs to study the influence, centrality, socialism and proactiveness of human beings in any social network.
Social competition is a widespread mechanism to figure out a best-suited group economically, politically or educationally.Social competition occurs when individual's opinions, decisions and behaviors are influenced by others.Graph theory is a conceptual framework to study and analyze the units that are intensely or frequently connected in a network.Fuzzy hypergraphs can be used to study the influence and competition between objects more precisely.The social influence and conflict between different communities can be studied using a fuzzy competition hypergraph known as fuzzy influence hypergraph.
The fuzzy influence hypergraph G = (µ, λ c ) has the same set of vertices as #» G and there is a hyperedge consisting of vertices x 1 , x 2 , . . ., x r if N − (x 1 ) ∩ N − (x 2 ) ∩ . . .∩ N − (x r ) = ∅.The degree of membership of hyperedge E = {x 1 , x 2 , . . ., x r } is defined as The strength of influence between different objects in a fuzzy influence hypergraph can be calculated by the method presented in Algorithm 2. The complexity of algorithm is O(n 2 ).

1.
Input the adjacency matrix Using fuzzy in neighbourhoods, construct the fuzzy influence hypergraph following Algorithm A3.
If x i belongs to the hyperedge E in fuzzy influence hypergraph then calculate the degree of each vertex x i as,
do i from 1 → n 7.
If A i > 1 then calculate the degree of influence of each vertex x i as,

end do
Consider a fuzzy social digraph of Florientine trading families Peruzzi, Lambertes, Bischeri, Strozzi, Guadagni, Tornabuon, Castellan, Ridolfi, Albizzi, Barbadori, Medici, Acciaiuol, Salviati, Ginori and Pazzi.The vertices in a fuzzy network represent the name of trading families.The degree of membership of each family represents the strength of centrality in that network.The directed edge # » xy indicates that the family x is influenced by y.The degree of membership of each directed edge indicates to what extent the opinions and suggestions of one family influence the other.The degree of membership of Medici is 0.9, which shows that Medici has a 90% central position in a trading network.The degree of membership between Redolfi and Medici is 0.6, which indicates that Redolfi follows 60% of the suggestions of Medici.The fuzzy social digraph is shown in Figure 16.The concept of fuzzy competition graphs presented in [16,17] can be utilized successfully in different domains of applications.In the existing methods, we usually consider fuzziness in pairwise competition and conflicts between objects.However, in these representations, we miss some information about whether there is a conflict or a relation among three or more objects.For example, Figure 15 shows the strong competition for profit among SONY, LR and AK.However, if we draw the fuzzy competition graph of Figure 14, we cannot discuss the group-wise conflict among companies.Sometimes, we are not only interested in pair-wise relations but also in group-wise conflicts, influence and relations.The novel notion of fuzzy competition hypergraphs are a mathematical tool to overcome this difficulty.We have presented different methods for solving decision-making problems.These methods not only generalize the existing ones but also give better results regarding uncertainty.

Conclusions
In this research paper, we have applied the powerful technique of fuzziness to generalize the notion of competition hypergraphs and fuzzy competition graphs.Fuzzy models give more precision, flexibility and compatibility to the system as compared to the crisp models.We have mainly discussed the construction methods of various types of fuzzy hypergraphs using open and closed neighbourhoods, strong hyperedges, kth strength of competition and symmetric properties.We have also established strong relations among fuzzy k-competition hypergraphs and underlying fuzzy graphs along with fuzzy digraphs having loops at vertices.We have applied fuzzy competition hypergraphs to real-world problems for representation of fuzziness in different domains including identification of predator-prey relations, competitions in the business market and social networks which motivate the idea introduced in this research paper.We have designed certain algorithms to solve these decision-making problems.Input the adjacency matrix A = [x ij ] n×n of a fuzzy digraph #» G.

3.
Define a relation f : X → X by f (x i ) = x j , if x ij > 0. 4.
do i from 1 → n 5.
do j from 1 → n 6.
If x ij > 0 then (x j , x ij ) belongs to the fuzzy out neighbourhood N + (x i ).

Example 3 .Example 2 . 3 .
Consider the example of fuzzy digraph shown in Figure 1.From Example 2, the fuzzy double competition hypergraph of Figure 1 is given in Figure 6.In addition, Figures 2, 3 and 6 show that the fuzzy double competition hypergraph is the intersection of fuzzy row hypergraph and fuzzy column hypergraph.It follows that fuzzy double competition hypergraph is the intersection of fuzzy row hypergraph and fuzzy column hypergraph.Consider the example of fuzzy digraph shown in Fig.2.1.From Example 2.2, the fuzzy double competition hypergraph of Fig. 2.1 is given in Fig. 2.6.Also Fig. 2.2, 2.3, 2.6

Lemma 2 . 3 .Example 2 . 4 .
The fuzzy niche hypergraph is the union of fuzzy row hypergraph and fuzzy column hypergraph.The fuzzy niche hypergraph of Fig. 2.1 is shown in Fig. 2.7 which is the union of Fig. 2.3 and Fig. 2.2.

Definition 14 .Definition 15 .
[16] The fuzzy closed neighbourhood N [y] of a vertex y in a fuzzy graph G = (µ, λ) is defined as N [y] = N (y) ∪ {(y, µ(y))}.The fuzzy open neighbourhood hypergraph of a fuzzy graph G = (µ, λ) is a fuzzy hypergraph N (G) = (µ, λ ) whose fuzzy vertex set is the same as G and there is a hyperedge E

Theorem 4 .
Let #» G = (C,#» D) be a symmetric fuzzy digraph having loops at every vertex; then, C k

Table 4 .
Fuzzy out neighbourhood and fuzzy in neighbouhood of vertices in Let H be a fuzzy hypergraph and t be the smallest non-negative number such that H ∪ I t is a fuzzy niche hypergraph of some fuzzy digraph #» G where, I t is a fuzzy set on t isolated vertices X t , then t is called fuzzy niche number of H denoted by n(H).Let H be a fuzzy hypergraph on a non-empty set X with n G, where I t is a fuzzy set on t isolated vertices X t ; then, t is called fuzzy niche number of H denoted by n(H).Let H be a fuzzy hypergraph on a non-empty set X with n Definition 10.Let H be a fuzzy hypergraph and t be the smallest non-negative number such that H ∪ I t is a fuzzy niche hypergraph of some fuzzy digraph #»

Table 5 .
Fuzzy open neighbourhood of vertices.

Table 6 .
Fuzzy closed neighbourhood of vertices.

Table 7 .
Fuzzy out neighbourhoods of vertices.

Table 8 .
Fuzzy out neighbourhoods of companies.

Table 9 .
Strength of rivalry between companies.

Table 11 .
Degree of influence of vertices.