This paper explores the defects in fuzzy (hyper) graphs (as complex (hyper) networks) and extends the fuzzy (hyper) graphs to fuzzy (quasi) superhypergraphs as a new concept. We have modeled the fuzzy superhypergraphs as complex superhypernetworks in order to make a relation between labeled objects in the form of details and generalities. Indeed, the structure of fuzzy (quasi) superhypergraphs collects groups of labeled objects and analyzes them in the form of the part to part of objects, the part of objects to the whole group of objects, and the whole to the whole group of objects at the same time. We have investigated the properties of fuzzy (quasi) superhypergraphs based on any positive real number as valued fuzzy (quasi) superhypergraphs, considering the complement of valued fuzzy (quasi) superhypergraphs, the notation of isomorphism of valued fuzzy (quasi) superhypergraphs based on the permutations, and we have presented the isomorphic conditions of (self complemented) valued fuzzy (quasi) superhypergraphs. The concept of impact membership value of fuzzy (quasi) superhypergraphs is introduced in this study and it is applied in designing the real problem in the real world. Finally, the problem of business superhypernetworks is presented as an application of fuzzy valued quasi superhypergraphs in the real world.

The theory of hypergraph as a model of hypernetwork has been introduced by Berge as a generalization of graph theory in 1960 [

Regarding these points, we introduce the concept of valued fuzzy superhypergraphs as a generalization of fuzzy hypergraphs. Valued fuzzy superhypergraphs are dependent on the concept of fuzzy supervertices and fuzzy superedges or fuzzy links, which are defined in this study. The motivation of valued fuzzy superhypergraphs is based on a design of real problems as complex superhypernetworks. Indeed, we modify a real problem as a fuzzy superhypergraph, and with respect to the notation of the impact membership value of modeled fuzzy superhypergraph, we have made the best decision. This paper presents the fuzzy algebraic structures on valued fuzzy superhypergraphs such as strong valued fuzzy superhypergraphs, isomorphic valued fuzzy superhypergraphs, the complement of fuzzy link, and self-complemented valued fuzzy superhypergraphs. Also, the relation between fuzzy hypergraphs and strongly valued fuzzy superhypergraphs has been investigated.

In this section, we recall some definitions and results, which we need as follows:

the quasi superhypergraph

The superhypergraph

Let

In this section, we introduce the novel concept of fuzzy supervertices, fuzzy superedges or fuzzy links, and fuzzy superhypergraph.

From now on, if there is not any fuzzy links from

Every fuzzy hypergraph is a fuzzy quasi superhypergraph.

Every fuzzy graph is a fuzzy quasi superhypergraph.

For all

If

If

(

It follows that

(

(

A bijective mapping

Let ^{′}-fuzzy quasi superhypergraph on quasi superhypergraphs

For any given fuzzy supervertex ^{′} and

For any

(

It follows that for any

Hence

It follows that

where

It follows that

It follows that

Thus for any

if

if

In this section, we apply the concept of fuzzy valued quasi superhypergraph in the real world. To simplify and better display the method of application of fuzzy

Step 1: Consider a real problem as a complex (super)hypernetwork,

Step 2: Separate the factors and components of this complex (super)hypernetwork according to the type of application,

Step 3: Put the factors and constituent factors classified according to their values in different tables,

Step 4: According to the tables of step 3, model the relationship between components in an optimal state through a fuzzy

Step 5: Compute the impact membership value of components of the

Step 6: Find the extremum case based on step 5 and the conditions we need to solve the real problem.

In the following section, we consider a real problem as a business (super)hypernetwork, and in this regard, first, we analyze and describe a business network. A

People | Amount of capital |
---|---|

Benjamin | 0.6 |

Oliver | 0.68 |

James | 0.7 |

William | 0.84 |

Lucas | 0.65 |

Machinery | High production quality |
---|---|

0.5 | |

0.55 | |

0.65 | |

0.75 | |

0.85 |

Shopping market | Liquidity |
---|---|

0.4 | |

0.35 | |

0.75 | |

0.45 | |

0.55 |

It follows that:

All these qualities will be represented in the fuzzy quasi superhypergraph, where the fuzzy supervertices are the people, Machinery and the fuzzy superedges list all the ability levels, communication levels, and service time (

(People, Machinery) | |||||
---|---|---|---|---|---|

(Amount of capital, High production quality) |

(People, Shopping market) | |||||
---|---|---|---|---|---|

(Amount of capital, Liquidity) |

(Machinery, Shopping market) | |||||
---|---|---|---|---|---|

(High production quality, Liquidity) |

In

Benjamin | Oliver | James | William | Lucas | ||||||
---|---|---|---|---|---|---|---|---|---|---|

The current paper has defined and explored the notion of fuzzy quasi superhypergraphs. This study has tried to prove that the fuzzy links in the concept of fuzzy quasi superhypergraphs are fundamental and play a main role in the impact value of quasi superhypergraphs. The main motivation of this work is to apply fuzzy quasi superhypergraphs in the real world and to generalize the application of fuzzy hypergraphs in the real world. Indeed, all results of fuzzy hypergraphs can be extended to fuzzy quasi superhypergraphs both theoretically and practically. The merit of the proposed method is to fix the defects of fuzzy hypergraph theory. Indeed, fuzzy hypergraph theory investigates the optimal case for limited elements, while fuzzy quasi superhypergraphs consider the optimal case for the set of elements or object (object can be set). Also, we have shown that:

The extension of a valued fuzzy quasi superhypergraph depends on its value.

Every fuzzy hypergraph is a fuzzy quasi superhypergraph.

The supremum of fuzzy superedges is less than or equal to the inverse of the real value.

Every fuzzy graph is a fuzzy quasi superhypergraph.

The concept of isomorphic valued fuzzy quasi superhypergraphs is introduced and proved that the necessary condition for isomorphism of two valued fuzzy quasi superhypergraphs is equality of their values.

The concept of the complement of valued fuzzy quasi superhypergraphs is introduced and proved that the complement of isomorphic valued fuzzy quasi superhypergraphs is isomorphic. Also, it is proved that the complement of valued fuzzy quasi superhypergraphs satisfies involution properties.

The concept of self complemented valued fuzzy quasi superhypergraphs is introduced and the conditions for a valued fuzzy quasi-superhypergraph to be self-complemented have been investigated.

The impacts membership value of any valued fuzzy quasi superhypergraph is introduced and the properties of impact membership value of strong valued fuzzy quasi superhypergraph have been explored. We have particularly proven that the impact membership value of strong valued fuzzy quasi superhypergraph is zero.

We proved that the sum of all impact membership values of isomorphic valued fuzzy quasi superhypergraphs is equal.

Based on the impact membership value of valued fuzzy quasi superhypergraph, we presented business superhypernetwork as a real-world problem and reached an optimal decision in this type of problem.

We hope that these results are helpful for further studies in complex (super)hypernetwork via algebraic structures and hyperstructures and fuzzy quasi superhypergraphs. In our future studies, we hope to obtain more results through the comparison of the suggested method with some existing methods and prove the effectiveness of the method, the fundamental relation on fuzzy quasi superhypergraphs, and their applications in other research. Also, we intend to work on the Plithogenics and new types of (hyper)soft sets and on superhypergraph and its relation to real-world problems.

The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.

The authors received no specific funding for this study.

The authors confirm contribution to the paper as follows: Idea of superhypergraph: Florentin Smarandache; data collection: Mohammad Hamidi; analysis and interpretation of results: Mohammad Hamidi, Mohadesheh Taghinezhad; draft manuscript preparation: Mohammad Hamidi. All authors reviewed the results and approved the final version of the manuscript.

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The authors declare that they have no conflicts of interest to report regarding the present study.