Granulation of Hypernetwork Models under the q -Rung Picture Fuzzy Environment

: In this paper, we deﬁne q -rung picture fuzzy hypergraphs and illustrate the formation of granular structures using q -rung picture fuzzy hypergraphs and level hypergraphs. Further, we deﬁne the q -rung picture fuzzy equivalence relation and q -rung picture fuzzy hierarchical quotient space structures. In particular, a q -rung picture fuzzy hypergraph and hypergraph combine a set of granules, and a hierarchical structure is formed corresponding to the series of hypergraphs. The mappings between the q -rung picture fuzzy hypergraphs depict the relationships among granules occurring at different levels. The consequences reveal that the representation of the partition of the universal set is more efﬁcient through q -rung picture fuzzy hypergraphs and the q -rung picture fuzzy equivalence relation. We also present an arithmetic example and comparison analysis to signify the superiority and validity of our proposed model. The positive T ( m ij ) , neutral N ( m ij ) , and negative F ( m ij ) degrees of each m ij entry of the above matrix describe the healthy, neutral, and rival relationships among the x i and x j corporations, respectively. The transitive closure of the above matrix is shown in the matrix below. The transitive closure of 5-RPFER is also an equivalence relation.


Introduction
Granular computing (GrC) is defined as an identification of techniques, methodologies, tools, and theories that yield the advantages of clusters, groups, or classes, i.e., the granules. The terminology was first introduced by Lin [1]. The fundamental concepts of GrC are utilized in various fields of research, including rough set theory, machine learning, fuzzy set theory, and artificial intelligence. Different models have been proposed to study the numerous issues occurring in GrC, including classification of the universe, illustration of granules, and the identification of relations among granules. For example, the procedure of problem solving through GrC can be considered as distinct descriptions of the problem at multiple levels, and these levels are linked together to construct a hierarchical space structure. Thus, this is a way of dealing with the formation of granules and the switching between different granularities. Here, the term "hierarchy" implies the methodology of hierarchical analysis in solving complicated problems and human diversions [2]. To understand this methodology, let us consider an example of national administration in which the complete nation is subdivided into various provinces. Further, divide every province into various divisions similarly. The human activities and problem solving involve the simplification of the original complicated problem by ignoring some details rather than thinking about all points of the problem. This rational model is then further sifted till the issue is completely resolved. Thus, we resolve and interpret the complex problems from the weaker grain to the stronger one, or from the highest rank to lowest, or from universal to particular. This technique is called hierarchical problem solving. It is further acknowledged that the hierarchical strategy is the only technique that is used by humans to deal with complicated problems, and it enhances the competency and efficiency. This strategy is also known as multi-GrC. Yang et al. [3] applications to decision-making were discussed by Akram et al. [18]. Recently, a novel description on edge-regular q-rung picture fuzzy graphs with applications was proposed by Akram et al. [19].
To combine the benefits of both fuzzy sets and hypergraphs, Kaufmann [20], in 1977, defined fuzzy hypergraphs. Fuzzy hypergraphs were redefined and generalized by Lee-Kwang and Keon-Myung [21]. A valuable contribution to fuzzy graphs and fuzzy hypergraphs was given in [22]. Many researchers have explored the formation of granules/granular structures using hypergraphs in various fields, but there are many graph theoretic problems that may contain uncertainty and vagueness. To overcome the problems of uncertainty in models of GrC, Wang and Gong [23] studied the construction of granular structures by means of fuzzy hypergraphs. They concluded that the representation of granules and partition is very efficient through the fuzzy hypergraphs. Gong and Wang [24] represented the connection of fuzzy hypergraph with the fuzzy information system and the fuzzy concept lattice. Parvathi et al. [25] originated the notion of IF hypergraphs. Later on, this idea was extended by Akram and Dudek [26]. Akram and Luqman [27,28] introduced the intuitionistic single-valued neutrosophic and bipolar neutrosophic hypergraphs. Recently, q-ROF hypergraphs with applications were studied by Luqman et al. [29]. Luqman et al. [30] also introduced an m-polar fuzzy hypergraph model of GrC.
The fundamental goal of this research work is to introduce the q-rung picture fuzzy hypergraphs and to construct a hypergraph model of GrC possessing the characteristics of PFSs, as well as q-ROFSs. In the proposed model, the vertex of the q-rung picture fuzzy hypergraph denotes an object, and a q-rung picture fuzzy hyperedge represents a granule. By applying the most effective and flexible theory of q-RPFSs to hypergraphs, we not only broaden the space of uncertain and vague data, but also deal with neutral relationships between granules. Our proposed model of GrC, based on q-rung picture fuzzy hypergraphs, is much closer to and considerable of human reasoning as compared to earlier concepts. In this model, the range of indication of decision data can be changed by changing the value of parameter q, q ≥ 1. The rest of the paper is arranged as follows: In Section 2, q-rung picture fuzzy hypergraphs are defined. A q-rung picture fuzzy hierarchical quotient space structure is developed based on the q-rung picture fuzzy equivalence relation. In Section 3, we construct a q-rung picture fuzzy hypergraph model of GrC. The partition and covering of granules are defined in the same section. The method of bottom-up construction is explained by an algorithm and an example. In Section 4, a model of GrC is constructed using level hypergraphs based on the q-rung picture fuzzy equivalence relation, and the construction of the level hypergraph model is illustrated through a concrete example. The "refinement" and "coarsening" operators are defined and their certain properties are discussed briefly. In Section 5, we compare the flexibility of our proposed model with existing models of q-ROFSs and PFSs. Section 6 deals with some concluding remarks and future studies.
2. q-Rung Picture Fuzzy Hierarchical Structure Definition 1. [14] A q-rung orthopair fuzzy set (q-ROFS) Q in the universe X is an object of the form, where T Q : X → [0, 1] represents the positive degree and F Q : X → [0, 1] represents the negative degree of the element a ∈ X such that, is called a q-ROF index or indeterminacy degree of a to the set Q. Definition 2. [16] A picture fuzzy set (PFS) P is an object of the form, P = {(a, T P (a), N P (a), F P (a))|a ∈ X}, where the mapping T P : X → [0, 1] represents the positive degree, N P : X → [0, 1] represents the neutral degree, and F P : X → [0, 1] represents the negative degree of the element a ∈ X such that, 0 ≤ T P (a) + N P (a) + F P (a) ≤ 1, for all a ∈ X.
The refusal degree of a in P is defined as (1 − T P (a) − N P (a) − F P (a)). Definition 3. [17] A q-rung picture fuzzy set (q-RPFS) R is an object having the form, where the function T R : X → [0, 1] represents the positive degree, N R : X → [0, 1] represents the neutral degree, and F R : X → [0, 1] represents the negative degree of the element a ∈ X such that, The refusal degree of a in R is defined as Definition 5. Let M and L be two q-RPFSs on X. The union and intersection of M and L are defined as, Definition 6. Let X be a set of universes. A q-rung picture fuzzy hypergraph (q-RPFH) on X is defined as an ordered pair H = (R, S), where R = {R 1 , R 2 , R 3 , . . . , R l } is a collection of non-trivial q-RPFSs on X and S is a q-RPFR on R i such that: If E (α,β,γ) is non-empty, then the (α, β, γ)-level hypergraph of H is defined as H (α,β,γ) = (X (α,β,γ) , E (α,β,γ) ), which is a crisp hypergraph.
We now discuss the q-rung picture fuzzy hierarchical quotient space structure. Different techniques have been proposed to deal with GrC. Quotient space (QS), FSs, and rough sets are three basic computing tools dealing with uncertainty. Based on FSs, the fuzzy equivalence relation (FER), as an extension of equivalence relation, was proposed by Zadeh [31]. The question of distinct membership degrees of the same object from different scholars has arisen because of various ways of thinking about the interpretation of different functions dealing with the same problem. To resolve this issue, FS was structurally defined by Zhang and Zhang [32], which was based on QS theory and FER [33]. This definition provides some new intuitiveness regarding the membership degree, called a hierarchical QS structure of an FER. By following the same concept, we develop a hierarchical QS structure of a q-RPFER. Definition 9. Let X 1 and X 2 be two finite sets, then the Cartesian product between X 1 and X 2 is X 1 × X 2 . Every q-RPF subset R of X 1 × X 2 is defined as a q-RPF binary relation from X 1 -X 2 . Let X 1 = {r 1 , r 2 , r 3 , · · · , r l } and X 2 = {s 1 , s 2 , s 3 , · · · , s m }, a q-RPF binary relation matrixM R , be given as follows, In general,M R is called q-RPF relation matrix of R, where d R (r, s) = (T R (r, s), N R (r, s), F R (r, s)), and T R : 1] represent the positive degree, neutral degree, and negative degree of r and s, respectively, such that: for all (r, s) ∈ X 1 × X 2 .
Definition 11. A q-RPFR is called a q-RPF equivalence relation (q-RPFER) if the following conditions are satisfied, 1. d R (r, r) = (1, 0, 0), for all r ∈ X, 2. d R (r, s) = d R (s, r), for all r, s ∈ X, 3. for all r, s, t ∈ X A q-RPF quotient space (q-RPFQS) is denoted by a triplet (X,Ã, R), where X is a finite domain,Ã represents the attributes of X, and R represents the q-RPF relationship between the objects of universe X, which is called the structure of the domain. Definition 12. Let x i and x j be two objects in the universe X. The similarity between x i , x j ∈ X having the attributeã k is defined as: whereã ik represents that object x i possesses the attributeã k andã jk represents that object x j possesses the attributeã k .
It is noted that the q-RPFR matrixM R is symmetric and reflexive q-RPFR, but in general, it does not satisfy the transitivity condition. Proposition 1. Let R be a q-RPFR on a finite domain X and R (α,β, 1]. Then, R (α,β,γ) is an ER on X and is said to be a cut-equivalence relation of R.
Proof. The terminology of HQSS implies that is an amalgam of sub-blocks of X(ρ i ). Without loss of generality, it is assumed that only one sub-block X i−1,j in X(ρ i−1 ) is formed by the combination of two sub-blocks X ir , X is in X(ρ i ), and all other remaining blocks are equal in both sequences. Thus, Since, Therefore, we have: Hence, I X (ρ 1 ) < I X (ρ 2 ) < I X (ρ 2 ) < · · · < I X (ρ j ).
Definition 16. Let X = {x 1 , x 2 , x 3 , · · · , x n } be a non-empty set of universes, and let P t (X) = {X 1 , X 2 , X 3 , · · · , X t } be a partition space (PS) of X, where |P t (X)| = t, then P t (X) is called the t-order partition space (t-OPS) on X.
which is also increasing and called a sub-block sequence of P t (X).
Note that two different t-order partition spaces on X may possess the similar sub-block sequence H(t).
An ω-displacement is obtained by subtracting one from some bigger term and adding one to the smaller one such that the sequence does not change its increasing property.

Construction of Model
GrC may utilize frameworks in terms of levels and granular structures, which are built on the basis of multiple representations and multiple levels. A granule is defined as a group of elements having the same attributes or properties and are taken as a whole.

Definition 19.
A system (X, R) is called an object space, where X is a universe of objects or elements and R = {r 1 , r 2 , r 3 , · · · , r k }, k = |X| is a family of relations between the elements of X. For n ≤ k, r n ∈ R, r n ⊆ X × X × X · · · × X, if (x 1 , x 2 , · · · , x n ) ⊆ r n , then there exists an n-array relation r n on (x 1 , x 2 , · · · , x n ).
The elements of an object space having some relation r i ∈ R can be assumed as a granule. A single object is considered as the smallest granule, and the set of all elements is said to be the largest granule in an object space.
We consider one vertex of a q-RPFH as a representation of an object, and the group of elements having some relationship s i is represented by the q-RPF hyperedge. The positive membership T of an element x i refers to the belonging; neutral membership N refers to the unbiased behavior; and negative membership F refers to the disconnection to the granule, where T( Example 2. Let X = {s 1 , s 2 , s 3 , s 4 , s 5 } be the set of objects and R = {r 1 , r 2 , r 3 , r 4 , r 5 , r 6 } be the set of relations. A hypergraph H = (R, S) representing the objects and the relations r i ∈ R(1 ≤ i ≤ 6) between them is shown in Figure 3. By assigning the positive membership T ∈ [0, 1], neutral membership N ∈ [0, 1], and negative membership F ∈ [0, 1] to each element s i , we form a seven-rung picture fuzzy hypergraph. Let X = {s 1 , s 2 , s 3 , s 4 , s 5 } and S = {S 1 , S 2 , S 3 , S 4 }, and the corresponding incidence matrix is given in Table 1.
Hypergraph representation of granules in a level is given in Figure 4. The relationships between the vertices can be obtained by following the present situation, through computing the mathematical function or by figuring out the internal, external, and contextual properties of the granule. After the relationship is computed, the set of vertices possessing the relationship among them can be combined into one module. By taking into account the actual condition, we can calculate or assign positive, neutral, and negative degrees through the membership functions T(s), N(s), and F(s), respectively. As a result, we have formed a granule that is also called the q-RPF hyperedge. The (α, β, γ)-cuts are computed for this q-RPF hyperedge. When all vertices under consideration are to be integrated into a single unit and the membership degrees are assigned or computed for the unit, we have constructed a single-level model of GrC.
The amalgamation of the elements or objects that are processed at the same time while resolving a problem is called a granule. It reflects the representation of the perceptions and the attributes of the integration. A granule may be a small part of another granule and can also be considered as the block to form larger granules. It can also be a collection of granules or may be thought of as a whole unit. Thus, a granule may play two distinct roles.

Definition 20.
A partition of a set X established on the basis of relations between objects is defined as a collection of non-empty subsets that are pairwise disjoint and whose union is the whole of X. These subsets that form the partition of X are called blocks. Every partition of a finite set X contains the finite number of blocks. Corresponding to the q-RPFH, the constraints of partition ψ = {E i |1 ≤ i ≤ n} can be stated as follows, a set X is defined as non-empty subsets of X such that, In a q-RPFH, if S i , S j ∈ S, and S i ∩ S j = ∅, i.e., S i and S j do not intersect each other, then these hyperedges form a partition of granules. Furthermore, if S i , S j ∈ S and S i ∩ S j = ∅, i.e., S i and S j intersect each other, then these hyperedges form a covering of granules at this level. Note that, the definition of q-RPFH concludes that the q-RPFH forms a covering of a set of universes X.
x 5 x 6 x 7 x 8 x 9 Figure 5. A partition of granules in a level.
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 Figure 6. A covering of granules in a level.
The elements belonging to the granule determine the internal property and study the relationships between objects. The internal property in this model incorporates the interaction of vertices. The external property reflects the relationships between granules. This property incorporates those vertices that belong to a q-RPF hyperedge under the membership functions T, N, and F. The existence of a granule is indicated by the contextual property in an environment.
A set-theoretic way to study the GrC model uses the following operators in a q-RPFH model.

Definition 22.
Let G 1 and G 2 be two granules in our model and the q-RPF hyperedges E 1 , E 2 represent their external properties. The union of two granules G 1 ∪ G 2 is defined as a larger q-RPF hyperedge that contains the vertices of both E 1 and E 2 . If x i ∈ G 1 ∪ G 2 , then the membership degrees of x i in larger granule G 1 ∪ G 2 are defined as follows, Definition 23. Let G 1 and G 2 be two granules in our model and the q-RPF hyperedges E 1 , E 2 represent their external properties. The intersection of two granules G 1 ∩ G 2 is defined as a smaller q-RPF hyperedge that contains those vertices belonging to both E 1 and E 2 . If x i ∈ G 1 ∩ G 2 , then the membership degrees of x i in smaller granule G 1 ∩ G 2 are defined as follows, Definition 24. Let G 1 and G 2 be two granules in our model and the q-RPF hyperedges E 1 , E 2 represent their external properties. The difference between two granules G 1 − G 2 is defined as a smaller q-RPF hyperedge that contains those vertices belonging to E 1 , but not to E 2 . Note that, if a vertex x i ∈ E 1 and x i ∈ E 2 , then:

Definition 25.
A granule G 1 is said to be the sub-granule of G 2 , if each vertex x i of E 1 also belongs to E 2 , i.e., E 1 ⊆ E 2 . In such a case, G 2 is called the super-granule of G 1 .
Note that, if E (x i ) = {0, 1}, then all above-described operators are reduced to the classical hypergraphs theory of GrC.

The Construction of Hierarchical Structures
As earlier, we have constructed a granular structure of a specific level as a q-RPFH. In this way, we can interpret a problem at distinct levels of granularities. Hence, these granular structures at different levels produce a set of q-RPFHs. The upper set of these hypergraphs constructs a hierarchical structure at distinct levels. The relationships between granules are expressed by the lower level, which represents the problem as a concrete example of granularity. The relationships between granule sets are expressed by the higher level, which represents the problem as an abstract example of granularity. Thus, the single-level structures can be constructed and then can be subdivided into hierarchical structures using the relational mappings between different levels.

Definition 26.
Let H 1 = (R 1 , S 1 ) and H 2 = (R 2 , S 2 ) be two q-RPFHs. In a hierarchy structure, their level cuts are H (α,β,γ) 1 and H The q-RPFH H 1 possesses a finer granularity than H 2 , so H 1 is referred to as the finer granularity and H 2 as the coarser granularity.
In a q-RPFH model, the mappings are used to describe the relations among different levels of granularities. At each distinct level, the problem is interpreted w.r.t. the q-RPF granularity of that level. The different descriptions of some problems are associated through mappings at distinct levels of granularities. There are two fundamental types to construct the hierarchical structures, the top-down construction procedure and the bottom-up construction procedure [34]. The method for the bottom-up construction is described in Algorithm 1.

Algorithm 1 The bottom-up construction.
Input: An undirected q-RPFH H.

Output:
The bottom-up construction of granular structures. 1: Combine the vertices having some relationship r ∈ R into a unit U j , according to the set of relations R = {r 1 , r 2 , r 3 , · · · , r n }. 2: Compute the membership degrees through the functions T(x), N(x), and F(x) of each unit U j , which are q-RPF hyperedges and regarded as granules. 3: Now, formulate the i-level of the model using the following steps. 4: Input the number of q-RPF hyperedges or granules k. 5: Fix parameters α, β, and γ. 6: for α, β, γ ∈ [0, 1] do 7: for s from 1-k do 8:  18: Perform the steps form 1-16 repeatedly to construct the i + 1-level of granularity. 19: Except the last level, each level is mapped to the next level using some operators. 20: Step 1-Step 16 are performed repeatedly until the complete set is formulated as a single granule.
x 1 x 3 x 5 x 8 x 9 By considering the fixed α, β, γ and following Algorithm 1, the bottom-up construction of this model is given in Figure 9.
In GrC, more than one hierarchical structure (HS) is considered to emphasize the multiple approaches. These different HSs can be formed by considering the different interpretations of the relation set R. Every HS is a distinct aspect of the problem. A multiple or a multilevel model of GrC based on the hypergraph is formed by combining these different hierarchical structures. In a hypergraph model, a series of correlated hypergraphs is used to present every hierarchical structure. The mapping relates the distinct levels in an HS, and each hypergraph denotes a particular level in that structure. Each specific level has multiple hyperedges, and the objects having similar attributes are contained in one hyperedge. We now extend the example given in [23] to construct a q-RPFH model of GrC to illustrate the validity and flexibility of our model.

Example 5.
Consider an express hypernetwork, where the vertices represent the express corporations. These vertices are combined together in one unit U according to the relation set R and possessing some type of relation among them. To form a 6-RPF hyperedge, which is also called a granule, we calculate and assign the membership degrees to each unit. Each 6-RPF hyperedge denotes a shop demanding express services, and the vertices contained in that hyperedge are the express corporations serving that shop. There are ten {c 1 , c 2 , c 3 , c 4 , c 5 , c 6 , c 7 , c 8 , c 9 , c 10 } express corporations, and the corresponding incidence matrix of this model is given in Table 2.
The corresponding 6-rung picture fuzzy hypergraph is shown in Figure 10. A 6-RPFH model of GrC illustrates an uncertain set of objects with specific positive, neutral, and negative degrees, which incorporate the undetermined behavior of an element with respect to its granule. In this example, there are five shops requiring the services of the express corporation. The membership degrees of each corporation to distinct shops is different because every shop selects a corporation by considering different factors, including consignment, life span, limited liability, scale of that express corporation, the distance of receiving and mailing parcels, and so on. Note that, using a q-RPFH in GrC is meaningful and more flexible as compared to the fuzzy hypergraph because it also deals with negative membership and the neutral behavior of objects toward their granules.

Zooming-in and Zooming-out Operators
The multi-level granularity of the problem is represented by a q-RPFH model, which allows the problem solvers to decompose it into various minor problems and transform it into other granularities. The transformation of the problem in other granularities is performed by using two operators, i.e, zooming-in and zooming-out operators. The transformation from the weaker level to the finer level of granularity is done by zoom-in and the zoom-out operators dealing with the shifting of the problem from coarser to finer granularity.

Definition 27.
Let H 1 = (R 1 , S 1 ) and H 2 = (R 2 , S 2 ) be two q-RPFHs, which are considered as two levels of hierarchical structures, and H 2 owns a coarser granularity than H 1 . Suppose H ) are the corresponding (α, β, γ)-level hypergraphs of H 1 and H 2 , respectively. Let j and x 2 m and is obtained by the characteristics of granules.

Definition 28.
Let the hyperedge φ − 1(x l ) be a vertex at a new level and the relation between hyperedges at this level be the same as that of the relationship between vertices at the previous level. This is called the zoom-in operator and transforms a weaker level to a stronger level. The function r(x 1 j , x 2 m ) defines the relation between vertices of the original level, as well as the new level.
Let the vertex φ(e i ) be a hyperedge at the next level and the relation between elements at this level be the same as that of the relationship between hyperedges at the corresponding level. This is called the zoom-out operator and transformation of a finer granularity to a coarser one.
By applying these operators, a problem can be viewed at multiple levels of granularities. These operations allow us to solve the problem more appropriately, and granularity can be switched easily at any level of problem solving.
In a q-RPFH model of GrC, the membership degrees of elements reflect the actual situation more efficiently, and a wide variety of complicated problems in uncertain and vague environments can be presented by means of q-RPFHs. The previous analysis concluded that this model of GrC generalized the classical hypergraph model and fuzzy hypergraph model.
We now construct a hypergraph model of GrC based on q-RPFER.

A Level Hypergraph Partition Model
In this partition model, the blocks are represented by hyperedges and each block contains the elements having some q-RPFER R. Let π R be the partition of X induced by R, then the q-RPFER R π is defined as, xR π y if and only if they are contained in a similar block/subset.

Definition 29.
A system (X, R ρ ), ρ = (α, β, γ) is called an object space, where R ρ is a non-empty collection of equivalence relations between the elements of X.
then there exists an i-array relation r ρ i on (x 1 , x 2 , · · · , x i ).
The q-RPFERs between the vertices are obtained from the actual situation, and then, the vertices having some relation are combined into a hyperedge. After the integration of all vertices into hyperedges is done, a single level of the GrC model is obtained. The construction of the level hypergraph model is illustrated through the following example, which is the extension of example given in [23].

Example 7.
Consider an express hypernetwork, as discussed in Example 5; we consider the same hypernetwork to illustrate the construction of the level hypergraph model. Ten express corporations are considered, and suppose that each corporation experiences six attributes. These attributes are delivery time, freight, tracking and circulation information, time window service, customer satisfaction, and flexible and customizable and are denoted by A 1 , A 2 , A 3 , A 4 , A 5 , and A 6 , respectively. To indicate the values of attributes, zero and one are used, as shown in Table 3. Table 3. Description of the attributes.

Attributes Characteristic Value Characteristic Value
Delivery time Short  1  Long  0  Freight  Low  1  High  0  Tracking and circulation information Existence  1  Non-existence 0  Time window service  Existence  1  Non-existence 0  Customer satisfaction  High  1  Low  0  Flexible and customizable  Large  1 Small 0 The information about express corporations having these attributes is given in Table 4. Table 4. Corporations having attributes.

Express Corporations
The positive membership function T(x i ) ∈ [0, 1] represents the beneficial and helpful relationships between these corporations, and this affirmative relationship is described through the following matrix.
The neutral membership function N(x i ) ∈ [0, 1] represents the unbiased and vague relationships between these corporations, and this indeterminate behavior is described through the following matrix.
The negative membership function F(x i ) ∈ [0, 1] represents the contrary and competent relationships between these corporations, and this opposing relationship is described through the following matrix.
A 5-RPFR matrix can be obtained by combining positive membership, neutral membership, and negative membership degrees, as shown in the following matrix. (0, 0, 1) (0, 0, 1) The positive T(m ij ), neutral N(m ij ), and negative F(m ij ) degrees of each m ij entry of the above matrix describe the healthy, neutral, and rival relationships among the x i and x j corporations, respectively. The transitive closure of the above matrix is shown in the matrix below. The transitive closure of 5-RPFER is also an equivalence relation. Let 0.20 < α ≤ 0.25, 0.12 < β ≤ 0.20, and 0.32 < γ ≤ 0.45 and their corresponding HQS be given as follows, Thus, we can conclude that r 1 = {∅}, Thus, we have constructed a single level of the hypergraph model, as shown in Figure 11. Figure 11. A single level of the hypergraph model.
It is noted that a q-RPFER determines the partition of the domain set into different layers. All q-RPFER, which are isomorphic, can also determine the same classification. for j from 1-m do 6: x i , y j ∈ X 7: end if 10: if (U (α,β,γ) s = ∅) then 11: end if 13: return D is the value domain of R. 14: Fix the parameters α, β, γ ∈ [0, 1], where α, β, γ ∈ D. 15:

19:
end for 20: end for 21: The subsets {X/R (α,β,γ) |(α, β, γ) ∈ D} of q-RPFHQSS correspond to the hyperedges or granules of the level hypergraph. 22: The k-level of granularity is constructed. 23: These hyperedges or granules at the k-level are mapped to (k + 1)-level. 24: Step 1-Step 22 are repeated until the whole universe is formulated as a single granule. 25: Except the last level, each level is mapped to the next level of granularity through different operators.

Definition 31.
Let R be a q-RPFER on X. A coarse grained universe X/R (α,β,γ) can be obtained by using q-RPFER, where [x i ] R α,β,γ = {x j ∈ X|x i Rx j }. This equivalence class [x i ] R (α,β,γ) is considered as a hyperedge in the level hypergraph.

Definition 32.
Let H 1 = (X 1 , E 1 ) and H 2 = (X 2 , E 2 ) be level hypergraphs of q-RPFHs, and H 2 has weaker granularity than H 1 . Suppose that e 1 i , e 2 j ∈ E 1 , and x 2 i , x 2 j ∈ X 2 , i, j = 1, 2, · · · , n. The zoom-in operator κ : The relations between the vertices of H 2 define the relationships among the hyperedges at the new level. The zoom-in operator of the two levels is shown in Figure 12.
H 1 Figure 12. The zoom-in operator.
(ii) As we know that for all x 2 i ∈ X 2 , we have κ(X 2 ) = (iii) Let [X 2 ] c = Z 2 and [X 2 ] c = Z 2 , then it is obvious that Z 2 ∩ X 2 = ∅ and Z 2 ∪ X 2 = X 2 . It follows from (ii) that κ(X 2 ) = E 1 , and we denote by W 1 that edge set of H 1 on which the vertex set Z 2 of H 2 is mapped under κ, i.e., κ( Since the relationship between hyperedges at the new level is the same as that of the relations among the vertices at the original level, we have Since the relationship between hyperedges at the new level is the same as that of the relations among the vertices at the original level, we have Since the relationship between hyperedges at the new level is the same as that of the relations among the vertices at the original level, we have E 1 ∪ E 1 =Ē 1 . Hence, we conclude that κ(X 2 ∪ X 2 ) = κ(X 2 ) ∪ κ(X 2 ). (vi) First, we show that X 2 ⊆ X 2 implies that κ(X 2 ) ⊆ κ(X 2 ). Since X 2 ⊆ X 2 , this implies that X 2 ∩ X 2 = X 2 and κ(X 2 ) = Since the relationship between hyperedges at the new level is the same as that of the relations among the vertices at the original level, we have We now prove that κ(X 2 ) ⊆ κ(X 2 ) implies that X 2 ⊆ X 2 . Suppose on the contrary that whenever κ(X 2 ) ⊆ κ(X 2 ), then there is at least one vertex x 2 i ∈ X 2 , but x 2 i ∈ X 2 , i.e., X 2 X 2 . Since, κ(x 2 i ) = e 1 i , and the relationship between hyperedges at the new level is the same as that of the relations among the vertices at the original level, we have e 1 i ∈ E 1 , but e 1 i ∈ E 1 ,.i.e., E 1 E 1 , which is a contradiction to the supposition. Thus, we have that κ(X 2 ) ⊆ κ(X 2 ) implies that X 2 ⊆ X 2 . Hence, X 2 ⊆ X 2 if and only if κ(X 2 ) ⊆ κ(X 2 ). Definition 33. Let H 1 = (X 1 , E 1 ) and H 2 = (X 2 , E 2 ) be level hypergraphs of q-RPFHs, and H 2 has weaker granularity than H 1 . Suppose that e 1 i , e 2 j ∈ E 1 , and x 2 i , x 2 j ∈ X 2 , i, j = 1, 2, · · · , n. The zoom-out operator σ : The zoom-out operator of two levels is shown in Figure 13.

Comparison Analysis
A q-ROFS is defined as an ordered pair (T, F) such that the sum of the qth power of the positive degree T and the qth power of negative degree F is no greater than one, i.e., T q + F q ≤ 1. Note that IFSs and PYFSs are special cases of q-ROFSs, i.e, every IF grade is also a PYF grade, as well as a q-ROF grade. Suppose that (x, y) is an IF grade, where x ∈ [0, 1], y ∈ [0, 1], and 0 ≤ x + y ≤ 1, since x q ≤ x, y q ≤ y, q ≥ 1, so we have 0 ≤ x q + y q ≤ 1. This implies that the class of q-ROFSs generalizes the classes of IFSs and PYFSs. In spite of the fact that IFSs, q-ROFSs, and PYFSs are applicable effectively in many decision making problems, still there are various circumstances that are difficult to handle using these models. For example, voters can be classified into four groups of people who: vote for, abstain, or refuse in a voting system, i.e., in many real-life situations, there are many answers other than yes or no, including, abstain and refusal. In such cases, the existing models of IFSs, q-ROFSs, and PYFSs fail to work effectively. Thus, Cuong [16] gave the idea of PFS, which is characterized by a positive degree T, a neutral degree N, and a negative degree F such that T + N + F ≤ 1. We have proposed a model of GrC based on q-RPFH and combined the advantages of two fruitful theories of q-ROFSs and PFSs. Our proposed q-RPF (T, N, F) model not only expresses the neutral degree, but also relaxes the condition of PFS that the sum of the three degrees must be less than one since T q + N q + F q ≤ 1, q ≥ 1 is the defining constraint of q-RPFS. Note that the proposed model extends the q-ROF and PF models by considering the neutral degree and by taking the constraint T q + N q + F q ≤ 1, q ≥ 1 into consideration; for example, if a decision maker assigns the positive, neutral, and negative degrees as 0.6, 0.3, and 0.5, respectively. Then, the triplet (0.6, 0.3, 0.5) cannot be handled through q-ROF or PF models, whereas the proposed q-RPF model can deal with this problem. This reveals that the q-RPF model contains a higher capacity in modeling of uncertain data than the q-ROF and PF models. It is worth noting that as we increase the value of parameter q, the space of uncertain data also increases, and the bounding constraint is satisfied by more triplets. Thus, a wider range of vague information can be expressed using q-RPFSs. Our proposed model is more generic as compared to the q-ROF and PF models, as when q = 1, the model reduces to the PF model, and when N = 0, q ≥ 1, it reduces to the q-ROF model. Hence, our approach is more flexible and generalized, and different values of q can be chosen by decision makers according to the different attitudes.

Conclusions and Future Directions
A q-rung picture fuzzy set is the generalization of the picture of the fuzzy set and the q-rung orthopair fuzzy set. This model deals with uncertainty and vagueness more practically and effectively because of its third component, i.e., the neutral part. The proposed q-rung picture fuzzy model owns the advantages of the q-rung orthopair fuzzy set, as well as the picture fuzzy set and proves to be a more powerful framework to analyze the qualitative and quantitative studies. Granular computing is a general criterion that emphasizes the exploitation of multi-level granular structures. GrC reflects and captures our ability to explore real-world problems at different levels of granularity and enables us to change these granularities during problem solving. In this paper, we have combined the q-RPFSs with hypergraphs to attain the advantages of both theories. A q-RPFS is the generalization of PFS and q-ROFS. This set deals with uncertainty and vagueness more practically and effectively because of its third component, i.e., the neutral part. It is worth noting that as we increase the value of parameter q, the space of uncertain data also increases, and the bounding constraint is satisfied by more triplets. Thus, a wider range of vague information can be expressed by using q-RPFSs. Our proposed model is more generic as compared to q-ROF and PF models, as when q = 1, the model reduces to the PF model, and when N = 0, q ≥ 1, it reduces to q-ROF model. We have defined q-RPFHs, and then, the proposed model has been used to examine the GrC. In this hypergraph model, we have obtained the granules as hyperedges from q-RPFER, and the partition of the universe has been defined by means of these granules. Further, we have defined the zoom-in and zoom-out operators to give the description of mappings between two hypergraphs. We have also discussed a concrete example to reveal the validity and applicability of our proposed model. We aim to extend our research: (1) q-rung picture fuzzy directed hypergraphs, (2) fuzzy rough soft directed hypergraphs, (3) granular computing based on fuzzy rough hypergraphs, (4) hesitant q-rung orthopair fuzzy hypergraphs, and (5) q-rung picture fuzzy concept lattice.