On various aggregation operators for picture fuzzy hypersoft information in decision making application

Abstract

Systematic assessment of insufficiencies and inexactness in the information along with parametrization of multi-sub attributes is one of the substantial features in the field of decision-making. In the present communication, a new way of defining Picture Fuzzy Hypersoft Set (PFHSS) has been presented which contains an additional capacity of accommodating the components of neutral membership (abstain) and refusal compared to Intuionistic Fuzzy Hypersoft Set (IFHSS). The main objective of the present study is to establish the novelty of PFHSS with some of the basic operations and introduce various important aggregation operators. Some of the important properties and operational laws related to the introduced picture fuzzy hypersoft weighted average/ordered weighted average operator (PFHSWA/PFHSOWA) and weighted geometric/ordered weighted geometric operator (PFHSWG/PFHSOWG) have been proved in detail. On the basis of these aggregation operators and obtained results, a new algorithm for solving a decision-making problem, involving the multi-sub attributes and their parametrization in the shade of abstain and refusal feature, has been proposed. A numerical example of the selection process of employees for a company has been solved in order to suitably ensure and validate the implementation of the proposed methodology. Some of the advantageous features of the proposed notions and algorithm have been listed along with the comparative analysis in contrast with the existing literature. Finally, the efficacy of the proposed notion and methodology has been duly concluded with the scope for future work.

Keywords

Picture fuzzy set soft set hypersoft set aggregation operators decision-making

1 Introduction

It has been paradigmatically pointed out that in various real life situations and physical problems, there is an ample amount of vagueness and uncertainties inherited which can not be adequately addressed with the help of the classical set theory and traditional mathematical theories. In view of this, the theory of multi-valent logic and consequently the vast literature on the fuzzy set theory [1] has been developed in due course of time by different researchers working in this field. This theory has brought an enormous archetypic change to enrich the multi-dimensional features of mathematics. In a real life problem where choosing an ideal alternative for the sake of satisfying any practical purpose is an important task, there the notion of decision-making serves as a delightful option. It may be noted that the membership degree of the elements in a set (classical set theory) is investigated in binary form (0 or 1) while in case of fuzzy set theory, membership categorization is in the entire continuous range of 0 to 1.

Further, in 1986, Atanassov [2] came up with the idea of Intuitionistic Fuzzy Set (IFS) to handle with the component of indeterminacy/heistation in the inexact information lying between membership degree and non-membership degree. However, the notion of intuitionistic fuzzy set also has some shortcomings, as it is not able to deal with the inconsistent and deceptive data in which alternatives have parametric values. Later, Yager [4] unveiled that the formal structure of fuzzy sets and intuitionistic fuzzy set are not able to portray the total flexibility and freedom of the decision maker’s opinion at its full. Subsequently, the notion of Pythagorean Fuzzy Set (PyFS) [4] was introduced which was found to be capable enough for adequately intensifying the spell of information by presenting a new condition. The main objective behind the introduction of PyFS was to handle those situations where the sum of membership and non-membership degrees exceeds one, wherein Yager modified the condition to squared sum less than or equal to 1.

In literature, different researchers have studied various other extensions of fuzzy sets and intuitionistic fuzzy sets based on the observed limitations/shortcomings of the existing notions. Foe example, consider a situation of the voting system where the voters can be classified into four different stratums: favor, against, abstain and one more important class is the one who refuses for voting (refusal). On the basis of this categorization and to address the additional component of refusal, Coung [6] introduced the notion of Picture Fuzzy Set (PFS) to deal with such situations which would be satisfactorily close to the adaptability of human nature, where all the four parameters taken into account. Further, Atanassov et al. [3] presented some other types of fuzzy sets and their generalizations where it may be noted that a picture fuzzy set can also be represented by an interval-valued intuitionistic fuzzy set. However, it may also be noted that all such sets stated above are unable to deal with alternatives which have parametric values.

In order to deal with such difficulties, Molodtsov [7] introduced the concept of soft set which has the capability to incorporate the parametric nature associated with the alternatives in a decision-making problem. Then, Maji et al. [8] introduced the concept of fuzzy soft set by making a suitable synchronization of fuzziness and soft set with parametrization. Also, in continuation Maji et al. [9] presented the notion of Intuitionistic Fuzzy Soft Set (IFSS) with the compatible combination of intuitionistic fuzzy set and soft set along with important properties and fundamental operations. In addition to PFS, Coung [6] also introduced the notion of Picture Fuzzy Soft Set (PFSS) by combining the four components of picture fuzzy information and parametrization feature of soft set theory.

In these extensions, Smarandache [15] observed that it is difficult to deal with some irreconcilable and inexplicit information where the involvement of attributes from a set of parameters containing further sub attributes is present and subsequently introduced the concept of Hypersoft Set (HSS). This proposed set counterfeits the one-parameter function to a multi-parameter (sub-attribute) function. According to Smarandache, HSS can adeptly handle such kind of uncertainty in contrast to soft set. In these days, the theory of HSS and its extensions have been gaining a significant amount of appreciation in the field of soft computing. Smarandache also introduced Fuzzy Hypersoft Sets (FHSS) [15] to deal with the uncertainties involving sub-attribute family of parameters. Also, in order to include the indeterminacy component in the sub-attribute family of parameters, the concept of Intuitionistic Fuzzy Hypersoft Set (IFHSS) has further been studied by Yolcu et al. [16]. The notion of hypersoft set has got many other important extensions with several aggregation operators [17 –23].

In the field of soft computing applications, the concept of aggregation operator (average/geometric/hybrid/ordered etc.) plays an important role in handling the interrelationships and concerns among various criteria. Garg [24] presented weighted aggregation operations on PFSSs and utilized them to solve multi-criteria decision making (MCDM) problems. Garg et al. [25] presented improved version of interactive aggregation operators by taking interval valued IFS in decision making process. Arora & Garg [26] have also given some robust aggregation operators for MCDM considering the intuitionistic fuzzy soft information. Garg and Arora [27] introduced the aggregation operators, correlation coefficient and the TOPSIS technique for IFSSs. Peng et al. [28] introduced the concept of Pythagorean Fuzzy Soft Set (PyFSS) by combining the previous two sets PyFS and soft set along with some fundamental properties. Zulqarnain et al. [29] devised the functional rules for Pythagorean Fuzzy soft Numbers (PyFSNs) and established various aggregation operators namely, Pythagorean fuzzy soft weighted average and geometric, with the help of the operators for PyFSNs. They also designed a decision-making process to find the solutions of the problems of MADM with the use of introduced operators. Peng [30] studied various important operations for interval-valued Pythagorean fuzzy sets. Riaz et al. [31] have developed aggregation operators by using Einstein operations and inspected their elementary properties along with a decision making problem.

On the basis of the above literature review, it is being observed that the existing extensions do not incorporate the degree of refusal in the sub-attributes parametrization of objects. However, Chinnadurai and Robin [32] have very recently initiated the study on Picture Fuzzy Hypersoft Set (PFHSS) with a different approach and applied it in TOPSIS method by making use of correlation coefficient. In order to deal with decision making problems where the flexibility of being choice specific, the above definition will encounter certain restrictions and limitations.

In order to deal with flexibility and matter of choice specific under sub-parametrization, we propose to introduce a new way of defining Picture Fuzzy Hypersoft Set (PFHSS) which would certainly improve the enumerating power and variability of the available information. The main objective behind the proposed way of defining PFHSS would be to devise a novel structure where the decision-makers/experts would gain an ample flexibility and freedom. In the present work, various fundamental operations and different aggregation operators (average/geometric/hybrid/ordered etc.) for the proposed way of defining PFHSS have been studied. An algorithm based on the new aggregation operators for solving a decision making problem has been presented with a numerical illustration.

The proposed way of defining PFHSS gives more flexibility to the decision-maker with the inclusion of the degree of refusal and degree of abstain.

It has an advantage to deal with any kind of sub-attribute family of parameters in decision-making problems.

With the help of the newly proposed PFHSS, Picture Fuzzy Hypersoft Weighted Average (PFHSWA) and Picture Fuzzy Hypersoft Geometric Average (PFHSGA) have been derived for application in the MCDM technique with the revised score and accuracy function.

The flexibility of the proposed approach has been described through the comparative analysis in contrast with some of the recent existing techniques.

The proposed study has been organized as follows: In Section 2, fundamental definitions related to the proposed notion have been presented to build the formal structure. We presented the novel definition of the proposed Picture Fuzzy Hypersoft Set (PFHSS) along with example and some basic operations in Section 3. In Section 4, new picture fuzzy hypersoft weighted average/ordered weighted average operator (PFHSWA/PFHSOWA) and weighted geometric/ordered weighted geometric operator (PFHSWG/PFHSOWG) have been presented with some important results. On the basis of the obtained results, an algorithm for solving a decision-making problem which involves the multi-sub attributes and their parametrization has been proposed in Section 5. Further, an illustrative example of a decision-making (selection process) problem has been solved for showing the computational process and implementation of the proposed algorithm. In Section 6, we precisely present the advantages and effectivity of the proposed algorithm and carry out the comparative analysis in contrast with the existing techniques. Finally, the paper has been duly concluded with the scope of future work in Section 7.

2 Preliminaries

In this section, we are presenting the basic notions and definitions of various other fundamental sets which are available in the literature. These preliminaries would help to understand the proposed notions of picture fuzzy hypersoft set and increase the readability for the researchers.

Definition 1. Intuitionistic Fuzzy Set (IFS) [2]. “An intuitionistic fuzzy Set R in V is given by $R = {v, ρ_{R} (v), ω_{R} (v) | v ɛ V};$ where ρ_R : V → [0, 1] is the degree of membership of v in R and ω_R : V → [0, 1] is the degree of non-membership of v in R and ρ_R, ω_R satisfies the constraint $0 \leq ρ_{R} (v) + ω_{R} (v) \leq 1 (\forall v \in V);$ and, ℶ_R (v) = (1 - (ρ_R (v) + ω_R (v))) is called the degree of indeterminacy of v in V. We denote the set of all intuitionistic fuzzy sets over V by IFS (V).”

Definition 2. Picture Fuzzy Set (PFS) [6]. “A picture fuzzy Set R in V is given by $R = {v, ρ_{R} (v), τ_{R} (v), ω_{R} (v) | v ɛ V};$ where ρ_R : V → [0, 1] is the degree of positive membership of v in R, τ_R : V → [0, 1] is the degree of neutral membership of v in R and ω_R : V → [0, 1] is the degree of negative membership of v in R and ρ_R, τ_R, ω_R satisfies the constraint $0 \leq ρ_{R} (v) + τ_{R} (v) + ω_{R} (v) \leq 1 (\forall v \in V);$ and, ℶ_R (v) = (1 - (ρ_R (v) + τ_R (v) + ω_R (v))) is called the degree of refusal membership of v in V. We denote the set of all the picture fuzzy sets over V by PFS (V).”

Definition 3. Soft Set (SS) [7]. “Let V be an initial universe and K be a set of parameters and B ⊆ K. A pair (R,B) is called a soft set over V, where R is a mapping from R: B→P (V). In other words, the soft set is a parameterized family of subsets of the set V. Here, P (V) is the set of all subsets of the set V.”

Definition 4. Fuzzy Soft Set (FSS) [8]. “Let V be an initial universe and K be a set of parameters and B ⊆ K. A pair (R, B) is called a fuzzy soft set over V, where R is a mapping from R : B→F (V). In other words, FSS is a parameterized family of fuzzy subsets of the set V. Here, F (V) is the set of all fuzzy subsets of V .”

Definition 5. Picture Fuzzy Soft Set (PFSS) [6]. “Let V be an initial universe and K be a set of parameters and B ⊆ K. A pair (R, B) is called a picture fuzzy soft set over V, where R is a mapping given R : B→PFS (V) for every b ∈ B, R(b) is a picture fuzzy soft set of V and is called Picture Fuzzy Value for the set of parameter b. Here, PFS (V) is the set of all picture fuzzy subsets of V and R(b) = {v, ρ_R(b) (v) , τ_R(b) (v) , ω_R(b) (v) |v ɛ V} where, ρ_R(b) (v) , τ_R(b) (v) , ω_R(b) (v) are the degrees of positive membership, neutral membership and negative membership respectively, with the constraint $0 \leq ρ_{R (b)} (v) + τ_{R (b)} (v) + ω_{R (b)} (v) \leq 1,$ and the degree of refusal is given by $ℶ_{R (b)} (v) = (1 - (ρ_{R (b)} (v) + τ_{R (b)} (v) + ω_{R (b)} (v)))$ (∀ v ∈ V) . "

Definition 6. Hypersoft Set (HSS) [15]. “Let V be the universal set and P(V) be the power set of V. Consider k₁, k₂, … . k_n for n ≥ 1, be n well-defined attributes, whose corresponding attribute values are the sets K₁, K₂, …, K_n with K_i ∩ K_j = φ for i ≠ j and i, j ∈ { 1, 2, …, n } . Let B_i be the non-empty subsets of K_i for each i = 1, 2, …, n . Then the pair (R, B₁ × B₂ × … . B_n) is said to be Hypersoft Set over V where R : B₁ × B₂ × … . × B_n → P (V). In other words, hypersoft set is a multi-parameterized family of subsets of the set V.”

Definition 7. Fuzzy Hypersoft Set (FHSS) [15]. “Let V be the universal set and F (V) be the set all fuzzy subsets of V. Consider k₁, k₂, … . k_n for n ≥ 1, be n well-defined attributes, whose corresponding attribute values are the sets K₁, K₂, …, K_n with K_i ∩ K_j = φ for i ≠ j and i, j ∈ { 1, 2, …, n } . Let B_i be the non-empty subsets of K_i for each i = 1, 2, …, n . Then the pair (R, B₁ × B₂ × … . B_n) is said to be Fuzzy Hypersoft Set over V where R : B₁ × B₂ × … . × B_n → F (V) and, R (b) = { v, R (b) (v) |v ∈ V }; b ∈ B₁ × B₂ × … B_n ⊆ K₁ × K₂ × … K_n .”

Definition 8. Intuitionistic Fuzzy Hypersoft Set (IFHSS) [15]. “Let V be the universal set and IFS(V) be the set of all intuitionistic fuzzy subsets of V.

Consider k₁, k₂, … . k_n for n ≥ 1, be n well –defined attributes, whose corresponding attribute values are the sets K₁, K₂, … , K_n with K_i ∩ K_j = φ for i ≠ j and i, j ∈ { 1, 2, …, n }. Let B_i be the non-empty subsets of K_i for each i = 1, 2, …, n . An Intuitionistic Fuzzy Hypersoft Set is defined as the pair, (R, B₁ × B₂ × … . × B_n) where ; R : B₁ × B₂ × … . × B_n → IFS (V) and $\begin{matrix} R (B_{1} \times B_{2} \times \dots . \times B_{n}) = \\ {< ϑ, (\frac{v}{ρ_{R (ϑ)} (v), ω_{R (ϑ)} (v)}) > | v ɛ V}; \end{matrix}$ where, ϑ ∈ B₁ × B₂ × … . × B_n ⊆ K₁ × K₂ × … . K_n. It may be noted that ρ and ω represent membership and non-membership degrees respectively, and satisfies the condition 0≤ ρ_R(ϑ) (v) + ω_R(ϑ) (v) ≤1 ; where ρ_R(ϑ) (v) , ω_R(ϑ)∈ [0, 1] ; and, complement_R(ϑ) (v) = 1 - ρ_R(ϑ) (v) - ω_R(ϑ) (v) is called the degree of indeterminacy.”

Definition 9. Pythagorean Fuzzy Hypersoft Set [20]. “Let V be the universal set and PyFS(V) be the set of all Pythagorean fuzzy subsets of V.

Consider k₁, k₂, …, k_n for n ≥ 1, be n well-defined attributes, whose corresponding attribute values are the sets K₁, K₂, … , K_n with K_i ∩ K_j = φ for i ≠ j and i, j ∈ { 1, 2, …, n }. Let B_i be the non-empty subsets of K_i for each i = 1, 2, …, n . A Pythagorean Fuzzy Hypersoft Set is defined as the pair, (R, B₁ × B₂ × … . × B_n) , where R : B₁ × B₂ × … . × B_n → PyFS (V) and $\begin{matrix} R (B_{1} \times B_{2} \times \dots . \times B_{n}) = \\ {< ϑ, (\frac{v}{ρ_{R (ϑ)} (v), ω_{R (ϑ)} (v)}) > | v ɛ V}; \end{matrix}$ where ϑ ∈ B₁ × B₂ × … . × B_n ⊆ K₁ × K₂ × … . K_n. It may be noted that ρ and ω represent membership and non-membership degrees respectively, and satisfies the condition $0 \leq ρ_{R (ϑ)}^{2} (v) + ω_{R (ϑ)}^{2} (v) \leq 1}; where ρ_{R (ϑ)} (v),$ ω_R(ϑ)∈ [0, 1] ; and, ${complement}_{R (ϑ)} (v) = \sqrt{1 - ρ_{R (ϑ)}^{2} (v) - ω_{R (ϑ)}^{2} (v)}$ is called the degree of indeterminacy.”

3 Picture fuzzy hypersoft set & operations

In this section, we are presenting the definition of Picture Fuzzy Hypersoft Set (PFHSS) in a different way along with various important properties and fundamental operations. However, Chinnadurai and Robin [32] have very recently defined Picture Fuzzy Hypersoft Set (PFHSS) which restrict the decision makers of being choice specific. In order to give the decision makers more flexibility, we defined the PFHSS with a different approach. The following definition of PFHSS (the parametrization of multi-sub attributes and all the four components of picture fuzzy information) is being proposed:

Definition 10. (Picture Fuzzy Hypersoft Set). Let V be the universal set and PFS(V) be the set of all picture fuzzy subsets of V. Consider k₁, k₂, … . k_n for n ≥ 1, be n well-defined attributes, whose corresponding attribute values are the sets K₁, K₂, … , K_n with K_i ∩ K_j = φ for i ≠ j and i, j ∈ { 1, 2, …, n }. Let B_i be the non-empty subsets of K_i for each i = 1, 2, …, n .

A Picture Fuzzy Hypersoft Set (PFHSS) is defined as the pair (R, B₁× B₂ × … . × B_n) ; where R : B₁ × B₂ × … × B_n → PFS (V) and $\begin{matrix} R (B_{1} \times B_{2} \times \dots . \times B_{n}) = \\ {< ϑ, (\frac{v}{ρ_{R (ϑ)} (v), τ_{R (ϑ)} (v), ω_{R (ϑ)} (v)}) > | v ɛ V} . \end{matrix}$ It may be noted that ϑ ∈ B₁ × B₂ × … . × B_n ⊆ K₁ × K₂ × … . K_n and ρ τ and ω represent the positive membership, neutral membership and negative membership degrees respectively, and satisfies the condition 0≤ ρ_R(ϑ) (v) + τ_R(ϑ) (v) + ω_R(ϑ) (v) ≤ 1 ; where ρ_R(ϑ) (v) , τ_R(ϑ) (v) , ω_R(ϑ) (v) ∈ [0, 1] . The term complement_R(ϑ) (v) = 1 - ρ_R(ϑ) (v) - τ_R(ϑ) (v) - ω_R(ϑ) (v) is called the degree of refusal membership of v in PFS (V). For the sake of simplicity, we denote K₁ × K₂ × … . K_n by Γ and B₁ × B₂ × … . × B_n by Λ. We denote the set of all PFHSSs over V by PFHSS (V).

Particular Case: In particular, the proposed definition also directs that every picture fuzzy hypersoft set is also a picture fuzzy soft set. If we select the parameters from only one attribute set, say, K₁, while forming the picture fuzzy hypersoft set, then the resulting set becomes the picture fuzzy soft set. In other words, picture fuzzy hypersoft set is the generalized version of the picture fuzzy soft set.

In view of the possible variability based on the extreme values of the four components of picture fuzzy information, we may categorize two sub-definitions of PFHSS as follows:

Definition 11. A picture fuzzy hypersoft set (R, Γ ) over the universe V is known as void picture fuzzy hypersoft set and denoted by 0_{(V_PFH,Γ)} if for all v ∈ V and ϑ ∈ Γ, .

Definition 12. A picture fuzzy hypersoft set (R, Γ ) over the universe V is known as absolute picture fuzzy hypersoft set and denoted by 1_{(V_PFH,Γ)} if for all v ∈ V and ϑ ∈ Γ, .

Example 1. Let V be the set of available four smart phones given as V ={ v₁, v₂, v₃, v₄ } and the set of attributes given as K₁ = Display, K₂ = Storage1 (ROM), K₃ = Storage2 (RAM), K₄ = Colour. Further, in order to understand the framework of proposed notion, assume that their respective sub-attributes are $\begin{matrix} K_{1} = Display = {OLED (α_{1}), AMOLED (α_{2}), \\ super AMOLED (α_{3})} \end{matrix}$ $\begin{matrix} K_{2} = Storage 1 = {32 GB (β_{1}), 64 GB (β_{2}), \\ 128 GB (β_{3})} \end{matrix}$ $\begin{matrix} K_{3} = Storage 2 = {4 GB (γ_{1}), 8 G B (γ_{2}), \\ 16 GB (γ_{3})} \end{matrix}$ $\begin{matrix} K_{4} = Colour = {Black (δ_{1}), Rose Gold (δ_{2}), \\ Space Grey (δ_{3})} \end{matrix}$ Now, consider two different subsets C₁ = {α₁, α₃} , C₂ = {β₁} , C₃ = {γ₁, γ₂ } , C₄ = {δ₁} of K_i for each i = 1, 2, 3, 4 . Then the picture fuzzy hypersoft set (R₁, Λ ) and (R₂, Λ′ ) may have the set-theoretic (Appendix II) and tabular (Table 1 & Table 2) representation.

Basic Operations on Picture Fuzzy Hypersoft Sets:

In view of the proposed definition of picture fuzzy hypersoft set above, we formally define some of the fundamental set-theoretic operations for the sake of understanding and better readability.

Let (R₁, Λ ) and (R₂, Λ′ ) be two picture fuzzy hypersoft sets on V and Λ, Λ′ ⊆ Γ be the set of multi-parameters.

Complement. The complement of picture fuzzy hypersoft set over V is denoted by (R₁, Λ ) ^c and defined as (R₁, Λ ) ^c = (R₁^c, Λ) , where R₁^c : Γ → PFS (V) is a mapping given by R₁^c (Λ) = (R₁ (Λ)) ^c ∀ Λ ⊆ Γ.

Thus if, $\begin{matrix} (R_{1}, Λ) = {< ϑ, (\frac{v}{ρ_{R (ϑ)} (v), τ_{R (ϑ)} (v), ω_{R (ϑ)} (v)}) \\ > | v ɛ V, ϑ \in Λ} \end{matrix}$ then $\begin{matrix} {(R_{1}, Λ)}^{c} = {< ϑ, (\frac{v}{ω_{R (ϑ)} (v), τ_{R (ϑ)} (v), ρ_{R (ϑ)} (v)}) \\ > | v ɛ V, ϑ \in Λ} \end{matrix}$ Subset. Let V be the universe of discourse and (R₁, Λ ), (R₂, Λ′ ) be any two picture fuzzy hypersoft sets over the set V . Then, (R₁, Λ ) is said to be a picture fuzzy hypersoft subset of (R₂, Λ′ ) and denoted by (R₁, Λ ) ⊆ (R₂, Λ′ ) if Λ ⊆ Λ′ and for any ϑ ∈ Λ, R₁ (ϑ) ⊆ R₂ (ϑ), i.e., ∀v ∈ V and ϑ ∈ Λ,

Equality. Let V be the universe of discourse and (R₁, Λ ) , (R₂, Λ′ ) be any two picture fuzzy hypersoft sets over the set V . Then, (R₁, Λ ) is said to be a picture fuzzy hypersoft equal (R₂, Λ′ ) and denoted by (R₁, Λ ) = (R₂, Λ′ ) if for all v ∈ V and ϑ ∈ Λ, and ω_R₁(ϑ) (v) = ω_R₂(ϑ) (v) .

Union. Let V be the universe of discourse, Λ, Λ′ ⊆ Atilde ; and (R₁, Λ ) , (R₂, Λ′ ) be any two picture fuzzy hypersoft sets over V. The union of (R₁, Λ ) and (R₂, Λ′ ) is denoted by (R₁, Λ ) ∪ (R₂, Λ′ ) = (R, Λ^′′), where Λ^′′ = Λ ∪ Λ′ and ϑ ∈ Λ^′′ $R (ϑ) = {\begin{matrix} R_{1} (ϑ) if ϑ \in Λ - Λ^{'} \\ R_{2} (ϑ) if ϑ \in Λ^{'} - Λ \\ R_{1} (ϑ) \cup R_{2} (ϑ) if ϑ \in Λ \cap Λ^{'} \end{matrix}$ In other words, ∀ ϑ ∈ Λ ∩ Λ′, we have $\begin{matrix} R (ϑ) = {v, \max (ρ_{R_{1} (ϑ)} (v)), \\ (ρ_{R_{2} (ϑ)} (v)), \min (τ_{R_{1} (ϑ)} (v), τ_{R_{2} (ϑ)} (v)), \\ {\min (ω}_{R_{1} (ϑ)} (v), ω_{R_{2} (ϑ)} (v))} . \end{matrix}$

Intersection. Let V be the universe of discourse, Λ, Λ′ ⊆ Γ and (R₁, Λ ) , (R₂, Λ′ ) be any two picture fuzzy hypersoft sets over V. The intersection of (R₁, Λ ) and (R₂, Λ′ ) is denoted by (R₁, Λ) ∩ (R₂, Λ′) = (R, Λ^*), where Λ^* = Λ ∩ Λ′ and ϑ ∈ Λ^* $R (ϑ) = {\begin{matrix} R_{1} (ϑ) if ϑ \in Λ - Λ^{'} \\ R_{2} (ϑ) if ϑ \in Λ^{'} - Λ \\ R_{1} (ϑ) \cap R_{2} (ϑ) if ϑ \in Λ \cap Λ^{'} . \end{matrix}$ In other words, ∀ϑ ∈ Λ ∩ Λ′, we have $\begin{matrix} R (ϑ) = {v, \min (ρ_{R_{1} (ϑ)} (v)), \\ (ρ_{R_{2} (ϑ)} (v)), \min (τ_{R_{1} (ϑ)} (v), τ_{R_{2} (ϑ)} (v)), \\ {\max (ω}_{R_{1} (ϑ)} (v), ω_{R_{2} (ϑ)} (v))} . \end{matrix}$ Remarks: Let (R, Λ) be a picture fuzzy hypersoft set over the universal set V. Then,

((R, Λ) ^c) ^c = (R, Λ)

$0_{(V_{PFH, Γ})}^{c} = 1_{(V_{PFH, Γ})}$

$1_{(V_{PFH, Γ})}^{c} = 0_{(V_{PFH, Γ})}$

Difference. Let V be the universe of discourse, Λ, Λ′ ⊆ Γ and (R₁, Λ ) , (R₂, Λ′ ) be any two picture fuzzy hypersoft sets over V. The difference of (R₁, Λ ) and (R₂, Λ′ ) is denoted by (R₁, Λ ) ∖ (R₂, Λ′ ) = (R, Λ^†), where (R₁, Λ ) ∩ (R₂, Λ′) ^c = (R, Λ^†).

Example 2. Consider the attributes and the picture fuzzy hypersoft sets taken in Example 1 over V ={ v₁, v₂, v₃, v₄ }. The computation for various proposed operations on the sets under consideration are provided as follows: $\begin{matrix} (R_{1}, Λ) \cup (R_{2}, Λ^{'}) = \\ {\begin{matrix} \begin{matrix} < {(α}_{3}, β_{1}, γ_{1}, δ_{1}), {\frac{v_{1}}{(0.1, 0.1, 0.5)}, \frac{v_{2}}{(0.2, 0.3, 0.2)}, \frac{v_{3}}{(0.1, 0.2, 0.2)}, \frac{v_{4}}{(0.1, 0, 0.6)}} > \\ < {(α}_{3}, β_{1}, γ_{3}, δ_{1}), {\frac{v_{1}}{(0.2, 0.3, 0.2)}, \frac{v_{2}}{(0.2, 0.1, 0.2)}, \frac{v_{3}}{(0.1, 0.3, 0.4)}, \frac{v_{4}}{(0, 0, 1)}} > \\ < {(α}_{3}, β_{2}, γ_{1}, δ_{1}), {\frac{v_{1}}{(0.1, 0.1, 0.5)}, \frac{v_{2}}{(0.2, 0.2, 0.3)}, \frac{v_{3}}{(0.1, 0.4, 0.4)}, \frac{v_{4}}{(0.1, 0.2, 0.3)}} > \end{matrix} \\ < {(α}_{3}, β_{2}, γ_{3}, δ_{1}), {\frac{v_{1}}{(0.1, 0.2, 0.5)}, \frac{v_{2}}{(0.2, 0.2, 0.2)}, \frac{v_{3}}{(0.2, 0.4, 0.4)}, \frac{v_{4}}{(0.1, 0.2, 0.6)}} > \\ < {(α}_{1}, β_{1}, γ_{1}, δ_{1}), {\frac{v_{1}}{(0.1, 0.3, 0.2)}, \frac{v_{2}}{(0.2, 0.4, 0.3)}, \frac{v_{3}}{(0.1, 0.2, 0.5)}, \frac{v_{4}}{(0, 0, 1)}} > \\ < {(α}_{1}, β_{1}, γ_{2}, δ_{1}), {\frac{v_{1}}{(0.2, 0.3, 0.1)}, \frac{v_{2}}{(0.2, 0.1, 0.1)}, \frac{v_{3}}{(0.1, 0.3, 0.5)}, \frac{v_{4}}{(0, 0, 1)}} > \\ < {(α}_{3}, β_{1}, γ_{2}, δ_{1}), {\frac{v_{1}}{(0.5, 0.2, 0.1)}, \frac{v_{2}}{(0.2, 0.4, 0.2)}, \frac{v_{3}}{(0.2, 0.4, 0.1)}, \frac{v_{4}}{(0.1, 0.2, 0.1)}} > \end{matrix}}; \end{matrix}$ $\begin{matrix} (R_{1}, Λ) \cap (R_{2}, Λ^{'}) = {< {(α}_{3}, β_{1}, γ_{1}, δ_{1}), {\frac{v_{1}}{(0.1, 0.3, 0.7)}, \\ \frac{v_{2}}{(0.2, 0.4, 0.2)}, \frac{v_{3}}{(0.1, 0.4, 0.4)}, \frac{v_{4}}{(0, 0.2, 1)}} >}; \end{matrix}$ and $\begin{matrix} (R_{1}, Λ) ∖ (R_{2}, Λ^{'}) = {< {(α}_{3}, β_{1}, γ_{1}, δ_{1}), {\frac{v_{1}}{(0.1, 0.3, 0.5)}, \\ \frac{v_{2}}{(0.2, 0.4, 0.2)}, \frac{v_{3}}{(0.1, 0.4, 0.4)}, \frac{v_{4}}{(0, 0.2, 1)}} >} . \end{matrix}$

Now, for the sake of methodological calculations and further simplifications, the notion of PFHSS can also be viewed as a Picture Fuzzy Hypersoft Number (PFHSN) given below: $R_{v_{i}} (ϑ_{j}) = {ρ_{R (ϑ_{j})} (v_{i}), τ_{R (ϑ_{j})} (v_{i}), ω_{R (ϑ_{j})} (v_{i}) | v_{i} \in V} .$

This structure would be known as Picture Fuzzy Hypersoft Number (PFHSN). Also, for the convenience, PFHSN can also be described as I_{ϑ_ij} = (ρ_{R(ϑ_ij)}, {τ_{R(ϑ_ij)} {ω_{R(ϑ_ij)} ) ; where the subscript ϑ_ij is used to establish the connection between the alternatives and the parameters for the calculation purposes.

In order to propose a new algorithm for ranking the alternatives based on the proposed PFHSS and their aggregation operators, we suitably reframe the notion of the score function and accuracy function for PFHSNs as follows:

Let I_{ϑ_ij} = (ρ_{R(ϑ_ij)}, τ_{R(ϑ_ij)} ω_{R(ϑ_ij)}) be a PFHSN. The score function of I_{ϑ_ij} is given by $S (I_{ϑ_{ij}}) = ρ_{R (ϑ_{ij})} - τ_{R (ϑ_{ij})} - ω_{R (ϑ_{ij})}; i, j \in {1, 2, \dots, n}$ and $S (I_{ϑ_{ij}}) \in [- 1, 1] .$

Remarks:

It may be noted that in some situations, the score function for two different PFHSNs may be the same, e.g., if we take I_ϑ₁₁ = (0.3, 0.5, 0.2) and I_ϑ₁₂ = (0.6, 0.4, 0.6) as two PFHSNs then as per the definition of the score function, the score value would be -0.4.

In such cases, it will not be easy to decide which one is the most appropriate I_ϑ₁₁ or I_ϑ₁₂. Therefore, in order to overcome such problems, the notion of accuracy function has to be further introduced.

The Accuracy function of I_{ϑ_ij} is given by $ℍ (I_{ϑ_{ij}}) = ρ_{R (ϑ_{ij})} + τ_{R (ϑ_{ij})} + ω_{R (ϑ_{ij})}; i, j \in {1, 2, \dots, n} .$

It may also be noted that $ℍ (I_{ϑ_{ij}}) \in [0, 1]$ and for the comparison of the two PFHSNs, I_{ϑ_ij} and $I_{ϑ_{ij}}^{'}$ , the following comparisons of the above defined functions have been done.

If $S (I_{ϑ_{ij}})$ > $S (J_{ϑ_{ij}})$ then I_{ϑ_ij} > $I_{ϑ_{ij}}^{'}$ .

If $S (I_{ϑ_{ij}})$ < $S (J_{ϑ_{ij}})$ then $I_{ϑ_{ij}} < I_{ϑ_{ij}}^{'}$ .

If $S (I_{ϑ_{ij}})$ = $S (J_{ϑ_{ij}})$ then $I_{ϑ_{ij}} \equiv I_{ϑ_{ij}}^{'}$ .

If $ℍ (I_{ϑ_{ij}})$ > $ℍ (J_{ϑ_{ij}})$ then $I_{ϑ_{ij}} > I_{ϑ_{ij}}^{'}$ .

If $ℍ (I_{ϑ_{ij}})$ < $ℍ (J_{ϑ_{ij}})$ then $I_{ϑ_{ij}} < I_{ϑ_{ij}}^{'}$ .

If $ℍ (I_{ϑ_{ij}}) = ℍ (J_{ϑ_{ij}})$ then $I_{ϑ_{ij}} \equiv I_{ϑ_{ij}}^{'}$ .

4 Average/geometric aggregation operators

In the process of information fusion, the mathematical notion of aggregation operator, which aggregates the interrelated multiple input values to solely one outturn value, is an essential tool and widely utilized for handling various decision making problems. The problems are not only limited to the field of mathematics but also widely spread in physics, economics, engineering, social and other sciences. In this section, we devise two types of aggregation operators (Averaging aggregation operators and Geometric aggregation operators) for Picture Fuzzy Hypersoft Numbers and discuss various results based on this.

For the sake of defining the picture fuzzy hypersoft weighted averaging operator/geometric operator, we first need to understand the notion of direct sum, direct product, scalar multiplication, exponent and complement of PFHSNs which have bene defined as below:

Let I_{ϑ_d} = (ρ_{ϑ_d}, τ_{ϑ_d}, ω_{ϑ_d}), I_ϑ₁₁ = (ρ_ϑ₁₁, τ_ϑ₁₁, ω_ϑ₁₁) and I_ϑ₁₂ = (ρ_ϑ₁₁, τ_ϑ₁₂, ω_ϑ₁₂) be three PFHSNs and κ be a positive real number. Then, the following operations are defined over three PFHSNs:

(Direct Sum) I_ϑ₁₁ ⊕ I_ϑ₁₂ $\begin{matrix} = 〈 ρ_{ϑ_{11}} + ρ_{ϑ_{12}} - ρ_{ϑ_{11}} ρ_{ϑ_{12}}, τ_{ϑ_{11}} τ_{ϑ_{12}}, \\ ω_{ϑ_{11}} ω_{ϑ_{12}} 〉 . \end{matrix}$

(Direct Product) I_ϑ₁₁ ⊗ I_ϑ₁₂ $\begin{matrix} = 〈 ρ_{ϑ_{11}} ρ_{ϑ_{12}}, τ_{ϑ_{11}} + τ_{ϑ_{12}} - τ_{ϑ_{11}} τ_{ϑ_{12}}, \\ ω_{ϑ_{11}} + ω_{ϑ_{12}} - ω_{ϑ_{11}} ω_{ϑ_{12}} 〉 . \end{matrix}$

(Scalar Multiplication) κI_{ϑ_d} $\begin{matrix} = 〈 [1 - {(1 - ρ_{ϑ_{d}})}^{κ},] \\ [{(τ_{ϑ_{d}})}^{κ}, {(ω_{ϑ_{d}})}^{κ}] 〉 . \end{matrix}$

(Exponent) $I_{ϑ_{d}}^{κ}$ $\begin{matrix} = 〈 [{(ρ_{ϑ_{d}})}^{κ}, {1 - (1 - τ_{ϑ_{d}})}^{κ}], \\ [{1 - (1 - ω_{ϑ_{d}})}^{κ}] 〉 . \end{matrix}$

(Complement) $I_{ϑ_{d}}^{c} = (ω_{ϑ_{d}}, τ_{ϑ_{d}}, ρ_{ϑ_{d}}) .$

Theorem 1. Let I_{ϑ_d} = (ρ_{ϑ_d}, τ_{ϑ_d}, ω_{ϑ_d}) and I_{ϑ_e} = (ρ_{ϑ_e}, τ_{ϑ_e}, ω_{ϑ_e}) be two PFHSNs. Suppose κ₁, κ₂, κ₃ be three positive real numbers. Then, the following functional laws hold:

I_{ϑ_d} ⊕ I_{ϑ_e} = I_{ϑ_e} ⊕ I_{ϑ_d}

I_{ϑ_d} ⊗ I_{ϑ_e} = I_{ϑ_e} ⊗ I_{ϑ_d}

κ (I_{ϑ_d} ⊕ I_{ϑ_e}) = κI_{ϑ_e} ⊕ κI_{ϑ_d}

${(I_{ϑ_{d}} \otimes I_{ϑ_{e}})}^{κ} = I_{ϑ_{e}}^{κ} \otimes I_{ϑ_{d}}^{κ}$

κ₁I_{ϑ_d} ⊕ κ₂I_{ϑ_d} = (κ₁ + κ₂) I_{ϑ_d}

$I_{ϑ_{d}}^{κ_{1}} \otimes I_{ϑ_{d}}^{κ_{2}} = {I_{ϑ_{d}}}^{(κ_{1} + κ_{2})}$

(I_{ϑ_d}^κ₁) ^κ₂ = I_{ϑ_d}^κ₁
κ₂

Proof 1: The proof of all these functional laws can easily be derived with the help of basic definitions (a)-(e).

4.1 Picture fuzzy hypersoft weighted averaging (PFHSWA) aggregation operator

Definition 13. Let I_{ϑ_d} = (ρ_{ϑ_d}, τ_{ϑ_d}, ω_{ϑ_d}) be a PFHSN, $J_{i}$ and $F_{j}$ represent weight vectors for expert’s and sub-attributes for selected parameters respectively with the constraint $J_{i} > 0, \sum_{i = 1}^{n} J_{i} = 1 and F_{j} > 0, \sum_{i = 1}^{n} F_{j} = 1$ . Then, the Picture Fuzzy Hypersoft Weighted Average (PFHSWA) aggregation operator is a mapping $N^{n} \to N$ given by

PFHSWA (I_ϑ₁₁, I_ϑ₁₂, …, I_{ϑ_nm})

$= \oplus_{j = 1}^{m} F_{j} (\oplus_{i = 1}^{n} J_{i} I_{ϑ_{ij}}) .$ (4.1) where $N^{n} = (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}})$ is a collection of all the PFHSNs.

Theorem 2. Let I_{ϑ_d} = (ρ_{ϑ_d}, τ_{ϑ_d}, ω_{ϑ_d}) be a PFHSN. Then from equation (4.1), the weighted average aggregation (fusion) of all the input values also gives a PFHSN, represented by

PFHSWA (I_ϑ₁₁, I_ϑ₁₂, … . , I_{ϑ_nm}) $\begin{matrix} = 〈 1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉 . \end{matrix}$ (4.2) where $J_{i}$ and $F_{j}$ represent weight vectors for expert’s and sub-attributes for selected parameters respectively with the constraint $J_{i} > 0, \sum_{i = 1}^{n} J_{i} = 1 and F_{j} > 0, \sum_{j = 1}^{n} F_{j} = 1$ .

Proof: Refer to Proof 2 in the Appendix.

Properties of PFHSWA Operator

Idempotency

If I_{ϑ_ij} = I_{ϑ_α} = (ρ_{ϑ_ij}, τ_{ϑ_ij}, ω_{ϑ_ij}) ∀i, j, then PFHSWA (I_ϑ₁₁, I_ϑ₁₂, … . , I_{ϑ_nm}) = I_{ϑ_α}.

Proof 3: Refer to Appendix for the detailed proof.

Boundedness

Suppose I_{ϑ_ij} be a collection of picture fuzzy hypersoft numbers.

Let $\begin{matrix} I_{ϑ_{ij}}^{-} = 〈 min_{j} min_{i} {ρ_{ϑ_{ij}}}, max_{j} max_{i} {τ_{ϑ_{ij}}}, \\ max_{j} max_{i} {ω_{ϑ_{ij}}} 〉 \end{matrix}$ and

$\begin{matrix} I_{ϑ_{ij}}^{+} = 〈 max_{j} max_{i} {ρ_{ϑ_{ij}}}, min_{j} min_{i} {τ_{ϑ_{ij}}}, \\ min_{j} \min_{i} {ω_{ϑ_{ij}}} 〉, \end{matrix}$ then $I_{ϑ_{ij}}^{-} \leq PFHSWA (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}} \leq I_{ϑ_{ij}}^{+}) .$ Proof 4: Refer to Appendix for the detailed proof.

Shift Invariance If I_{ϑ_α} = (ρ_{ϑ_α}, τ_{ϑ_α}, ω_{ϑ_α}) be a PFHSN, then,

$\begin{matrix} PFHSWA (I_{ϑ_{11}} \oplus I_{ϑ_{α}}, I_{ϑ_{12}} \oplus I_{ϑ_{α}}, \dots, I_{ϑ_{nm}} \oplus I_{ϑ_{α}}) = \\ PFHSWA (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots, I_{ϑ_{nm}}) \oplus I_{ϑ_{α}} . \end{matrix}$

Proof 5: Refer to Appendix for the detailed proof.

Homogeneity For any positive real number κ, $\begin{matrix} PFHSWA (κ I_{ϑ_{11}}, κ I_{ϑ_{12}}, \dots ., {κ I}_{ϑ_{nm}}) \\ = κ PFHSWA (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) . \end{matrix}$ Proof 6: Refer to Appendix for the detailed proof.

Monotonicity Let I_{ϑ_ij}and $I_{ϑ ij}^{'}$ be the collection of two PFHSNs. If I_{ϑ_ij} ≤ $I_{ϑ ij}^{'}$ then, $\begin{matrix} PFHSWA (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) \\ \leq PFHSWA (I_{ϑ_{11}^{'}}, I_{ϑ_{12}^{'}}, \dots . I_{ϑ_{nm}^{'}}) . \end{matrix}$

Proof 7. The proof clearly follows by making use of the operational laws stated above.

Further, we introduce another type of average aggregation operator called as the ordered weighted averaging operator for Picture Fuzzy Hypersoft Numbers as follows:

Definition 14. Let I_{ϑ_d} = (ρ_{ϑ_d}, τ_{ϑ_d}, ω_{ϑ_d}) be a PFHSN, $J_{i}$ and $F_{j}$ represent weight vectors for expert’s and sub-attributes for the selected parameters respectively with the constraint $J_{i} > 0, \sum_{i = 1}^{n} J_{i} = 1 and F_{j} > 0, \sum_{i = 1}^{n} F_{j} = 1$ . Then, Picture Fuzzy Hypersoft Ordered Weighted Average (PFHSOWA) aggregation operator is a mapping $N^{n} \to N$ given by $\begin{matrix} PFHSOWA (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) \\ = \oplus_{j = 1}^{m} F_{j} (\oplus_{i = 1}^{n} J_{i} I_{ϑ_{σ (ij)}}) . \end{matrix}$ (4.3) where $N^{n} = (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}})$ is a collection of Picture Fuzzy Hypersoft Numbers and σ (11), σ (12) , σ (21) , …… σ (nm) is a possible permutation of i and j with i = 1, 2, … . n and j = 1, 2, … . m.

Remarks:

On similar lines, we can establish the above important properties of idempotency, boundedness, shift invariance, homogeneity and monotonicity for Picture Fuzzy Hypersoft Ordered Weighted Average (PFHSOWA) aggregation operators by possible rotation of the permutations needed.

Implementation of these operators will be applicable in those real-world problem where ordered position of weights will play a crucial role for the decision making.

In case of weighted averaging operator, the weights of picture fuzzy hypersoft numbers are being taken into consideration while in case of the picture fuzzy hypersoft ordered weighted averaging operator, the weights of the ordered positions of PFHSNs have been taken into account.

In this way, it gives rise to a limitation of taking only one feature at a time. This limitation can be ruled out if the notion of picture fuzzy hypersoft hybrid averaging (PFHSHA) operator gets introduced in due course of time. For the sake of getting an outline to pursue further detailing over this, we can refer to reference [33].

4.2 Geometric aggregation operators

In this subsection, we subsequently study and define the new geometric aggregation operators for the proposed picture fuzzy hypersoft numbers as follows:

Definition 15. Let I_{ϑ_d} = (ρ_{ϑ_d}, τ_{ϑ_d}, ω_{ϑ_d}) be a PFHSN, $J_{i}$ and $F_{j}$ represent weight vectors for expert’s and sub-attributes for selected parameters respectively with the constraint $J_{i} > 0, \sum_{i = 1}^{n} J_{i} = 1 and F_{j} > 0, \sum_{i = 1}^{n} F_{j} = 1$ . Then the Picture Fuzzy Hypersoft Weighted Geometric (PFHSWG) aggregation operator is a mapping $N^{n} \to N$ given by

$\begin{matrix} PFHSWG (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) \\ = \otimes_{j = 1}^{m} {(\otimes_{i = 1}^{n} {I_{ϑ_{ij}}}^{J_{i}})}^{F_{j}}; \end{matrix}$ (4.4) where $N^{n} = (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}})$ is a collection of PFHSNs.

Theorem 3. Let I_{ϑ_d} = (ρ_{ϑ_d}, τ_{ϑ_d}, ω_{ϑ_d}) be a PFHSN. Then from equation (4.1), the weighted geometric aggregation(fusion) of all the input values also gives a PFHSN given by $\begin{matrix} PFHSWG (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) \\ = 〈 \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ 1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ 1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉; \end{matrix}$ (4.5) where $J_{i}$ and $F_{j}$ represent weight vectors for expert’s and sub-attributes for selected parameters respectively with the constraint $J_{i} > 0, \sum_{i = 1}^{n} J_{i} = 1 and F_{j} > 0, \sum_{j = 1}^{n} F_{j} = 1$ .

Proof: Refer to Proof 8 in the Appendix.

Now, we devise some properties for the collection of PFHSNs based on Theorem 3, by making use of introduced PFHSWG operator.

Properties of PFHSWG Operator

Idempotency

If I_{ϑ_ij} = I_{ϑ_α} = (ρ_{ϑ_ij}, τ_{ϑ_ij}, ω_{ϑ_ij}) ∀ i, j, sthen PFHSWG(I_ϑ₁₁, I_ϑ₁₂, … . , I_{ϑ_nm}) = I_{ϑ_α}.

Proof 9: Refer to Appendix for the detailed proof.

Boundedness

Suppose I_{ϑ_ij} be a collection of picture fuzzy hypersoft numbers.

Let $\begin{matrix} I_{ϑ_{ij}}^{-} = 〈 min_{j} min_{j} {ρ_{ϑ_{ij}}}, max_{j} max_{j} {τ_{ϑ_{ij}}}, \\ max_{j} max_{j} {ω_{ϑ_{ij}}} 〉 \end{matrix}$ and

$\begin{matrix} I_{ϑ_{ij}}^{+} = 〈 max_{j} max_{j} {ρ_{ϑ_{ij}}}, min_{j} min_{j} {τ_{ϑ_{ij}}}, \\ min_{j} min_{j} {ω_{ϑ_{ij}}} 〉, \end{matrix}$ then

$I_{ϑ_{ij}}^{-} \leq PFHSWG (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) \leq I_{ϑ_{ij}}^{+} .$

Proof 10: Refer to Appendix for the detailed proof.

Shift Invariance

Let I_{ϑ_α} = (ρ_{ϑ_α}, τ_{ϑ_α}, ω_{ϑ_α}) be a PFHSN. Then,

PFHSWG(I_ϑ₁₁ ⊗ I_{ϑ_α}, I_ϑ₁₂ ⊗ I_{ϑ_α}, … . , I_{ϑ_nm} ⊗ I_{ϑ_α}) = PFHSWA(I_ϑ₁₁, I_ϑ₁₂, … . , I_{ϑ_nm}) ⊗ I_{ϑ_α} .

Proof 11: Refer to Appendix for the detailed proof.

Homogeneity For any positive real number κ,

$\begin{matrix} PFHSWG (κ I_{ϑ_{11}}, κ I_{ϑ_{12}}, \dots ., {κ I}_{ϑ_{nm}}) \\ = κ PFHSWG (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) . \end{matrix}$

Proof 12: Refer to Appendix for the detailed proof.

Monotonicity

Let I_{ϑ_ij}and $I_{ϑ ij}^{'}$ be the collection of two PFHSNs. If I_{ϑ_ij} ≤ $I_{ϑ ij}^{'}$ then, $\begin{matrix} PFHSWG (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) \\ \leq PFHSWG (I_{ϑ_{11}^{'}}, I_{ϑ_{12}^{'}}, \dots . I_{ϑ_{nm}^{'}}) . \end{matrix}$

Proof 13. On similar lines, proof of monotonicity can be given using functional laws.

5 Methodology for MCDM model under PFHSS information

In this section, we propose a new methodology for solving a multi-criteria decision making problem based on the picture fuzzy hypersoft information by utilizing the proposed aggregation operators (PFHSWA / PFHSWG) and subsequently illustrate the methodology with the help of a numerical example.

5.1 Articulation of problem

Consider $Z = {Z^{1}, Z^{2}, \dots . Z^{w}}$ to be a collection of w experts and $Y = {Y^{1}, Y^{2}, \dots . Y^{n}}$ to be a collection of n alternatives. The expert’s weights are given by $J = {(J_{1}, J_{2}, J_{3}, \dots ., J_{w})}^{T}$ and $J_{i} > 0, \sum_{i = 1}^{n} J_{i} = 1 .$ Let Ω ={ ϑ₁, ϑ₂, ϑ₃, … . ϑ_m } be a collection of attributes with their corresponding multi sub-attributes are $\begin{matrix} Ω^{'} = {(ϑ_{1 η} \times ϑ_{2 η} \times ϑ_{3 η} \times \dots . \times ϑ_{m η}) \forall η \in {1, 2, \dots ., s}} \end{matrix}$ with weights $F = {(F_{1 η}, F_{2 η}, F_{3 η}, \dots ., F_{m η})}^{T}$ with $F_{η} > 0, \sum_{η = 1}^{s} F_{η} = 1$ .

It may be noted that in the set of sub-attributes, the components may be multi-valued and for the sake of simplicity, the components of Ω^′ can be restated as $Ω^{'} = {\underset{p}{ϑ^{'}} : p \in {1, 2, \dots, k}$ . Now, the expert’s team ${Z^{(i)} : i = 1, 2, \dots, w}$ assess the alternatives ${Y^{(q)} : q = 1, 2, \dots, n}$ under the chosen sub-attributes of the selected parameters $\begin{matrix} {\underset{p}{ϑ^{'}} : p \in {1, 2, \dots, k}} \end{matrix}$ which is given in the form of PFHSNs as ${(I_{ϑ_{i j}^{'}}^{(q)})}_{w \times p} = {(ρ_{ϑ_{i j}^{'}}^{(q)}, τ_{ϑ_{i j}^{'}}^{(q)}, ω_{ϑ_{i j}^{'}}^{(q)},)}_{w \times p};$

where $\begin{matrix} {0 \leq ρ}_{{ϑ_{ij}}^{'}}^{(q)}, {τ_{{ϑ_{ij}}^{'}}}^{(q)}, {ω_{{ϑ_{ij}}^{'}}}^{(q)} \leq 1 \end{matrix}$ and $\begin{matrix} {0 \leq ρ}_{{ϑ_{ij}}^{'}}^{(q)} + {τ_{{ϑ_{ij}}^{'}}}^{(q)} + {ω_{{ϑ_{ij}}^{'}}}^{(q)} \leq 1 \forall i, j . \end{matrix}$

5.2 Algorithm for proposed articulation and devised aggregation operators under the surroundings of PFHSS

For the sake of solving the proposed articulation outlined above, we present a new algorithm and list out the necessary steps with the help of Fig. 1.

Fig. 1

Flowchart of the proposed algorithm.

The detailing of the outlined steps of the proposed methodology is also being presented.

Step 1: Assemble the data related to each alternative in the form of Picture Fuzzy Hypersoft Number in accordance with several conditions of multi-parameterizations and rearrange them to construct a Picture Fuzzy Hypersoft decision matrix for the available experts provided with respect to each alternative ${Y^{(q)} : q = 1, 2, \dots, n}$ as follows:

$(Y^{(q)}, Ω^{'})_{w \times p} =$ $\begin{matrix} Z^{1} \\ Z^{2} \\ ⋮ \\ Z^{ω} \end{matrix} (\begin{matrix} {ϑ^{'}}_{1} & {ϑ^{'}}_{2} & \dots & {ϑ^{'}}_{p} \\ (ρ_{{ϑ^{'}}_{11}}^{(q)}, τ_{{ϑ^{'}}_{11}}^{(q)}, ω_{{ϑ^{'}}_{11}}^{(q)}) & (ρ_{{ϑ^{'}}_{12}}^{(q)}, τ_{{ϑ^{'}}_{12}}^{(q)}, ω_{{ϑ^{'}}_{12}}^{(q)}) & \dots & (ρ_{{ϑ^{'}}_{1 p}}^{(q)}, τ_{{ϑ^{'}}_{1 p}}^{(q)}, ω_{{ϑ^{'}}_{1 p}}^{(q)}) \\ (ρ_{{ϑ^{'}}_{21}}^{(q)}, τ_{{ϑ^{'}}_{21}}^{(q)}, ω_{{ϑ^{'}}_{21}}^{(q)}) & (ρ_{{ϑ^{'}}_{22}}^{(q)}, τ_{{ϑ^{'}}_{22}}^{(q)}, ω_{{ϑ^{'}}_{22}}^{(q)}) & \dots & (ρ_{{ϑ^{'}}_{2 p}}^{(q)}, τ_{{ϑ^{'}}_{2 p}}^{(q)}, ω_{{ϑ^{'}}_{2 p}}^{(q)}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ (ρ_{{ϑ^{'}}_{n 1}}^{(q)}, τ_{{ϑ^{'}}_{n 1}}^{(q)}, ω_{{ϑ^{'}}_{n 1}}^{(q)}) & (ρ_{{ϑ^{'}}_{n 2}}^{(q)}, τ_{{ϑ^{'}}_{n 2}}^{(q)}, ω_{{ϑ^{'}}_{n 2}}^{(q)}) & \dots & (ρ_{{ϑ^{'}}_{n p}}^{(q)}, τ_{{ϑ^{'}}_{n p}}^{(q)}, ω_{{ϑ^{'}}_{n p}}^{(q)}) \end{matrix})$

Step 2: In the case of inconsistent sub-attributes, transformation of cost and benefit type sub-attributes is required. This can be done with the help of the normalization rule and the resulting normalized decision matrix is as below: $\begin{matrix} ς_{ij} = {\begin{matrix} {I_{{ϑ_{ij}}^{'}}}^{c} = ({ω_{{ϑ_{ij}}^{'}}}^{(q)}, {τ_{{ϑ_{ij}}^{'}}}^{(q)}, {ρ_{{ϑ_{ij}}^{'}}}^{(q)}); cost type \\ I_{{ϑ_{ij}}^{'}} = ({ρ_{{ϑ_{ij}}^{'}}}^{(q)}, {τ_{{ϑ_{ij}}^{'}}}^{(q)}, {ω_{{ϑ_{ij}}^{'}}}^{(q)}); benefit type \end{matrix} \end{matrix}$ If the data is consistent then move to Step 3.

Step 3: Now with the use of the devised aggregation operators, we get a collective decision matrix I_{ϑ_ij} for each alternative $Y = {Y^{1}, Y^{2}, \dots . Y^{n}} .$

Step 4: For the collection of alternatives

$Y = {Y^{1}, Y^{2}, \dots . Y^{n}}$ ,

we compute the scoring values with the help of the formulae of scoring function.

Step 5: Choose the alternative with maximum score value & then rank the alternatives.

Table 1

Tabular form of PFHSS (R₁, Λ )

(R₁, Λ )	v ₁	v ₂	v ₃	v ₄
(α₃, β₁, γ₁, δ₁)	(0.1, 0.3, 0.5)	(0.2, 0.4, 0.2)	(0.1, 0.2, 0.4)	(0, 0, 1)
(α₃, β₁, γ₃, δ₁)	(0.2, 0.3, 0.2)	(0.2, 0.1, 0.2)	(0.1, 0.3, 0.4)	(0, 0, 1)
(α₃, β₂, γ₁, δ₁)	(0.1, 0.1, 0.5)	(0.2, 0.2, 0.3)	(0.1, 0.4, 0.4)	(0.1, 0.2, 0.3)
(α₃, β₂, γ₃, δ₁)	(0.1, 0.2, 0.5)	(0.2, 0.2, 0.2)	(0.2, 0.4, 0.4)	(0.1, 0.2, 0.6)

Table 2

Tabular form of PFHSS (R₂, Λ′ )

(R₂, Λ′ )	v ₁	v ₂	v ₃	v ₄
(α₁, β₁, γ₁, δ₁)	(0.1, 0.3, 0.2)	(0.2, 0.4, 0.3)	(0.1, 0.2, 0.5)	(0, 0, 1)
(α₁, β₁, γ₂, δ₁)	(0.2, 0.3, 0.1)	(0.2, 0.1, 0.1)	(0.1, 0.3, 0.5)	(0, 0, 1)
(α₃, β₁, γ₁, δ₁)	(0.1, 0.1, 0.7)	(0.2, 0.3, 0.2)	(0.1, 0.4, 0.2)	(0.1, 0.2, 0.6)
(α₃, β₁, γ₂, δ₁)	(0.5, 0.2, 0.1)	(0.2, 0.4, 0.2)	(0.2, 0.4, 0.1)	(0.1, 0.2, 0.1)

5.3 Numerical illustration and computation

In order to solve a multi-criteria decision-making problem based on the proposed methodology, a numerical problem of selecting the most suitable employee for a multi-national company from the set of employees by taking into account the choice of parameterizations is presented below:

Let $Y = {Y^{1}, Y^{2}, Y^{3}, Y^{4}}$ be a set of employees and Ω be the set of attributes given in the form of a hypersoft set as Ω = {ϑ₁ = Age, ϑ₂ = ForeignLanguageknowledge, ϑ₃ = Academicqualification, ϑ₄ = Workexperience} and their further sub-parameters given by

Age = ϑ₁ = {ϑ₁₁ = 21 - 35, ϑ₁₂ = 35 - 52} ,

ForeignLanguageKnowledge = ϑ₂ = { ϑ₂₁ = English, ϑ₂₂ = French } ,

AcademicQualification = ϑ₃ = { ϑ₃₁ = UG, ϑ₃₂ = PG } ,

WorkExperience = ϑ₄ = { ϑ₄₁ = atleastoneyear } .

Let Ω^′ = ϑ₁ × ϑ₂ × ϑ₃ × ϑ₄ be a collection of sub-attributes.

For the sake of simplicity collection of all sub-attributes can be restated as

$Ω^{'} = {ϑ_{1}^{'}, ϑ_{2}^{'}, ϑ_{3}^{'}, ϑ_{4}^{'}, ϑ_{5}^{'}, ϑ_{6}^{'}, ϑ_{7}^{'}, ϑ_{8}^{'}}$ and their respective weights are (0.12,0.18,0.1,0.15,0.22,0.08,0.1)^T.

Consider $Z = {Z^{1}, Z^{2}, Z^{3}, Z^{4}}$ be a collection of experts with weight’s (0.2, 0.3, 0.4, 0.1) ^T to examine the suitable alternative. The preferences are supposed to be given by experts in terms of PFHSNs by using multi sub-attributes. In order to obtain the most suitable choice, we go through the following process.

Productive Employees Selection Using Picture Fuzzy Hypersoft Weighted Average Operator

Step 1: The situations are examined by the experts in terms of PFHSNs. The multi-subattributes of the selected attributes, along with computation of score values are given in the Tables 3 –6.

Table 3
Decision Matrix given by Experts for Alternative $Y^{1}$

$Y^{1}$ $ϑ_{1}^{'}$ $ϑ_{2}^{'}$ $ϑ_{3}^{'}$ $ϑ_{4}^{'}$ $ϑ_{5}^{'}$ $ϑ_{6}^{'}$ $ϑ_{7}^{'}$ $ϑ_{8}^{'}$

$Z^{1}$ (0.2,0.5,0.1) (0.3,0.4,0.2) (0.4,0.1,0.2) (0.3,0.5,0.1) (0.4,0.1,0.2) (0.3,0.2,0.1) (0.3,0.2,0.4) (0.1,0.2,0.3)

$Z^{2}$ (0.4,0.3,0.2) (0.2,0.4,0.1) (0.1,0.2,0.3) (0.3,0.2,0.1) (0.4,0.2,0.3) (0.1,0.3,0.5) (0.1,0.2,0.4) (0.2,0.3,0.4)

$Z^{3}$ (0.1,0.2,0.5) (0.4,0.1,0.2) (0.3,0.2,0.1) (0.2,0.4,0.3) (0.2,0.1,0.5) (0.7,0.1,0.1) (0.4,0.2,0.3) (0.1,0.5,0.3)

$Z^{4}$ (0.3,0.5,0.1) (0.2,0.4,0.1) (0.1,0.2,0.4) (0.3,0.4,0.2) (0.1,0.3,0.5) (0.2,0.4,0.1) (0.4,0.1,0.2) (0.4,0.3,0.2)

$Y^{1}$	$ϑ_{1}^{'}$	$ϑ_{2}^{'}$	$ϑ_{3}^{'}$	$ϑ_{4}^{'}$	$ϑ_{5}^{'}$	$ϑ_{6}^{'}$	$ϑ_{7}^{'}$	$ϑ_{8}^{'}$
$Z^{1}$	(0.2,0.5,0.1)	(0.3,0.4,0.2)	(0.4,0.1,0.2)	(0.3,0.5,0.1)	(0.4,0.1,0.2)	(0.3,0.2,0.1)	(0.3,0.2,0.4)	(0.1,0.2,0.3)
$Z^{2}$	(0.4,0.3,0.2)	(0.2,0.4,0.1)	(0.1,0.2,0.3)	(0.3,0.2,0.1)	(0.4,0.2,0.3)	(0.1,0.3,0.5)	(0.1,0.2,0.4)	(0.2,0.3,0.4)
$Z^{3}$	(0.1,0.2,0.5)	(0.4,0.1,0.2)	(0.3,0.2,0.1)	(0.2,0.4,0.3)	(0.2,0.1,0.5)	(0.7,0.1,0.1)	(0.4,0.2,0.3)	(0.1,0.5,0.3)
$Z^{4}$	(0.3,0.5,0.1)	(0.2,0.4,0.1)	(0.1,0.2,0.4)	(0.3,0.4,0.2)	(0.1,0.3,0.5)	(0.2,0.4,0.1)	(0.4,0.1,0.2)	(0.4,0.3,0.2)

Table 4

Decision Matrix given by Experts for Alternative $Y^{2}$

$Y^{2}$	$ϑ_{1}^{'}$	$ϑ_{2}^{'}$	$ϑ_{3}^{'}$	$ϑ_{4}^{'}$	$ϑ_{5}^{'}$	$ϑ_{6}^{'}$	$ϑ_{7}^{'}$	$ϑ_{8}^{'}$
$Z^{1}$	(0.4,0.2,0.3)	(0.1,0.2,0.6)	(0.2,0.5,0.1)	(0.1,0.2,0.3)	(0.3,0.2,0.1)	(0.7,0.1,0.1)	(0.2,0.4,0.1)	(0.2,0.5,0.2)
$Z^{2}$	(0.1,0.2,0.5)	(0.2,0.4,0.1)	(0.3,0.5,0.1)	(0.2,0.4,0.3)	(0.1,0.3,0.5)	(0.2,0.5,0.1)	(0.1,0.5,0.3)	(0.4,0.1,0.2)
$Z^{3}$	(0.4,0.1,0.2)	(0.1,0.2,0.4)	(0.3,0.5,0.1)	(0.4,0.3,0.2)	(0.1,0.5,0.3)	(0.3,0.2,0.1)	(0.3,0.4,0.2)	(0.2,0.5,0.1)
$Z^{4}$	(0.4,0.3,0.2)	(0.2,0.1,0.5)	(0.1,0.3,0.5)	(0.1,0.2,0.4)	(0.3,0.4,0.2)	(0.2,0.6,0.1)	(0.1,0.7,0.1)	(0.3,0.2,0.1)

Table 5

Decision Matrix given by Experts for Alternative $Y^{3}$

$Y^{3}$	$ϑ_{1}^{'}$	$ϑ_{2}^{'}$	$ϑ_{3}^{'}$	$ϑ_{4}^{'}$	$ϑ_{5}^{'}$	$ϑ_{6}^{'}$	$ϑ_{7}^{'}$	$ϑ_{8}^{'}$
$Z^{1}$	(0.1,0.3,0.4)	(0.4,0.3,0.2)	(0.2,0.6,0.1)	(0.3,0.2,0.1)	(0.2,0.5,0.1)	(0.4,0.1,0.2)	(0.1,0.2,0.4)	(0.2,0.4,0.1)
$Z^{2}$	(0.3,0.5,0.1)	(0.4,0.3,0.2)	(0.1,0.5,0.3)	(0.2,0.4,0.1)	(0.4,0.2,0.3)	(0.4,0.3,0.2)	(0.1,0.2,0.3)	(0.2,0.3,0.5)
$Z^{3}$	(0.4,0.1,0.2)	(0.1,0.2,0.3)	(0.2,0.1,0.5)	(0.7,0.1,0.1)	(0.4,0.1,0.2)	(0.1,0.3,0.1)	(0.1,0.2,0.3)	(0.1,0.3,0.6)
$Z^{4}$	(0.3,0.5,0.1)	(0.2,0.4,0.1)	(0.1,0.2,0.4)	(0.3,0.4,0.2)	(0.1,0.3,0.5)	(0.2,0.4,0.1)	(0.4,0.1,0.2)	(0.4,0.3,0.2)

Table 6

Decision Matrix given by Experts for Alternative $Y^{4}$

$Y^{4}$	$ϑ_{1}^{'}$	$ϑ_{2}^{'}$	$ϑ_{3}^{'}$	$ϑ_{4}^{'}$	$ϑ_{5}^{'}$	$ϑ_{6}^{'}$	$ϑ_{7}^{'}$	$ϑ_{8}^{'}$
$Z^{1}$	(0.3,0.5,0.1)	(0.2,0.3,0.4)	(0.1,0.3,0.2)	(0.3,0.2,0.1)	(0.2,0.4,0.3)	(0.2,0.6,0.1)	(0.5,0.3,0.2)	(0.1,0.3,0.4)
$Z^{2}$	(0.2,0.3,0.4)	(0.1,0.2,0.3)	(0.3,0.4,0.2)	(0.1,0.2,0.6)	(0.1,0.2,0.2)	(0.2,0.4,0.3)	(0.2,0.1,0.4)	(0.3,0.2,0.1)
$Z^{3}$	(0.2,0.1,0.5)	(0.2,0.5,0.2)	(0.1,0.5,0.2)	(0.3,0.2,0.1)	(0.1,0.4,0.2)	(0.2,0.5,0.1)	(0.1,0.3,0.5)	(0.2,0.4,0.3)
$Z^{4}$	(0.1,0.5,0.2)	(0.2,0.3,0.1)	(0.2,0.5,0.1)	(0.1,0.2,0.3)	(0.3,0.4,0.2)	(0.1,0.2,0.3)	(0.3,0.2,0.1)	(0.2,0.4,0.1)

Step 2: Since all attributes are identical, so there is no need for normalization.

Step 3: By using equation (4.2), the opinion of expert’s can be summarized as $\begin{matrix} J_{1} = 〈 0.309967, 0.231837, 0.200275 〉, \\ J_{2} = 〈 0.269288, 0.275446, 0.196839 〉, \\ J_{3} = 〈 0.288827, 0.238588, 0.212493 〉, \\ J_{4} = 〈 0.198194, 0.304841, 0.213194 〉 . \end{matrix}$

Step 4: Now compute the scoring values by using the formulae of scoring functions. $\begin{matrix} S (J_{1}) = - 0.122145, S (J_{2}) = - 0.202997, \\ S (J_{3}) = - 0.162254, S (J_{4}) = - 0.319841 . \end{matrix}$ Step 5: Finally, on the basis of the obtained values of the score function, we observe that $S (J_{1}) > S (J_{3}) > S (J_{2}) > S (J_{4}) .$ So, $Y^{1} > Y^{3} > Y^{2} > Y^{4}$ . Hence, the alternative $Y^{1}$ is the most appropriate one.

Productive Employees Selection Using Picture Fuzzy Hypersoft Weighted Geometric Operator

Step 1: Experts inspected the situations in terms of PFHSNs. The summary of multi sub-attributes of the considered attributes, along with score values are given in Tables 3 –6.

Step 2: Since all attributes are identical, so there is no need for normalization.

Step 3: By using equation (4.5), opinion of expert’s can be summarized as $\begin{matrix} J_{1} = 〈 0.239208, 0.278116, 0.254269 〉, \\ J_{2} = 〈 0.216339, 0.335441, 0.259252 〉, \\ J_{3} = 〈 0.217776, 0.284027, 0.27034 〉, \\ J_{4} = 〈 0.178452, 0.353784, 0.287766 〉 . \end{matrix}$ Step 4: Now compute the scoring values by using the formulae of scoring functions. $\begin{matrix} S (J_{1}) = - 0.293177, S (J_{2}) = - 0.378354, \\ S (J_{3}) = - 0.336591, S (J_{4}) = - 0.463098 \end{matrix}$ Step 5: Finally, on the basis of the obtained values of the score function, we observe that $S (J_{1}) > S (J_{3}) > S (J_{2}) > S (J_{4}) .$ So, $Y^{1} > Y^{3} > Y^{2} > Y^{4}$ . Hence, the alternative $Y^{1}$ is the most appropriate one.

6 Comparative analysis, advantages and discussions

In this section, we discuss the functionality, receptiveness, and conformity of the proposed notion and methodology in contrast with the existing techniques. In addition to this, some advantages and discussions over the obtained results have also been presented for better understanding and readability.

In view of the literatures available on various types of generalizations of fuzzy sets and information, we tabulate the comparative study in terms of advantages, applicability and flexibility as shown in Table 7.

Table 7
Superiority of PFHSSs with some existing types of sets

Zadeh [1] Maji et al. [8] Coung [6] Coung [6] Proposed Approach

Categorization of FS FSS PFS PFSS Proposed Set - PFHSS

Uncertainty Component

Membership ✓ ✓ ✓ ✓ ✓

Non-membership × × ✓ ✓ ✓

Attributes ✓ ✓ ✓ ✓ ✓

Sub-Attributes × × × × ✓

Loss of Information × × × × ×

Parametrization × ✓ × ✓ ✓

Abstain &Refusal × × ✓ ✓ ✓

Advantages in terms of dealing factors of uncertainty Uses Fuzzy Interval Uses Fuzzy Soft Intervals Utilize Membership, Non-membership, refusal and degree of abstain Uses Parametrization of Attributes Utilizing Parametrization of Multi sub-attributes with the Inclusion of Degree of Refusal and Abstain

	Zadeh [1]	Maji et al. [8]	Coung [6]	Coung [6]	Proposed Approach
Categorization of	FS	FSS	PFS	PFSS	Proposed Set - PFHSS
Uncertainty Component
Membership	✓	✓	✓	✓	✓
Non-membership	×	×	✓	✓	✓
Attributes	✓	✓	✓	✓	✓
Sub-Attributes	×	×	×	×	✓
Loss of Information	×	×	×	×	×
Parametrization	×	✓	×	✓	✓
Abstain &Refusal	×	×	✓	✓	✓
Advantages in terms of dealing factors of uncertainty	Uses Fuzzy Interval	Uses Fuzzy Soft Intervals	Utilize Membership, Non-membership, refusal and degree of abstain	Uses Parametrization of Attributes	Utilizing Parametrization of Multi sub-attributes with the Inclusion of Degree of Refusal and Abstain

The notion of fuzzy set along with its various extensions available in literature handles the uncertainty/inexactness of the information of sub-attributes of the considered attributes for individual alternatives of decision making process. But these concepts and their related components stated above are not able to handle the degree of abstain and refusal component along with parametrization of sub-attributes of the considered attributes. However, our proposed set, i.e., PFHSS, has the additional capability to handling the further parametrization of the sub-attributes with the important components of refusal and abstain. This certainly provides more freedom and flexibility to account the opinion of the decision-makers/experts mathematically. In view of the numerical example under consideration and the results obtained through the existing techniques utilizing the intuitionistic fuzzy soft/hypersoft aggregation operators, we present the following Table 8, stating the ranking of the alternatives for the decision-making problem.

Table 8

Results of Comparative Analysis with Some Existing Aggregation Operators

Method	$Y^{1}$	$Y^{2}$	$Y^{3}$	$Y^{4}$	Ranking Order
IFSWA	0.08158	0.07674	0.14762	0.09959	$Y^{3}$ > $Y^{4}$ > $Y^{1}$ > $Y^{2}$
IFSWG [26]	0.49830	0.41735	0.40935	0.46175	$Y^{1}$ > $Y^{4}$ > $Y^{2}$ > $Y^{3}$
IFHSWA [34]	-0.195086	-0.124363	0.084652	0.095501	$Y^{4}$ > $Y^{3}$ > $Y^{2}$ > $Y^{1}$
IFHSWG [34]	-0.259867	-0.242376	-0.141950	-0.035913	$Y^{4}$ > $Y^{3}$ > $Y^{2}$ > $Y^{1}$

Subsequently, on the basis of the obtained results by utilizing the proposed methodology involving the introduced PFHSWA/PFHSWG aggregation operators, we present the following respective computed values: $\begin{matrix} S (J_{1}) = - 0.122145, S (J_{2}) = - 0.202997, \\ S (J_{3}) = - 0.162254, S (J_{4}) = - 0.319841 . \end{matrix}$ and $\begin{matrix} S (J_{1}) = - 0.293177, S (J_{2}) = - 0.378354, \\ S (J_{3}) = - 0.336591, S (J_{4}) = - 0.463098 \end{matrix}$ On the basis of the computed score values, we finally conclude the following ranking of the alternatives (employee) which is certainly different due to the additional parametrization of sub-attributes: $Y^{1} > Y^{3} > Y^{2} > Y^{4} .$ Remark: Here, the score values are distinct in this example and hence we did not utilized the concept of accuracy function to determine the ranking.

Important Remarks and Advantages:

Finally, we are able to state that the proposed notion of picture fuzzy hypersoft set (PFHSS) is a novel concept and a valid extension of fuzzy set/hypersoft set theories. The PFHSS has an added advantage to deal with the wider sense of applicability in uncertain situations with the incorporation of degree of refusal and abstain.

The existing types of hypersoft sets - intuitionistic fuzzy hypersoft set [15], Pythagorean fuzzy hypersoft set [20], Neutrosophic hypersoft set [15] have their own limitations because of the exclusion of refusal and abstain component.

It may be noted that the categorically designed information having the picture fuzzy relation would not be possible to address with the help of existing hypersoft set theory in order to ensure a kind of parametrization in the relation.

The methodology implementing the proposed PFHSWA/PFHSWG aggregation operators can be well utilized for various group strategic MCDM models in a generalized framework effectively and consistently.

As an overall critical aspect, we observe that eventually with the picture fuzzy information, it won’t be possible to suitably address those membership values (given by the decision-makers/experts) whose sum exceeds one. Such restrictions in respect of decision-maker’s opinion can be eradicated with the notion of T-spherical fuzzy information.

7 Conclusions & future work

The processing of uncertain information in terms of multi sub-attributes parametrization with the help of the proposed way of defining picture fuzzy hypersoft set is a novel and useful concept. PFHSSs and their aggregation operators can be a strong mathematical tool to handle incomplete and inexact information with vagueness where the components of neutral membership (abstain) and refusal have been additionally addressed along with sub-parametrization. Various important properties and operational laws have been studied which are helpful in decision making problems. The concept of picture fuzzy hypersoft weighted average/ordered weighted average operator (PFHSWA/PFHSOWA) and weighted geometric/ordered weighted geometric operator (PFHSWG/PFHSOWG) have been proved and studied in detail. Based on this, we successfully presented a new methodology for MCDM problem where the multi-sub attributes and their parametrization (considering of abstain and refusal feature) are involved. The numerical example under consideration has been solved showing the implementation of the proposed technique with necessary computations. Comparative analysis and discussions over the obtained results clearly shows the efficacy of the introduced notion in the literature. In order to overcome the limitations (sum exceeding 1) pointed in the above section, the concept of T-spherical fuzzy hypersoft set may further be put forward in future to complement this study. Also, on the basis of recent research going on, the concept of Frank triangle norms ([35 –37]) can be implemented for introducing the aggregation operators for the desired sets along with applications. In addition to this, the proposed methodology can further be extended in the area of multi-granular linguistic information based decision-making problems [38] and also for the consensus reaching for group decision problems [39].

Appendix I

Proof 2. The proof of the theorem follows from the principle of mathematical induction.

For n = 1, we get $J_{1} = 1$ . (because $\sum_{i = 1}^{n} J_{i} = 1 .$ )

Then, from equation (4.1), we have

$PFHSWA (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) = \oplus_{j = 1}^{m} F_{j} I_{ϑ_{1 j}}$ .

Now, using the above stated functional laws (a)-(e), we get

$\begin{matrix} PFHSWA (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) = \\ 〈 1 - \prod_{j = 1}^{m} {(1 - ρ_{ϑ_{1 j}})}^{F_{j}}, \prod_{j = 1}^{m} {(τ_{ϑ_{1 j}})}^{F_{j}}, \prod_{j = 1}^{m} {(ω_{ϑ_{1 j}})}^{F_{j}} 〉 \end{matrix}$ $\begin{matrix} = 〈 1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{1} {(1 - ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \prod_{j = 1}^{m} {(\prod_{i = 1}^{1} {(τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ \prod_{j = 1}^{m} {(\prod_{i = 1}^{1} {(ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉 . \end{matrix}$

Also, For m = 1, we get $F_{1} = 1$ . (because $\sum_{j = 1}^{m} F_{j} = 1 .$ )

Then, from equation (4.1), we have

$PFHSWA (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) = \oplus_{i = 1}^{n} J_{i} I_{ϑ_{i 1}}$ . Again, using the above stated functional laws (a)-(e), we get

$\begin{matrix} PFHSWA (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) = \\ 〈 1 - \prod_{i = 1}^{n} {(1 - ρ_{ϑ_{i 1}})}^{J_{i}}, \prod_{i = 1}^{n} {(τ_{ϑ_{i 1}})}^{J_{i}}, \prod_{i = 1}^{n} {(ω_{ϑ_{i 1}})}^{J_{i}} 〉 \end{matrix}$ $\begin{matrix} = 〈 1 - \prod_{j = 1}^{1} {(\prod_{i = 1}^{n} {(1 - ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \prod_{j = 1}^{1} {(\prod_{i = 1}^{n} {(τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ \prod_{j = 1}^{1} {(\prod_{i = 1}^{n} {(ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉 . \end{matrix}$ This shows that equation (4.2) satisfies for n = 1 and m = 1. As per induction hypothesis, assume that equation (4.2) holds for m = α₁ + 1, n = α₂ and m = α₁, n = α₂ + 1; i.e., $\begin{matrix} \oplus_{j = 1}^{α_{1} + 1} F_{j} (\oplus_{i = 1}^{α_{2}} J_{i} I_{ϑ_{ij}}) = 〈 1 - \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2}} {(1 - ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2}} {(τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2}} {(ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉; \end{matrix}$ $\begin{matrix} \oplus_{j = 1}^{α_{1}} F_{j} (\oplus_{i = 1}^{α_{2} + 1} J_{i} I_{ϑ_{ij}}) = 〈 1 - \prod_{j = 1}^{α_{1}} {(\prod_{i = 1}^{α_{2} + 1} {(1 - ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ \prod_{j = 1}^{α_{1}} {(\prod_{i = 1}^{α_{2} + 1} {(τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \prod_{j = 1}^{α_{1}} {(\prod_{i = 1}^{α_{2} + 1} {(ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉 . \end{matrix}$

Now for m = α₁ + 1, n = α₂ + 1, we get $\begin{matrix} \oplus_{j = 1}^{α_{1} + 1} F_{j} (\oplus_{i = 1}^{α_{2} + 1} J_{i} I_{ϑ_{ij}}) = \oplus_{j = 1}^{α_{1} + 1} F_{j} (\oplus_{i = 1}^{α_{2}} J_{i} I_{ϑ_{ij}} \oplus J_{α_{2} + 1} I_{ϑ_{(α_{2} + 1) j}}) \\ in = \oplus_{j = 1}^{α_{1} + 1} \oplus_{i = 1}^{α_{2}} F_{j} J_{i} I_{ϑ_{ij}} \oplus_{j = 1}^{α_{1} + 1} F_{j} J_{α_{2} + 1} I_{ϑ_{(α_{2} + 1) j}} \end{matrix}$

$\begin{matrix} = 〈 (1 - \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2}} {(1 - ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}) \\ \oplus (1 - \prod_{j = 1}^{α_{1} + 1} {({(1 - ρ_{ϑ_{(α_{2} + 1) j}})}^{J_{(α_{2} + 1)}})}^{F_{j}}), \\ \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2}} {(τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} \\ \oplus \prod_{j = 1}^{α_{1} + 1} {({(τ_{ϑ_{(α_{2} + 1) j}})}^{J_{(α_{2} + 1)}})}^{F_{j}}, \\ \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2}} {(ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} \\ \oplus \prod_{j = 1}^{α_{1} + 1} {({(ω_{ϑ_{(α_{2} + 1) j}})}^{J_{(α_{2} + 1)}})}^{F_{j}} 〉 \end{matrix}$ $\begin{matrix} = 〈 1 - \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2} + 1} {(1 - ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2} + 1} {(τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2} + 1} {(ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉 . \end{matrix}$ Therefore, the result is true for m = α₁ + 1, n = α₂ + 1 and the theorem is proved.

Proof 3. LetI_{ϑ_ij} = I_{ϑ_α} = (ρ_{ϑ_ij}, τ_{ϑ_ij}, ω_{ϑ_ij}) be a collection of PFHSNs, then with the use of equation (4.2), we get $\begin{matrix} PFHSWA (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) \\ = 〈 1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉 \end{matrix}$ $\begin{matrix} = 〈 1 - {({(1 - ρ_{ϑ_{ij}})}^{\sum_{i = 1}^{n} J_{i}})}^{\sum_{j = 1}^{m} F_{j}}, \\ {({(τ_{ϑ_{ij}})}^{\sum_{i = 1}^{n} J_{i}})}^{\sum_{j = 1}^{m} F_{j}}, \\ {({(ω_{ϑ_{ij}})}^{\sum_{i = 1}^{n} J_{i}})}^{\sum_{j = 1}^{m} F_{j}} 〉 \end{matrix}$ $\begin{matrix} = 〈 1 - ((1 - ρ_{ϑ_{ij}}), (τ_{ϑ_{ij}}), ω_{ϑ_{ij}} 〉 = (ρ_{ϑ_{ij}}, τ_{ϑ_{ij}}, ω_{ϑ_{ij}}) \\ = I_{ϑ_{α}} . \end{matrix}$

Hence, the idempotency holds.

Proof 4. Let I_{ϑ_ij} = (ρ_{ϑ_ij}, τ_{ϑ_ij}, ω_{ϑ_ij}) be a PFHSN, then $min_{j} min_{i} {ρ_{ϑ_{ij}}} \leq {ρ_{ϑ_{ij}}} \leq max_{j} max_{i} {ρ_{ϑ_{ij}}}$ $\Rightarrow 1 - max_{j} max_{i} {ρ_{ϑ_{ij}}} \leq {1 - ρ_{ϑ_{ij}}} \leq 1 - min_{j} min_{i} {ρ_{ϑ_{ij}}}$ $\begin{matrix} \Leftrightarrow {(1 - max_{j} max_{i} {ρ_{ϑ_{ij}}})}^{J_{i}} \leq {(1 - ρ_{ϑ_{ij}})}^{J_{i}} \\ \leq {(1 - min_{j} min_{i} {ρ_{ϑ_{ij}}})}^{J_{i}} \end{matrix}$ $\begin{matrix} \Leftrightarrow {(1 - max_{j} max_{i} {ρ_{ϑ_{ij}}})}^{\sum_{i = 1}^{n} J_{i}} \leq \prod_{i = 1}^{n} {(1 - ρ_{ϑ_{ij}})}^{J_{i}} \\ \leq {(1 - min_{j} min_{i} {ρ_{ϑ_{ij}}})}^{\sum_{i = 1}^{n} J_{i}} \end{matrix}$ $\begin{matrix} \Leftrightarrow (1 - max_{j} max_{i} {ρ_{ϑ_{ij}}}) \leq \prod_{i = 1}^{n} {(1 - ρ_{ϑ_{ij}})}^{J_{i}} \\ \leq (1 - min_{j} min_{i} {ρ_{ϑ_{ij}}}) \end{matrix}$

(as $\sum_{i = 1}^{n} J_{i} = 1 .)$ $\begin{matrix} \Leftrightarrow {(1 - max_{j} max_{i} {ρ_{ϑ_{ij}}})}^{\sum_{j = 1}^{m} F_{j}} \leq {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} \\ \leq {(1 - min_{j} min_{i} {ρ_{ϑ_{ij}}})}^{\sum_{j = 1}^{m} F_{j}} \end{matrix}$ $\begin{matrix} \Leftrightarrow (1 - max_{j} max_{i} {ρ_{ϑ_{ij}}}) \leq {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} \\ \leq (1 - min_{j} min_{i} {ρ_{ϑ_{ij}}}) \end{matrix}$ (as $\sum_{j = 1}^{m} F_{j} = 1$ .)

$\begin{matrix} \Leftrightarrow (min_{j} min_{i} {ρ_{ϑ_{ij}}}) \leq {1 - \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} \\ \leq (max_{j} max_{i} {ρ_{ϑ_{ij}}}) . \end{matrix}$ (6) Similarly,

$\begin{matrix} (min_{j} min_{i} {τ_{ϑ_{ij}}}) \leq {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} \\ \leq (max_{j} max_{i} {τ_{ϑ_{ij}}}) . \end{matrix}$ (7)

$\begin{matrix} (min_{j} min_{i} {ω_{ϑ_{ij}}}) \leq {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} \\ \leq (max_{j} max_{i} {ω_{ϑ_{ij}}}) . \end{matrix}$ (8) Let PFHSWA (I_ϑ₁₁, I_ϑ₁₂, … . , I_{ϑ_nm})= (ρ_{ϑ_α}, τ_{ϑ_α}, ω_{ϑ_α}) = I_{ϑ_α}, so that the inequalities (8.1), (8.2) and (8.3) could be transformed into following forms: $\begin{matrix} min_{j} min_{i} {ρ_{ϑ_{ij}}} \leq {ρ_{ϑ_{ij}}} \leq {ρ_{ϑ_{α}}} \\ \leq max_{j} max_{i} {ρ_{ϑ_{ij}}}; \end{matrix}$ $\begin{matrix} min_{j} min_{i} {τ_{ϑ_{ij}}} \leq {τ_{ϑ_{ij}}} \leq {τ_{ϑ_{α}}} \\ \leq max_{j} max_{i} {τ_{ϑ_{ij}}}; \end{matrix}$ and $\begin{matrix} min_{j} min_{i} {ω_{ϑ_{ij}}} \leq {ω_{ϑ_{ij}}} \leq {ω_{ϑ_{α}}} \\ \leq max_{j} max_{i} {ω_{ϑ_{ij}}} \end{matrix}$ respectively. So, by making use of the earlier defined score function, we obtain the following values: $\begin{matrix} S (I_{ϑ_{α}}) = ρ_{ϑ_{α}} - τ_{ϑ_{α}} - ω_{ϑ_{α}} \leq max_{j} max_{i} {ρ_{ϑ_{ij}}} \\ - min_{j} min_{i} {τ_{ϑ_{ij}}} - min_{j} min_{i} {ω_{ϑ_{ij}}} = S (I_{ϑ_{ij}}^{+}) . \end{matrix}$ Also, $\begin{matrix} S (I_{ϑ_{α}}) = ρ_{ϑ_{α}} - τ_{ϑ_{α}} - ω_{ϑ_{α}} \geq min_{j} min_{i} {ρ_{ϑ_{ij}}} \\ - max_{j} max_{i} {τ_{ϑ_{ij}}} - max_{j} max_{i} {ω_{ϑ_{ij}}} = S (I_{ϑ_{ij}}^{-}) . \end{matrix}$ Now, by making use of order relation between these two PFHSNs, we get

$I_{ϑ_{ij}}^{-}$ ≤PFHSWA( $I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}} \leq I_{ϑ_{ij}}^{+}$ which is the proof of the boundedness.

Proof 5. Let I_{ϑ_α} and I_{ϑ_ij} be two PFHSNs. Then, by the operational law of direct sum defined above in (a), we get I_{ϑ_α} ⊕ I_{ϑ_ij} = 〈 ρ_{ϑ_α} + ρ_{ϑ_ij} - ρ_{ϑ_α} ρ_{ϑ_ij}, τ_{ϑ_α} τ_{ϑ_ij}, ω_{ϑ_α} ω_{ϑ_ij} 〉 . Therefore,

PFHSWA( $I_{ϑ_{11}} \oplus I_{ϑ_{α}}, I_{ϑ_{12}} \oplus I_{ϑ_{α}}, \dots ., I_{ϑ_{nm}} \oplus I_{ϑ_{α}}) = \oplus_{j = 1}^{m} F_{j} (\oplus_{i = 1}^{n} J_{i} (I_{ϑ_{ij}} \oplus I_{ϑ_{α}}))$

$\begin{matrix} = 〈 {1 - \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - ρ_{ϑ_{ij}})}^{J_{i}} {(1 - ρ_{ϑ_{α}})}^{J_{i}})}^{F_{j}}, \\ {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(τ_{ϑ_{ij}})}^{J_{i}} {(τ_{ϑ_{α}})}^{J_{i}})}^{F_{j}}, \\ {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(ω_{ϑ_{ij}})}^{J_{i}} {(ω_{ϑ_{α}})}^{J_{i}})}^{F_{j}} 〉 \end{matrix}$ $\begin{matrix} = 〈 {1 - (1 - ρ_{ϑ_{α}}) \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ {(τ_{ϑ_{α}}) \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ {(ω_{ϑ_{α}}) \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉; \end{matrix}$ (as $\sum_{i = 1}^{n} J_{i} = 1$ .) $\begin{matrix} = 〈 {1 - \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉 \oplus 〈 ρ_{ϑ_{α}}, τ_{ϑ_{α}}, ω_{ϑ_{α}} 〉 \end{matrix}$ = PFHSWA (I_ϑ₁₁, I_ϑ₁₂, …, I_{ϑ_nm}) ⊕ I_{ϑ_α}; which completes the proof of operator being shift invariant.

Proof 6. Let I_{ϑ_ij} be a PFHSN and κ > 0 be a real number, then by the operational law of scalar multiplication defined above in (c), we get $\begin{matrix} κ I_{v_{ij}} = 〈 [1 - {(1 - ρ_{ϑ_{ij}})}^{κ}, {(τ_{ϑ_{ij}})}^{κ}, {(ω_{ϑ_{ij}})}^{κ}] 〉 . \end{matrix}$ Thus, $\begin{matrix} PFHSWA (κ I_{ϑ_{11}}, κ I_{ϑ_{12}}, \dots ., {κ I}_{ϑ_{nm}}) = \\ in 〈 {1 - \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - ρ_{ϑ_{ij}})}^{κ J_{i}})}^{F_{j}}, \\ {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(τ_{ϑ_{ij}})}^{κ J_{i}})}^{F_{j}}, {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(ω_{ϑ_{ij}})}^{κ J_{i}})}^{F_{j}} 〉 \end{matrix}$

$\begin{matrix} = 〈 1 - {({\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}})}^{κ}, \\ {({\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}})}^{κ}, \\ {({\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}})}^{κ} 〉 \end{matrix}$ = κ PFHSWA(I_ϑ₁₁, I_ϑ₁₂, … . , I_{ϑ_nm}).

Proof 8. The proof of the theorem follows from the principle of mathematical induction.

For n = 1, we get $J_{1} = 1$ . (because $\sum_{i = 1}^{n} J_{i} = 1 .$ )

Then, from (4.4), we have

$PFHSWG (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) = \otimes_{j = 1}^{m} {I_{ϑ_{1 j}}}^{F_{j}}$ .

Now, using the above stated functional laws (a)-(e), we get

$\begin{matrix} PFHSWG (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) = \\ 〈 \prod_{j = 1}^{m} {(ρ_{ϑ_{1 j}})}^{F_{j}}, 1 - \prod_{j = 1}^{m} {(1 - τ_{ϑ_{1 j}})}^{F_{j}}, \\ 1 - \prod_{j = 1}^{m} {(1 - ω_{ϑ_{1 j}})}^{F_{j}} 〉 \end{matrix}$

$\begin{matrix} = 〈 \prod_{j = 1}^{m} {(\prod_{i = 1}^{1} {(ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, 1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{1} {(1 - τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ 1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{1} {(1 - ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉 . \end{matrix}$

Also, for m = 1, we get $F_{1} = 1$ . (because $\sum_{j = 1}^{m} F_{j} = 1$ .)

Then, from (4.4), we have

$PFHSWG (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) = \otimes_{i = 1}^{n} {I_{ϑ_{i 1}}}^{J_{i}}$ .

Again, using the above stated functional laws (a)-(e), we get $\begin{matrix} PFHSWG (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) \\ = 〈 \prod_{i = 1}^{n} {(ρ_{ϑ_{i 1}})}^{J_{i}}, 1 - \prod_{i = 1}^{n} {(1 - τ_{ϑ_{i 1}})}^{J_{i}}, \\ 1 - \prod_{i = 1}^{n} {(1 - ω_{ϑ_{i 1}})}^{J_{i}} 〉 \end{matrix}$ $\begin{matrix} = 〈 \prod_{j = 1}^{1} {(\prod_{i = 1}^{n} {(ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, 1 - \prod_{j = 1}^{1} {(\prod_{i = 1}^{n} {(1 - τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ 1 - \prod_{j = 1}^{1} {(\prod_{i = 1}^{n} {(1 - ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉 . \end{matrix}$

This shows that equation (4.4) satisfies for n = 1 and m = 1. Assume that equation (4.4) holds for m = α₁ + 1, n = α₂ and m = α₁, n = α₂ + 1; i.e., $\begin{matrix} \otimes_{j = 1}^{α_{1} + 1} {(\otimes_{i = 1}^{α_{2}} {I_{ϑ_{ij}}}^{J_{i}})}^{F_{j}} = 〈 \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2}} {(ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ 1 - \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2}} {(1 - τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ 1 - \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2}} {(1 - ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉; \end{matrix}$ $\begin{matrix} \otimes_{j = 1}^{α_{1}} {(\otimes_{i = 1}^{α_{2} + 1} {I_{ϑ_{ij}}}^{J_{i}})}^{F_{j}} = 〈 \prod_{j = 1}^{α_{1}} {(\prod_{i = 1}^{α_{2} + 1} {(ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ 1 - \prod_{j = 1}^{α_{1}} {(\prod_{i = 1}^{α_{2} + 1} {(1 - τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ 1 - \prod_{j = 1}^{α_{1}} {(\prod_{i = 1}^{α_{2} + 1} {(1 - ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉 . \end{matrix}$

Now for m = α₁ + 1, n = α₂ + 1, we get $\begin{matrix} \otimes_{j = 1}^{α_{1} + 1} {(\otimes_{i = 1}^{α_{2} + 1} {I_{ϑ_{ij}}}^{J_{i}})}^{F_{j}} = \otimes_{j = 1}^{α_{1} + 1} {(\otimes_{i = 1}^{α_{2}} {I_{ϑ_{ij}}}^{J_{i}} \otimes {I_{ϑ_{(α_{2} + 1) j}}}^{J_{α_{2} + 1}})}^{F_{j}} \\ = \otimes_{j = 1}^{α_{1} + 1} {(\otimes_{i = 1}^{α_{2}} {I_{ϑ_{ij}}}^{J_{i}})}^{F_{j}} \otimes_{j = 1}^{α_{1} + 1} {({I_{ϑ_{(α_{2} + 1) j}}}^{J_{α_{2} + 1}})}^{F_{j}} \end{matrix}$ $= 〈 \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{^{'} a_{2}} {(ρ_{ϑ_{i j}})}^{J_{i}})}^{F_{j}} \otimes \prod_{j = 1}^{α_{1} + 1} {({(ρ_{ϑ_{(α_{2} + 1) j}})}^{J_{(α_{2} + 1)}})}^{F_{j}}, 1 - \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2}} {(1 - τ_{ϑ_{i j}})}^{J_{i}})}^{F_{j}} \otimes 1 - \prod_{j = 1}^{α_{1} + 1} {({(1 - τ_{ϑ_{(α_{2} + 1) j}})}^{J_{(α_{2} + 1)}})}^{F_{j}}, 1 - \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2}} {(1 - ω_{ϑ_{i j}})}^{J_{i}})}^{F_{j}} \otimes 1 - \prod_{j = 1}^{α_{1} + 1} {({(1 - ω_{ϑ_{(α_{2} + 1) j}})}^{J_{(α_{2} + 1)}})}^{F_{j}} 〉$ $\begin{matrix} = 〈 \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2} + 1} {(ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ 1 - \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2} + 1} {(1 - τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ 1 - \prod_{j = 1}^{α_{1} + 1} {(\prod_{i = 1}^{α_{2} + 1} {(1 - ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉 . \end{matrix}$ Therefore, the result is true for m = α₁ + 1, n = α₂ + 1 and the theorem is proved.

Proof 9. Let I_{ϑ_ij} = I_{ϑ_α} = (ρ_{ϑ_ij}, τ_{ϑ_ij}, ω_{ϑ_ij}) be a collection of PFHSNs, then with the use of equation (4.2), we get $\begin{matrix} PFHSWG (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) \\ = 〈 \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, 1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ 1 - \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} {(1 - ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉 \end{matrix}$ $\begin{matrix} = 〈 {({(ρ_{ϑ_{ij}})}^{\sum_{i = 1}^{n} J_{i}})}^{\sum_{j = 1}^{m} F_{j}}, \\ 1 - {({(1 - τ_{ϑ_{ij}})}^{\sum_{i = 1}^{n} J_{i}})}^{\sum_{j = 1}^{m} F_{j}}, \\ - {({(1 - ω_{ϑ_{ij}})}^{\sum_{i = 1}^{n} J_{i}})}^{\sum_{j = 1}^{m} F_{j}} 〉 \end{matrix}$ $\begin{matrix} = 〈 ρ_{ϑ_{ij}}, 1 - (1 - τ_{ϑ_{ij}}), 1 - (1 - ω_{ϑ_{ij}}) 〉 \\ = (ρ_{ϑ_{ij}}, τ_{ϑ_{ij}}, ω_{ϑ_{ij}}) = I_{ϑ_{α}} . \end{matrix}$ Hence, the idempotency holds.

Proof 10. Let I_{ϑ_ij} = (ρ_{ϑ_ij}, τ_{ϑ_ij}, ω_{ϑ_ij}) be a PFHSN, then

$min_{j} min_{i} {ρ_{ϑ_{ij}}} \leq {ρ_{ϑ_{ij}}} \leq max_{j} max_{i} {ρ_{ϑ_{ij}}}$ $\begin{matrix} \Leftrightarrow {(min_{j} min_{i} {ρ_{ϑ_{ij}}})}^{J_{i}} \leq {(ρ_{ϑ_{ij}})}^{J_{i}} \leq {(max_{j} max_{i} {ρ_{ϑ_{ij}}})}^{J_{i}} \end{matrix}$ $\begin{matrix} \Leftrightarrow {(min_{j} min_{i} {ρ_{ϑ_{ij}}})}^{\sum_{i = 1}^{n} J_{i}} \leq \prod_{i = 1}^{n} {(ρ_{ϑ_{ij}})}^{J_{i}} \\ \leq {(max_{j} max_{i} {ρ_{ϑ_{ij}}})}^{\sum_{i = 1}^{n} J_{i}} \end{matrix}$ $\begin{matrix} \Leftrightarrow (min_{j} min_{i} {ρ_{ϑ_{ij}}}) \leq \prod_{i = 1}^{n} {(ρ_{ϑ_{ij}})}^{J_{i}} \\ \leq (max_{j} max_{i} {ρ_{ϑ_{ij}}}) \end{matrix}$

(as $\sum_{i = 1}^{n} J_{i} = 1)$ $\begin{matrix} \Leftrightarrow {(min_{j} min_{i} {ρ_{ϑ_{ij}}})}^{\sum_{j = 1}^{m} F_{j}} \leq {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} \\ \leq {(max_{j} max_{i} {ρ_{ϑ_{ij}}})}^{\sum_{j = 1}^{m} F_{j}} \end{matrix}$ $\begin{matrix} \Leftrightarrow (min_{j} min_{i} {ρ_{ϑ_{ij}}}) \leq {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} \\ \leq (max_{j} max_{i} {ρ_{ϑ_{ij}}}) \end{matrix}$ (as $\sum_{j = 1}^{m} F_{j}$ = 1) $\begin{matrix} \Leftrightarrow (min_{j} min_{i} {ρ_{ϑ_{ij}}}) \leq {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} \\ \leq (max_{j} max_{i} {ρ_{ϑ_{ij}}}) \end{matrix}$ (9) Similarly, $\begin{matrix} (min_{j} min_{i} {τ_{ϑ_{ij}}}) \leq {1 - \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} \\ \leq (max_{j} max_{i} {τ_{ϑ_{ij}}}) . \end{matrix}$ (10) $\begin{matrix} (min_{j} min_{i} {ω_{ϑ_{ij}}}) \leq {1 - \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} \\ \leq (max_{j} max_{i} {ω_{ϑ_{ij}}}) . \end{matrix}$ (11) Let PFHSWG (I_ϑ₁₁, I_ϑ₁₂, … . , I_{ϑ_nm})= (ρ_{ϑ_α}, τ_{ϑ_α}, ω_{ϑ_α}) = I_{ϑ_α}, so that inequalities (8.4),(8.5) and (8.6) could be transformed into following forms: $\begin{matrix} min_{j} min_{i} {ρ_{ϑ_{ij}}} \leq {ρ_{ϑ_{ij}}} \leq {ρ_{ϑ_{α}}} \leq max_{j} max_{i} {ρ_{ϑ_{ij}}}; \end{matrix}$ and $\begin{matrix} max_{j} min_{i} {τ_{ϑ_{ij}}} \leq {τ_{ϑ_{ij}}} \leq {τ_{ϑ_{α}}} \leq max_{j} max_{i} {τ_{ϑ_{ij}}}; \end{matrix}$ and $\begin{matrix} min_{j} min_{i} {ω_{ϑ_{ij}}} \leq {ω_{ϑ_{ij}}} \leq {ω_{ϑ_{α}}} \leq max_{j} max_{i} {ω_{ϑ_{ij}}} \end{matrix}$ respectively. So, by making use of the earlier defined score function, we obtain the following values: $\begin{matrix} S (I_{ϑ_{α}}) = ρ_{ϑ_{α}} - & ocirc; ϑ_{α} - ω_{ϑ_{α}} \leq max_{j} max_{i} {ρ_{ϑ_{ij}}} \\ - min_{j} min_{i} {τ_{ϑ_{ij}}} - min_{j} min_{i} {ω_{ϑ_{ij}}} = S (I_{ϑ_{ij}}^{+}) . \end{matrix}$ Also, $\begin{matrix} S (I_{ϑ_{α}}) = ρ_{ϑ_{α}} - τ_{ϑ_{α}} - ω_{ϑ_{α}} \geq min_{j} min_{i} {ρ_{ϑ_{ij}}} \\ - max_{j} max_{i} {τ_{ϑ_{ij}}} - max_{j} max_{i} {ω_{ϑ_{ij}}} = S (I_{ϑ_{ij}}^{-}) . \end{matrix}$

Now, by making use of order relation between these two PFHSNs, we get

$I_{ϑ_{ij}}^{-}$ ≤PFHSWG( $I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}} \leq I_{ϑ_{ij}}^{+}$ which is the proof of the boundedness.

Proof 11. Let I_{ϑ_α} and I_{ϑ_ij} be two PFHSNs. Now, by using the functional law of direct product stated above, we get

$\begin{matrix} I_{ϑ_{α}} \otimes I_{ϑ_{ij}} = 〈 ρ_{ϑ_{α}} ρ_{ϑ_{ij}}, τ_{ϑ_{α}} + τ_{ϑ_{ij}} - τ_{ϑ_{α}} τ_{ϑ_{ij}}, \\ ω_{ϑ_{α}} + ω_{ϑ_{ij}} - ω_{ϑ_{α}} ω_{ϑ_{ij}} 〉 . \end{matrix}$ Therefore, $\begin{matrix} PFHSWG (I_{ϑ_{11}} \otimes I_{ϑ_{α}}, I_{ϑ_{12}} \otimes I_{ϑ_{α}}, \dots ., I_{ϑ_{nm}} \otimes I_{ϑ_{α}}) \\ = \otimes_{j = 1}^{m} F_{j} (\otimes_{i = 1}^{n} J_{i} (I_{ϑ_{ij}} \otimes I_{ϑ_{α}})) \end{matrix}$ $\begin{matrix} = 〈 {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(ρ_{ϑ_{ij}})}^{J_{i}} {(ρ_{ϑ_{α}})}^{J_{i}})}^{F_{j}}, \\ {1 - \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - τ_{ϑ_{ij}})}^{J_{i}} {(1 - τ_{ϑ_{α}})}^{J_{i}})}^{F_{j}}, \\ 1 - {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - ω_{ϑ_{ij}})}^{J_{i}} {(1 - ω_{ϑ_{α}})}^{J_{i}})}^{F_{j}} 〉 \end{matrix}$ $\begin{matrix} = 〈 {(ρ_{ϑ_{α}}) \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ {(1 - τ_{ϑ_{α}}) \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ {(1 - ω_{ϑ_{α}}) \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉 \end{matrix}$ (because $\sum_{i = 1}^{n} J_{i} = 1$ ) $\begin{matrix} = 〈 {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ {1 - \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}}, \\ {1 - \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}} 〉 \\ \otimes 〈 ρ_{ϑ_{α}}, τ_{ϑ_{α}}, ω_{ϑ_{α}} 〉 \end{matrix}$ $\begin{matrix} = PFHSWG (I_{ϑ_{11}}, I_{ϑ_{12}}, \dots ., I_{ϑ_{nm}}) \otimes I_{ϑ_{α}} . \end{matrix}$ This completes the proof of shift invariance.

Proof 12. Let I_{ϑ_ij} be a PFHSN and κ > 0 be a real number, then by making use of scalar multiplication functional law stated above, we get $\begin{matrix} κ I_{v_{ij}} = 〈 [{(ρ_{ϑ_{ij}})}^{κ}, 1 - {({1 - τ}_{ϑ_{ij}})}^{κ},] \\ [{1 - (1 - ω_{ϑ_{ij}})}^{κ}] 〉 . \end{matrix}$ So, $\begin{matrix} PFHSWG (κ I_{ϑ_{11}}, κ I_{ϑ_{12}}, \dots ., {κ I}_{ϑ_{nm}}) = \\ 〈 {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(ρ_{ϑ_{ij}})}^{κ J_{i}})}^{F_{j}}, \\ {1 - \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - τ_{ϑ_{ij}})}^{κ J_{i}})}^{F_{j}}, \\ {1 - \prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - ω_{ϑ_{ij}})}^{κ J_{i}})}^{F_{j}} 〉 \end{matrix}$ $\begin{matrix} = 〈 {({\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(ρ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}})}^{κ}, \\ {(1 - {\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - τ_{ϑ_{ij}})}^{J_{i}})}^{F_{j}})}^{κ}, \\ 1 - {({\prod_{j = 1}^{m} (\prod_{i = 1}^{n} {(1 - ω_{ϑ_{ij}})}^{J_{i}})}^{F_{j}})}^{κ} 〉 \end{matrix}$ = κ PFHSWG(I_ϑ₁₁, I_ϑ₁₂, … . , I_{ϑ_nm}). Hence proved.

Appendix II

$(R_{1}, Λ) = {\begin{matrix} \begin{matrix} < {(α}_{3}, β_{1}, γ_{1}, δ_{1}), {\frac{v_{1}}{(0.1, 0.3, 0.5)}, \frac{v_{2}}{(0.2, 0.4, 0.2)}, \frac{v_{3}}{(0.1, 0.2, 0.4)}, \frac{v_{4}}{(0, 0, 1)}} >, \\ < {(α}_{3}, β_{1}, γ_{3}, δ_{1}), {\frac{v_{1}}{(0.2, 0.3, 0.2)}, \frac{v_{2}}{(0.2, 0.1, 0.2)}, \frac{v_{3}}{(0.1, 0.3, 0.4)}, \frac{v_{4}}{(0, 0, 1)}} >, \\ < {(α}_{3}, β_{2}, γ_{1}, δ_{1}), {\frac{v_{1}}{(0.1, 0.1, 0.5)}, \frac{v_{2}}{(0.2, 0.2, 0.3)}, \frac{v_{3}}{(0.1, 0.4, 0.4)}, \frac{v_{4}}{(0.1, 0.2, 0.3)}} > \end{matrix} \\ < {(α}_{3}, β_{2}, γ_{3}, δ_{1}), {\frac{v_{1}}{(0.1, 0.2, 0.5)}, \frac{v_{2}}{(0.2, 0.2, 0.2)}, \frac{v_{3}}{(0.2, 0.4, 0.4)}, \frac{v_{4}}{(0.1, 0.2, 0.6)}} > \end{matrix}} .$

$(R_{2}, Λ^{'}) = {\begin{matrix} \begin{matrix} < {(α}_{1}, β_{1}, γ_{1}, δ_{1}), {\frac{v_{1}}{(0.1, 0.3, 0.2)}, \frac{v_{2}}{(0.2, 0.4, 0.3)}, \frac{v_{3}}{(0.1, 0.2, 0.5)}, \frac{v_{4}}{(0, 0, 1)}} >, \\ < {(α}_{1}, β_{1}, γ_{2}, δ_{1}), {\frac{v_{1}}{(0.2, 0.3, 0.1)}, \frac{v_{2}}{(0.2, 0.1, 0.1)}, \frac{v_{3}}{(0.1, 0.3, 0.5)}, \frac{v_{4}}{(0, 0, 1)}} >, \\ < {(α}_{3}, β_{1}, γ_{1}, δ_{1}), {\frac{v_{1}}{(0.1, 0.1, 0.7)}, \frac{v_{2}}{(0.2, 0.3, 0.2)}, \frac{v_{3}}{(0.1, 0.4, 0.2)}, \frac{v_{4}}{(0.1, 0.2, 0.6)}} > \end{matrix} \\ < {(α}_{3}, β_{1}, γ_{2}, δ_{1}), {\frac{v_{1}}{(0.5, 0.2, 0.1)}, \frac{v_{2}}{(0.2, 0.4, 0.2)}, \frac{v_{3}}{(0.2, 0.4, 0.1)}, \frac{v_{4}}{(0.1, 0.2, 0.1)}} > \end{matrix}} .$

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