Novel MCGDM with q-rung orthopair fuzzy soft sets and TOPSIS approach under q-Rung orthopair fuzzy soft topology

Abstract

In this article, we study some concepts related to q-rung orthopair fuzzy soft sets (q-ROFS sets), together with their algebraic structure. We present operations on q-ROFSSs and their specific properties and elaborate them with real-life examples and tabular representations to develop influx of linguistic variables based on q-rung orthopair fuzzy soft (q-ROFS) information. We present an application of q-ROFS sets to multi-criteria group decision-making (MCGDM) process related to the university choice, accompanied by algorithm and flowchart. We develop q-ROFS TOPSIS method and q-ROFS VIKOR method as extensions of TOPSIS (a technique for ordering preference through the ideal solution) and VIKOR (Vlse Kriterijumska Optimizacija Kompromisno Resenje), respectively. Finally, we tackle a problem of construction business utilizing q-ROFS TOPSIS and q-ROFS VIKOR methods.

Keywords

MCGDM aggregation operator TOPSIS VIKOR

1 Introduction

Mathematical modeling on real world problems often involve multi-attribute, multi-polar information, uncertainties, hesitancy and vagueness. Fuzzy sets introduced by Zadeh [58], soft set introduced by Molodtsov [29] and rough sets introduced by Pawlak [31] are strong mathematical models for solving many problems in social sciences, engineering, artificial intelligence, computational intelligence, image processing and medical diagnosis. Zadeh [58] initiated the idea of fuzzy sets by means of membership function as an extension of characteristic function associated with traditional crisp set. Atanassov [10] established the idea of intuitionistic fuzzy sets as an extension of fuzzy set by introducing the concepts of membership and non-membership grades along with the restriction that sum of these two grades must not exceed unity. Atanassov [12] portrayed geometrical elucidation of the elements of intuitionistic fuzzy entities. Molodtsov [29] originated the notion of a new kind of sets conventionally known as soft sets, as a mathematical model for sorting out uncertainties by means of parameterizations. Topological structure on these sets have many applications to group decision making problems. Soft topology was introduced by Bashir and Sabir [2], Cagman et al. [13], Hazra et al. [27], Roy and Samanta [50], Shabir and Naz [51], and Varol et al. [53]. Aygunoglu et al. [1] introduced fuzzy soft topological spaces. Riaz and Masooma [37] and Zorlutuna and Atmaca [65] introduced fuzzy parametrized fuzzy soft topology (FPFS-topology). Osmanoglu and Tokat [30] and Li and Cui [28] introduced intuitionistic fuzzy soft topology (IFS-topology). Riaz et al. [41] introduced Hesitant fuzzy soft topology and its applications to multi-attribute group decision-making (MAGDM).

Yager [54 –56] introduced Pythagorean fuzzy set as an extension of Atanassov’s intuitionistic fuzzy set and presented Pythagorean membership grades with applications to the multi-criteria decision making (MCDM). Ali [9] presented another view on q-rung orthopair fuzzy sets. Recently, Riaz et al. [45] presented a robust q-rung orthopair fuzzy information aggregation using Einstein operations with application to sustainable energy planning decision management.

Fuzzy sets, Pythagorean fuzzy sets, soft sets, rough sets, neutrosophic sets and their hybrid structures have been studied by many researchers including Akram et al. [3 –6], Eraslan and Karaaslan [16], Feng et al. [17, 18], Garg [20, 21], Hashmi et al. [22], Hashmi and Riaz [23], Kumar and Garg [24], Naeem et al. [25, 26], Peng and Yang [32], Peng and Selvachandran [34], Peng and Liu [36], Riaz et al. [38 –41], Riaz and Hashmi [42 –44], Riaz and Tehrim [46 –49], Tehrim and Riaz [52], Zhang and Xu [59], Zhan et al. [60 –62], Zhang and Zhan [63], and Zhang et al. [64].

The goal of paper is to introduce q-rung orthopair fuzzy soft sets and q-rung orthopair fuzzy soft topology. The existing intuitionistic fuzzy sets and Pythagorean fuzzy sets fail to handle the uncertain information when the sum of membership degree (μ_P) and non-membership degree (ν_P) corresponding to an alternative (ρ) is larger than 1 (μ_P (ρ) + ν_P (ρ) >1) as well as the sum of squares of these degrees is larger than 1 ( $μ_{P}^{2} (ρ) + ν_{P}^{2} (ρ) > 1$ ). A q-rung orthopair fuzzy set (q-ROFS) has the ability to deal with such problems. The eminent characteristic of q-ROFS is that the degree of membership (μ_P) and degree of non-membership (ν_P) satisfy the condition $0 \leq μ_{P}^{q} (x) + ν_{P}^{q} (x) \leq 1 (q \geq 1)$ , so the space of uncertain information they can describe is broader. Moreover, in order to make this model more powerful with addition of parametrization, we propose the idea of q-rung orhtopair fuzzy soft set (q-ROFSS) as a hybrid model of q-rung orthopair fuzzy set and soft set. A q-ROFSS and q-ROFS topology provide a large number of applications to the multi-criteria group decision making (MCGDM) in a comprehensive way.

Rest of the paper is organized as follows. In Section 2, some rudiments of fuzzy set, soft set, IFS, PFS and q-ROFS are given. In Section 3, the concept of q-rung orthopair fuzzy soft set (q-ROFSS) is introduced. In Section 4, the construction of q-rung orthopair fuzzy topology is expressed. Meanwhile, many topological properties and their illustrative examples are presented. Separation axioms, regular space and normal space in the context of q-rung orthopair fuzzy topology are defined. In Section 5, q-ROFSS based choice value method for MCGDM is developed. An application of the proposed method for the optimum results in school education systems is presented. In Section 6, a robust extension of TOPSIS approach with q-ROFS topology is elaborated for money investment. Finally, in Section 7, the conclusion of this research work is summarized.

2 Preliminaries

In this section, we briefly review some basic concepts including fuzzy set, soft set, intuitionistic fuzzy set, Pythagorean fuzzy set and q-rung orthopair fuzzy sets that are very helpful in analysis of rest of the paper.

Definition 2.1. [58] Let X be a crisp set and μ_A : X → [0, 1] be the membership function. Then the fuzzy set is an object of the form, $A = {(ρ, μ_{A} (ρ)) : ρ \in X},$ the value μ_A (ρ) denotes the membership degree of ρ to the fuzzy set A.

Definition 2.2. [10, 11] An intuitionistic fuzzy set (IFS) $A$ over the universe X is an object of the form, $A = {(ρ, μ_{A} (ρ), ν_{A} (ρ)) : ρ \in X},$ where the mappings $μ_{A}$ and $ν_{A}$ map elements of X to the unit closed interval [0, 1] with the restriction that $0 \leq μ_{A} (ρ) + ν_{A} (ρ) \leq 1 (ρ \in X)$ . The values $μ_{A} (ρ)$ and $ν_{A} (ρ)$ denote the membership degree and non-membership degree, respectively.

Definition 2.3. [29] Let X to be a crisp set and E be the set of attributes or parameters. Let A ⊆ E, then the soft can be defined as $(ψ, A) = {(e, ψ (e)) : e \in A, ψ (e) \in P (X)},$ where ψ : A → P (X) be the set-valued mapping. The pair (ψ, A) can be shortly written ψ_A.

Definition 2.4. [54 –56] A Pythagorean Fuzzy Set (PFS) is an object of the form, $P = {< ρ, μ_{P} (ρ), ν_{P} (ρ) > : ρ \in X},$ where μ_P and ν_P are mappings from X to the unit closed interval [0, 1] with the restriction that $0 \leq μ_{P}^{2} (ρ) + ν_{P}^{2} (ρ) \leq 1 (ρ \in X)$ . The pair (μ_p (ρ) , ν_p (ρ) ) is known as Pythagorean Fuzzy Number (PFN).

Yager [57] introduced the idea of q-rung orthopair fuzzy set (q-ROFS) as an extension of both IFS and PFS.

Definition 2.5. [57] Let X be the universal set, a q-rung orthopair fuzzy set (q-ROF set) is defined as, $R = {< ϱ, μ_{R} (ϱ), ν_{R} (ϱ) > : ϱ \in X},$ where μ_R (ϱ) ∈ [0, 1] and ν_R (ϱ) ∈ [0, 1] represent the degree of membership and degree of non membership of ϱ ∈ R that fulfil the requirement $0 \leq μ_{R} (ϱ)^{q} + ν_{R} (ϱ)^{q} \leq 1$ for ϱ ∈ R and q > 2. In general, $Π_{R} = \sqrt[q]{1 - μ_{R} (ϱ)^{q} - ν_{R} (ϱ)^{q}}$ is known as the degree of hesitancy for ϱ to R.

To make it more simple ℵ = (μ_ℵ, ν_ℵ), with 0 ≤ μ_ℵ (ϱ) ^q + μ_ℵ (ϱ) ^q ≤ 1, ϱ ∈ R, q > 2, is said to be a q-rung orthopair fuzzy number (q-ROFN). A q-ROFN can be regarded as different kinds of fuzzy numbers with the help of different values of q. When q = 1 it becomes intuitionistic fuzzy number (IFN) and when q = 2 it becomes Pythagorean fuzzy number (PFN). We observe that IFNs and PFNs are special cases of q-ROFNs. The space for IFN, PFN, q-ROFN is shown in Fig. 1.

Fig. 1

Space for IFN (μ_α, ν_α), PFN (μ_p, ν_p) & q-ROFN (μ_τ, ν_τ). The green line in this figure may be considered as dotted line.

3 q-Rung orthopair fuzzy soft sets

In this section, we introduce the novel concept of q-rung orthopair fuzzy soft set (q-ROFSS) as a hybrid model of q-rung orthopair fuzzy set and soft set. We introduce some elementary operations on q-rung orthopair fuzzy soft sets and their related results.

Definition 3.1. Let X be the universal set and E be the set of attributes. Assume further that A ⊆ E and qROF^X represents the class of all q-rung orthopair fuzzy subsets over X and κ : A → qROF^X be a mapping. A q-rung orthopair fuzzy soft set (q-ROFSS) on X is represented by (κ, A) or κ_A and defined by $\begin{matrix} (κ, A) & = & {(e, {ϱ, μ_{κ_{A}} (ϱ), ν_{κ_{A}} (ϱ)}) : e \in A, ϱ \in X} \\ = & {(e, {\frac{ϱ}{(μ_{κ_{A}} (ϱ), ν_{κ_{A}} (ϱ))}}) : e \in A, ϱ \in X} \\ = & {(e, {\frac{(μ_{κ_{A}} (ϱ), ν_{κ_{A}} (ϱ))}{ϱ}}) : e \in A, ϱ \in X} \end{matrix}$ -9.7ptwhere μ_{κ
_A} : X → [0, 1] and ν_{κ
_A} : X → [0, 1] describe maps called membership map and non-membership map, respectively. Specifically, μ_{κ
_A} (ϱ) denotes the grade of membership and ν_{κ
_A} (ϱ) represents grade of non-membership of the alternative ϱ ∈ X to the set (κ, A) having the quality that sum of the qth power of these grades must not exceed unity.

For each parameter e, the value κ (e) expresses κ (e)-approximate point.

Table 1
Tabular array of q-ROFSS κ_A

κ _A e ₁ e ₂ ⋯ e _n

ρ ₁ (μ₁₁, ν₁₁) (μ₁₂, ν₁₂) ⋯ (μ_1n, ν_1n)

ρ ₂ (μ₂₁, ν₂₁) (μ₂₂, ν₂₂) ⋯ (μ_2n, ν_2n)

⋮ ⋮ ⋮ ⋱ ⋮

ρ _m (μ_m1, ν_m1) (μ_m2, ν_m2) ⋯ (μ_mn, ν_mn)

κ _A	e ₁	e ₂	⋯	e _n
ρ ₁	(μ₁₁, ν₁₁)	(μ₁₂, ν₁₂)	⋯	(μ_1n, ν_1n)
ρ ₂	(μ₂₁, ν₂₁)	(μ₂₂, ν₂₂)	⋯	(μ_2n, ν_2n)
⋮	⋮	⋮	⋱	⋮
ρ _m	(μ_m1, ν_m1)	(μ_m2, ν_m2)	⋯	(μ_mn, ν_mn)

The multitude of every q-ROFSS over X taken from E is known as q-ROFS class and is denoted as q - ROFS (X, E).

Let us take μ_ij = μ_{κ
_A} (e_j) (ϱ_i) and ν_ij = ν_{κ
_A} (e_j) (ϱ_i) where i and j run, respectively, from 1 to m and from 1 to n. Thus the q-ROFSS κ_A may be written in tabular form as cited in Table 1:

The corresponding matrix form is $\begin{matrix} (κ, A) & = & [(μ_{ij}, ν_{ij})]_{m \times n} \\ = & (\begin{matrix} (μ_{11}, ν_{11}) & (μ_{12}, ν_{12}) & \dots & (μ_{1 n}, ν_{1 n}) \\ (μ_{21}, ν_{21}) & (μ_{22}, ν_{22}) & \dots & (μ_{2 n}, ν_{2 n}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ (μ_{m 1}, ν_{m 1}) & (μ_{m 2}, ν_{m 2}) & \dots & (μ_{mn}, ν_{mn}) \end{matrix}) \end{matrix}$ The matrix shown above is known as q-rung orthopair fuzzy soft matrix or simply q-ROFS matrix.

Definition 3.2. We consider $κ_{A_{1}}^{(1)}$ and $κ_{A_{2}}^{(2)}$ be q-rung orthopair fuzzy soft sets (q-ROFSSs) on X. We say that $κ_{A_{1}}^{(1)}$ is q-ROFS subset of $κ_{A_{2}}^{(2)}$ i.e. $κ_{A_{1}}^{(1)} \tilde{\subseteq} κ_{A_{2}}^{(2)}$ , if

A₁ ⊆ A₂, and

κ⁽¹⁾ (e) is q-ROFS subset of κ⁽²⁾ (e) for all e ∈ A₁.

It’s more important that

κ_{A_{1}}^{(1)} \tilde{\subseteq} κ_{A_{2}}^{(2)}

may not require that each element of

κ_{A_{1}}^{(1)}

must be in

κ_{A_{2}}^{(2)}

, as we must have in classical set theory.

Definition 3.3. We consider (κ₁, A₁) and (κ₂, A₂) be q-ROFSSs over X. Then the union of these sets may be discussed as $(κ, A) = (κ_{1}, A_{1}) \tilde{\cup} (κ_{2}, A_{2}),$ where A = A₁ ∪ A₂ and for all e ∈ A, $κ (e) = {\begin{matrix} κ_{1} (e), & if e \in A_{1} ∖ A_{2} \\ κ_{2} (e), & if e \in A_{2} ∖ A_{1} \\ κ_{1} (e) \cup κ_{2} (e), & if e \in A_{1} \cap A_{2} \end{matrix}$ where κ₁ (e) ∪ κ₂ (e) is the union of two q-ROFSSs.

Definition 3.4. The intersection of two q-ROFSSs (κ₁, A₁) and (κ₂, A₂) is also a q-ROFSS $(κ, A) = (κ_{1}, A_{1}) \tilde{\cap} (κ_{2}, A_{2})$ , where A = A₁ ∩ A₂ ≠ φ and κ (e) = κ₁ (e) ∩ κ₂ (e) for all e ∈ A.

Definition 3.5. The difference (κ, A) of two q-ROFSSs (κ₁, A₁) and (κ₂, A₂) over X is defined by (κ, e) = (κ₁, e) \ (κ₂, e) ; ∀ e ∈ E and is discussed as $(κ_{1}, A) \tilde{\} (κ_{2}, A)$ . Hence, $\begin{matrix} (κ_{1}, A_{1}) \tilde{\} (κ_{2}, A_{2}) = {(e, {ϱ, min {μ_{κ_{1} (e)} (ϱ), ν_{κ_{2} (e)} (ϱ)}, \\ max {ν_{κ_{1} (e)} (ϱ), μ_{κ_{2} (e)} (ϱ)}}) : ϱ \in X, e \in E} \end{matrix}$

Definition 3.6. The complement of a q-ROFSS (κ, A) is a map κ^c : A → q - ROFS^X given by κ^c (e) = [κ (e)] ^c, for all e ∈ A. We can defined it as (κ, A) ^c or it may also by (κ^c, A). Thus, if $κ (e) = {(ϱ, μ_{κ (e)} (ϱ), ν_{κ (e)} (ϱ)) : ϱ \in X}$ thus $κ^{c} (e) = {(ϱ, ν_{κ (e)} (ϱ), μ_{κ (e)} (ϱ)) : ϱ \in X}$ for every e ∈ A.

Example 3.7. Let X = {ζ₁, ζ₂, ⋯ , ζ₆} and E = {e₁, e₂, ⋯ , e₆}. Suppose that A₁ = {e₁, e₃} and A₂ = {e₂, e₃, e₄}. Consider the q-ROFS sets (κ₁, A₁) = {(e₁, {ζ₁, 0.71, 0.36} , {ζ₃, 0.24, 0.96} , {ζ₆, 0.34, 0.49}) , (e₃, {ζ₂, 0.61, 0.82} , {ζ₆, 0.83, 0.41})}, and (κ₂, A₂) = {(e₂, {ζ₃, 0.46, 0.51} , {ζ₄, 0.61, 0.49} , {ζ₅, 0.94, 0.32 } , {ζ₆, 0.87, 0.45}) , (e₃, {ζ₁, 0.56, 0.66} , {ζ₆, 0.71, 0.55}) , (e₄, {ζ₂, 0.36, 0.87} , {ζ₃, 0.71, 0.39} , {ζ₆, 0.46, 0.63})} with q = 3.

1) Their union is $(κ_{1}, A_{1}) \tilde{\cup} (κ_{2}, A_{2}) = {(e_{1}, {ζ_{1}, 0.71, 0.36}$ , {ζ₃, 0.24, 0.96}, {ζ₆, 0.34, 0.49}), (e₂, {ζ₃, 0.46, 0.51} , {ζ₄, 0.61, 0.49} , {ζ₅, 0.94, 0.32} , {ζ₆, 0.87, 0.45}), (e₃, {ζ₁, 0.56, 0.66 } , {ζ₂, 0.61, 0.82 } , {ζ₆, 0.83, 0.41 }), (e₄, {ζ₂, 0.36, 0.87} , {ζ₃, 0.71, 0.39} , {ζ₆, 0.46, 0.63})}

2) Their intersection is $(κ_{1}, A_{1}) \tilde{\cap} (κ_{2}, A_{2}) = {(e_{3}, {ζ_{6}, 0.71, 0.55})}$

3) The complements of (κ₁, A₁) and (κ₂, A₂) are (κ₁, A₁) ^c = {(e₁, {ζ₁, 0.36, 0.71} , {ζ₃, 0.96, 0.24} , {ζ₆, 0.49, 0.34 } ) , (e₃, {ζ₂, 0.82, 0.61 } , { ζ₆, 0.41, 0.83})} and (κ₂, A₂) ^c = {(e₂, {ζ₃, 0.51, 0.46}, {ζ₄, 0.49, 0.61}, {ζ₅, 0.32, 0.94 } , {ζ₆, 0.45, 0.87 }) , (e₃, {ζ₁, 0.66, 0.56}, {ζ₆, 0.55, 0.71}) , (e₄, {ζ₂, 0.87, 0.36 } , {ζ₃, 0.39, 0.71 }, {ζ₆, 0.63, 0.46})} respectively.

4) The difference of (κ₁, A₁) and (κ₂, A₂) is $(κ_{1}, A_{1}) \tilde{\} (κ_{2}, A_{2}) = {(e_{1}, {ζ_{1}, 0.71, 0.36}, {ζ_{3}, 0.24,$ 0.96 } , {ζ₆, 0.34, 0.49 }) , (e₃, {ζ₂, 0.61, 0.82 } , {ζ₆, 0.71, 0.55})}

We can easily digest these notions with the help of q-ROFS matrices as given below: $\begin{matrix} (κ_{1}, A_{1}) & = & (\begin{matrix} (0.71, 0.36) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.61, 0.82) & (0, 1) & (0, 1) & (0, 1) \\ (0.24, 0.96) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0.34, 0.49) & (0, 1) & (0.83, 0.41) & (0, 1) & (0, 1) & (0, 1) \end{matrix}) \\ (κ_{2}, A_{2}) & = & (\begin{matrix} (0, 1) & (0, 1) & (0.56, 0.66) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0.36, 0.87) & (0, 1) & (0, 1) \\ (0, 1) & (0.46, 0.51) & (0, 1) & (0.71, 0.39) & (0, 1) & (0, 1) \\ (0, 1) & (0.61, 0.49) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0.94, 0.32) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0.87, 0.45) & (0.71, 0.55) & (0.46, 0.63) & (0, 1) & (0, 1) \end{matrix}) \end{matrix}$ $\begin{matrix} (κ_{1}, A_{1}) \tilde{\cup} (κ_{2}, A_{2}) & = & (\begin{matrix} (0.71, 0.36) & (0, 1) & (0.56, 0.66) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.61, 0.82) & (0.36, 0.87) & (0, 1) & (0, 1) \\ (0.24, 0.96) & (0.46, 0.51) & (0, 1) & (0.71, 0.39) & (0, 1) & (0, 1) \\ (0, 1) & (0.61, 0.49) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0.94, 0.32) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0.34, 0.49) & (0.87, 0.45) & (0.83, 0.41) & (0.46, 0.63) & (0, 1) & (0, 1) \end{matrix}) \\ (κ_{1}, A_{1}) \tilde{\cap} (κ_{2}, A_{2}) & = & (\begin{matrix} (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.71, 0.55) & (0, 1) & (0, 1) & (0, 1) \end{matrix}) \\ (κ_{1}, A_{1})^{c} & = & (\begin{matrix} (0.36, 0.71) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.82, 0.61) & (0, 1) & (0, 1) & (0, 1) \\ (0.96, 0.24) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0.49, 0.36) & (0, 1) & (0.41, 0.83) & (0, 1) & (0, 1) & (0, 1) \end{matrix}) \\ (κ_{2}, A_{2})^{c} & = & (\begin{matrix} (0, 1) & (0, 1) & (0.66, 0.56) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0.87, 0.36) & (0, 1) & (0, 1) \\ (0, 1) & (0.51, 0.46) & (0, 1) & (0.39, 0.71) & (0, 1) & (0, 1) \\ (0, 1) & (0.49, 0.61) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0.32, 0.94) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0.45, 0.87) & (0.55, 0.71) & (0.63, 0.46) & (0, 1) & (0, 1) \end{matrix}) \end{matrix}$ $\begin{matrix} (κ_{1}, A_{1}) \tilde{\} (κ_{2}, A_{2}) & = & (\begin{matrix} (0.71, 0.36) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.61, 0.82) & (0, 1) & (0, 1) & (0, 1) \\ (0.24, 0.96) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0.34, 0.49) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.71, 0.55) & (0, 1) & (0, 1) & (0, 1) \end{matrix}) \end{matrix}$

Definition 3.8. A q-ROFSS (κ, E) upon X is known as null q-ROFSS, written as κ_φ or Φ, if ∀ e ∈ E we have $κ (e) = \tilde{0}$ , where $\tilde{0}$ express null q-rung orthopair fuzzy set. So, $κ_{φ} = {(e, {ϱ, 0, 1}) : e \in E, ϱ \in X}$

Definition 3.9. A q-ROFSS (κ, E) upon X is known as absolute q-ROFSS, represented by $\overset{˘}{X}$ , if $κ (e) = \tilde{1}, \forall e \in E$ . Here $\tilde{1}$ denotes absolute q-rung orthopair fuzzy soft set. Thus, $\overset{˘}{X} = {(e, {ϱ, 1, 0}) : e \in E, ϱ \in X}$

Definition 3.10. A q-rung orthopair fuzzy soft set q-ROFSS (κ, A) is called q-rung orthopair fuzzy soft point (q-ROFS-point), expressed as ϑ_κ, if for the element ϑ ∈ A we have

κ (ϑ) ≠ κ_φ, and

κ (ϑ′) = κ_φ, for all ϑ′ ∈ A - {ϑ}.

Definition 3.11. A q-ROFS point ϑ_κ is said to be in q-ROFSS (κ₁, A) i.e. $ϑ_{κ} \tilde{\in} (κ_{1}, A)$ if ϑ ∈ A ⇒ κ (ϑ) ⊆ κ₁ (ϑ).

Example 3.12. Let X = {c, m, t} and A = {ϑ₁, ϑ₂}, then $ϑ_{κ}^{(1)} = {(ϑ_{1}, {(c, 0.49, 0.41), (m, 0.56, 0.21)})},$ and $ϑ_{κ}^{(2)} = {(ϑ_{2}, {(c, 0.68, 0.53), (t, 0.77, 0.48)})}$ are two distinct q-ROFS points contained in the following q-ROFSS $\begin{matrix} κ_{A} = {(ϑ_{1}, {(c, 0.49, 0.41), (m, 0.56, 0.21)}), \\ (ϑ_{2}, {(c, 0.68, 0.53), (t, 0.77, 0.48)})} \end{matrix}$

It is to be noted that $κ_{A} = ϑ_{κ}^{(1)} \tilde{\cup} ϑ_{κ}^{(2)}$ . i.e. a q-rung orthopair fuzzy soft set (q-FOFSS) is union of its q-rung orthopair fuzzy soft (q-ROFS) points.

It is worth mentioning that any q-ROFS point $ϑ_{κ}^{(1)}$ (assume) is said to be a singleton subset of q-ROFSS κ_A as it has a parameter and its κ-approximate element.

4 q-rung orthopair fuzzy soft topology

In this section, the notion of q-rung orthopair fuzzy soft topology (q-ROFS topology) based on on q-ROFSSs is defined. Certain properties of q-ROFS topology are expressed and their related results are well proved.

Definition 4.1. Let q - ROFS (X, E) be a set of all the q-ROFS subsets of the absolute q-ROFSS ${\overset{˘}{X}}_{E}$ . For A, B ⊆ E, a sub collection $\tilde{τ}$ of q - ROFS (X, E) is called q-rung orthopair fuzzy soft topology (q-ROFS topology) on ${\overset{˘}{X}}_{E}$ if it satisfy these axioms:

$φ_{E}, {\overset{˘}{X}}_{E} \tilde{\in} \tilde{τ}$ ,

$κ_{A}, κ_{B} \tilde{\in} \tilde{τ}$ then $κ_{A} \tilde{\cap} κ_{B} \tilde{\in} \tilde{τ}$ ,

If $κ_{A_{i}} \tilde{\in} \tilde{τ}, \forall i \in I$ , then ${\tilde{\cup}}_{i \in I} κ_{A_{i}} \tilde{\in} \tilde{τ}$ .

The doublet or 2-tuples $({\tilde{X}}_{E}, \tilde{τ})$ or simply ${\tilde{X}}_{E}$ is called q-ROFS topological space. The members of $\tilde{τ}$ are known as q-ROFS open sets and their complements are known as q-ROFS closed sets.

Definition 4.2. Assume that $({\tilde{X}}_{E}, {\tilde{τ}}_{X})$ is a q-ROFS topology. For some Y ⊆ X equipped with E as set of parameters, ${\tilde{τ}}_{Y}$ is a q-ROFS topology on Y whose q-ROFS open sets are $κ_{B} = κ_{A} \tilde{\cap} {\overset{˘}{Y}}_{E}$ , where κ_A are q-ROFS open sets of ${\tilde{τ}}_{X}$ , κ_B are q-ROFS open sets of ${\tilde{τ}}_{Y}$ and ${\overset{˘}{Y}}_{E}$ is absolute q-ROFSS on Y. Then ${\tilde{τ}}_{Y}$ is taken as the q-ROFS subspace of ${\tilde{τ}}_{X}$ . It can be written as ${\tilde{τ}}_{Y} = {κ_{B} : κ_{B} = κ_{A} \tilde{\cap} {\overset{˘}{Y}}_{E}, κ_{A} \in {\tilde{τ}}_{X}} .$

Example 4.3. Let $X = {ϱ_{1}, ϱ_{2}, ϱ_{3}}$ be the universe and E = {e_i : i = 1, 2, ⋯ , 4} be the collection of decision variables. Take two sub collections A = {e₁, e₂} and B = {e₁, e₂, e₃} of E. Assume that $\begin{matrix} κ_{A}^{(1)} = {(e_{1}, {\frac{ϱ_{1}}{(0.76, 0.51)}}, {\frac{ϱ_{2}}{(0.79, 0.62)}}), \\ (e_{2}, {\frac{ϱ_{2}}{(0.35, 0.83)}}, {\frac{ϱ_{3}}{(0.37, 0.83)}})}, \end{matrix}$ $\begin{matrix} κ_{B}^{(2)} = {(e_{1}, {\frac{ϱ_{1}}{(0.85, 0.33)}}, {\frac{ϱ_{2}}{(0.81, 0.45)}}), \\ (e_{2}, {\frac{ϱ_{2}}{(0.54, 0.63)}}, {\frac{ϱ_{3}}{(0.81, 0.31)}}), \end{matrix}$ $\begin{matrix} (e_{3}, {\frac{ϱ_{1}}{(0.64, 0.63)}}, {\frac{ϱ_{2}}{(0.41, 0.83)}})} . \end{matrix}$ Then, ${\tilde{τ}}_{X} = {φ_{E}, {\overset{˘}{X}}_{E}, κ_{A}^{(1)}, κ_{B}^{(2)}}$ is a q-ROFS topology on X.

Now, absolute q-ROFSS on Y = {ϱ₂, ϱ₃} ⊆ X is ${\overset{˘}{Y}}_{E} = {(e_{i}, {\frac{ϱ_{2}}{(1, 0)}}, {\frac{ϱ_{3}}{(1, 0)}}) : 1 \leq i \leq 4}$ Since $\begin{matrix} {\overset{˘}{Y}}_{E} \tilde{\cap} φ_{E} = φ_{E}, \\ {\overset{˘}{Y}}_{E} \tilde{\cap} κ_{A}^{(1)} = {(e_{1}, {\frac{ϱ_{2}}{(0.79, 0.62)}}), \\ (e_{2}, {\frac{ϱ_{2}}{(0.35, 0.83)}}, {\frac{ϱ_{3}}{(0.37, 0.83)}})} = Ψ_{A}^{(1)}, \end{matrix}$ $\begin{matrix} {\overset{˘}{Y}}_{E} \tilde{\cap} κ_{B}^{(2)} = {(e_{1}, {\frac{ϱ_{2}}{(0.81, 0.45)}}, \\ (e_{2}, {\frac{ϱ_{2}}{(0.54, 0.63)}}, {\frac{ϱ_{3}}{(0.81, 0.31)}}), \\ (e_{3}, {\frac{ϱ_{2}}{(0.41, 0.83)}})} = Ψ_{B}^{(2)}, {\overset{˘}{Y}}_{E} \tilde{\cap} {\overset{˘}{X}}_{E} = {\overset{˘}{Y}}_{E} \end{matrix}$ so ${\tilde{τ}}_{Y} = {φ_{E}, Ψ_{A}^{(1)}, Ψ_{B}^{(2)}, {\overset{˘}{Y}}_{E}}$ is a q-ROFS sub-topology of ${\tilde{τ}}_{X}$ .

Definition 4.4. Let $({\tilde{X}}_{E}, \tilde{τ})$ be a q-ROFS topological space and $κ_{A} \tilde{\subseteq} {\tilde{X}}_{E}$ .

The interior $κ_{A}^{\circ}$ of κ_A is q-ROFS union of all q-ROFS open subsets of κ_A. Then $κ_{A}^{\circ}$ is the greatest q-ROFS open subset of κ_A.

The closure $\bar{κ_{A}}$ of κ_A is the q-ROFS intersection of all q-ROFS closed supersets of κ_A. Further, $\bar{κ_{A}}$ is the smallest q-ROFS closed superset of κ_A.

The boundary or frontier Fr (κ_A) of κ_A is defined as $Fr (κ_{A}) = \bar{κ_{A}} \tilde{\cap} \bar{κ_{A}^{c}}$

The exterior Ext (κ_A) of κ_A is defined as $Ext (κ_{A}) = (κ_{A}^{c})^{\circ}$

These notions can be discussed with the help of this example.

Example 4.5. Consider X = {ϱ₁, ϱ₂} a universe set with A = {e₁, e₂} ∈ E is the gathering of characteristics. If $\begin{matrix} κ_{A}^{(1)} = {(e_{1}, {\frac{ϱ_{1}}{(0.47, 0.53)}}, {\frac{ϱ_{2}}{(0.63, 0.27)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.33, 0.79)}}, {\frac{ϱ_{2}}{(0.46, 0.44)}})} \\ κ_{A}^{(2)} = {(e_{1}, {\frac{ϱ_{1}}{(0.67, 0.29)}}, {\frac{ϱ_{2}}{(0.71, 0.17)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.37, 0.63)}}, {\frac{ϱ_{2}}{(0.58, 0.39)}})} \\ κ_{A}^{(3)} = {(e_{1}, {\frac{ϱ_{1}}{(0.74, 0.21)}}, {\frac{ϱ_{2}}{(0.80, 0.07)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.64, 0.52)}}, {\frac{ϱ_{2}}{(0.60, 0.31)}})} \end{matrix}$ then the collection $\tilde{τ} = {φ_{A}, {\overset{˘}{X}}_{A}, κ_{A}^{(1)}, κ_{A}^{(2)}, κ_{A}^{(3)}}$ of q-ROFS subsets of ${\tilde{X}}_{A}$ is a q-ROFS topology. The elements of $\tilde{τ}$ are now q-ROFS open sets. Its related closed q-ROFSS s are $\begin{matrix} (φ_{A})^{c} = {\overset{˘}{X}}_{A} ({\overset{˘}{X}}_{A})^{c} = φ_{A} \\ (κ_{A}^{(1)})^{c} = {(e_{1}, {\frac{ϱ_{1}}{(0.53, 0.47)}}, {\frac{ϱ_{2}}{(0.27, 0.63)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.79, 0.33)}}, {\frac{ϱ_{2}}{(0.44, 0.46)}})} \\ (κ_{A}^{(2)})^{c} = {(e_{1}, {\frac{ϱ_{1}}{(0.29, 0.67)}}, {\frac{ϱ_{2}}{(0.17, 0.71)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.63, 0.37)}}, {\frac{ϱ_{2}}{(0.39, 0.58)}})} \end{matrix}$ $\begin{matrix} (κ_{A}^{(3)})^{c} = {(e_{1}, {\frac{ϱ_{1}}{(0.21, 0.74)}}, {\frac{ϱ_{2}}{(0.07, 0.80)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.52, 0.64)}}, {\frac{ϱ_{2}}{(0.31, 0.60)}})} \end{matrix}$ Let’s take the q-ROFSS $\begin{matrix} κ_{A} = {(e_{1}, {\frac{ϱ_{1}}{(0.58, 0.32)}}, {\frac{ϱ_{2}}{(0.73, 0.29)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.71, 0.25)}}, {\frac{ϱ_{2}}{(0.78, 0.09)}})} \end{matrix}$ such that $\begin{matrix} (κ_{A})^{c} = {(e_{1}, {\frac{ϱ_{1}}{(0.32, 0.58)}}, {\frac{ϱ_{2}}{(0.29, 0.73)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.25, 0.71)}}, {\frac{ϱ_{2}}{(0.09, 0.78)}})} \end{matrix}$ The q-ROFS interior of κ_A is $\begin{matrix} κ_{A}^{\circ} = φ_{A} \tilde{\cup} κ_{A}^{(1)} \tilde{\cup} κ_{A}^{(3)} = {(e_{1}, {\frac{ϱ_{1}}{(0.47, 0.53)}}, \\ {\frac{ϱ_{2}}{(0.80, 0.07)}}), (e_{2}, {\frac{ϱ_{1}}{(0.73, 0.39)}}, \\ {\frac{ϱ_{2}}{(0.60, 0.31)}})} \end{matrix}$ The q-ROFS closure of κ_A is $\bar{κ_{A}} = {\overset{˘}{X}}_{E}$ Now, $\begin{matrix} \bar{κ_{A}^{c}} & = & {\overset{˘}{X}}_{A} \tilde{\cap} (κ_{A}^{(1)})^{c} \tilde{\cap} (κ_{A}^{(3)})^{c} \\ = & {(e_{1}, {\frac{ϱ_{1}}{(0.53, 0.47)}}, {\frac{ϱ_{2}}{(0.07, 0.80)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.39, 0.73)}}, {\frac{ϱ_{2}}{(0.31, 0.60)}})} \end{matrix}$ Hence q-ROFS frontier of κ_A is $\begin{matrix} Fr (κ_{A}) & = & \bar{κ_{A}} \tilde{\cap} \bar{κ_{A}^{c}} \\ = & {(e_{1}, {\frac{ϱ_{1}}{(0.32, 0.58)}}, {\frac{ϱ_{2}}{(0.29, 0.73)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.25, 0.71)}}, {\frac{ϱ_{2}}{(0.09, 0.78)}})} \end{matrix}$ Finally, the q-ROFS exterior of κ_A is $\begin{matrix} Ext (κ_{A}) & = & (κ_{A}^{c})^{\circ} \\ = & φ_{A} \end{matrix}$

Theorem 4.6. Let $({\tilde{X}}_{E}, \tilde{τ})$ be a q-ROFS top. space and $κ_{A} \tilde{\subseteq} {\tilde{X}}_{E}$ , then

$(κ_{A}^{\circ})^{c} = \bar{(κ_{A}^{c})}$ , and

$(\bar{κ_{A}})^{c} = (κ_{A}^{c})^{\circ}$ .

Theorem 4.7. Let $({\tilde{X}}_{E}, \tilde{τ})$ be a q-ROFS top. space and $κ_{A} \tilde{\subseteq} {\tilde{X}}_{E}$ , then $Fr (κ_{A}) = Fr (κ_{A}^{c})$ .

Proof. By using the definition, $\begin{matrix} Fr (κ_{A}) & = & \bar{(κ_{A})} \tilde{\cap} \bar{(κ_{A}^{c})} \\ = & \bar{(κ_{A}^{c})} \tilde{\cap} \bar{(κ_{A})} \\ = & \bar{(κ_{A}^{c})} \tilde{\cap} \bar{[(κ_{A}^{c})]^{c}} \\ = & Fr (κ_{A}^{c}) \end{matrix}$ □

Definition 4.8. We are taking X a crisp set having E as the gathering of attributes then ${\tilde{τ}}_{discrete} =$ q - ROFS (X, E) and ${\tilde{τ}}_{indiscrete} = {φ_{E}, {\overset{˘}{X}}_{E}}$ are termed as discrete and indiscrete q-ROFS topologies conformably. Concentrate that ${\tilde{τ}}_{discrete}$ is the biggest whereas ${\tilde{τ}}_{indiscrete}$ is the smallest q-ROFS topology on ${\tilde{X}}_{E}$ .

Remark. The intersection of two q-ROFS topology is always a q-ROFS topology but their union may not be q-ROFS topology. The following example illustrate this idea.

Example 4.9. Let X = {ϱ₁, ϱ₂} be the universal set and E = {e_i : i = 1 . . . .4} be the assembly of characteristics. Let A = {e₁, e₂} , B = {e₃, e₄} ⊆ E in correspondence to q-ROFSSs $\begin{matrix} κ_{A} = {(e_{1}, {\frac{ϱ_{1}}{(0.41, 0.71)}}, {\frac{ϱ_{2}}{(0.76, 0.29)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.76, 0.35)}}, {\frac{ϱ_{2}}{(0.58, 0.43)}})} \\ κ_{B} = {(e_{3}, {\frac{ϱ_{1}}{(0.83, 0.37)}}, {\frac{ϱ_{2}}{(0.16, 0.63)}}), \\ (e_{4}, {\frac{ϱ_{1}}{(0.59, 0.32)}}, {\frac{ϱ_{2}}{(0.79, 0.45)}})} \end{matrix}$ then ${\tilde{τ}}_{1} = {φ_{E}, κ_{A}, {\overset{˘}{X}}_{E}}$ and ${\tilde{τ}}_{2} = {φ_{E}, κ_{B}, {\overset{˘}{X}}_{E}}$ are q-ROFS topologies on X but ${\tilde{τ}}_{1} \tilde{\cup} {\tilde{τ}}_{2} = {φ_{E}, {\overset{˘}{X}}_{E}, κ_{A}, κ_{B}}$ is not q-ROFS topology.

Definition 4.10. If we consider ${\tilde{τ}}_{1}$ and ${\tilde{τ}}_{2}$ be q-ROFS topological spaces and ${\tilde{τ}}_{1} \tilde{\subseteq} {\tilde{τ}}_{2}$ then ${\tilde{τ}}_{1}$ is known as weaker or coarser than ${\tilde{τ}}_{2}$ and ${\tilde{τ}}_{2}$ is said to as stronger or finer than ${\tilde{τ}}_{1}$ . In both cases ${\tilde{τ}}_{1}$ and ${\tilde{τ}}_{2}$ are said to be comparable.

Example 4.11. We consider X = {ϱ₁, ϱ₂} the universal set and E = {e₁, e₂, e₃} be the set of attributes with A = {e₁, e₂} ⊂ E. Consider q-ROFSS s $\begin{matrix} κ_{A}^{(1)} = {(e_{1}, {\frac{ϱ_{1}}{(0.41, 0.67)}}, {\frac{ϱ_{2}}{(0.61, 0.47)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.51, 0.87)}}, {\frac{ϱ_{2}}{(0.61, 0.77)}})} \\ κ_{A}^{(2)} = {(e_{1}, {\frac{ϱ_{1}}{(0.71, 0.31)}}, {\frac{ϱ_{2}}{(0.72, 0.37)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.67, 0.66)}}, {\frac{ϱ_{2}}{(0.72, 0.66)}})} \\ κ_{A}^{(3)} = {(e_{1}, {\frac{ϱ_{1}}{(0.79, 0.27)}}, {\frac{ϱ_{2}}{(0.84, 0.25)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.86, 0.53)}}, {\frac{ϱ_{2}}{(0.84, 0.55)}})} \end{matrix}$

It is clear now that ${\tilde{τ}}_{1} = {φ_{E}, κ_{A}^{(1)}, {\overset{˘}{X}}_{E}}$ and ${\tilde{τ}}_{2} = {φ_{E}, κ_{A}^{(1)}, κ_{A}^{(2)}, κ_{A}^{(3)}, {\overset{˘}{X}}_{E}}$ are two q-ROFS topologies. Since ${\tilde{τ}}_{1} \tilde{\subset} {\tilde{τ}}_{2}$ so ${\tilde{τ}}_{1}$ is weaker than ${\tilde{τ}}_{2}$ .

Remark. It is evident that"law of excluded middle" and "law of contradiction" existing in crisp set theory are not applicable in IFSSs and PFSSs. This property may be verified in q-ROFSS theory due to this few results in q-ROFS topology differ from classical topology. Here is some evidence in the favor of our wording.

Example 4.12. Let X = {ϱ₁, ϱ₂, ϱ₃} be a universal set and E = {e_i : i = 1, 2, 3, 4} be a class of characteristics with A = {e₁, e₂} ⊆ E. Now consider $\begin{matrix} κ_{A} = {(e_{1}, {\frac{ϱ_{1}}{(0.51, 0.27)}}, {\frac{ϱ_{2}}{(0.61, 0.47)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.51, 0.27)}}, {\frac{ϱ_{2}}{(0.61, 0.47)}})} \end{matrix}$ Obviously $\tilde{τ} = {φ_{E}, {\overset{˘}{X}}_{E}, κ_{A}, κ_{A}^{c}}$ does not form a q-ROFS topology on X as neither $κ_{A} \tilde{\cap} κ_{A}^{c} \tilde{\in} \tilde{τ}$ nor $κ_{A} \tilde{\cup} κ_{A}^{c} \tilde{\in} \tilde{τ}$ .

Definition 4.13. Let $({\tilde{X}}_{E}, \tilde{τ})$ be a q-ROFS topological space. Then $β \tilde{\subseteq} \tilde{τ}$ is a -basis for $\tilde{τ}$ if for each $κ_{A} \tilde{\in} \tilde{τ}$ , there exists $B \in β$ such that $κ_{A} = \tilde{\cup} B$ .

Example 4.14. We consider X = {ϱ₁, ϱ₂} the universal set and E = {e₁, e₂} be characteristics’ set with A ⊆ E, by taking q-ROFSS s $\begin{matrix} κ_{A}^{(1)} = {(e_{1}, {\frac{ϱ_{1}}{(0.26, 0.55)}}, {\frac{ϱ_{2}}{(0.17, 0.72)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.37, 0.86)}}, {\frac{ϱ_{2}}{(0.37, 0.70)}})} \\ κ_{A}^{(2)} = {(e_{1}, {\frac{ϱ_{1}}{(0.48, 0.34)}}, {\frac{ϱ_{2}}{(0.17, 0.72)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.49, 0.67)}}, {\frac{ϱ_{2}}{(0.50, 0.64)}})} \\ κ_{A}^{(3)} = {(e_{1}, {\frac{ϱ_{1}}{(0.63, 0.24)}}, {\frac{ϱ_{2}}{(0.54, 0.60)}}), \\ (e_{2}, {\frac{ϱ_{1}}{(0.69, 0.60)}}, {\frac{ϱ_{2}}{(0.54, 0.60)}})} \end{matrix}$ This set $β = {κ_{A}^{(1)}, κ_{A}^{(2)}, κ_{A}^{(3)}}$ provide a q-ROFS basis for the q-ROFS topology $\tilde{τ} = {φ_{E}, {\overset{˘}{X}}_{E}, κ_{A}^{(1)}, κ_{A}^{(2)}, κ_{A}^{(3)}}$ .

Theorem 4.15. If β is a q-ROFS basis for $\tilde{τ}$ then, for each e ∈ E, β_e = {κ (e) : κ_E ∈ β} serves as a q-ROFS basis for the q-ROFS topology $\tilde{τ} (e) = {κ (e) : κ_{E} \in \tilde{τ}}$ .

Proof. Let κ⁽¹⁾ (e) ∈ τ (e) for some $κ_{E}^{(1)} \in τ$ . Since β is a q-ROFS basis for $\tilde{τ}$ so,by def. there is β′ ∈ β such that $κ_{E}^{(1)} = \cup β^{'}$ and hence $κ^{(1)} (e) = \cup β_{e}^{'}$ , where $β_{e}^{'} = {κ (e) : κ_{E} \in β^{'}} \subseteq β_{e}$ . □

Theorem 4.16. Let $({\tilde{X}}_{E}, \tilde{τ})$ be a q-ROFS topological space. A collection $β = {κ_{B_{α}} | α \in Ω} \tilde{\subseteq} \tilde{τ}$ is a q-ROFS basis for $\tilde{τ}$ if and only if for any q-ROFSS κ_U and any q-ROFS point ς_κ ∈ κ_U, there is a κ_{B
_α} ∈ β such that $ς_{κ} \in κ_{B_{α}} \tilde{\subseteq} κ_{U}$ .

Proof. Let β be a q-ROFS basis for $\tilde{τ}$ . Now for every q-ROFS open set κ_U of $\tilde{τ}$ , there are q-ROFS sets κ_{B
_γ}, $γ \in Ω^{'} \tilde{\subseteq} Ω$ such that $κ_{U} = \tilde{\cup} κ_{B_{γ}}$ . Therefor, for any $P_{e}^{x} \in κ_{U}$ , we have a B_γ such that $ς_{κ} \in κ_{B_{γ}} \tilde{\subseteq} κ_{U}$ On the other hand, we consider that for every q-ROFS open set κ_U and ς_κ ∈ κ_U, we have κ_B in β such that $ς_{κ} \in κ_{B} \tilde{\subseteq} κ_{U}$ . Then $κ_{U} = {\tilde{\cup}}_{ς_{κ} \in κ_{U}} {x} \tilde{\subseteq} {\tilde{\cup}}_{ς_{κ} \in κ_{U}} {κ_{B}} \tilde{\subseteq} κ_{U}$ so that $κ_{U} = {\tilde{\cup}}_{ς_{κ} \in κ_{U}} {κ_{B}}$ which is union of q-ROFSSs in β. □

4.1 q-ROFS separation axioms

Definition 4.17. A q-ROFS topological space $({\tilde{X}}_{E}, \tilde{τ})$ is said to be a q-ROFS T₀-space if for each pair of different q-ROFS points $ς_{κ}^{(1)}, ς_{κ}^{(2)}$ there is an q-ROFS open set κ_A which contain only one of the q-ROFS points.

Example 4.18. Each discrete q-ROFS topological space is a q-ROFS T₀-space because there is an open set ${ς_{κ}^{(1)}}$ which have $ς_{κ}^{(1)}$ but not $ς_{κ}^{(2)}$ .

Remark. The property of being a q-ROFS T₀-space is a q-ROFS topological property i.e. every subspace of a q-ROFS T₀-space is q-ROFS T₀-space.

Definition 4.19. A q-ROFS topological space is said to be a q-ROFS T₁-space if for any two q-ROFS points $ς_{κ}^{(1)}, ς_{κ}^{(2)}$ of ${\tilde{X}}_{E}$ , there exists two q-ROFS open sets κ_C and κ_D such that $ς_{κ}^{(1)} \tilde{\in} κ_{C}$ , $ς_{κ}^{(2)} \tilde{\notin} κ_{C}$ and $ς_{κ}^{(1)} \tilde{\notin} κ_{D}$ , $ς_{κ}^{(2)} \tilde{\in} κ_{D}$ .

Example 4.20. Each discrete q-ROFS topological space is a q-ROFS T₁-space. If $ς_{κ}^{(1)}$ and $ς_{κ}^{(2)}$ are 2 different q-ROFS points in ${\tilde{X}}_{E}$ , then we have q-ROFS open sets ${ς_{κ}^{(1)}}$ and ${ς_{κ}^{(2)}}$ such that $ς_{κ}^{(1)} \in {ς_{κ}^{(1)}}$ whereas $ς_{κ}^{(2)} \notin {ς_{κ}^{(1)}}$ .

Theorem 4.21. The following statements for a q-ROFS topological space $({\tilde{X}}_{E}, \tilde{τ})$ are equivalent.

$({\tilde{X}}_{E}, \tilde{τ})$ is a q-ROFS T₁-space.

Each q-ROFS singleton subset of ${\tilde{X}}_{E}$ is q-ROFS closed.

Proof. its very simple to do. □

Remark. The property of being a q-ROFS T₁-space is a q-ROFS topological property i.e. every subspace of a q-ROFS T₁-space is q-ROFS T₁-space.

Definition 4.22. A q-ROFS space $({\tilde{X}}_{E}, \tilde{τ})$ is is said to be q-ROFS T₂-space or q-ROFS Hausdorff space if for any two different q-ROFS points $ς_{κ}^{(1)}$ and $ς_{κ}^{(2)}$ of ${\tilde{X}}_{E}$ , there exist two q-ROFS open sets κ_C and κ_D s.t. $ς_{κ}^{(1)} \tilde{\in} κ_{C}$ , $ς_{κ}^{(2)} \tilde{\in} κ_{D}$ and $κ_{C} \tilde{\cap} κ_{D} = φ_{E}$ .

Example 4.23. We take a discrete q-ROFS topological space $({\tilde{X}}_{E}, \tilde{τ})$ . If $ς_{κ}^{(1)}$ and $ς_{κ}^{(2)}$ be two different q-ROFS points in ${\tilde{X}}_{E}$ , so obviously ${ς_{κ}^{(1)}}$ and ${ς_{κ}^{(2)}}$ are disjoint q-ROFS open sets, s. t. $ς_{κ}^{(1)} \in {ς_{κ}^{(1)}}$ and $ς_{κ}^{(2)} \in {ς_{κ}^{(2)}}$ . Hence $({\tilde{X}}_{E}, \tilde{τ})$ is a q-ROFS T₂-space.

Theorem 4.24. A q-ROFS topological space $({\tilde{X}}_{E}, \tilde{τ})$ is a q-ROFS T₂-space iff for two different q-ROFS points $ς_{κ}^{(1)}$ and $ς_{κ}^{(2)}$ , there we have q-ROFS closed sets κ_{C
₁} and κ_{C
₂} s. t. $ς_{κ}^{(1)} \tilde{\in} κ_{C_{1}}$ , $ς_{κ}^{(2)} \tilde{\notin} κ_{C_{1}}$ , $ς_{κ}^{(1)} \tilde{\notin} κ_{C_{2}}$ , $ς_{κ}^{(2)} \tilde{\in} κ_{C_{2}}$ and $κ_{C_{1}} \tilde{\cup} κ_{C_{2}} = {\overset{˘}{X}}_{E}$ .

Proof. We consider that $({\tilde{X}}_{E}, \tilde{τ})$ be a q-ROFS T₂-space and also $ς_{κ}^{(1)}$ and $ς_{κ}^{(2)}$ be two different q-ROFS points of ${\tilde{X}}_{E}$ . Now, with the help of definition, there will exist two q-ROFS open sets κ_{U
₁} and κ_{U
₂} such that $ς_{κ}^{(1)} \tilde{\in} κ_{U_{1}}$ , $ς_{κ}^{(2)} \tilde{\notin} κ_{U_{1}}$ , $ς_{κ}^{(1)} \tilde{\notin} κ_{U_{2}}$ , $ς_{κ}^{(2)} \tilde{\in} κ_{U_{2}}$ and $κ_{U_{1}} \tilde{\cap} κ_{U_{2}} = φ_{E}$ . But then, ${κ_{U_{1}}}^{c} \tilde{\cup} {κ_{U_{2}}}^{c} = {\overset{˘}{X}}_{E}$ and $ς_{κ}^{(1)} \tilde{\notin} {κ_{U_{1}}}^{c} = κ_{C_{2}}$ , $ς_{κ}^{(2)} \tilde{\in} {κ_{U_{1}}}^{c} = κ_{C_{2}}$ . $ς_{κ}^{(1)} \tilde{\in} {κ_{U_{2}}}^{c} = κ_{C_{1}}$ , $ς_{κ}^{(2)} \tilde{\notin} {κ_{U_{2}}}^{c} = κ_{C_{1}}$ .

On the other hand, we consider that for any two different q-ROFS points $ς_{κ}^{(1)}, ς_{κ}^{(2)} \tilde{\in} {\tilde{X}}_{E}$ , we have q-ROFS closed sets κ_{C
₁} and κ_{C
₂} s. t. $ς_{κ}^{(1)} \tilde{\in} κ_{C_{1}}$ , $ς_{κ}^{(2)} \tilde{\notin} κ_{C_{1}}$ , $ς_{κ}^{(1)} \tilde{\notin} κ_{C_{2}}$ , $ς_{κ}^{(2)} \tilde{\in} κ_{C_{2}}$ and $κ_{C_{1}} \tilde{\cup} κ_{C_{2}} = {\overset{˘}{X}}_{E}$ . So {κ_{C
₁}} ^c and {κ_{C
₂}} ^c are q-ROFS open sets such that $ς_{κ}^{(1)} \tilde{\notin} {κ_{C_{1}}}^{c}$ , $ς_{κ}^{(2)} \tilde{\in} {κ_{C_{1}}}^{c}$ , $ς_{κ}^{(1)} \tilde{\in} {κ_{C_{2}}}^{c}$ , $ς_{κ}^{(2)} \tilde{\notin} {κ_{C_{2}}}^{c}$ and ${κ_{C_{1}}}^{c} \tilde{\cap} {κ_{C_{2}}}^{c} = {{\overset{˘}{X}}_{E}}^{c} = φ_{E}$ . Thus, ${\tilde{X}}_{E}$ is a q-ROFS T₂-space. □

Remark. A q-ROFS T₂-space satisfies the hereditary property i.e. every subspace of a q-ROFS T₂-space is q-ROFS T₂-space.

Definition 4.25. A q-ROFS topological space $({\tilde{X}}_{E}, \tilde{τ})$ is said to be a q-ROFS regular space if for any q-ROFS closed set κ_A and any q-ROFS point ς_κ ∉ κ_A, ∃ q-ROFS open sets κ_C and κ_D such that $ς_{κ} \tilde{\in} κ_{C}, κ_{A} \tilde{\subseteq} κ_{D}$ and $κ_{C} \tilde{\cap} κ_{D} = φ_{E}$ , where φ_E is the null q-rung orthopair fuzzy soft set.

Definition 4.26. A q-ROFS topological space $({\tilde{X}}_{E}, \tilde{τ})$ is said to be a q-ROFS T₃-space if it is both q-ROFS regular and q-ROFS T₁-space.

Definition 4.27. A q-ROFS topological space $({\tilde{X}}_{E}, \tilde{τ})$ is known as a q-ROFS normal space if there are two q-ROFS closed disjoint subsets κ_A and κ_B of $({\tilde{X}}_{E}, \tilde{τ})$ , ∃ two q-ROFS open sets κ_C and κ_D such that $κ_{A} \tilde{\subseteq} κ_{C}$ , $κ_{B} \tilde{\subseteq} κ_{D}$ and $κ_{C} \tilde{\cap} κ_{D} = φ_{E}$ . A q-ROFS normal T₁-space is said to be a q-ROFS T₄-space.

Theorem 4.28. Each q-ROFS T₄-space is q-ROFS regular, that is every q-ROFS normal T₁-space is q-ROFS regular.

Proof. We consider that $({\tilde{X}}_{E}, \tilde{τ})$ be a q-ROFS T₄-topological space. We say that ς_κ be a q-ROFS point in ${\tilde{X}}_{E}$ . Now using Theorem 4.21 {ς_κ} is a closed q-ROFSS in $({\tilde{X}}_{E}, \tilde{τ})$ . We suppose that κ_A is a closed q-ROFSS does not contain ς_κ. As $({\tilde{X}}_{E}$ is q-ROFS normal, so we have an open q-ROFSS namely κ_U, κ_V s.t. ${ς_{κ}} \tilde{\subseteq} κ_{U}, κ_{A} \tilde{\subseteq} κ_{V}, and κ_{U} \cap κ_{V} = κ_{φ}$ so that $ς_{κ} \tilde{\in} κ_{U}, κ_{A} \tilde{\subseteq} κ_{V}, and κ_{U} \cap κ_{V} = κ_{φ}$ Hence $({\tilde{X}}_{E}, \tilde{τ})$ is a q-ROFS regular. □

5 MCGDM with choice value method under q-rung orthopair fuzzy soft sets

This section gives multi-criteria group decision making (MCGDM) to rank the alternative from high preference to low preference. In MCGDM the alternatives are evaluated by a group of decision makers (DMs) in the presence of multiple criterion to choose the best alternative from a set of feasible alternatives under a particular situation. Although there exist numerous aggregation methods, in the current context we propose weighted choice values method under q-ROFSSs in order to calculate the final decision because they are a keystone in well-established mechanisms from the soft sets and fuzzy sets literature. Meanwhile, we now develop respective algorithm for problems that are characterized by q-rung orthopair fuzzy soft sets.

Algorithm 1. Choice Value Method Step 1: Input X = {ϱ_i : i = 1, 2, ⋯ , n} as a set of objects and E = {e_i : i = 1, 2, ⋯ , k} as a set of attributes. Then compute the q-rung orthopair fuzzy soft sets.

Step 2: Compute the corresponding q-ROFS matrix.

Step 3: Assign weights to each attribute.

Step 4: Compute the matrix of choice values using $C = \frac{1}{Σ W (e_{i})} (κ_{E} \times W^{t})$ .

Step 5: Compute the score value for each ϱ_i by using score function S = μ^q - ν^q.

Step 6: The ϱ_i for which s (ϱ_i) is maximum is the desired alternative.

Step 7: In case of a tie, compute the values of accuracy function a (ϱ_i).

Step 8: The ϱ_i for which a (ϱ_i) is maximum is the requisite choice.

5.1 Selection of school education system

The selection of the best school education system for children in a newly established housing society is always a difficult task. Since each individuals has different priorities when searching for a school. The teaching staff of the school is something that is very important to gain quality education. There are a number of factors to take into consideration for selecting the best school education system, whether we are looking for a great building, learning environment and communication skills, effective teaching methods, economical fee structure and carrier counseling.

Example 5.1. Let X = {ϱ_i : i = 1, 2, ⋯ , 6} be the set of well known school education systems existing in the city. In order to get the best school education system in the society, the board of housing society consider E = {e_i : i = 1, 2, ⋯ , 7} as set of attributes, where $\begin{matrix} e_{1} = effective teaching methods, \\ e_{2} = best teaching staff and carrier counseling, \\ e_{3} = economical fee structure, \\ e_{4} = learning environment and communication skills, \\ e_{5} = latest curriculum and latest assessment standards, \\ e_{6} = monitoring of teaching and learning, \\ e_{7} = great building and big grounds . \end{matrix}$

By viewing the track record of the education school systems in the city, the board of housing society prepares q-rung orthopair fuzzy soft information as given in Table 2.

Table 2
q-ROFSS information

κ _E e ₁ e ₂ e ₃ e ₄ e ₅ e ₆ e ₇

ϱ ₁ (0.73,0.26) (0.80,0.34) (0.83,0.22) (0.65,0.11) (0.82,0.30) (0.82,0.34) (0.78,0.28)

ϱ ₂ (0.70,0.30) (0.73,0.47) (0.77,0.31) (0.61,0.28) (0.75,0.37) (0.64,0.45) (0.71,0.33)

ϱ ₃ (0.65,0.38) (0.59,0.54) (0,65,0.40) (0.59,0.33) (0.67,0.44) (0.61,0.59) (0.67,0.44)

ϱ ₄ (0.61,0.46) (0.48,0.60) (0.60,0.43) (0.49,0.41) (0.60,0.55) (0.59,0.62) (0.62,0.55)

ϱ ₅ (0.54,0.58) (0.39,0.67) (0.53,0.58) (0.45,0.56) (0.58,0.66) (0.50,0.68) (0.60,0.59)

ϱ ₆ (0.46,0.61) (0.33,0.71) (0.22,0.69) (0.38,0.74) (0.52,0.72) (0.46,0.75) (0.53,0.71)

κ _E	e ₁	e ₂	e ₃	e ₄	e ₅	e ₆	e ₇
ϱ ₁	(0.73,0.26)	(0.80,0.34)	(0.83,0.22)	(0.65,0.11)	(0.82,0.30)	(0.82,0.34)	(0.78,0.28)
ϱ ₂	(0.70,0.30)	(0.73,0.47)	(0.77,0.31)	(0.61,0.28)	(0.75,0.37)	(0.64,0.45)	(0.71,0.33)
ϱ ₃	(0.65,0.38)	(0.59,0.54)	(0,65,0.40)	(0.59,0.33)	(0.67,0.44)	(0.61,0.59)	(0.67,0.44)
ϱ ₄	(0.61,0.46)	(0.48,0.60)	(0.60,0.43)	(0.49,0.41)	(0.60,0.55)	(0.59,0.62)	(0.62,0.55)
ϱ ₅	(0.54,0.58)	(0.39,0.67)	(0.53,0.58)	(0.45,0.56)	(0.58,0.66)	(0.50,0.68)	(0.60,0.59)
ϱ ₆	(0.46,0.61)	(0.33,0.71)	(0.22,0.69)	(0.38,0.74)	(0.52,0.72)	(0.46,0.75)	(0.53,0.71)

The corresponding q-ROFS matrix is $\begin{matrix} κ_{E} = (\begin{matrix} (0.73, 0.26) & (0.80, 0.34) & (0.83, 0.22) & (0.65, 0.11) & (0.82, 0.30) & (0.82, 0.34) & (0.78, 0.28) \\ (0.70, 0.30) & (0.73, 0.47) & (0.77, 0.31) & (0.61, 0.28) & (0.75, 0.37) & (0.64, 0.45) & (0.71, 0.33) \\ (0.65, 0.38) & (0.59, 0.54) & (0, 65, 0.40) & (0.59, 0.33) & (0.67, 0.44) & (0.61, 0.59) & (0.67, 0.44) \\ (0.61, 0.46) & (0.48, 0.60) & (0.60, 0.43) & (0.49, 0.41) & (0.60, 0.55) & (0.59, 0.62) & (0.62, 0.55) \\ (0.54, 0.58) & (0.39, 0.67) & (0.53, 0.58) & (0.45, 0.56) & (0.58, 0.66) & (0.50, 0.68) & (0.60, 0.59) \\ (0.46, 0.61) & (0.33, 0.71) & (0.22, 0.69) & (0.38, 0.74) & (0.52, 0.72) & (0.46, 0.75) & (0.53, 0.71) \end{matrix}) \end{matrix}$ Next, the weights are assigned to each parameter as follows. $W (e_{1}) = 0.33, W (e_{2}) = 0.45, W (e_{3}) = 0.92, W (e_{4}) = 0.65, W (e_{5}) = 0.71, W (e_{6}) = 0.50, W (e_{7}) = 0.60$ The sum of the weights is ΣW (e_i) =4.16. Then the weight vector is given by, $\begin{matrix} W & = & (\begin{matrix} 0.33 & 0.45 & 0.92 & 0.65 & 0.71 & 0.50 & 0.60 \end{matrix}) \end{matrix}$ Thus, the q-ROFS matrix for choice values is

$\begin{matrix} C & = & \frac{1}{Σ W (e_{i})} (κ_{E} \times W^{t}) \\ = & \frac{1}{4.16} (\begin{matrix} (0.73, 0.26) & (0.80, 0.34) & (0.83, 0.22) & (0.65, 0.11) & (0.82, 0.30) & (0.82, 0.34) & (0.78, 0.28) \\ (0.70, 0.30) & (0.73, 0.47) & (0.77, 0.31) & (0.61, 0.28) & (0.75, 0.37) & (0.64, 0.45) & (0.71, 0.33) \\ (0.65, 0.38) & (0.59, 0.54) & (0, 65, 0.40) & (0.59, 0.33) & (0.67, 0.44) & (0.61, 0.59) & (0.67, 0.44) \\ (0.61, 0.46) & (0.48, 0.60) & (0.60, 0.43) & (0.49, 0.41) & (0.60, 0.55) & (0.59, 0.62) & (0.62, 0.55) \\ (0.54, 0.58) & (0.39, 0.67) & (0.53, 0.58) & (0.45, 0.56) & (0.58, 0.66) & (0.50, 0.68) & (0.60, 0.59) \\ (0.46, 0.61) & (0.33, 0.71) & (0.22, 0.69) & (0.38, 0.74) & (0.52, 0.72) & (0.46, 0.75) & (0.53, 0.71) \end{matrix}) (\begin{matrix} 0.33 \\ 0.45 \\ 0.92 \\ 0.65 \\ 0.71 \\ 0.50 \\ 0.60 \end{matrix}) \\ = & (\begin{matrix} (0.7806, 0.4811) \\ (0.9437, 0.5789) \\ (0.8693, 0.6669) \\ (0.8050, 0.7377) \\ (0.7495, 0.8460) \\ (0.6261, 0.0433) \end{matrix}) \end{matrix}$ The score values with q = 3, q = 4 and q = 9 are demonstrated in Table 3.

Table 3 demonstrates the final ranking as follows $ϱ_{2} ≻ ϱ_{3} ≻ ϱ_{1} ≻ ϱ_{4} ≻ ϱ_{5} ≻ ϱ_{6} .$ From the final ranking it is clear that the first preference is ϱ₂. The rankings of schools for different values of q chosen is depicted in the Fig. 2.

Fig. 2

Ranking of alternatives for different values of q.

Table 3

Score values for alternatives with q = 3, q = 4 and q = 9

X	S = μ³ - ν³	S = μ⁴ - ν⁴	S = μ⁹ - ν⁹	Ranking
ϱ ₁	0.3643	0.3177	0.1062	3
ϱ ₂	0.6456	0.6798	0.5846	1
ϱ ₃	0.3663	0.3732	0.2574	2
ϱ ₄	0.2458	0.1238	0.0773	4
ϱ ₅	-0.1845	-0.1967	-0.1474	5
ϱ ₆	-0.5939	-0.6381	-0.5766	6

The proposed Algorithm 1 is used in the environment of q-ROFS information for the selection of school education systems and the results are compared as indicated in the Table 6 listing the results of final ranking of seven alternatives.

Table 4

Comparison analysis of final ranking with existing methods using Algorithm 1

Methods	The optimal alternative
Algorithm 1 (Proposed)	ϱ ₂
Algorithm (Çağman et al.)	ϱ ₂
Guleria and Bajaj [19]	ϱ ₂
Algorithm (Naeem et al. [25])	ϱ ₂
Algorithm (Peng et al. [35])	ϱ ₂
Algorithm (Riaz et al. [41])	ϱ ₂

6 MCGDM with TOPSIS approach under q-ROFS topology

q-ROFS topology is a new approach towards uncertainty in this approach a set of objects, a set of attributes and a team of decision makers (DMs) are considered. The opinion of each DM is expressed by a q-ROFS open set. Additionally, to find the unanimous decision, the q-ROFS topology is constructed with the help of these q-ROFS open sets which describe the opinion of DMs. This is a robust MCGDM approach in addition to other existing MCGDM approaches. The linguistic terms for deciding alternatives are given in Table 5.

Table 5
Linguistic terms for deciding alternatives

Linguistic terms Fuzzy weights

Highly valuable (HV) 0.90

Much valuable (MV) 0.75

Valuable (V) 0.55

Less valuable (LV) 0.30

Least valuable (LtV) 0.15

Linguistic terms	Fuzzy weights
Highly valuable (HV)	0.90
Much valuable (MV)	0.75
Valuable (V)	0.55
Less valuable (LV)	0.30
Least valuable (LtV)	0.15

Algorithm 2. Extension of TOPSIS to q-ROFS Topology

Step 1: Identifying the problem. Assume that $D = {D_{i} : i \in NI}$ is the set of decision makers, $V = {{\ddot{v}}_{i} : i \in NI}$ is the objects and P = {p_i : i ∈ NI} is the set of parameters/criterion.

Step 2: By selecting the linguistic terms from Table 5, constructing weighted parameter matrix $P$ as $P = [w_{ij}]_{n \times m} = (\begin{matrix} w_{11} & w_{12} & \dots & w_{1 m} \\ w_{21} & w_{22} & \dots & w_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ w_{i 1} & w_{i 2} & \dots & w_{im} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ w_{n 1} & w_{n 2} & \dots & w_{nm} \end{matrix})$

where w_ij is the weight assigned by the expert $D_{i}$ to the alternative $P_{j}$ by considering linguistic variables as given (for example) in Table 5.

Step 3: Construct normalized weighted matrix $\hat{N} = [{\hat{n}}_{ij}]_{n \times m} = (\begin{matrix} {\hat{n}}_{11} & {\hat{n}}_{12} & \dots & {\hat{n}}_{1 m} \\ {\hat{n}}_{21} & {\hat{n}}_{22} & \dots & {\hat{n}}_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\hat{n}}_{i 1} & {\hat{n}}_{i 2} & \dots & {\hat{n}}_{im} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\hat{n}}_{n 1} & {\hat{n}}_{n 2} & \dots & {\hat{n}}_{nm} \end{matrix})$

where ${\hat{n}}_{ij} = \frac{w_{ij}}{\sqrt{Σ_{i = 1}^{n} w_{ij}^{2}}}$ and obtaining the weighted vector $W = (w_{1}, w_{2}, \dots, w_{m})$ , where $w_{i} = \frac{w_{i}}{Σ_{α = 1}^{m} w_{i α}}$ and $w_{j} = \frac{Σ_{i = 1}^{n} {\hat{n}}_{ij}}{n}$ . Notice that $Σ_{α = 1}^{m} w_{i α}$ represents sum of weights of i^th row in the matrix $P$ cited at step 2 above.

Step 4: Construct q-ROFS decision matrix $D_{i} = [v_{jk}^{i}]_{l \times m} = (\begin{matrix} v_{11}^{i} & v_{12}^{i} & \dots & v_{1 m}^{i} \\ v_{21}^{i} & v_{22}^{i} & \dots & v_{2 m}^{i} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ v_{j 1}^{i} & v_{j 2}^{i} & \dots & v_{jm}^{i} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ v_{l 1}^{i} & v_{l 2}^{i} & \dots & v_{lm}^{i} \end{matrix})$

where $v_{jk}^{i}$ is a q-ROFS element, for i^th decision maker so that D_i makes q-ROFS topolgy for each i. Then obtain the aggregating matrix $A = \frac{D_{1} + D_{2} + \dots + D_{n}}{n} = [{\dot{v}}_{jk}]_{l \times m}$ Step 5: Obtain the weighted q-ROFS decision matrix $B = [{\ddot{v}}_{jk}]_{l \times m} = (\begin{matrix} {\ddot{v}}_{11} & {\ddot{v}}_{12} & \dots & {\ddot{v}}_{1 m} \\ {\ddot{v}}_{21} & {\ddot{v}}_{22} & \dots & {\ddot{v}}_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\ddot{v}}_{j 1} & {\ddot{v}}_{j 2} & \dots & {\ddot{v}}_{jm} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\ddot{v}}_{l 1} & {\ddot{v}}_{l 2} & \dots & {\ddot{v}}_{lm} \end{matrix})$

where ${\ddot{v}}_{jk} = w_{k} \times {\dot{v}}_{jk}$ .

Step 6: In general, two sorts of criteria are thought of in daily life decision making which are named benefit (D₁) and the cost (D₂). In order to obtain q-ROFS valued positive ideal solution (q-ROFSV-PIS) and q-ROFS valued negative ideal solution (q-ROFSV-NIS), we employ in order $\begin{matrix} q - ROFSV - PIS & = & {{\ddot{v}}_{1}^{+}, {\ddot{v}}_{2}^{+}, \dots, {\ddot{v}}_{m}^{+}} \\ = & {(\lor_{j} {\ddot{v}}_{jk} : k \in D_{1}), \\ (\land_{j} {\ddot{v}}_{jk} : k \in D_{2}); j \in l} \end{matrix}$ and $\begin{matrix} q - ROFSV - NIS & = & {{\ddot{v}}_{1}^{-}, {\ddot{v}}_{2}^{-}, \dots, {\ddot{v}}_{m}^{-}} \\ = & {(\land_{j} {\ddot{v}}_{jk} : k \in D_{1}), \\ (\lor_{j} {\ddot{v}}_{jk} : k \in D_{2}); j \in l} \end{matrix}$ -3ptwhere ∨ stands for q-ROFS union and ∧ represents q-ROFS intersection.

Step 7: Compute q-ROFS Euclidean distances of each alternative from q-ROFSV-PIS and q-ROFSV-NIS, respectively, utilizing $\begin{matrix} d_{j}^{+} = \sqrt{Σ_{k = 1}^{m} {(μ_{jk}^{+} -^{+} μ_{k}^{+})^{2} + (ν_{jk}^{+} -^{+} ν_{k}^{+})^{2} + (μ_{jk}^{-} -^{+} μ_{k}^{-})^{2} + (ν_{jk}^{-} -^{+} ν_{k}^{-})^{2}}} \\ and \\ d_{j}^{-} = \sqrt{Σ_{k = 1}^{m} {(μ_{jk}^{+} -^{-} μ_{k}^{+})^{2} + (ν_{jk}^{+} -^{-} ν_{k}^{+})^{2} + (μ_{jk}^{-} -^{-} μ_{k}^{-})^{2} + (ν_{jk}^{-} -^{-} ν_{k}^{-})^{2}}} \end{matrix}$ Step 8: The closeness coefficient of each alternative with ideal solution by making use of $C_{j}^{-} ({\ddot{v}}_{j}) = \frac{d_{j}^{-}}{d_{j}^{+} + d_{j}^{-}} \in [0, 1]$ /.

Step 9: Rank the alternatives in decreasing (or increasing) order to get the preference order of the alternatives.

The picture of procedural steps of Algorithm 2 is shown in Fig. 3.

Fig. 3

The picture of Algorithm 2.

6.1 Numerical example

Assume that a firm plans to invest some money in stock exchange by purchasing some shares from anyone of four best companies. We compute the ranking methodology by using TOPSIS method based on q-ROFS topology.

Step 1: Let $X = {{\ddot{v}}_{i} : i = 1, 2, 3, 4}$ be the set four companies for money investment, P = {p_i : i = 1, 2, 3, 4}, be the set of parameters/attributes. Let D = {D_i : i = 1, 2, 3, 4} be the set of decision makers.

Step 2: Choose linguistic terms from Table 5, construct weighted parameter matrix $\begin{matrix} P = [w_{ij}]_{4 \times 4} = (\begin{matrix} LV & HV & MV & V \\ V & LtV & LV & MV \\ LtV & MV & LV & HV \\ LV & V & HV & MV \end{matrix}) = (\begin{matrix} 0.30 & 0.90 & 0.75 & 0.55 \\ 0.55 & 0.15 & 0.30 & 0.75 \\ 0.15 & 0.75 & 0.30 & 0.90 \\ 0.30 & 0.55 & 0.90 & 0.75 \end{matrix}) \end{matrix}$ where w_ij is the weight adjusted by the board of governors of the society. D_i (row-wise) to each parameter p_j (column-wise).

Step 3: The normalized weighted matrix is $\begin{matrix} \hat{N} = [{\hat{n}}_{ij}]_{4 \times 4} = (\begin{matrix} 0.4218 & 0.6908 & 0.6019 & 0.3677 \\ 0.7732 & 0.1151 & 0.2408 & 0.5014 \\ 0.2109 & 0.5757 & 0.2408 & 0.6017 \\ 0.4218 & 0.4222 & 0.7224 & 0.5014 \end{matrix}) \end{matrix}$ and hence the weighted vector is $W = (0.2466, 0.2435, 0.2437, 0.2662)$ .

Step 4: The q-ROFS decision matrix D_i of each member is placed, in which alternatives are expressed row-wise and parameters are expressed column-wise such that the aggregate of all D_i’s makes a q-ROFS topology. We consider the following matrices $\begin{matrix} D_{1} & = & (\begin{matrix} (0.90, 0.17) & (0.36, 0.77) & (0.95, 0.31) & (0.88, 0.46) \\ (0.10, 0.70) & (0.56, 0.66) & (0.58, 0.49) & (0.77, 0.62) \\ (0.63, 0.27) & (0.49, 0.46) & (0.44, 0.40) & (0.66, 0.46) \\ (0.52, 0.38) & (0.40, 0.66) & (0.69, 0.56) & (0.55, 0.56) \end{matrix}) \end{matrix}$ $\begin{matrix} D_{2} & = & (\begin{matrix} (1.00, 0.00) & (1.00, 0.00) & (1.00, 0.00) & (1.00, 0.00) \\ (1.00, 0.00) & (1.00, 0.00) & (1.00, 0.00) & (1.00, 0.00) \\ (1.00, 0.00) & (1.00, 0.00) & (1.00, 0.00) & (1.00, 0.00) \\ (1.00, 0.00) & (1.00, 0.00) & (1.00, 0.00) & (1.00, 0.00) \end{matrix}) \\ D_{3} & = & (\begin{matrix} (0.65, 0.84) & (0.25, 0.88) & (0.81, 0.58) & (0.79, 0.68) \\ (0.07, 0.78) & (0.34, 0.79) & (0.52, 0.69) & (0.57, 0.71) \\ (0.55, 0.61) & (0.29, 0.55) & (0.33, 0.56) & (0.49, 0.53) \\ (0.38, 0.43) & (0.37, 0.73) & (0.60, 0.74) & (0.30, 0.65) \end{matrix}) \\ D_{4} & = & (\begin{matrix} (0.00, 1.00) & (0.00, 1.00) & (0.00, 1.00) & (0.00, 1.00) \\ (0.00, 1.00) & (0.00, 1.00) & (0.00, 1.00) & (0.00, 1.00) \\ (0.00, 1.00) & (0.00, 1.00) & (0.00, 1.00) & (0.00, 1.00) \\ (0.00, 1.00) & (0.00, 1.00) & (0.00, 1.00) & (0.00, 1.00) \end{matrix}) \end{matrix}$ Hence, the aggregated decision matrix comes out to be $\begin{matrix} A & = & \frac{D_{1} + D_{2} + D_{3} + D_{4}}{4} \\ = & (\begin{matrix} (0.6375, 0.5025) & (0.4025, 0.6625) & (0.6900, 0.4725) & (0.6675, 0.5350) \\ (0.2925, 0.6200) & (0.4750, 0.6125) & (0.5250, 0.5450) & (0.5850, 0.5825) \\ (0.5450, 0.4700) & (0.4450, 0.5025) & (0.4425, 0.4900) & (0.5375, 0.4975) \\ (0.4750, 0.4525) & (0.4425, 0.5975) & (0.5725, 0.5750) & (0.4625, 0.5525) \end{matrix}) \\ = & [{\dot{v}}_{jk}]_{4 \times 4} \end{matrix}$

Step 5: The weighted q-ROFS decision matrix $\begin{matrix} B & = & [{\ddot{v}}_{jk}]_{4 \times 4} \\ = & (\begin{matrix} (0.1572, 0.1239) & (0.0980, 0.1613) & (0.1682, 0.1151) & (0.1777, 0.1424) \\ (0.0721, 0.1529) & (0.1157, 0.1491) & (0.1279, 0.1328) & (0.1557, 0.1551) \\ (0.1344, 0.1159) & (0.1084, 0.1224) & (0.1078, 0.1194) & (0.1431, 0.1324) \\ (0.1171, 0.1116) & (0.1077, 0.1455) & (0.1395, 0.1401) & (0.1231, 0.1471) \end{matrix}) \end{matrix}$ where ${\ddot{v}}_{jk} = w_{k} \times {\dot{v}}_{jk}$ .

Step 6: Here, we have to calculate q-ROFS valued positive ideal solution (q-ROFSV-PIS) and q-ROFS valued negative ideal solution (q-ROFSV-NIS). As our benefit is with the good selection of an institution which provides better education at reasonable price. $\begin{matrix} PFSV - PIS = {{\ddot{v}}_{1}^{+}, {\ddot{v}}_{2}^{+}, {\ddot{v}}_{3}^{+}, {\ddot{v}}_{4}^{+}} = {(0.1572, 0.1116), (0.1157, 0.1224), (0.1682, 0.1151), (0.1777, 0.1324)} \end{matrix}$ and $\begin{matrix} PFSV - NIS = {{\ddot{v}}_{1}^{-}, {\ddot{v}}_{2}^{-}, {\ddot{v}}_{3}^{-}, {\ddot{v}}_{4}^{-}} = {(0.0721, 0.1529), (0.0980, 0.1613), (0.1078, 0.1401), (0.1231, 0.1551)} \end{matrix}$

Step 7: The q-ROFS Euclidean distances of each alternative from q-ROFSV-PIS and q-ROFSV-NIS, respectively, are computed as $\begin{matrix} d_{1}^{+} = d^{+} ({\ddot{v}}_{1}, q - ROFSV - PIS) & = & 0.1153 \\ d_{2}^{+} = d^{+} ({\ddot{v}}_{2}, q - ROFSV - PIS) & = & 0.0566 \\ d_{3}^{+} = d^{+} ({\ddot{v}}_{3}, q - ROFSV - PIS) & = & 0.0831 \\ d_{4}^{+} = d^{+} ({\ddot{v}}_{4}, q - ROFSV - PIS) & = & 0.0742 \end{matrix}$ and $\begin{matrix} d_{1}^{-} = d^{-} ({\ddot{v}}_{1}, q - ROFSV - NIS) & = & 0.1304 \\ d_{2}^{-} = d^{-} ({\ddot{v}}_{2}, q - ROFSV - NIS) & = & 0.0480 \\ d_{3}^{-} = d^{-} ({\ddot{v}}_{3}, q - ROFSV - NIS) & = & 0.0781 \\ d_{4}^{-} = d^{-} ({\ddot{v}}_{4}, q - ROFSV - NIS) & = & 0.0728 \end{matrix}$

Step 8: The closeness coefficient of each alternative with ideal solution is computed as $\begin{matrix} C_{1}^{-} ({\ddot{v}}_{1}) & = & 0.5307 \\ C_{2}^{-} ({\ddot{v}}_{2}) & = & 0.4589 \\ C_{3}^{-} ({\ddot{v}}_{3}) & = & 0.4845 \\ C_{4}^{-} ({\ddot{v}}_{4}) & = & 0.4952 \end{matrix}$

Step 9: The preference order of the alternatives, therefore, is given by ${\ddot{v}}_{1} ≻ {\ddot{v}}_{4} ≻ {\ddot{v}}_{3} ≻ {\ddot{v}}_{2}$ The final ranking is portrayed in Fig. 4.

Fig. 4

3D Pie chart of ranking.

Table 6

Comparison analysis of final ranking with existing methods using Algorithm 2

Methods	The optimal alternative
Algorithm 2 (Proposed)	${\ddot{v}}_{1}$
Algorithm (Eraslan and Karaaslan [16])	${\ddot{v}}_{1}$
Algorithm Naeem et al. [25]	${\ddot{v}}_{1}$
Algorithm (Tehrim and Riaz [52])	${\ddot{v}}_{1}$
Algorithm (Zhang and Xu [59])	${\ddot{v}}_{1}$

As we see that, rank of ${\ddot{v}}_{1}$ is maximum, so the optimal alternative is ${\ddot{v}}_{1}$ . The remaining alternative are also very close to the optimal alternative. The best possible ranking of feasible alternatives is computed by proposed MCGDM method. The proposed Algorithm 2 is used in the environment of q-ROFS topology based TOPSIS for the ranking of alternatives and the results are compared with some existing decision making methods as indicated in the Table 6 listing the optimal alternative.

7 Conclusion

Multi-criteria group decision making (MCGDM) has been intensively studied by numerous researchers. The techniques developed for this task mainly depend on the type of decision problem under consideration. Most of the real-life situations are uncertain, imperfect, imprecise and vague. The existing models of fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets, q-rung orthopair fuzzy sets are very helpful in solving real world problems under uncertainty. Nevertheless, all these notions have some limitations imposed on the membership and non-membership grades. In this paper, the concept of q-rung orthopair fuzzy soft set (q-ROFSS) as a hybrid model of q-rung orthopair fuzzy set and soft set is introduced. Some elementary operations on q-rung orthopair fuzzy soft sets and their key properties are defined. The idea of q-rung orthopair fuzzy soft topology (q-ROFS topology) on q-rung orthopair fuzzy soft set (q-ROFSS) is presented. Certain topological properties like basis, subspace, interior, closure and boundary of a q-ROFSS are given. Separation axioms, regular spaces and normal spaces in the context of q-ROFS topology are studied. Additionally, the choice value method based on q-ROFSs and extension of TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) to q-ROFS topology for multi-criteria group decision making (MCGDM) are developed. An application of q-ROFS information to the MCGDM by using choice value method for the optimum results in school education systems is given. The ranking of companies for money investment is presented by using TOPSIS method with q-ROFS topology. The comparison analysis of proposed approaches with some existing MCGDM approaches are given to illustrate the feasibility and validity of the proposed approaches.

Footnotes

Acknowledgment

The authors are highly thankful to the Editor-in-chief and the referees for their valuable comments and suggestions for improving the quality of our paper.

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Novel MCGDM with q-rung orthopair fuzzy soft sets and TOPSIS approach under q-Rung orthopair fuzzy soft topology

Abstract

Keywords

1 Introduction

2 Preliminaries

Table 1 Tabular array of q-ROFSS κ A κ A e 1 e 2 ⋯ e n ρ 1 (μ11, ν11) (μ12, ν12) ⋯ (μ1n, ν1n) ρ 2 (μ21, ν21) (μ22, ν22) ⋯ (μ2n, ν2n) ⋮ ⋮ ⋮ ⋱ ⋮ ρ m (μm1, νm1) (μm2, νm2) ⋯ (μ mn , ν mn )

4.1 q-ROFS separation axioms

5 MCGDM with choice value method under q-rung orthopair fuzzy soft sets

5.1 Selection of school education system

Table 5 Linguistic terms for deciding alternatives Linguistic terms Fuzzy weights Highly valuable (HV) 0.90 Much valuable (MV) 0.75 Valuable (V) 0.55 Less valuable (LV) 0.30 Least valuable (LtV) 0.15

Footnotes

Acknowledgment

References

Table 1
Tabular array of q-ROFSS κ_A

κ _A e ₁ e ₂ ⋯ e _n

ρ ₁ (μ₁₁, ν₁₁) (μ₁₂, ν₁₂) ⋯ (μ_1n, ν_1n)

ρ ₂ (μ₂₁, ν₂₁) (μ₂₂, ν₂₂) ⋯ (μ_2n, ν_2n)

⋮ ⋮ ⋮ ⋱ ⋮

ρ _m (μ_m1, ν_m1) (μ_m2, ν_m2) ⋯ (μ_mn, ν_mn)

Table 5
Linguistic terms for deciding alternatives

Linguistic terms Fuzzy weights

Highly valuable (HV) 0.90

Much valuable (MV) 0.75

Valuable (V) 0.55

Less valuable (LV) 0.30

Least valuable (LtV) 0.15