TOPSIS,VIKOR and aggregation operators based on q-rung orthopair fuzzy soft sets and their applications

Abstract

In this article, we study some concepts related to q-rung orthopair fuzzy soft sets (q-ROFSSs) 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 an influx of linguistic variables based on q-rung orthopair fuzzy soft (q-ROFS) information. We present an application of q-ROFSSs 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 utilizing q-ROFS TOPSIS and q-ROFS VIKOR methods.

Keywords

q-ROFNs TOPSIS aggregation operators VIKOR soft sets

1 Introduction

The fuzzy linguistic framework offers favorable outcomes in a number of areas, the description of which is relatively qualitative. The incentive to use phrases rather than just numbers seems to be that linguistic characterizations or categories are generally even less accurate than algebraic or arithmetic characterizations. Issues associated with unstable situations typically arise in decision-making and, furthermore, they are making demands on the complex problem of modeling and manipulation that appears as a side product in these uncertainties. Often we face problems in coping with daily life problems, which are usually complex in nature. The main factor behind these hindrances is that the methodologies commonly selected for use in conventional mathematics are not always of assistance due to the many types of uncertainties and vagueness present in these problems. To deal with uncertainties and vagueness Zadeh initiated fuzzy set (FS) theory [52], Atanassov [6] introduced intuitionistic fuzzy sets (IFSs), Molodtsov [25] introduced soft set theory, Zhang et al. [59 –61] presented bipolar fuzzy sets. Yager [46, 47] and Yager and Abbasov [48] introduced the notion of Pythagorean fuzzy sets (PFSs) as an extension of IFSs.

Applications of fuzzy sets, rough sets, soft sets and their hybrid structures have been studied by many researchers; Ali et al. [1], Ali [2], Akram et al. [3 –5], Chen et al. [7], Chi and Lui [8], Çağman et al. [9], Eraslan and Karaaslan [10], Feng et al. [11 –14], Garg and Arora [15 –17], Garg and Nancy [18], Jose and Kuriaskose [19], Kaur and Garg [20], Kumar and Garg [21], Karaaslan [22], Liu and Wang [23], Liu et al. [24], Naeem et al. [26 –28], Peng et al. [29 –32], Riaz and Hashmi [33, 34], Riaz et al. [35 –37], Riaz and Tehrim [38 –40], Shabir and Naz [41], Wang et al. [42], Xu [43], Xu and Cai [44], Xu [45], Ye [50, 51], Zhang and Xu [53], Zhan et al. [54, 55] and Zhang et al. [56 –58].

Yager initiated the idea of q-rung orthopair fuzzy set (q-ROFS) as an extension of PFS [49], in which the sum of membership degree (MD) $F_{A} (x)$ and non membership degree (NMD) $T_{A} (x)$ satisfy the condition $0 \leq (F_{A} (x))^{q} + (T_{A} (x))^{q} \leq 1, (q \geq 1)$ . The degree of indeterminacy is given by $π_{A} (x) = (F_{A}^{q} (x) + T_{A}^{q} (x) - F_{A}^{q} (x) T_{A}^{q} (x))^{1 / q}$ . There is no condition on q other than q ≥ 1. Although q is real number, but if q is integral value It’s quite easy to know the area from which MD and NMD are selected. we can easily check that 99% area is covered when we put q = 10 of unit square [0, 1] × [0, 1].

The rest of this paper is organized as: Section 2 consists of the basic concepts of fuzzy sets and their extensions. Section 3 sets out the vital operations and fundamental characteristics of q-rung orthopair fuzzy soft sets (q-ROFSSs). Section 4 represents an excellent implementation of multi-criteria group decision-making (MCGDM) to q-ROF TOPSIS. In Section 5, we presented the VIKOR approach and implemented that to the same example in section four. Section 6 consists on MCGDM based on aggregation operators of q-ROFSSs. Lastly, Section 7 sums up the entire article.

2 Preliminaries

In this section, we present some basics of the fuzzy set, soft set, IFS and q-ROFS that would be helpful for better study of this paper.

Definition 2.1. [52] Assume that $H$ is the universe of discourse and $F_{F}$ a map that sends members of $H$ to [0, 1]. Then an FS F is of the form $F = {(x, F_{F} (x)) : x \in H}$ $F_{F} (x)$ denotes the MD of x to F. The collection of all FSs in $H$ is denoted by $F (H)$ .

Definition 2.2. [6] An IFS $\ddot{A}$ on $H$ is object of the form $\ddot{A} = {(ζ, F_{\ddot{A}} (ζ), T_{\ddot{A}} (ζ)) : ζ \in H}$ The mappings $F_{\ddot{A}}$ and $T_{\ddot{A}}$ map elements of $H$ to interval [0, 1], with the constraint that the sum of MD and NMD does not surpass unity, the mappings is referred to respectively as MD and NMD.

Definition 2.3. [25] Let $H$ be universe and $E$ a non-empty set of attributes. Let $2^{H}$ is the power set of $H$ and $A$ is a subset of $E$ . An object of the form $(φ, A)$ , where $φ : A \to 2^{H}$ is a mapping, is regarded as soft set (SS) on $H$ . Thus an SS may be expressed as $(φ, A) = {(e, φ (e)) : e \in A, φ (e) \in 2^{H}}$ $φ_{A}$ is an alternative way to represent $(φ, A)$ .

Definition 2.4. [9] Let $H$ be the universe and E be the assemblage of parameters with $A \subseteq E$ . Consider $F (H)$ collection of all FSs on $H$ . An item of the form $Γ_{A} = {(e, γ_{A} (e)) : e \in E, γ_{A} \in F (H)}$ is called fuzzy soft set over $H$ .

Definition 2.5. [49] Let $H$ be the universe. A q-rung orthopair fuzzy set, written as q-ROF set has the form $P = {< ζ, F_{p} (ζ), T_{p} (ζ) > ∣ ζ \in H}$ where $F_{p}$ and $T_{P}$ are two maps that send elements of $H$ to unit closed interval. $F_{p} (ζ)$ denotes the MD and $T_{P} (ζ)$ denotes the NMD of the element $ζ \in H$ to the set P, respectively, with the condition that $0 \leq F_{P}^{q} (ζ) + T_{P}^{q} (ζ) \leq 1, (q \geq 1)$ . The degree of indeterminacy is given as $π_{P} (ζ) = (F_{P}^{q} (ζ) + T_{P}^{q} (ζ) - F_{P}^{q} (ζ) T_{P}^{q} (ζ))^{1 / q}$ .

For convenience, a basic element $〈 F_{P} (ζ), T_{P} (ζ) 〉$ in a q-ROF set is denoted by $α = 〈 F_{P}, T_{P} 〉$ for short and is called q-rung orthopair fuzzy number (q-ROFN).

2.1 Operational laws of q-ROFs

Let $A_{1} = 〈 F_{A_{1}} (ζ), T_{A_{1}} (ζ) 〉$ and $A_{2} = 〈 F_{A_{2}} (ζ), T_{A_{2}} (ζ) 〉$ be q-ROF sets on a $H$ . Then the following operations are valid [49]:

$\bar{A_{1}} = 〈 T_{A_{1}} (ζ), F_{A_{1}} (ζ) 〉$ .

A₁ ⊆ A₂ iff $F_{A_{1}} (ζ) \leq F_{A_{2}} (ζ)$ and $T_{A_{2}} (ζ) \leq T_{A_{1}} (ζ)$ .

A₁ = A₂ iff A₁ ⊆ A₂ and A₂ ⊆ A₁.

$A_{1} \cup A_{2} = {〈 ζ, max {F_{A_{1}} (ζ), F_{A_{2}} (ζ)}, min {T_{A_{1}} (ζ), T_{A_{2}} (ζ)} 〉 ∣ ζ \in H}$ .

$A_{1} \cap A_{2} = {〈 ζ, min {F_{A_{1}} (ζ), F_{A_{2}} (ζ)}, max {T_{A_{1}} (ζ), T_{A_{2}} (ζ)} 〉 ∣ ζ \in H}$ .

$A_{1} + A_{2} = {〈 ζ, (F_{A_{1}}^{q} (ζ) + F_{A_{2}}^{q} (ζ) - F_{A_{1}}^{q} (ζ) F_{A_{2}}^{q} (ζ))^{1 / q}, T_{A_{1}} (ζ) T_{A_{2}} (ζ) 〉 ∣ ζ \in H}$ .

$A_{1} . A_{2} = {〈 ζ, (F_{A_{1}} (ζ) F_{A_{2}} (ζ), T_{A_{1}}^{q} (ζ) + T_{A_{2}}^{q} (ζ) - T_{A_{1}}^{q} (ζ) T_{A_{2}}^{q} (ζ))^{1 / q} 〉 ∣ ζ \in H}$ .

$λ A_{1} = {〈 ζ, (1 - (1 - F_{A_{1}} (ζ)^{q})^{λ}))^{1 / q}, (T_{A_{1}} (ζ))^{λ} 〉}$ .

$A_{1}^{λ} = {〈 ζ, (F_{A_{1}} (ζ))^{λ}, (1 - (1 - T_{A_{1}}^{q} (ζ))^{λ}))^{1 / q} 〉} .$

2.2 Operational laws of q-ROFNs

Let $α_{1} = 〈 F_{1}, T_{1} 〉$ and $α_{2} = 〈 F_{2}, T_{2} 〉$ be q-ROFNs on a $H$ . Then [23]

$\bar{α_{1}} = 〈 T_{1}, F_{1} 〉$

$α_{1} \lor α_{2} = 〈 max {F_{1}, T_{1}}, min {F_{2}, T_{2}} 〉$

$α_{1} \land α_{2} = 〈 min {F_{1}, T_{1}}, max {F_{2}, T_{2}} 〉$

$α_{1} \oplus α_{2} = 〈 (F_{1}^{q} + F_{2}^{q} - F_{1}^{q} F_{2}^{q})^{1 / q}, T_{1} T_{2} 〉$

$α_{1} \otimes α_{2} = 〈 F_{1} F_{2}, (T_{1}^{q} + T_{2}^{q} - T_{1}^{q} T_{2}^{q})^{1 / q} 〉$

$λ α_{1} = 〈 (1 - (1 - F_{1}^{q})^{λ})^{1 / q}, T_{1}^{λ} 〉$

$α_{1}^{λ} = 〈 F_{1}^{λ}, (1 - (1 - T_{1}^{q})^{λ})^{1 / q} 〉$

Definition 2.6. Suppose $α = 〈 F, T 〉$ is a q-ROFN, then a score function S of α is defined as $S (α) = F^{q} - T^{q} \in [- 1, 1]$ Score function may not be useful in some cases. For example, consider two q-ROFNs $α_{1} = 〈 \frac{\sqrt{3}}{2}, \frac{1}{2} 〉$ and $α_{2} = 〈 \sqrt{\frac{5}{6}}, \frac{1}{\sqrt{3}} 〉$ and take q = 2, then $S (α_{1}) = S (α_{2}) = \frac{1}{2}$ . Thus, score function is not enough to compare q-ROFNs. In the description below we add another function to solve this problem.

Definition 2.7. Suppose $α = 〈 F, T 〉$ is a q-ROFN, then the accuracy function R of α is defined as $R (α) = F^{q} + T^{q} \in [0, 1]$

Definition 2.8. Let $★_{1} = 〈 F_{★_{1}}, T_{★_{1}} 〉$ and $★_{2} = 〈 F_{★_{2}}, T_{★_{2}} 〉$ be two q-ROFNs. Then

If S (★ ₁) > S (★ ₂), then ★₁ ≻ ★ ₂

If S (★ ₁) = S (★ ₂), and

If R (★ ₁) > R (★ ₂) then ★₁ ≻ ★ ₂,

If R (★ ₁) = R (★ ₂), then ★₁ ∼ ★ ₂.

3 q-Rung orthopair fuzzy soft sets

We propose, in this section, the idea of q-rung orthopair fuzzy soft set (q-ROFSS) along with some basic notions and basic characteristics.

Definition 3.1. Let $H$ be a finite underlying set accompanied by E as the assemblage of attributes. Let H ⊆ E be non empty and q- ${ROF}^{H}$ represent the aggregate of all q-ROF subsets over $H$ . A q-ROFSS on $H$ is denoted as (φ, H) or φ_H, where φ : H → q- ${ROF}^{H}$ is a mapping, and is given as $\begin{matrix} (φ, H) & = & {(e, {ζ, F_{φ_{H}} (ζ), T_{φ_{H}} (ζ)}) : e \in H, ζ \in H} \\ = & {(e, {\frac{ζ}{(F_{φ_{H}} (ζ), T_{φ_{H}} (ζ))}}) : e \in H, ζ \in H} \\ = & {(e, {\frac{(F_{φ_{H}} (ζ), T_{φ_{H}} (ζ))}{ζ}}) : e \in H, ζ \in H} \end{matrix}$ where $F_{φ_{H}} : H \to [0, 1]$ and $T_{φ_{H}} : H \to [0, 1]$ are two mappings along with the property $0 \leq F_{φ_{H}}^{q} (ζ) + T_{φ_{H}}^{q} (ζ) \leq 1 (q \geq 1)$ The values of all of these maps $ζ \in H$ are named as q-ROFNs. Here, $F_{φ_{H}} (ζ)$ shows the MD and $T_{φ_{H}} (ζ)$ represents NMD of the element $ζ \in H$ to the set $(φ, H)$ .

If we write ${\ddot{ϖ}}_{ij} = F_{φ_{H}} (e_{j}) (ζ_{i})$ and ${\hat{R}}_{ij} = T_{φ_{H}} (e_{j}) (ζ_{i})$ q-ROFS set φ_H can be viewed as a tabular form, where i varies from 1 to m and j differs from 1 to n.

φ _H	e ₁	e ₂	⋯	e _n
ζ ₁	$({\ddot{ϖ}}_{11}, {\hat{R}}_{11})$	$({\ddot{ϖ}}_{12}, {\hat{R}}_{12})$	⋯	$({\ddot{ϖ}}_{1 n}, {\hat{R}}_{1 n})$
ζ ₂	$({\ddot{ϖ}}_{21}, {\hat{R}}_{21})$	$({\ddot{ϖ}}_{22}, {\hat{R}}_{22})$	⋯	$({\ddot{ϖ}}_{2 n}, {\hat{R}}_{2 n})$
⋮	⋮	⋮	⋱	⋮
ζ _m	$({\ddot{ϖ}}_{m 1}, {\hat{R}}_{m 1})$	$({\ddot{ϖ}}_{m 2}, {\hat{R}}_{m 2})$	⋯	$({\ddot{ϖ}}_{mn}, {\hat{R}}_{mn})$

and in matrix form as $\begin{matrix} (φ, H) & = & [({\ddot{ϖ}}_{ij}, {\hat{R}}_{ij})]_{m \times n} \\ = & (\begin{matrix} ({\ddot{ϖ}}_{11}, {\hat{R}}_{11}) & ({\ddot{ϖ}}_{12}, {\hat{R}}_{12}) & \dots & ({\ddot{ϖ}}_{1 n}, {\hat{R}}_{1 n}) \\ ({\ddot{ϖ}}_{21}, {\hat{R}}_{21}) & ({\ddot{ϖ}}_{22}, {\hat{R}}_{22}) & \dots & ({\ddot{ϖ}}_{2 n}, {\hat{R}}_{2 n}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ ({\ddot{ϖ}}_{m 1}, {\hat{R}}_{m 1}) & ({\ddot{ϖ}}_{m 2}, {\hat{R}}_{m 2}) & \dots & ({\ddot{ϖ}}_{mn}, {\hat{R}}_{mn}) \end{matrix}) \end{matrix}$ The matrix is referred to as the q-rung orthopair fuzzy soft matrix or shortly the matrix q-ROFS.

Example 3.2. Consider $H = {ζ_{i} : i = 1, 2, \dots, 6}$ and E = {e_i : i = 1, 2, ⋯ , 5}, also assume that H = {e₁, e₃} and q = 3 then, φ_H = {(e₁, {(ζ₂, 0.57, 0.68), (ζ₄, 0.36, 0.49) , (ζ₆, 0.60, 0.13)}), (e₃, {(ζ₁, 0.76, 0.23) , (ζ₃, 0.66, 0.53), (ζ₅, 0.62, 0.76)})} is a q-ROFSS over $H$ . The tabular representation of φ_H is

φ _H	e ₁	e ₂	e ₃	e ₄	e ₅
ζ ₁	(0,1)	(0,1)	(0.76, 0.23)	(0,1)	(0,1)
ζ ₂	(0.57, 0.68)	(0,1)	(0,1)	(0,1)	(0,1)
ζ ₃	(0,1)	(0,1)	(0.66, 0.53)	(0,1)	(0,1)
ζ ₄	(0.36, 0.49)	(0,1)	(0,1)	(0,1)	(0,1)
ζ ₅	(0,1)	(0,1)	(0.62, 0.76)	(0,1)	(0,1)
ζ ₆	(0.60, 0.13)	(0,1)	(0,1)	(0,1)	(0,1)

The corresponding q-ROFS matrix is $\begin{matrix} (φ, H) & = & [a_{ij}, b_{ij}]_{6 \times 5} \\ = & (\begin{matrix} (0, 1) & (0, 1) & (0.76, 0.23) & (0, 1) & (0, 1) \\ (0.57, 0.68) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.66, 0.53) & (0, 1) & (0, 1) \\ (0.36, 0.49) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.62, 0.76) & (0, 1) & (0, 1) \\ (0.60, 0.13) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \end{matrix}) \end{matrix}$

Definition 3.3. Let φ_H and ϱ_K be q-ROFSS over $H$ . Then φ_H is q-ROFS subset of ϱ_K i.e. $φ_{H} \tilde{\subseteq} ϱ_{K}$ , if

H ⊆ K, and

φ (e) is q-ROFS subset of ϱ (e) for all e ∈ H.

There is something worth noting that

φ_{H} \tilde{\subseteq} ϱ_{K}

does not vital, every element of φ_H is also in ϱ_K.

Definition 3.4. Assume (φ₁, H₁) and (φ₂, H₂) be q-ROFS defined over $H$ . Then union will be defined as $(φ, H) = (φ_{1}, H_{1}) \tilde{\cup} (φ_{2}, H_{2}),$ where H = H₁ ∪ H₂ and for all e ∈ H, $φ (e) = {\begin{matrix} φ_{1} (e), & if e \in H_{1} ∖ H_{2} \\ φ_{2} (e), & if e \in H_{2} ∖ H_{1} \\ φ_{1} (e) \cup φ_{2} (e), & if e \in H_{1} \cap H_{2} \end{matrix}$

Definition 3.5. Intersection of two q-ROFSSs (φ₁, H₁) and (φ₂, H₂) is a q-ROFS set $(φ, H) = (φ_{1}, H_{1}) \tilde{\cap} (φ_{2}, H_{2})$ where H = H₁ ∩ H₂ ≠ φ and φ (e) = φ₁ (e) ∩ φ₂ (e), ∀e ∈ H.

Definition 3.6. The difference (φ, H) of two q-ROFSSs (φ₁, H₁) and (φ₂, H₂) over $H$ is defined by

(φ, e) = (φ₁, e) \ (φ₂, e); ∀ e ∈ E and is represented as $(φ_{1}, H) \tilde{\} (φ_{2}, H)$ . Hence, $(φ_{1}, H_{1}) \tilde{\} (φ_{2}, H_{2})$ $= {(e, {ζ, min {F_{φ_{1} (e)} (ζ), T_{φ_{2} (e)} (ζ)}, max {T_{φ_{1} (e)} (ζ), F_{φ_{2} (e)} (ζ)}}) : ζ \in H, e \in E}$

Definition 3.7. The complement of a q-ROFSS (φ, H) is a mapping φ^c : H → q- ${ROF}^{H}$ defined by φ^c (e) = [φ (e)] ^c, ∀e ∈ H. It is illustrated as (φ, H) ^c or (φ^c, H). Thus, if $φ (e) = {(ζ, F_{φ (e)} (ζ), T_{φ (e)} (ζ)) : ζ \in H}$ then $φ^{c} (e) = {(ζ, T_{φ (e)} (ζ), F_{φ (e)} (ζ)) : ζ \in H}$ for all e ∈ H.

Definition 3.8. A q-ROFSS (φ, H) over $H$ is null q-ROFSS, illustrated as φ_φ or Φ, φ maps each member of H to null q-ROFS i.e. $φ (e \in H) = \tilde{0}$ . Hence, $φ_{φ} = {(e, {ζ, 0, 1}) : e \in H, ζ \in H}$

Definition 3.9. A q-ROFSS (φ, H) over $H$ is called absolute q-ROFSS, denoted by $\overset{˘}{H}$ , if $φ (e) = \tilde{1}, \forall e \in H$ . Here $\tilde{1}$ represents absolute q-ROFS. Hence, $\overset{˘}{H} = {(e, {ζ, 1, 0}) : e \in H, ζ \in H}$

With the assistance of illustration below we explain some of these concepts.

Example 3.10. Let $H = {ζ_{1}, ζ_{2}, \dots, ζ_{6}}$ and E = {e₁, e₂, ⋯ , e₆}. Suppose that H₁ = {e₁, e₃} and H₂ = {e₂, e₃, e₄}. Consider the q-ROFSSs (φ₁, H₁) = {(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 (φ₂, H₂) = {(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 $(φ_{1}, H_{1}) \tilde{\cup} (φ_{2}, H_{2})$ = {(e₁, {ζ₁, 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 $(φ_{1}, H_{1}) \tilde{\cap} (φ_{2}, H_{2}) = {(e_{3}, {ζ_{6}, 0.71, 0.55})}$ .

(3) The complements of (φ₁, H₁) and (φ₂, H₂) are (φ₁, H₁) ^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 (φ₂, H₂) ^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 (φ₁, H₁) and (φ₂, H₂) is $(φ_{1}, H_{1}) \tilde{\} (φ_{2}, H_{2})$ = {(e₁, {ζ₁, 0.71, 0.36} , {ζ₃, 0.24, 0.96} , {ζ₆, 0.34, 0.49}) , (e₃, {ζ₂, 0.61, 0.82} , {ζ₆, 0.71, 0.55})}.

Proposition 3.11. If φ_H is any q-ROFS set over $H$ , then

$φ_{φ} \tilde{\subseteq} φ_{H}$ .

$φ_{H} \tilde{\subseteq} \overset{˘}{H}$ .

Proof.

As we know, MD and NMD always lay in [0, 1], so $0 \leq F (ζ)$ and $1 \geq T (ζ)$ for all $ζ \in H$ . It is therefore based on the interpretation of a subset and φ_φ that $φ_{φ} \tilde{\subseteq} φ_{H}$ .

As we know, MD and NMD always lay in [0, 1], so $F (ζ) \leq 1$ and $T (ζ) \geq 0$ for all $ζ \in H$ . It is therefore based on the interpretation of a subset and $\overset{˘}{H}$ that $φ_{H} \tilde{\subseteq} \overset{˘}{H}$ .□

Proposition 3.12. If φ_H and φ_K are two q-ROFSS over $H$ , then

$φ_{H} \tilde{\cap} φ_{H} = φ_{H}$ .

$φ_{H} \tilde{\cup} φ_{H} = φ_{H}$ .

$φ_{H} \tilde{\cap} φ_{K} = φ_{K} \tilde{\cap} φ_{H}$ .

$φ_{H} \tilde{\cup} φ_{K} = φ_{K} \tilde{\cup} φ_{H}$ .

$φ_{H} \tilde{\cap} (φ_{K} \tilde{\cap} φ_{C}) = (φ_{H} \tilde{\cap} φ_{K}) \tilde{\cap} φ_{C}$ .

$φ_{H} \tilde{\cup} (φ_{K} \tilde{\cup} φ_{C}) = (φ_{H} \tilde{\cup} φ_{K}) \tilde{\cup} φ_{C}$ .

Proof. The proof is obvious.□

Proposition 3.13. If φ_H and φ_K are two q-ROFSSs over $H$ , then

$φ_{H} \tilde{\cap} φ_{K} \tilde{\subseteq} φ_{H} \tilde{\subseteq} φ_{H} \tilde{\cup} φ_{K}$

$φ_{H} \tilde{\cap} φ_{K} \tilde{\subseteq} φ_{K} \tilde{\subseteq} φ_{H} \tilde{\cup} φ_{K}$ .

Proof. We only prove (i) The (ii) proof can be accommodated along parallel track.

Since $min {F_{φ_{H}} (ζ), F_{φ_{K}} (ζ)} \leq F_{φ_{H}} (ζ) \leq$ $max {F_{φ_{H}} (ζ), F_{φ_{K}} (ζ)}$ and $max {T_{φ_{H}} (ζ), T_{φ_{K}} (ζ)} \geq T_{φ_{H}} (ζ)$ $\geq min {T_{φ_{H}} (ζ), T_{φ_{K}} (ζ)}$ for all $ζ \in H$ , so (i) follows.□

Remark. Let $H = {ζ_{1}, ζ_{2}, \dots, ζ_{6}}$ . Take two subsets H₁ = {e₁, e₃} and H₂ = {e₃, e₅} of E = {e₁, e₂, ⋯ , e₆}. Consider the q-ROFSSs (φ₁, H₁) and (φ₂, H₂) given by

(φ₁, H₁) = {(e₁, {ζ₁, 0.37, 0.91} , {ζ₃, 0.93, 0.21} ,

{ζ₆, 0.87, 0.41}) , (e₃, {ζ₂, 0.39, 0.85} , {ζ₆, 0.6, 0}) }, and (φ₂, H₂) = {(e₃, {ζ₁, 0.73, 0.46} , {ζ₆, 0.66, 0.76}) ,

(e₅, {ζ₂, 0.74, 0.53} , {ζ₃, 0.76, 0.56} , {ζ₆, 0.47, 0.23}) }

Then,

$(φ_{1}, H_{1}) \tilde{\cup} (φ_{2}, H_{2})$ = {(e₁, {ζ₁, 0.37, 0.91} , {ζ₃, 0.93, 0.21} , {ζ₆, 0.87, 0.41}) , (e₃, {ζ₁, 0.73, 0.46} ,

{ζ₂, 0.39, 0.85} , {ζ₆, 0.66, 0}) , (e₅, {ζ₂, 0.74, 0.53} , {ζ₃, 0.76, 0.56} , {ζ₆, 0.47, 0.23}) }

so that $((φ, H_{1}) \tilde{\cup} (φ_{2}, H_{2}))^{c}$ = {(e₁, {ζ₁, 0.91, 0.37} , {ζ₃, 0.21, 0.93} , {ζ₆, 0.41, 0.87}) , (e₃, {ζ₁, 0.46, 0.73} ,

{ζ₂, 0.85, 0.39} , {ζ₆, 0, 0.66}) , (e₅, {ζ₂, 0.53, 0.74} , {ζ₃, 0.56, 0.76} , {ζ₆, 0.23, 0.47}) }

Also,

(φ₁, H₁) ^c = {(e₁, {ζ₁, 0.91, 0.37} , {ζ₃, 0.21, 0.93} ,

{ζ₆, 0.41, 0.87}) , (e₃, {ζ₂, 0.85, 0.39} , {ζ₆, 0, 0.6}) }, and (φ₂, H₂) ^c = {(e₃, {ζ₁, 0.46, 0.73} , {ζ₆, 0.76, 0.66}) ,

(e₅, {ζ₂, 0.53, 0.74} , {ζ₃, 0.56, 0.76} , {ζ₆, 0.23, 0.47}) } so that $(φ_{1}, H_{1})^{c} \tilde{\cap} (φ_{2}, H_{2})^{c} = {(e_{3}, {ζ_{6}, 0, 0.66})}$ We conclude, therefore, that the laws of De Morgan do not hold to q-ROFSS.

Theorem 3.14. If (φ₁, H₁) and (φ₂, H₂) are two q-ROFS sets over $H$ , then

$((φ_{1}, H_{1}) \tilde{\cup} (φ_{2}, H_{2}))^{c} \neq (φ_{1}, H_{1})^{c} \tilde{\cap} (φ_{2}, H_{2})^{c}$ , and

$((φ_{1}, H_{1}) \tilde{\cap} (φ_{2}, H_{2}))^{c} \neq (φ_{1}, H_{1})^{c} \tilde{\cup} (φ_{2}, H_{2})^{c}$ .

Remark. Let $H = {ζ_{1}, ζ_{2}, \dots, ζ_{6}}$ and H = {e₁, e₃, e₆} ⊆ E where E = {e₁, e₂, ⋯ , e₆}. Consider the q-ROFS matrix

(φ, H) = $(\begin{matrix} (0.85, 0.52) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0.13, 0.98) \\ (0, 1) & (0, 1) & (0.51, 0.63) & (0, 1) & (0, 1) & (0.33, 0.75) \\ (0.83, 0.1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0.57, 0.41) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0.77, 0.39) \\ (0.68, 0.52) & (0, 1) & (0.79, 0.38) & (0, 1) & (0, 1) & (0, 1) \end{matrix})$

so that

(φ, H) ^c = $(\begin{matrix} (0.52, 0.85) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (0.98, 0.13) \\ (1, 0) & (0, 1) & (0.63, 0.51) & (0, 1) & (0, 1) & (0.75, 0.33) \\ (0.1, 0.83) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (1, 0) \\ (1, 0) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (0.41, 0.57) \\ (1, 0) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (0.39, 0.77) \\ (0.52, 0.68) & (0, 1) & (0.38, 0.79) & (0, 1) & (0, 1) & (1, 0) \end{matrix})$

We observe that

$(φ, H) \tilde{\cup} (φ, H)^{c} =$ $(\begin{matrix} (0.85, 0.52) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (0.98, 0.13) \\ (1, 0) & (0, 1) & (0.63, 0.51) & (0, 1) & (0, 1) & (0.75, 0.33) \\ (0.83, 0.1) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (1, 0) \\ (1, 0) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (0.57, 0.41) \\ (1, 0) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (0.77, 0.39) \\ (0.68, 0.52) & (0, 1) & (0.79, 0.38) & (0, 1) & (0, 1) & (1, 0) \end{matrix})$ $\neq \overset{˘}{H}$ and $(φ, H) \tilde{\cap} (φ, H)^{c} =$ $(\begin{matrix} (0.33, 0.52) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0.19, 0.98) \\ (0, 1) & (0, 1) & (0.43, 0.51) & (0, 1) & (0, 1) & (0.22, 0.75) \\ (0.01, 0.72) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0.37, 0.51) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0.07, 0.99) \\ (0.18, 0.52) & (0, 1) & (0.39, 0.88) & (0, 1) & (0, 1) & (0, 1) \end{matrix})$ ≠φ_φ .

Theorem 3.15. Let (φ, ) be a q-ROFSS over $H$ , then

Definition 3.16. The cardinal set of the q-ROFSS φ_H over $H$ is again a q-ROFSS over E and is defined as $c φ_{H} = {\frac{(F_{c λ_{H}} (e), T_{c T_{H}} (e))}{e} : e \in E}$ where $F_{c λ_{H}}$ and $T_{c T_{H}}$ are mappings E → [0, 1], respectively denoted as $F_{c λ_{H}} (e) = \frac{| λ_{H} (e) |}{| H |}$ and $T_{c T_{H}} (e) = \frac{| T_{H} (e) |}{| H |}$ where |λ_H (e) | and $| T_{H} (e) |$ denote respectively the scalar cardinalities of the q-ROFSSs λ_H (e) and $T_{H} (e)$ , and $| H |$ represents the universal cardinality $H$ .

Collection of all q-ROFSS cardinal sets over $H$ is represented as cq- ${ROFS}^{H}$ . If H ⊆ E = {e_i : i = 1, 2, ⋯ , n}, then cφ_H ∈ cq- ${ROFS}^{H}$ may be represented in tabular form as

E	e ₁	e ₂	⋯	e _n
cφ _H	$({\ddot{ϖ}}_{11}, {\hat{R}}_{11})$	$({\ddot{ϖ}}_{12}, {\hat{R}}_{12})$	⋯	$({\ddot{ϖ}}_{1 n}, {\hat{R}}_{1 n})$

and in matrix form as $\begin{matrix} [({\ddot{ϖ}}_{1 j}, {\hat{R}}_{1 j})]_{1 \times n} \\ = [\begin{matrix} ({\ddot{ϖ}}_{11}, {\hat{R}}_{11}) & ({\ddot{ϖ}}_{12}, {\hat{R}}_{12}) & \dots & ({\ddot{ϖ}}_{1 n}, {\hat{R}}_{1 n}) \end{matrix}] \end{matrix}$ where $({\ddot{ϖ}}_{1 j}, {\hat{R}}_{1 j}) = F_{c φ_{H}} (e_{j})$ , for j = 1, ⋯ , n. This matrix is known as cardinal matrix of cφ_H over E.

Definition 3.17. Let φ_H be a q-ROFSS over $H$ and $c φ_{H} \in {cPF}^{H}$ . The q-ROFS aggregation operator q-ROFS_agg : cq- ${ROFS}^{H} \times q$ - ${ROFS}^{H} \to q$ - $ROFS (H, E)$ is defined as $q - ROFS agg (c φ_{H}, φ_{H}) = {\frac{F_{φ_{H}^{*}} (ζ)}{ζ} : ζ \in H}$ $= {\frac{(F_{λ_{H}^{*}} (ζ), T_{T_{H}^{*}} (ζ))}{ζ} : ζ \in H}$ .

This collection is called aggregate q-rung orthopair fuzzy set of q-ROFSS φ_H. The membership function $F_{λ_{H}^{*}} : H \to [0, 1]$ is given by $F_{λ_{H}^{*}} (ζ) = \frac{1}{| E |} Σ_{e \in E} (F_{c λ_{H}} (e), F_{λ_{H}} (e)) (ζ)$ and the non-membership function $T_{T_{H}^{*}} : H \to [0, 1]$ is given by $T_{T_{H}^{*}} (ζ) = \frac{1}{| E |} Σ_{e \in E} (T_{c T_{H}} (e), T_{T_{H}} (e)) (ζ)$ The set q-ROFS_agg (cφ_H, φ_H) may be expressed in tabular form as

φ _H	$F_{φ_{H}^{*}}$
ζ ₁	$({\ddot{ϖ}}_{11}, {\hat{R}}_{11})$
ζ ₂	$({\ddot{ϖ}}_{21}, {\hat{R}}_{21})$
⋮	⋮
ζ _m	$({\ddot{ϖ}}_{m 1}, {\hat{R}}_{m 1})$

and in matrix form as $[{\ddot{ϖ}}_{i 1}, {\hat{R}}_{i 1}]_{m \times 1} = [\begin{matrix} ({\ddot{ϖ}}_{11}, {\hat{R}}_{11}) \\ ({\ddot{ϖ}}_{21}, {\hat{R}}_{21}) \\ ⋮ \\ ({\ddot{ϖ}}_{m 1}, {\hat{R}}_{m 1}) \end{matrix}]$ where $[({\ddot{ϖ}}_{i 1}, {\hat{R}}_{i 1})] = F_{φ_{H}^{*}} (ζ_{i})$ , where i runs from 1 to m. This matrix is known as q-ROFS aggregate matrix of q-ROFS_agg (cφ_H, φ_H) over $H$ .

Theorem 3.18. Let φ_H be a q-ROFS set. Suppose that M_{φ
_H}, M_{cφ
_H} and $M_{φ_{H}}^{*}$ are respectively representative matrices of φ_H, cφ_H and $φ_{H}^{*}$ , then $M_{φ_{H}} \times M_{c φ_{H}}^{t} = M_{φ_{H}}^{*} \times | E |$

4 MCDM using q-ROFSS TOPSIS methodology

TOPSIS is being utilized to choose on a supreme elective to compromise solutions. The solution closest to the ideal solution and the furthest from the negative ideal solution is recognized as a compromise solution. In this part, we’re studying how q-ROFSSs can be used in MCDM by making use of TOPSIS. Firstly, we will expand TOPSIS to the q-ROFSSs and afterwards study a problem related to construction company. Amid the gargantuan practices brought into being in the texts, TOPSIS is widely used approach in addressing such issues. Depending on the nature of the problem considered, each technique has its own advantages and disadvantages.

Algorithm-1 (q-ROFS TOPSIS)

Step 1: Understand the problem: Let $D = {M_{i} : i \in N}$ be the set of decision makers, $A = {{\ddot{z}}_{i} : i \in N}$ is some assemblage of alternatives and C = {q_i : i ∈ mathdsN} is a family of criterion.

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

Table 1
Linguistic terms for judging alternatives

Semantic Terms Fuzzy Weights

Very Highly Important (VHI) 0.95

Highly Important (HI) 0.80

Averagely Important (AI) 0.50

Less Important (LI) 0.35

Semantic Terms	Fuzzy Weights
Very Highly Important (VHI)	0.95
Highly Important (HI)	0.80
Averagely Important (AI)	0.50
Less Important (LI)	0.35

where y_ij is the weight assigned by the expert $M_{i}$ to the alternative $Q_{j}$ by considering linguistic variables

Step 3: Here, in this step we design a normalized weighted (NW) matrix

$\tilde{N} = [{\tilde{n}}_{ij}]_{n \times m} = (\begin{matrix} {\tilde{n}}_{11} & {\tilde{n}}_{12} & \dots & {\tilde{n}}_{1 m} \\ {\tilde{n}}_{21} & {\tilde{n}}_{22} & \dots & {\tilde{n}}_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{n}}_{i 1} & {\tilde{n}}_{i 2} & \dots & {\tilde{n}}_{im} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{n}}_{n 1} & {\tilde{n}}_{n 2} & \dots & {\tilde{n}}_{nm} \end{matrix})$

where ${\tilde{n}}_{ij} = \frac{y_{ij}}{\sqrt{Σ_{i = 1}^{n} y_{ij}^{2}}}$ and obtaining the weighted vector $W = (y_{1}, y_{2}, \dots, y_{m})$ , where $y_{i} = \frac{y_{i}}{Σ_{α = 1}^{n} y_{α i}}$ and $y_{j} = \frac{Σ_{i = 1}^{n} {\tilde{n}}_{ij}}{n}$ .

Step 4: Acquire q-ROFS decision matrix

$Z_{i} = [z_{jk}^{i}]_{l \times m} = (\begin{matrix} z_{11}^{i} & z_{12}^{i} & \dots & z_{1 m}^{i} \\ z_{21}^{i} & z_{22}^{i} & \dots & z_{2 m}^{i} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ z_{j 1}^{i} & z_{j 2}^{i} & \dots & z_{jm}^{i} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ z_{l 1}^{i} & z_{l 2}^{i} & \dots & z_{lm}^{i} \end{matrix})$

where $z_{jk}^{i}$ is a q-ROFS-element, for i^th decision maker. And get the aggregating matrix $A = \frac{Z_{1} + Z_{2} + \dots + Z_{n}}{n} = [{\dot{z}}_{jk}]_{l \times m}$

Step 5: Acquire the weighted q-ROFS decision matrix

$J = [{\overset{˘}{z}}_{jk}]_{l \times m} = (\begin{matrix} {\overset{˘}{z}}_{11} & {\overset{˘}{z}}_{12} & \dots & {\overset{˘}{z}}_{1 m} \\ {\overset{˘}{z}}_{21} & {\overset{˘}{z}}_{22} & \dots & {\overset{˘}{z}}_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\overset{˘}{z}}_{j 1} & {\overset{˘}{z}}_{j 2} & \dots & {\overset{˘}{z}}_{jm} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\overset{˘}{z}}_{l 1} & {\overset{˘}{z}}_{l 2} & \dots & {\overset{˘}{z}}_{lm} \end{matrix})$

where ${\overset{˘}{z}}_{jk} = y_{k} \times {\dot{z}}_{jk}$ .

Step 6: Locate q-ROFS-valued positive ideal solution (q-ROFSV-PIS) and q-ROFS-valued negative ideal solution (q-ROFSV-NIS), employing in order $q - ROFSV - PIS = {{\ddot{z}}_{1}^{+}, {\ddot{z}}_{2}^{+}, \dots, {\ddot{z}}_{l}^{+}}$ $= {(\lor_{k} {\ddot{z}}_{jk}, \land_{k} {\ddot{z}}_{jk}) : k = 1, 2, \dots, m}$ and $q - ROFSV - NIS = {{\ddot{z}}_{1}^{-}, {\ddot{z}}_{2}^{-}, \dots, {\ddot{z}}_{l}^{-}}$ $= {(\land_{k} {\ddot{z}}_{jk}, \lor_{k} {\ddot{z}}_{jk}) : k = 1, 2, \dots, m}$ where ∨ stands for q-ROFS union and ∧ represents q-ROFS intersection.

Step 7: Find q-ROFS Euclidean distances to every alternative from q-ROFSV-PIS and q-ROFSV-NIS, respectively, $(t_{j}^{+})^{2} = Σ_{k = 1}^{m} {\frac{1}{2} (F_{jk}^{+} -^{+} F_{k}^{+})^{2} + \frac{1}{2} (T_{jk}^{+} -^{+} T_{k}^{+})^{2} + \frac{1}{2} (F_{jk}^{-} -^{+} F_{k}^{-})^{2} + \frac{1}{2} (T_{jk}^{-} -^{+} T_{k}^{-})^{2}}$ and $(t_{j}^{-})^{2} = Σ_{k = 1}^{m} {\frac{1}{2} (F_{jk}^{+} -^{-} F_{k}^{+})^{2} + \frac{1}{2} (T_{jk}^{+} -^{-} T_{k}^{+})^{2} + \frac{1}{2} (F_{jk}^{-} -^{-} F_{k}^{-})^{2} + \frac{1}{2} (T_{jk}^{-} -^{-} T_{k}^{-})^{2}}$

Step 8: Evaluate the coefficient of relative nearness related to every choice under discussion to best plausible solution by using $R^{*} ({\ddot{v}}_{j}) = \frac{t_{j}^{-}}{t_{j}^{+} + t_{j}^{-}} \in [0, 1]$

Step 9: Evaluate the choices in decrease (or increase) arrangement to acquire the alternatives preferential array.

The process diagram is shown in the Fig. 1:

Fig. 1

Procedural steps of Algorithm-1.

We apply this algorithm in the following example to demonstrate its effectiveness.

Example 4.1. Suppose a construction company intends to buy raw materials for the subway train project. For this, the company wants to choose the best supplier in the country.

Step 1: Understand the problem: Let $D = {M_{i} : i = 1, 2, 3, 4}$ be the set of civil engineers, $A = {{\ddot{z}}_{i} : i = 1, 2, \dots, 10}$ be the some collection of suppliers and C = {q_i : i = 1, 2, ⋯ , 6} is a assemblage of criterion, where $\begin{matrix} q_{1} & = & Standing reputation of the supplier \\ q_{2} & = & Modern equipment and technology \\ q_{3} & = & Strongand positive leadership \\ q_{4} & = & Production capability \\ q_{5} & = & Accountability for quality issues \\ q_{6} & = & Remains in compliance with industry \\ standards \end{matrix}$ Step 2: Taking into account the track history of these vendors, the company’s technical team recommends weights from each parameter by considering the fuzzy weights pre-decided in the form of linguistic terms rendered in tabular form in Table 1 $\begin{matrix} Q & = & [w_{ij}] \\ = & (\begin{matrix} HI & AI & VHI & LI & HI & AI \\ AI & LI & VHI & AI & LI & LI \\ LI & LI & HI & LI & VHI & HI \\ AI & HI & VHI & LI & LI & AI \end{matrix}) \\ = & (\begin{matrix} 0.80 & 0.50 & 0.95 & 0.35 & 0.80 & 0.50 \\ 0.50 & 0.35 & 0.95 & 0.50 & 0.35 & 0.35 \\ 0.35 & 0.35 & 0.50 & 0.35 & 0.95 & 0.80 \\ 0.50 & 0.80 & 0.95 & 0.35 & 0.35 & 0.50 \end{matrix}) \end{matrix}$ where w_ij is the weight assigned by the civil engineer D_i to every parameter p_j.

Step 3: The NW matrix is $\begin{matrix} \hat{N} & = & [{\hat{n}}_{ij}]_{4 \times 6} \\ = & (\begin{matrix} 0.712 & 0.469 & 0.552 & 0.445 & 0.598 & 0.445 \\ 0.445 & 0.329 & 0.552 & 0.636 & 0.262 & 0.312 \\ 0.311 & 0.329 & 0.291 & 0.445 & 0.711 & 0.712 \\ 0.445 & 0.751 & 0.552 & 0.445 & 0.262 & 0.445 \end{matrix}) \end{matrix}$ then, the weight vector is $W = (0.17, 0.18, 0.11, 0.25, 0.12, 0.17)$ .

Step 4: Especially considering the previous record of respective companies, the q-ROFS decision matrix Z_i of every specialist is given in which alternatives are shown row-wise whereas parameters are represented column-wise.

$A = \frac{Z_{1} + Z_{2} + Z_{3} + Z_{4}}{4} = [{\dot{z}}_{jk}]_{10 \times 6}$ Now, the aggregated decision matrix can be written as

(0.45,0.72)	(0.77,0.34)	(0.94,0.18)	(0.76,0.23)	(0.36,0.46)	(0.75,0.59)
(0.73,0.19)	(0,1)	(0.69,0.38)	(0.68,0.45)	(0.87,0.17)	(0,1)
(0.23,0.75)	(0.89,0.15)	(0.56,0.76)	(0.67,0.48)	(0.66,0.44)	(0.55,0.64)
(0.65,0.62)	(0.29,0.34)	(0.67,0.77)	(0.46,0.33)	(0.64,0.77)	(0.81,0.25)
(0.46,0.67)	(0.44,0.77)	(0.45,0.71)	(0.56,0.45)	(0,1)	(0.45,0.36)
(0.92,0.12)	(0.74,0.46)	(0.67,0.61)	(0.67,0.44)	(0.98,0.23)	(0.23,0.46)
(0.87,0.41)	(0.91,0.28)	(0,1)	(0.55,0.44)	(0.27,0.32)	(0.25,0.36)
(0.97,0.29)	(0.97,0.32)	(0.67,0.55)	(0,1)	(0.66,0.76)	(0.22,0.22)
(0.74,0.27)	(0.55,0.71)	(0.56,0.67)	(0.78,0.39)	(0.17,0.44)	(0.71,0.38)
(0.33,0.61)	(0.83,0.29)	(0.23,0.67)	(0.17,0.34)	(0.36,0.16)	(0.37,0.22)

Step 5: The weighted q-ROFS decision matrix $J = [{\ddot{z}}_{jk}]_{10 \times 6}$ can be written as

(0.08,0.12)	(0.14,0.06)	(0.10,0.02)	(0.19,0.06)	(0.04,0.06)	(0.13,0.10)
(0.12,0.03)	(0.00,0.18)	(0.08,0.04)	(0.18,0.11)	(0.10,0.02)	(0.00,0.17)
(0.04,0.13)	(0.16,0.03)	(0.06,0.08)	(0.17,0.12)	(0.08,0.05)	(0.09,0.11)
(0.12,0.11)	(0.05,0.06)	(0.07,0.08)	(0.12,0.08)	(0.08,0.09)	(0.14,0.04)
(0.08,0.11)	(0.08,0.14)	(0.05,0.08)	(0.14,0.11)	(0.00,0.12)	(0.08,0.06)
(0.15,0.02)	(0.13,0.08)	(0.07,0.07)	(0.17,0.11)	(0.12,0.03)	(0.04,0.08)
(0.15,0.07)	(0.16,0.05)	(0.00,0.11)	(0.13,0.11)	(0.03,0.04)	(0.04,0.06)
(0.13,0.05)	(0.17,0.06)	(0.07,0.06)	(0.00,0.25)	(0.08,0.09)	(0.08,0.08)
(0.10,0.05)	(0.10,0.13)	(0.06,0.07)	(0.2,0.1)	(0.02,0.05)	(0.12,0.06)
(0.56,0.10)	(0.15,0.05)	(0.01,0.07)	(0.04,0.09)	(0.04,0.02)	(0.06,0.04)

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

Step 6: Now, we evaluate q-ROFS valued positive ideal solution (q-ROFSV-PIS) and q-ROFS-valued negative ideal solution (q-ROFSV-NIS) and are listed, respectively, as $q - ROFSV - PIS = {{\ddot{z}}_{1}^{+}, \dots, {\ddot{z}}_{10}^{+}} = {(0.19, 0.02), (0.18, 0.02), (0.17, 0.03), (0.14, 0.04), (0.14, 0.06), (0.17, 0.02), (0.16, 0.04), (0.17, 0.05), (0.13, 0.05), (0.15, 0.02)}$ and $q - ROFSV - NIS = {{\ddot{z}}_{1}^{-}, \dots, {\ddot{z}}_{10}^{-}} = {(0.04, 0.12), (0.00, 0.18), (0.04, 0.13), (0.05, 0.11), (0.00, 0.14), (0.04, 0.11), (0.00, 0.11), (0.00, 0.25), (0.02, 0.13), (0.01, 0.10)}$

Step 7, 8: The q-ROFS distances of every choice from q-ROFSV-PIS and q-ROFSV-NIS including the closeness coefficients can be seen in Table 2:

Table 2

Distance measures & closeness coefficient of each alternative

Alternative $({\ddot{z}}_{i})$	$t_{i}^{+}$	$t_{i}^{-}$	$R_{i}^{*}$
${\ddot{z}}_{1}$	0.187083	0.18303	0.4945
${\ddot{z}}_{2}$	0.265989	0.260672	0.4950
${\ddot{z}}_{3}$	0.188414	0.166132	0.4686
${\ddot{z}}_{4}$	0.119164	0.120000	0.5017
${\ddot{z}}_{5}$	0.405172	0.404418	0.4995
${\ddot{z}}_{6}$	0.157003	0.150665	0.4899
${\ddot{z}}_{7}$	0.185876	0.200125	0.5185
${\ddot{z}}_{8}$	0.221246	0.345615	0.6097
${\ddot{z}}_{9}$	0.127083	0.205791	0.6182
${\ddot{z}}_{10}$	0.193778	0.141244	0.4216

Step 9: Therefore, the preference initial order of the alternatives is ${\ddot{z}}_{9} ≻ {\ddot{z}}_{8} ≻ {\ddot{z}}_{7} ≻ {\ddot{z}}_{4} ≻ {\ddot{z}}_{5} ≻ {\ddot{z}}_{2} ≻ {\ddot{z}}_{1} ≻ {\ddot{z}}_{6} ≻ {\ddot{z}}_{3} ≻ {\ddot{z}}_{10}$ So, ${\ddot{z}}_{9}$ is the best alternative.

5 MCDM with q-ROFSS VIKOR approach

VIKOR (Vlse Kriterijumska Optimizacija Kompromisno Resenje) is a Serbian term that refers to different parameters of optimization and compromise. It was initially conceived by Serafim Opricovic to solve decision-making problems with contrasting and non-commensurable (different units) requirements, presuming that compromise is appropriate for conflict resolution, the decision-maker seems to want a solution that is the nearest to the ideal, and the alternatives are analyzed according to all indicators. VIKOR ranks the alternatives and calculates the solution referred to as a compromise that is the closest to the ideal.

We shall begin by illustrating the suggested technique step by step:

Algorithm-2 (q-ROFSS VIKOR)

Step 1: Analyzing the issue: Let $D = {M_{i} : i \in N}$ be the set of decision makers, $A = {{\ddot{k}}_{i} : i \in N}$ be the finite collection of alternatives and C = {q_i : i ∈ mathdsN} be the assemblage of parameters.

Step 2: Choosing the linguistic terms as well from the Table 1, acquiring weighted parameter matrix $Q$ as

$Q = [w_{ij}]_{n \times m} = (\begin{matrix} y_{11} & y_{12} & \dots & y_{1 m} \\ y_{21} & y_{22} & \dots & y_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ y_{i 1} & y_{i 2} & \dots & y_{im} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ y_{n 1} & y_{n 2} & \dots & y_{nm} \end{matrix})$

Step 3: Acquired NW matrix

where ${\tilde{n}}_{ij} = \frac{y_{ij}}{\sqrt{Σ_{i = 1}^{n} y_{ij}^{2}}}$ and obtaining the weighted vector $W = (y_{1}, y_{2}, \dots, y_{m})$ , where $y_{i} = \frac{y_{i}}{Σ_{α = 1}^{n} w_{α i}}$ and $y_{j} = \frac{Σ_{i = 1}^{n} {\tilde{n}}_{ij}}{n}$ .

Step 4: Acquire q-ROFS decision matrix

where $z_{jk}^{i}$ is a q-ROFS-element, for i^th decision maker. Then acquire the aggregating matrix $A = \frac{Z_{1} + Z_{2} + \dots + Z_{n}}{n} = [{\dot{z}}_{jk}]_{l \times m}$ Step 5: Find the weighted q-ROFS decision matrix

$J = [{\overset{˘}{℧}}_{jk}]_{l \times m} = [y_{k} \times {\dot{z}}_{jk}]_{l \times m}$

Step 6: Mark q-ROFS-valued +ve ideal solution (q-ROFSV-PIS) and q-ROFS-valued -ve ideal solution (q-ROFSV-NIS), employing in order $\begin{matrix} {q - ROFSV - PIS \end{matrix} = {{\ddot{℧}}_{1}^{+}, {\ddot{℧}}_{2}^{+}, \dots, {\ddot{℧}}_{l}^{+}} = {(max_{j} {\ddot{℧}}_{jk}, min_{k} {\ddot{℧}}_{jk}) : j = 1, \dots, l}$ and $\begin{matrix} {q - ROFSV - NIS \end{matrix} = {{\ddot{℧}}_{1}^{-}, {\ddot{℧}}_{2}^{-}, \dots, {\ddot{℧}}_{l}^{-}} = {(min_{j} {\ddot{℧}}_{jk}, max_{k} {\ddot{℧}}_{jk}) : j = 1, \dots, l}$ Step 7: Evaluate the group utility value $S_{i}$ for each alternative, the individual regret value $R_{i}$ , and compromise value $L_{i}$ by making use of $\begin{matrix} S_{i} & = & Σ_{j = 1}^{m} w_{j} (\frac{d ({\ddot{℧}}_{j}^{+}, {\ddot{℧}}_{ij})}{d ({\ddot{℧}}_{j}^{+}, {\ddot{℧}}_{j}^{-})}) \\ R_{i} & = & {max}_{j = 1}^{m} w_{j} (\frac{d ({\ddot{℧}}_{j}^{+}, {\ddot{℧}}_{ij})}{d ({\ddot{℧}}_{j}^{+}, {\ddot{℧}}_{j}^{-})}) \\ L_{i} & = & K (\frac{S_{i} - S^{-}}{S^{+} - S^{-}}) + (1 - K) (\frac{R_{i} - R^{-}}{R^{+} - R^{-}}) \end{matrix}$ where $S^{+} = max_{i} S_{i}$ , $S^{-} = min_{i} S_{i}$ , $R^{+} = max_{i} R_{i}$ , and $R^{-} = min_{i} R_{i}$ . The value of the coefficient of decision mechanism $K$ is is decided such that if compromise solution is to be selected by majority, we choose $K > 0.5$ ; for consensus we use $K = 0.5$ , and $K < 0.5$ represents veto. $w_{j}$ represents the weight of the j^th criteria, which expresses its relative importance. Step 8: Evaluate the choices and find a compromise solution. Arrange $S_{i}$ , $R_{i}$ , and $L_{i}$ in increasing order to make three ranking lists $S_{[.]}$ , $R_{[.]}$ , and $L_{[.]}$ . The alternative ${\ddot{℧}}_{α}$ will be declared compromise solution if it ranks the best in $L_{[.]}$ (having least value) and satisfies the following two requirements simultaneously:

(1) Feasible benefit:

If top most two alternatives in $L_{i}$ are ${\ddot{℧}}_{α}$ and ${\ddot{℧}}_{β}$ , then $L ({\ddot{℧}}_{β}) - L ({\ddot{℧}}_{α}) \geq \frac{1}{n - 1}$ where n stands for the cardinality of the set of attributes.

(2) Acceptance stability:

The choice ${\ddot{℧}}_{α}$ must be top ranked by at least one of $S_{i}$ and $R_{i}$ .

If the requirement (1) and (2) are not met at the same time, then several compromise solutions exist:

(i) If only (1) is met then both alternatives ${\ddot{℧}}_{α}$ and ${\ddot{℧}}_{β}$ will serve as the compromise solutions.

(ii) If (1) is not met then there will be a series of compromise solutions, which are ${\ddot{℧}}_{α}, {\ddot{℧}}_{β}, \dots, {\ddot{℧}}_{T}$ . ${\ddot{℧}}_{T}$ may be located by making use of $L ({\ddot{℧}}_{T}) - L ({\ddot{℧}}_{α}) \geq \frac{1}{n - 1}$ for the maximum.

The process diagram is shown in the Fig. 2.

Fig. 2

Procedural steps of Algorithm 2.

Example 5.1. We again solve Example 4.1 usage of VIKOR process using Algorithm 2. First five steps are similar to those given in Example 4.1. So we begin with step 6.

Step 6: We find q-ROFSV-PIS and q-ROFSV-NIS, listed as {q-ROFSV - PIS = { ${\overset{..}{℧}}_{1}^{+}, {\overset{..}{℧}}_{2}^{+}, {\overset{..}{℧}}_{3}^{+}, {\overset{..}{℧}}_{4}^{+}, {\overset{..}{℧}}_{5}^{+}, {\overset{..}{℧}}_{6}^{+}} = {$ (0.16, 0.02), (0.17, 0.03), (0.10, 0.02), (0.19, 0.06), (0.12, 0.02), (0.14, 0.04) }and{q-ROFSV-NIS} = { ${\overset{..}{℧}}_{1}^{-}, {\overset{..}{℧}}_{2}^{-}, {\overset{..}{℧}}_{3}^{-}, {\overset{..}{℧}}_{4}^{-}, {\overset{..}{℧}}_{5}^{-}, {\overset{..}{℧}}_{6}^{-}} = {$ (0.06, 0.13), (0.00, 0.18), (0.00, 0.11), (0.00, 0.25), (0.00, 0.12), (0.00, 0.17) }.

Step7 : Let′schoose $K$ = 0.5. Then the value of group utility $S_{i}$ , the individual regret value $R_{i}$ , and compromise value $L_{i}$ for each alternative ${\overset{..}{℧}}_{i}$ are computed by making use of $\begin{matrix} S_{i} & = & Σ_{j = 1}^{m} y_{j} (\frac{t ({\ddot{℧}}_{j}^{+}, {\ddot{℧}}_{ij})}{t ({\ddot{℧}}_{j}^{+}, {\ddot{℧}}_{j}^{-})}) \\ R_{i} & = & {max}_{j = 1}^{m} y_{j} (\frac{t ({\ddot{℧}}_{j}^{+}, {\ddot{℧}}_{ij})}{t ({\ddot{℧}}_{j}^{+}, {\ddot{℧}}_{j}^{-})}) \\ L_{i} & = & K (\frac{S_{i} - S^{-}}{S^{+} - S^{-}}) + (1 - K) (\frac{R_{i} - R^{-}}{R^{+} - R^{-}}) \end{matrix}$ and are given in Table 3 below:

Table 3

Values of $S_{i}, R_{i}$ and $L_{i}$ $ for all alternatives

Alternative ( ${\overset{..}{℧}}_{i}$ )	$S_{i}$	$R_{i}$	$L_{i}$
${\overset{..}{℧}}_{1}$	0.3029	0.1464	0.4116
${\overset{..}{℧}}_{2}$	0.4918	0.1800	0.8702
${\overset{..}{℧}}_{3}$	0.4376	0.1862	0.8043
${\overset{..}{℧}}_{4}$	0.4078	0.1126	0.4488
${\overset{..}{℧}}_{5}$	0.5686	0.1377	0.8263
${\overset{..}{℧}}_{6}$	0.2728	0.0594	0.0000
${\overset{..}{℧}}_{7}$	0.4337	0.1100	0.4818
${\overset{..}{℧}}_{8}$	0.5214	0.0619	0.4306
${\overset{..}{℧}}_{9}$	0.3487	0.0969	0.2838
${\overset{..}{℧}}_{10}$	0.5545	0.1687	0.9293

Step 8: The choices are ranked as solution. Initially given below: By $L_{i}$ : ${\overset{..}{℧}}_{6} ≺ {\overset{..}{℧}}_{9} ≺ {\overset{..}{℧}}_{1} ≺ {\overset{..}{℧}}_{8} ≺ {\overset{..}{℧}}_{4} ≺ {\overset{..}{℧}}_{7} ≺ {\overset{..}{℧}}_{3} ≺ {\overset{..}{℧}}_{5} ≺ {\overset{..}{℧}}_{2} ≺ {\overset{..}{℧}}_{10} By S_{i}$ : ${\overset{..}{℧}}_{6} ≺ {\overset{..}{℧}}_{1} ≺ {\overset{..}{℧}}_{9} ≺ {\overset{..}{℧}}_{4} ≺ {\overset{..}{℧}}_{7} ≺ {\overset{..}{℧}}_{3} ≺ {\overset{..}{℧}}_{2} ≺ {\overset{..}{℧}}_{8} ≺ {\overset{..}{℧}}_{10} ≺ {\overset{..}{℧}}_{5}$ By $R_{i}$ : ${\overset{..}{℧}}_{6} ≺ {\overset{..}{℧}}_{8} ≺ {\overset{..}{℧}}_{9} ≺ {\overset{..}{℧}}_{7} ≺ {\overset{..}{℧}}_{4} ≺ {\overset{..}{℧}}_{5} ≺ {\overset{..}{℧}}_{1} ≺ {\overset{..}{℧}}_{10} ≺ {\overset{..}{℧}}_{2} ≺ {\overset{..}{℧}}_{3}$ Since $L ({\overset{..}{℧}}_{9}) - L ({\overset{..}{℧}}_{6}$ ) = 0.2838 ≥ $\frac{1}{5}$ so the condition C-1 is satisfied. So, the alternative ${\overset{..}{℧}}_{9}$ is the best one and it is to be selected.

6 Application of q-ROFSS aggregation operators

Decision making is a part of our everyday lives. Decision-making is a process of choosing the best among the various alternatives. Every person in the world needs to make decisions. Whatever you are; whether you are a teacher, a management officer, a student or even a child; you are making decisions in your daily routine. Any policy or plan is established through decision making. No plan can fulfil its requirement without decision making. Therefore, decision making is very useful to achieve goals in planning. Decision making is also used to choose the best alternatives among all considered alternatives. So decision makers evaluate various advantages and disadvantages of every alternative and select the best alternative. Decision making is used in group behavior (sociology) and social psychology, utility and probability (economics/statistics), individual behavior (psychology), modeling and simulation (mathematics), environment (law anthropology political science) and values and ethics (philosophy).

We can make a MCGDM based on q-ROFSSs using the following algorithm:

6.1 Algorithm-3 (q-ROFSS aggregation operators)

Step 1: Consider input, ℋ = { ${\overset{..}{▾}}_{i}$ : i = 1, 2, ..., r} as collection of objects, E = { ${\overset{..}{e}}_{i}$ : i = 1, 2, ..., s } as set of attributes and a team of decision expert D = {D_i: i = 1, 2, ..., t}.

Step 2: Acquired q-ROFSS ¥_z: i = 1, 2, ..., t by each decision expert D = {D_i: i = 1, 2, ..., t}over the universe ℋ.

Step 3: Find intersection of all ¥_z: i = 1, 2, ..., t for unanimous decision by team of decision experts and obtain q-ROFSS φ_Hover ℋ.

Step 4: Find the cardinalities and cardinal set c φ_Hof φ_H.

Step 5: Find aggregate q-ROFSS $φ_{H}^{*}$ of φ_H.

Step 6: Evaluate the score values corresponding to each member of ℋ.

Step 7: Select the best alternative by using max operation.

The process diagram is shown in the Fig. 3.

We apply Algorithm-1 in the following example of MCDM.

Fig. 3

Procedural steps of Algorithm 3.

Example 6.1. Assume that a student wishes to get admission in university. The list of universities under consideration is ℋ = { ${\overset{..}{▾}}_{i}$ : i = 1, 2, ..., 10}. After meeting with some educational consults. He considers the features from the set E = { ${\overset{..}{e}}_{i}$ : i = 1, 2, ..., 5 }, where $\begin{matrix} {\ddot{e}}_{1} & = & Highly qualified faculty \end{matrix} {\ddot{e}}_{2} = Good financial aid {\ddot{e}}_{3} = Travel opportunities for students {\ddot{e}}_{4} = Internships and research opportunities {\ddot{e}}_{5} = Strong alumni network$ After pondering upon the pros and cons, each university is assessed according to a subset H = { ${\overset{..}{e}}_{1}, {\overset{..}{e}}_{3}, {\overset{..}{e}}_{4}$ } ⊆ E of parameters, and take q=3. Also consider a team of decision expert D = {D_i: i = 1, 2, 3}. He applies the above algorithm as follows:

Step-1: Take input ℋ = { ${\overset{..}{▾}}_{i}$ : i = 1, 2, ..., 10} as collection of objects, E = { ${\overset{..}{e}}_{1} {\overset{..}{e}}_{3}, {\overset{..}{e}}_{4}$ , } as set of attributes and a team of decision expert D = {D₁, D₂, D₃ }.

Step-2: Acquired q-ROFSS ¥_z: i = 1, 2, ..., t by each decision expert.

¥₁ =

{ ( ${\overset{..}{e}}_{1}, {\frac{(0.70, 0.40)}{{\overset{..}{▾}}_{1}}, \frac{(0.90, 0.15)}{{\overset{..}{▾}}_{2}}, \frac{(0.25, 0.75)}{{\overset{..}{▾}}_{3}}, \frac{(0.20, 0.45)}{{\overset{..}{▾}}_{4}}, \frac{(0.65, 0.40)}{{\overset{..}{▾}}_{7}}, \frac{(0.60, 0.25)}{{\overset{..}{▾}}_{8}}, \frac{(0.60, 0.35)}{{\overset{..}{▾}}_{7}}$ }), ( ${\overset{..}{e}}_{3}, {\frac{(0.90, 0.25)}{{\overset{..}{▾}}_{1}}, \frac{(0.75, 0.45)}{{\overset{..}{▾}}_{2}}, \frac{(0.40, 0.65)}{{\overset{..}{▾}}_{6}}, \frac{(0.30, 0.65)}{{\overset{..}{▾}}_{9}}, \frac{(0.75, 0.25)}{{\overset{..}{▾}}_{10}}$ }), ( ${\overset{..}{e}}_{4}, {\frac{(0.90, 0.15)}{{\overset{..}{▾}}_{1}}, \frac{(0.65, 0.35)}{{\overset{..}{▾}}_{2}}, \frac{(0.60, 0.75)}{{\overset{..}{▾}}_{3}}, \frac{(0.80, 0.35)}{{\overset{..}{▾}}_{5}}, \frac{(0.30, 0.25)}{{\overset{..}{▾}}_{7}}$ }) }. ¥₂ = { ( ${\overset{..}{e}}_{1}, {\frac{(0.95, 0.20)}{{\overset{..}{▾}}_{2}}, \frac{(0.20, 0.50)}{{\overset{..}{▾}}_{4}}, \frac{(0.35, 0.60)}{{\overset{..}{▾}}_{7}}, \frac{(0.75, 0.30)}{{\overset{..}{▾}}_{8}}, \frac{(0.85, 0.40)}{{\overset{..}{▾}}_{10}}$ }), ( ${\overset{..}{e}}_{3}, {\frac{(0.95, 0.30)}{{\overset{..}{▾}}_{1}}, \frac{(0.55, 0.70)}{{\overset{..}{▾}}_{6}}, \frac{(0.65, 0.80)}{{\overset{..}{▾}}_{9}}$ }), ( ${\overset{..}{e}}_{4}, {\frac{(0.95, 0.20)}{{\overset{..}{▾}}_{1}}, \frac{(0.65, 0.80)}{{\overset{..}{▾}}_{3}}, \frac{(0.85, 0.40)}{{\overset{..}{▾}}_{5}}, \frac{(0.65, 0.40)}{{\overset{..}{▾}}_{7}}$ }) }.

and

¥₃ =

{ ( ${\overset{..}{e}}_{1}, {\frac{(0.75, 0.55)}{{\overset{..}{▾}}_{1}}, \frac{(0.85, 0.35)}{{\overset{..}{▾}}_{3}}, \frac{(0.65, 0.45)}{{\overset{..}{▾}}_{5}}, \frac{(0.20, 0.55)}{{\overset{..}{▾}}_{9}},$ }), ( ${\overset{..}{e}}_{3}, {\frac{(0.45, 0.75)}{{\overset{..}{▾}}_{2}}, \frac{(0.55, 0.75)}{{\overset{..}{▾}}_{3}}, \frac{(0.55, 0.75)}{{\overset{..}{▾}}_{5}}, \frac{(0.45, 0.65)}{{\overset{..}{▾}}_{7}}$ , $\frac{(0.40, 0.65)}{{\overset{..}{▾}}_{10}}$ }),

( ${\overset{..}{e}}_{4}, {\frac{(0.55, 0.75)}{{\overset{..}{▾}}_{2}}, \frac{(0.40, 0.55)}{{\overset{..}{▾}}_{6}}, \frac{(0.70, 0.45)}{{\overset{..}{▾}}_{9}}, \frac{(0.75, 0.15)}{{\overset{..}{▾}}_{10}},$ }) }.

Step-3: Compute intersection of all ¥_z and obtain q-ROFSS φ_Hover the universe ℋ as

φ_H =

{ ( ${\overset{..}{e}}_{1}, {\frac{(0.9, 0.2)}{{\overset{..}{▾}}_{2}}, \frac{(0.8, 0.5)}{{\overset{..}{▾}}_{4}}, \frac{(0.2, 0.6)}{{\overset{..}{▾}}_{7}}, \frac{(0.6, 0.3)}{{\overset{..}{▾}}_{8}}, \frac{(0.6, 0.4)}{{\overset{..}{▾}}_{10}}$ }), ( ${\overset{..}{e}}_{3}, {\frac{(0.9, 0.3)}{{\overset{..}{▾}}_{1}}, \frac{(0.4, 0.7)}{{\overset{..}{▾}}_{6}}, \frac{(0.3, 0.8)}{{\overset{..}{▾}}_{9}}$ }),

( ${\overset{..}{e}}_{4}, {\frac{(0.9, 0.2)}{{\overset{..}{▾}}_{1}}, \frac{(0.6, 0.8)}{{\overset{..}{▾}}_{3}}, \frac{(0.8, 0.4)}{{\overset{..}{▾}}_{5}}, \frac{(0.3, 0.4)}{{\overset{..}{▾}}_{7}}$ }) }.

Step-4: The cardinal set of φ_His

c φ_H = { $\frac{(0.31, 0.20)}{{\overset{..}{e}}_{1}}, \frac{(0.16, 0.18)}{{\overset{..}{e}}_{3}}, \frac{(0.26, 0.18)}{{\overset{..}{e}}_{4}}$ }.

Step-5: By virtue of Theorem 3.18, the aggregate q-ROFSS $φ_{H}^{*}$ of φ_His $\begin{matrix} M_{φ_{H}}^{*} & = & \frac{M_{φ_{H}} \times M_{c φ_{H}}^{t}}{| E |} \\ = & {(\begin{matrix} 0.0756 \\ 0.0558 \\ 0.0312 \\ 0.0496 \\ 0.0416 \\ 0.0128 \\ 0.028 \\ 0.0372 \\ 0.0096 \\ 0.0372 \end{matrix}), (\begin{matrix} 0.018 \\ 0.008 \\ 0.0288 \\ 0.02 \\ 0.0014 \\ 0.0252 \\ 0.0384 \\ 0.012 \\ 0.0288 \\ 0.016 \end{matrix})} \end{matrix}$ Hence, $φ_{H}^{*} = {\frac{(0.0756, 0.018)}{{\overset{..}{▾}}_{1}}, \frac{(0.0558, 0.008)}{{\overset{..}{▾}}_{2}}, \frac{(0.0312, 0.0288)}{{\overset{..}{▾}}_{3}}, \frac{(0.0496, 0.02)}{{\overset{..}{▾}}_{4}}, \frac{(0.0416, 0.0144)}{{\overset{..}{▾}}_{5}}, \frac{(0.0128, 0.0252)}{{\overset{..}{▾}}_{6}}, \frac{(0.028, 0.0384)}{{\overset{..}{▾}}_{7}}, \frac{(0.0372, 0.012)}{{\overset{..}{▾}}_{8}}, \frac{(0.0096, 0.0288)}{{\overset{..}{▾}}_{9}}, \frac{(0.0372, 0.016)}{{\overset{..}{▾}}_{10}}} .$

Step-6: The values of the score function S( ${\overset{..}{▾}}_{i}$ ), using Definition 2.6, for each element of ℋ are tabulated in Table 4:

Table 4

Score values for each member of ℋ

$\overset{..}{▾}$	S( ${\overset{..}{▾}}_{i}$ )
${\overset{..}{▾}}_{1}$	0.000426
${\overset{..}{▾}}_{2}$	0.000173
${\overset{..}{▾}}_{3}$	0.0000065
${\overset{..}{▾}}_{4}$	0.000114
${\overset{..}{▾}}_{5}$	0.000069
${\overset{..}{▾}}_{6}$	-0.0000139
${\overset{..}{▾}}_{7}$	-0.0000347
${\overset{..}{▾}}_{8}$	0.000050
${\overset{..}{▾}}_{9}$	-0.000023
${\overset{..}{▾}}_{10}$	0.000047

Step-7: We observe that max {S( ${\overset{..}{▾}}_{i})} = 0.000426 that corresponds to {\overset{..}{▾}}_{1}, so the university {\overset{..}{▾}}_{1}$ should be chosen for admission.

6.2 Comparison analysis

The proposed approaches of TOPSIS, VIKOR and aggregation operators are compared as presented in Table 5. It can be noted in the comparison Table 5, the selected alternative given by any one MCDM approach acknowledges the authenticity and effectiveness of the proposed algorithms.

Table 5
Final classification review relative to current methods

Methods The optimal alternative

Algorithm-1 (q-ROFS aggregation operators) ${\overset{..}{℧}}_{9}$

Algorithm-2 (q-ROFSS TOPSIS) ${\overset{..}{℧}}_{9}$

Algorithm-3 (q-ROFSS VIKOR) ${\overset{..}{℧}}_{9}$

Algorithm (Eraslan and Karaaslan [10]) ${\overset{..}{℧}}_{9}$

Algorithm (Naeem et al. [26]) ${\overset{..}{℧}}_{9}$

Algorithm (Peng and Dai [32]) ${\overset{..}{℧}}_{9}$

Algorithm (Tehrim and Riaz [40]) ${\overset{..}{℧}}_{9}$

Algorithm (Zhang and Xu [53]) ${\overset{..}{℧}}_{9}$

Methods	The optimal alternative
Algorithm-1 (q-ROFS aggregation operators)	${\overset{..}{℧}}_{9}$
Algorithm-2 (q-ROFSS TOPSIS)	${\overset{..}{℧}}_{9}$
Algorithm-3 (q-ROFSS VIKOR)	${\overset{..}{℧}}_{9}$
Algorithm (Eraslan and Karaaslan [10])	${\overset{..}{℧}}_{9}$
Algorithm (Naeem et al. [26])	${\overset{..}{℧}}_{9}$
Algorithm (Peng and Dai [32])	${\overset{..}{℧}}_{9}$
Algorithm (Tehrim and Riaz [40])	${\overset{..}{℧}}_{9}$
Algorithm (Zhang and Xu [53])	${\overset{..}{℧}}_{9}$

7 Conclusion

We introduced the novel concept of q-rung orthopair fuzzy soft set (q-ROFSS). We established some fundamental properties of q-ROFSSs and explained them with the help of some examples. The proposed model of q-ROFSS is an important technique to deal with uncertainties by means of parametrization along with membership and non-membership in the larger space. We proposed three algorithms for MCDM under q-ROFSS environment. One of them is the use of the q-ROFS aggregation operator and the technique based on the function values. We presented ranking methodology by using q-ROFS aggregation operator for the university selection. Two other proposed algorithms are designed for MCDM named as q-ROFS TOPSIS and q-ROFS VIKOR, respectively. We tackled the problem of supplier selection for a construction company using these two techniques. The comparison analysis of TOPSIS, VIKOR and aggregation operators for q-rung orthopair fuzzy soft sets is presented with some existing MCDM methods to justify the selection of optimal alternative.

In the future, we will extend our work to solve other MCDM problems by using different approaches like AHP, COMET, ELECTRE family and PROMETHEE family based on different extensions of fuzzy sets like Pythagorean m-polar fuzzy sets, linear Diophantine fuzzy sets and cubic m-polar fuzzy sets.

Footnotes

Acknowledgment

The authors would like to thank the Editor-in-Chief and the referees for their valuable suggestions and recommendations for making our results more accurate.

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TOPSIS,VIKOR and aggregation operators based on q-rung orthopair fuzzy soft sets and their applications

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Operational laws of q-ROFs

2.2 Operational laws of q-ROFNs

3 q-Rung orthopair fuzzy soft sets

4 MCDM using q-ROFSS TOPSIS methodology

Table 1 Linguistic terms for judging alternatives Semantic Terms Fuzzy Weights Very Highly Important (VHI) 0.95 Highly Important (HI) 0.80 Averagely Important (AI) 0.50 Less Important (LI) 0.35

6.1 Algorithm-3 (q-ROFSS aggregation operators)

Footnotes

Acknowledgment

References

Table 1
Linguistic terms for judging alternatives

Semantic Terms Fuzzy Weights

Very Highly Important (VHI) 0.95

Highly Important (HI) 0.80

Averagely Important (AI) 0.50

Less Important (LI) 0.35