Covering based q-rung orthopair fuzzy rough set model hybrid with TOPSIS for multi-attribute decision making

Abstract

In this manuscript we develop a comprehensive model to tackle decision making problems where strong points of view are in the favor and against some projects, entities or plans. Therefore a new approach is adopted to hybrid q-rung orthopair fuzzy sets with notions of covering rough set and TOPSIS. This model has stronger capability than intuitionistic fuzzy set and Pythagorean fuzzy set to manage the uncertainty. Furthermore an example is given to demonstrate how the proposed method helps us in decision making problems.

Keywords

Fuzzy sets q-rung orthopair fuzzy sets covering based q-rung orthopair fuzzy rough set q-rung orthopair fuzzy TOPSIS methodology

1. Introduction

Notion of fuzzy sets introduced by Zadeh [29] revolutionized not only mathematics and logic but also science and technology. It is a very nice tool to handle uncertainty. Here some membership grade is assigned to an object of a fuzzy sets. In many situations of real world, apart from the grade of membership, the grade of nonmembership is also required. To handle such conditions, Atanassov in [2] initiate the notion of intuitionistic fuzzy sets (IFSs), which is the significant improvement of fuzzy sets. In IFSs the sum of membership grade and nonmembership grade of an object is always from the unit interval [0, 1]. However, the fascinating scenario emerges when the membership and nonmembership of an object is given from the unit interval [0, 1] but their sum exceeds 1 Ordinary IFSs fail to handle such situations. Therefore a more comprehensive model is required for such situations.

Yager enquired this scenario in [23, 24] and improved the concept of IFSs to Pythagorean fuzzy sets (PFSs), which could be considered as a generalization of IFSs. The main difference between IFSs and PFSs is that, in IFSs the sum of membership and nonmembership which is always from the unit closed interval [0, 1], but in PFSs the sum of squares of membership grade and nonmembership grade are real numbers between 0 and 1 In many situations of real life the decision makers are bound to the constraints of PFSs and they did not assign values to the membership and nonmembership grades freely on their own choices, so some more comprehensive model is needed which cope these restrictions for the decision makers. Therefore, Yager [25, 26] investigated a more generalized concept than both IFSs and PFSs. This new concept is called q-rung orthopair fuzzy sets (q-ROFSs in short). Here the sum of qth power of membership grade and qth power of nonmembership grade belongs to the unit interval [0, 1]. Ali [1] presented the notion of orbits in q-ROFSs. Concept of q-ROFSs gives more space to the decision makers for the selection of membership and nonmembership.

TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) is one of the standard decision making methods, having simple mathematical calculations. Hwang and Yoon [28] presented that TOPSIS can handle multi-attribute decision making (MADM) problem, where the target is to get an object which has highest score value known as a positive ideal solution (PIS) and least score value known as a negative ideal solution (NIS). Zhang and Xu [33] generalized the notion of (TOPSIS) for PFSs and for details see [9 , 14–16] Zhang et al. [32] proposed the study of a novel optimization-based consensus model with heterogeneous preference structures for group decision making and for more study see [19, 21]

The fundamental notion of rough sets, is initiated by Pawalk [15] Dubios and Prade [6] presented the fuzzy rough sets (FRSs) and rough fuzzy sets. Liu and Lin [10] studied roughness in IFS on the bases of conflict distance. Nowadays many researchers are working on covering based rough sets (CRSs) models. Zakowski [30] presented the notion of CRSs, which is a generalization of Pawlak rough sets. Moreover Xu and Zhang [22] put forward a new CRSs models which is based on the measure of roughness. Liu and Sai [11] made the comparison between different CRSs models presented in [22] and [34, 35] Wang et al. [17] studied and improved attribute reduction scheme with the help of CRSs. Many researchers have studied covering based fuzzy rough sets (CFRSs). D’eer et al. [4, 5] presented the concept of fuzzy neighborhoods and fuzzy β-neighborhoods. Ma [12] introduced the generalized structure of CFRSs. Then we have presented the q-rung orthopair fuzzy TOPSIS (q-ROF-TOPSIS) methodology for the MADM problem which depend on the cover based q-rung orthopair fuzzy sets (Cq-ROFRSs) model. The remaining of the manuscript is organized as.

The arrangement of the paper is summarized as following: Section 2, presents some basic concepts of IFSs and their generalization. In Section 3, the notion of Cq-ROFRSs models by means of q-ROF β-neighborhoods (q-ROF β-neighborhoods) is presented. In Section 4, based on the analysis of Cq-ROFRS models, the q-ROF-TOPSIS method is introduced to solve MADM problem by applying q-rung orthopair fuzzy sets. In Section 5, illustrative example is given to demonstrate how q-ROF-TOPSIS methodology works in decision making problems.

1.1. Problem statement

In many situation of real life, there exist many cases where people have quite different strong points of view about certain situations, projects, plans or entities. These points of view are diverse and opposite to each other. For example, in a certain country government starts a project to portray his performance. Leaders of the ruling party may rate their project highly by giving a membership grade about 0.9 whereas the opposition considers the same project as a wastage of money and try to defame it by providing strongly opposite points of view. So a nonmembership grade suggested may be 0.8. In this case (0.9) + (0.8) >1 but (0.9) ^q + (0.8) ^q ≤ 1 for q ≥ 5.

Similarly, in the field of medicine some situation occurs, when there arises the question of using antibiotics. Antibiotics work against bacteria. Bacteria called pseudomonas aeruginosa show resistance against all available drugs except colistin (drug) [8]. This means that colistin is effective against these bacteria. So, due to effectiveness of colistin against above mentioned bacteria, membership grade for the use of this drug may be 0.9 On the other hand the life threatening infection of nephrotoxicity and neurotoxicity associated with colistin [13] compel to limit its use. Therefore a nonmembership grade given to this drug may be 0.45 so in this case 0.9 + 0.45 > 1 but (0.9) ^q + (0.45) ^q ≤ 1 for q ≥ 3 In order to tackle such situations, the need for a more comprehensive model was felt.

To cope such circumstances Yager [25, 26] initiated the study of q-ROFSs. Here the sum of qth power of membership grade and qth power of nonmembership grade are real numbers between 0 and 1 The concept of q-ROFSs gives more space and freedom to the decision makers for the selection of membership and nonmembership grades.

According to best of our knowledge there does not exist any notion of q-ROF rough sets via q-ROF β-neighborhoods system in q-ROF environments. To fulfil the space in research this motivate the current paper to Cq-ROFRSs model via q-ROF β-neighborhoods system. Therefore, a new approach is adopted to hybrid q-rung orthopair fuzzy sets with notions of covering rough set and TOPSIS and present their application in MADM. In real life the Cq-ROFRSs model is a significant tools to cope with complexities and uncertainties. The idea of Cq-ROFRSs model via q-ROF β-neighborhoods has been investigated from the hybridization of the prominent concepts of CRSs, q-ROFSs and FRSs. Further it has been observed that the Cq-ROFRSs is an important generalization of cover based intuitionistic fuzzy rough sets by adjusting the value of parameter q = 1 and cover based Pythagorean fuzzy rough sets by adjusting the value of parameter q = 2 This show that Cq-ROFRSs model has stronger capability than IFSs and PFSs to manage the uncertainty.

2. Preliminaries

This section presents some basic notions related to IFSs, PFSs, q-rung orthpair fuzzy sets and fuzzy covering approximation space.

2.1. Definition [2]

Let us consider a set U, an IFS A in U is a set represented by the following form:

A = {< k, μ_A (k) , η_A (k) >/k ∈ U} where μ_A : U → [0, 1] and η_A : U → [0.1] represents membership mapping and nonmembership mapping of an object k =∈ to the set A with 0 ≤ μ_A (k) + η_A (k) ≤1. Moreover π = 1 - μ_A (k) - η_A (k) denotes the hesitancy or degree of indeterminacy.

Recently Yager studied the notion of PFSs, which could be considered as a new expansion of IFSs, the representation of which is given in the following definition.

2.2. Definition [23, 24]

Let U be a universal set, a PFSs A in U is an object having the form: $\begin{matrix} A = {< k, μ_{A} (k), μ_{A} (k) > / k \in U} \end{matrix}$ where μ_A : U → [0, 1] and η_A : U represents membership grade and nonmembership grade of an object k ∈ U to the set A. Furthermore it satisfies the condition that 0 ≤ (μ_A (k)) ² + η_A (k)) ² ≤ 1 The $π_{A} (k) = (1 - (μ_{A} (k))^{2} - η_{A} (k))^{2})^{\frac{1}{2}}$ is called hesitancy or indeterminacy. The family of PFSs on U is represented by PF(U).

In the following, a brief introduction of q-ROFSs is given.

2.3. Definition [25, 26]

Let U be any set. A q-ROFSs A on U is an object having the form:

$A = {< k, μ_{A} (k), η_{A} (k) > q | k \in U and q \geq 1}$ where μ_A : U → [0, 1] denotes the membership grade of k in A and η_A : U → [0, 1] denotes the non-membership grade of k in A, with the conditions that 0 ≤ (μ_A (k)) ² + η_A (k)) ² ≤ 1 such that q ≥ 1 The $π_{A} (k) = (1 - (μ_{A} (k))^{q} - (μ_{A} (k))^{q})^{\frac{1}{q}}$ for q ≥ 1 is called hesitancy or indeterminacy.

Let a = (μ_a, η_a) then a is said to be a q-rung orthopair fuzzy number (q-ROFN), where $0 \leq (μ_{a}^{q} + η_{a}^{q} \leq 1, for q \geq 1$ .

Here it is stated that if q = 1 then (μ_a + η_a) is an IF number and if q = 2 then it is a PF number (PFN). The family of q-ROFSs on U is denoted by q-ROF(U).

2.4. Definition [25]

Let a₁ = (μ_{a
₁}, η_{a
₁}) and a₂ = (μ_{a
₂}, η_{a
₂}). Then a₁ ≻ a₂ if and only if μ_{a
₁} ≥ μ_{a
₂} and η_{a
₁} ≤ η_{a
₂}.

Let A₁ = {< k, μ_{A
₁} (k) > q/k ∈ U} and A₂ = {< k, μ_{A
₂} (k) , η_{A
₂} (k) > q/k ∈ U} be two q-ROFS. Yager [25] defined the basic operations on these as follows:

A₁ ∪ A₂ = {< k, max(μ_{A
₁} (k) , μ_{A
₂} (k) , min(η_{A
₁} (k) , η_{A
₂} (k)) >/k ∈ U};

A₁ ∪ A₂ = {< k, min(μ_{A
₁} (k) , μ_{A
₂} (k) , max(η_{A
₁} (k) , η_{A
₂} (k)) >/k ∈ U}

A₁ ∪ A₂ if and only if (μ_{A
₁} (k) , μ_{A
₂} (k) and (η_{A
₁} (k) ≥ η_{A
₂} (k);

A₁ = A₂ and if and only if μ_{A
₁} (k) = μ_{A
₁} (k) and (η_{A
₁} (k) = η_{A
₂} (k);

$A_{1}^{c} = {< k, η_{A_{1}} (k), μ_{A_{1}} (k) > q / k \in U$ .

2.5. Definition [34]

Let U be a universal set. A set K = {A ≠ φ/A ⊆ U} is said to be covering of U, if ∪A = U Then the pair (U, K) is said to be covering approximation space.

2.6. Definition [34]

Let (U, K) be a covering approximation space. Then $H_{K} (k) = \cap {A / A \in K and k \in A}$ is known as the neighborhood of k ∈ U w.r.t (U, K).

2.7. Definition [20]

Let U be a universal set. Let $L = {L_{1}, L_{2}, . . ., L_{m}}$ with $L_{i} \in I (U) (i = 1, . . ., m)$ where I (U) represents the fuzzy power set of U. Then $L$ is known to be a fuzzy covering of U if $⋁_{L_{i \in L}} L_{i} (k) = 1$ for each k ∈ U. The pair $(U, L)$ is known to be a fuzzy covering approximation space (FCAS).

2.8. Definition [12]

Let us consider a universal set U. For any β ∈ (0, 1] and $L = {L_{1}, L_{2}, . . ., L_{m}}$ with $L_{m} \in I (U) (i = 1, . . ., m}$ then $L$ is known to be a fuzzy β-covering of U if $\cup_{i = 1}^{m} L_{i} (k) ≻ β$ for each k ∈ U. The pair (U, L) is said to be fuzzy β-covering approximation space.

2.9. Definition [12]

Consider a fuzzy cover approximation space $(U, L)$ and $L = {L_{1}, L_{2}, . . ., L_{m}}$ be a fuzzy β-covering of U for some β ∈ (0.1]. Then $H_{k^{β}} = \cap {L_{m} \in L / L_{i} (k) ≻ β (i = 1, 2, . . ., m)}$ is said to be fuzzy β-neighborhood of k ∈ U.

2.10. Definition [12]

Let $U_{L}$ be a fuzzy covering approximation space and $L = {L_{1}, L_{2}, . . ., L_{m}}$ be a fuzzy β-covering of U, for some β ∈ (0.1]. Then $(H_{y}^{B})^{*} = {k \in U / H_{y}^{B} (k) ≽ β}$ is said to be β-neighborhood of y for each y ∈ U.

3. Covering based q-rung orthopair fuzzy rough set

Here in this section we are going to investigate the hybrid structure of q-rung orthopair fuzzy set, fuzzy covering approximation space and fuzzy rough sets to get the generalized structure of covering based q-rung orthopair fuzzy rough sets. First we will briefly define the Pythagorean fuzzy covering approximation space (PFCAS).

3.1. Definition

Let U be any set and $C = {C_{1}, C_{2}, . . ., C_{m}}$ where $C_{i} \in PF (U)$ and i = 1, 2, . . . , m. For any PFN β = (μ_β, η_β) $C$ is said to be Pythagorean fuzzy β-covering (PF β-covering) of U, if $\cup_{i = 1}^{m} C_{i}) (k) ≻ β$ for all k ∈ U. Here $(U, C)$ is called a PFCAS.

Suppose that $(U, C)$ be a PFCAS and $C = {C_{1}, C_{2}, . . ., C_{m}}$ be a PFβ-covering of U for some β = (μ_β, η_β). Then $H_{C (k)}^{β} = {C_{j} \in C / C_{j} ≻ β, j = 1, 2, . . ., m}$ PF β-covering neighborhood of U.

Let $H_{C}^{β} = {H_{C (k)}^{β} / k \in U}$ represent a PFβ-neighborhood system induced by PF β-covering $C$ . With the help of PF matrix to represent a PFβ-neighborhood system as follows.

M_{C}^{β} = [H_{C (k_{i})}^{β}]_{k_{i} \times k_{j} \in U \times U}

3.2. Remarks

By taking β = (1, 0) then PF β-covering degenerate into a crisp covering.

By taking β = (1, 0) then PF β-neighborhood degenerate into a crisp neighborhood.

By taking β = (1, 0) where 0 < a < 1 then PFβ-covering degenerate into a fuzzy covering.

By taking β = (1, 0) then PFβ-neighborhood degenerate into a fuzzy β-neighborhood respectively.

3.3. Definition

Let U be any set and $C = {C_{1}, C_{2}, . . ., C_{m}}$ where $C_{i} \in q - ROF (U)$ and i = 1, 2, . . . , m. For any $q - ROFN β = (μ_{β}, η_{β}), C$ is said to be q-rung orthopair fuzzy β-covering (q - ROFβ-covering) of U, if $\cup_{i = 1}^{m} C_{i} (k) ≻ β$ for all k ∈ U. Here $(U, C)$ is called a q-rung orthopair fuzzy covering approximation space (q - ROFCAS).

Suppose that $(U, C)$ be a (q - ROFCAS) and $C = {C_{1}, C_{2}, . . ., C_{m}}$ be a q-ROF β-covering of U for some β = (μ_β, η_β). Then $H_{C (k)}^{β} = \cap {C_{i} \in C / C_{j} (k) ≻ β, j = 1, 2$ ,. . . , m} q-ROFβ-covering neighborhood of U.

Let $H_{C}^{β} = {H_{C (k)}^{β} / k \in U}$ represent a q-ROF β-neighborhood system induced by q-ROF β-covering $C$ . With the help of q-ROF matrix to represent a q-ROF β neighborhood system as follows.

M_{C}^{β} = [H_{C (k_{i})}^{β}]_{k_{i} \times k_{j} \in U \times U}

3.4. Remark

By taking β = (1, 0) then q-ROF β-covering degenerate into a crisp covering.

By taking β = (1, 0) then q-ROF β-neighborhood degenerate into a crisp neighborhood.

By taking β = (1, 0) where 0 < a < 1 then q-ROF β-covering degenerate into a fuzzy covering.

By taking β = (1, 0) then q-ROF β-neighborhood degenerate into a fuzzy β-neighborhood respectively.

Proof. (1) Let $C = {C_{1}, C_{2}, . . ., C_{m}}$ q-ROF β-covering. Then by definition $\cup_{i = 1}^{m} C_{i} (k) ≻ β$ for all k ∈ U. If β = (1, 0), then there exist at least a q-ROFN a = (μ_a, η_a) = (1, 0) such that $(1, 0) = C_{i} (k)$ for some i = 1, 2, . . . , m for k ∈ U. Thus $\cup_{C_{i} \in C} C_{i} = U$ . Therefore, if β = (1, 0), then q-ROF β-covering degenerate into crisp cover.

(2) Consider $H_{C (k)}^{β} = \cap {C_{i} \in C / C_{j} (k) ≻ β, j = 1, 2, . . ., m}$ be a q-ROF β-covering neighborhood of U. If β = (1, 0) then there exist at least a q-ROFN $a = (μ_{a}, η_{a}) = (1, 0) = C_{i} (k)$ such that $a = C_{i} (k) ≻ β$ for k ∈ U. Then each $H_{C (k)}^{β}$ contain at least a q-ROFN a = (μ_a, η_a) = (1, 0) for k ∈ U. Thus $ℋ_{k}^{β} = \cap {C_{j} / C_{j} \in C$ and $k \in C_{j}, j = 1, 2, . . ., m}$ . Therefore if β = (1, 0) then q-ROF β-covering neighborhood degenerate into crisp neighborhood.

Proofs of (3) and (4) are similar to (1) and (2).□

3.5. Definition [18]

For any q-ROFN A = (μ_A, η_A) the score function of A is defined as $\begin{matrix} S (A) = \frac{1}{2} (1 + μ_{A}^{q} - μ_{A}^{q}), S (A) \in [0, 1], \\ for q \geq 1 . \end{matrix}$

The larger the value of score function, the better the orthopair is.

3.6. Example

Let us consider that $(U, C)$ be a q-ROFCAS and $C = {C_{1}, C_{2}, C_{3}, C_{4}, C_{5},}$ be a set of q-ROFSs of U such that U = {k₁, k₂, . . . , k₆} with β = (0.8, 0.7) as given in Table 1,

Table 1
Tabular representation of q-ROF-covering c in Example 3.6

$U_{C}$ $C_{1}$ $C_{2}$ $C_{3}$ $C_{4}$ $C_{5}$

k ₁ (0.9, 0.5) (0.85, 0.65) (0.7, 0.8) (0.5, 0.9) (0.9, 0.3)

k ₂ (0.89, 0.7) (0.93, 0.45) (0.79, 0.65) (0.69, 0.8) (0.65, 0.95)

k ₃ (0.95, 0.6) (0.69, 0.85) (0.98, 0.63) (0.7, 0.4) (0.5, 0.9)

k ₄ (0.85, 0.7) (0.6, 0.9) (0.55, 0.85) (0.97, 0.3) (0.89, 0.4)

k ₅ (0.6, 0.87) (0.9, 0.45) (0.69, 0.85) (0.92, 0.6) (0.3, 0.75)

k ₆ (0.88, 0.55) (0.6.0.9) (0.9, 0.63) (0.8, 0.7) (0.5, 0.89)

$U_{C}$	$C_{1}$	$C_{2}$	$C_{3}$	$C_{4}$	$C_{5}$
k ₁	(0.9, 0.5)	(0.85, 0.65)	(0.7, 0.8)	(0.5, 0.9)	(0.9, 0.3)
k ₂	(0.89, 0.7)	(0.93, 0.45)	(0.79, 0.65)	(0.69, 0.8)	(0.65, 0.95)
k ₃	(0.95, 0.6)	(0.69, 0.85)	(0.98, 0.63)	(0.7, 0.4)	(0.5, 0.9)
k ₄	(0.85, 0.7)	(0.6, 0.9)	(0.55, 0.85)	(0.97, 0.3)	(0.89, 0.4)
k ₅	(0.6, 0.87)	(0.9, 0.45)	(0.69, 0.85)	(0.92, 0.6)	(0.3, 0.75)
k ₆	(0.88, 0.55)	(0.6.0.9)	(0.9, 0.63)	(0.8, 0.7)	(0.5, 0.89)

Hence $C$ is a q-ROF β-covering of U. Then $\begin{matrix} H_{C (k_{1})}^{(0.8, 0.7)} = C_{1} \cap C_{2} \cap C_{5}, H_{C (k_{2})}^{(0.8, 0.7)} = C_{1} \cap C_{2}, \\ H_{C (k_{3})}^{(0.8, 0.7)} = C_{1} \cap C_{3}, H_{C (k_{4})}^{(0.8, 0.7)} = C_{1} \cap C_{4} \cap C_{5}, \\ H_{C (k_{5})}^{(0.8, 0.7)} = C_{2} \cap C_{4}, H_{C (k_{4})}^{(0.8, 0.7)} = C_{1} \cap C_{3} \end{matrix}$

From $H_{C}^{β} = {H_{C (k)}^{β} / k \in U}$ Table 2. is obtained,

Table 2

Tabular representation of $H_{C}^{(0.8, 0.4)}$ in Example 3.6

$U_{C}$	$C_{1}$	$C_{2}$	$C_{3}$	$C_{4}$	$C_{5}$	$C_{6}$
k ₁	(0.35, 0.65)	(0.65, 0.95)	(0.5, 0.9)	(0.6, 0.9)	(0.6, 0.87)	(0.5, 0.9)
k ₂	(0.35, 0.65)	(0.39, 0.7)	(0.69, 0.35)	(0.6, 0.9)	(0.6, 0.87)	(0.6, 0.9)
k ₃	(0.7, 0.8)	(0.79, 0.7)	(0.95, 0.6)	(0.55, 0.85)	(0.6, 0.87)	(0.88, 0.63)
k ₄	(0.5, 0.9)	(0.65, 0.93)	(0.5, 0.9)	(0.85, 0.7)	(0.6, 0.87)	(0.5, 0.89)
k ₅	(0.5, 0.9)	(0.69, 0.3)	(0.69, 0.85)	(0.92, 0.6)	(0.6, 0.9)	(0.6, 0.9)
k ₆	(0.7, 0.8)	(0.79, 0.7)	(0.95, 0.6)	(0.55, 0.85)	(0.6, 0.8)	(0.88, 0.63)

Therefore $\begin{matrix} M_{C}^{(0.8, 0.7)} = \\ [\begin{matrix} (0.85, 0.65) & (0.65, 0.95) & (0.5, 0.9) & (0.6, 0.9) & (0.6, 0.87) & (0.5, 0.9) \\ (0.85, 0.65) & (0.89, 0.7) & (0.69, 0.85) & (0.6, 0.9) & (0.6, 0.87) & (0.9, 0.9) \\ (0.7, 0.8) & (0.79, 0.7) & (0.95, 0.6) & (0.55, 0.85) & (0.6, 0.87) & (0.88, 0.63) \\ (0.5, 0.9) & (0.65, 0.95) & (0.5, 0.9) & (0.85, 0.7) & (0.6, 0.87) & (0.5, 0.89) \\ (0.5, 0.9) & (0.69, 0.8) & (0.69, 0.85) & (0.6, 0.9) & (0.9, 0.6) & (0.6, 0.9) \\ (0.7, 0.8) & (0.79, 0.7) & (0.95, 0.6) & (0.55, 0.85) & (0.6, 0.87) & (0.88, 0.63) \end{matrix}] \end{matrix}$

3.7. Definition

Consider a q-ROFCAS (U, C) where $C = {C_{1}, C_{2}, . . ., C_{m}}$ is a set of q-ROF β-covering of U for some β = (μ_β, η_β) and U = {k₁, . . . , k_m}. Consider that the neighborhood system $H_{C}^{β} = {H_{C (k)}^{β} / k \in U)$ induced by q-ROF β-covering of $C$ such that $H_{C (k_{i})}^{β} - {< k_{i}, μ_{H_{C (k_{i})}^{β}} (k_{i}, k_{i}), η_{H_{C (k_{i})}^{β}} (k_{i}, k_{j}) > q / j = 1, . . ., m}$ , for all i = 1, . . . , n. Now for any A∈q-ROF(U) where A = {< μ_A (k_j) , η_A (k_j)) > _q/j = 1, . . . , m} the lower and upper approximations of A with respect to $H_{C (k)}^{β}$ is represented and defined by $\begin{matrix} H_{C}^{β} (A) = (\underline{H_{C}^{β}} (A), \bar{H_{C}^{β}} (A) \end{matrix}$ where $\begin{matrix} \underline{H_{C}^{β}} (A) = {< k_{i}, μ_{\underline{H_{C}^{β}} (A)} (k_{i}), η_{\underline{H_{C}^{β}} (A)} (k_{i}) \\ >_{q} / i = 1, . . ., n} \end{matrix}$ and $\bar{H_{C}^{β}} (A) = {< k_{i}, μ_{\bar{H_{C}^{β}} (A)} (k_{i}), η_{\bar{H_{C}^{β}} (A)} (k_{i}) >_{q} / i = 1, . . ., n}$ such that $\begin{matrix} μ_{\underline{H_{C}^{β}} (A)} (k_{i}) = \land_{j = 1}^{m} {μ_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \land μ_{A} (k_{j})} \end{matrix}$ $\begin{matrix} μ_{\underline{H_{C}^{β}} (A)} (k_{i}) = \land_{j = 1}^{m} {μ_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \land η_{A} (k_{j})} \end{matrix}$ and $\begin{matrix} μ_{\bar{H_{C}^{β}} (A)} (k_{i}) = \land_{j = 1}^{m} {μ_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \land μ_{A} (k_{j})} \end{matrix}$ $\begin{matrix} μ_{\bar{H_{C}^{β}} (A)} (k_{i}) = \land_{j = 1}^{m} {μ_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \land η_{A} (k_{j})} \end{matrix}$

So the operators $\underline{ℋ_{C}^{β}} (A),_{\bar{ℋ_{C}^{β}} (A)}$ q-ROF (U) → q-ROF(U) are said to be lower and upper q-rung orthopair fuzzy rough approximation operators (q-ROFRAOs) with respect to $H_{C}^{β}$ .

Hence the covering base q-rung orthopair fuzzy rough sets (Cq-ROFRSs) is the pair $H_{C}^{β} (A) = (\underline{H_{C}^{β}} (A), \bar{H_{C}^{β}} (A))$ , when ever $\underline{H_{C}^{β}} (A) \neq, \bar{H_{C}^{β}} (A)$ .

3.8. Remark

The notion of Cq-ROFRSs is the generalization of intuitionistic fuzzy covering rough set model given in [3] and Pythagorean fuzzy covering rough set model (CPFRSs).

3.9. Remark

If the value of q = 1 then the notion of Cq-ROFRSs is reduced to CIFRSs.

If the value of q = 2 then the notion of Cq-ROFRSs is reduced to CPFRSs.

3.10. Example

Consider that A∈q-ROF(U), that is $A = {\begin{matrix} (k_{1}, 0.91, 0.62), (k_{2}, 0.58, 0.83), (k_{3}, 0.8, 0.75), \\ (k_{4}, 0.95, 0.35), (k_{5}, 0.8, 0.7), (k_{6}, 0.98, 0.37) \end{matrix}}$ and if we consider $M_{C}^{β} = [H_{C (k_{i})}^{β}]_{k_{i} \times k_{j} \in U \times U}$ as from Example 3.6, where β = (0.8, 0.7). Then $\begin{matrix} \underline{H_{C}^{β}} (A) = \\ {\begin{matrix} (k_{1}, 0.5, 0.95), (k_{2}, 0.85, 0.9), (k_{3}, 0.55, 0.87) \\ (k_{4}, 0.5, 0.95), (k_{5}, 0.5, 0.9), (k_{6}, 0.55, 0.87) \end{matrix}} \end{matrix}$ $\begin{matrix} \bar{H_{C}^{β}} (A) = \\ {\begin{matrix} (k_{1}, 0.98, 0.35), (k_{2}, 0.98, 0.35), (k_{3}, 0.98, 0.35) \\ (k_{4}, 0.98, 0.35), (k_{5}, 0.98, 0.35), (k_{6}, 0.98, 0.35) \end{matrix}} \end{matrix}$

3.11. Definition

Let us consider that A₁ (μ_{A
₁}, η_{A
₁}) and A₂ = (μ_{A
₂}, η_{A
₂}) be two q-ROFSs. Then the distance between A₁ and A₂ are define as follows: $\begin{matrix} D (A_{1}, A_{2}) = {\frac{1}{2 n} \sum_{k \in U} | μ_{(A_{1})} (k) - μ_{(A_{2})} (k) |^{p} \\ {+ \frac{1}{2 n} \sum_{k \in U} | η_{(A_{1})} (k) - η_{(A_{2})} (k) |^{p}}}^{\frac{1}{p}}, \end{matrix}$ where p ≥ 1

3.12. Theorem

Let $(U, C)$ be a q-ROFCAS and $C = {C_{1}, C_{2}$ , $. . ., C_{m}}$ be a q-ROF β-covering of U for some β = (μ_β, η_β). Consider that the neighborhood system $H_{C}^{β} = {H_{C (k)}^{β} / k \in U)$ induced by q-ROF β-covering $C$ . Now for any A₁, A₂∈ q-ROF(U), we have the following:

$\underline{H_{C}^{β}} (A) \subseteq A \subseteq \bar{H_{C}^{β}} (A)$ ;

If A₁ ⊆ A₂ then $\underline{H_{C}^{β}} (A_{1}) \subseteq \underline{H_{C}^{β}} (A_{2})$ and $\bar{H_{C}^{β}} (A_{1}) \subseteq \bar{H_{C}^{β}} (A_{2})$ ;

$\sim \underline{H_{C}^{β}} (A_{1}) = \bar{H_{C}^{β}} (\sim A_{1})$ and $\sim \bar{H_{C}^{β}} (A_{1}) = \underline{H_{C}^{β}} (\sim A_{1})$ ;

$\underline{H_{C}^{β}} (A_{1} \cap A_{2}) = \underline{H_{C}^{β}} (A_{1}) \cap \underline{H_{C}^{β}} (A_{2})$ ;

$\underline{H_{C}^{β}} (A_{1} \cap A_{2}) \supseteq \underline{H_{C}^{β}} (A_{1}) \cup \underline{H_{C}^{β}} (A_{2})$ ;

$\bar{H_{C}^{β}} (A_{1} \cup A_{2}) = \bar{H_{C}^{β}} (A_{1}) \cup \bar{H_{C}^{β}} (A_{2})$ ;

$\bar{H_{C}^{β}} (A_{1} \cap A_{2}) \subseteq \bar{H_{C}^{β}} (A_{1}) \cap \bar{H_{C}^{β}} (A_{2})$

Proof: Proof of i. to iii. are straightforward and follows from Definition 14.□

iv: As we know that $\begin{matrix} \underline{H_{C}^{β}} (A_{1} \cap A_{2}) = {< k_{i}, μ_{\underline{H_{C}^{β}} (A_{1} \cap A_{2})} (k_{i}), \\ η_{\underline{H_{C}^{β}} A_{1} \cap A_{2}} (k_{i})_{q} / i = 1, . . ., n} \end{matrix}$ and $\begin{matrix} \underline{H_{C}^{β}} (A_{1}) = {< k_{i}, μ_{\underline{H_{C}^{β}} (A_{1})} (k_{i}), η_{\underline{H_{C}^{β}} (A_{1})} (k_{i}) \\ >_{q} / i = 1, 2, . . ., n} \end{matrix}$ $\begin{matrix} \underline{H_{C}^{β}} (A_{1}) = {< k_{i}, μ_{\underline{H_{C}^{β}} (A_{1})} (k_{i}), η_{\underline{H_{C}^{β}} (A_{1})} (k_{i}) \\ >_{q} / i = 1, 2, . . ., n} \end{matrix}$

In order to show $\underline{H_{C}^{β}} (A_{1} \cap A_{2}) = \underline{H_{C}^{β}} (A_{1}) \cap \underline{H_{C}^{β}} (A_{1})$ , we have to prove $\begin{matrix} μ_{\underline{H_{C}^{β}} (A_{1} \cap A_{2})} (k_{i}) = {μ_{\underline{H_{C}^{β}} (A_{1})} (k_{i}) \cap μ_{\underline{H_{C}^{β}} (A_{2})} (k_{i})} \end{matrix}$ and $\begin{matrix} η_{\underline{H_{C}^{β}} (A_{1} \cap A_{2})} (k_{i}) = {η_{\underline{H_{C}^{β}} (A_{1})} (k_{i}) \cap η_{\underline{H_{C}^{β}} (A_{2})} (k_{i})} \end{matrix}$

Consider $\begin{matrix} μ_{\underline{H_{C}^{β}} (A_{1} \cap A_{2})} (k_{i}) = \land_{j = 1}^{m} \\ {μ_{H_{C (k_{i})}^{β}} (k_{i}, k_{j}) \land μ_{A_{1} \cap A_{2}} (k_{i})} \\ = \land_{j = 1}^{m} {μ_{H_{C (k_{i})}^{β}} (k_{i}, k_{j}) \land {μ_{A_{1}} (k_{i}) \cap μ_{A_{2}} (k_{i})}} \\ = \land_{j = 1}^{m} {μ_{H_{C (k_{i})}^{β}} (k_{i}, k_{j}) \land {μ_{A_{1}} (k_{j})} \land \\ \land_{j = 1}^{m} {μ_{H_{C (k_{i})}^{β}} (k_{i}, k_{j}) \land {μ_{A_{2}} (k_{j})} \\ = η_{\underline{H_{C}^{β}} (A_{1})} (k_{i}) \lor η_{\underline{H_{C}^{β}} (A_{2})} (k_{i}) \end{matrix}$

Next $\begin{matrix} = η_{\underline{H_{C}^{β}} (A_{1} \cap A_{2})} (k_{i}) = \lor_{j = 1}^{m} \\ {η_{\underline{H_{C}^{β}} (k_{i})} (k_{i}, k_{j}) \lor {η_{A_{1}} (k_{i}) \cap η_{A_{2}} (k_{j})}} \\ = \lor_{j = 1}^{m} {η_{\underline{H_{C}^{β}} (k_{i})} (k_{i}, k_{j}) \lor {η_{A_{1}} (k_{i}) \cap η_{A_{2}} (k_{j})}} \\ = \lor_{j = 1}^{m} {η_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \lor η_{A_{2}} (k_{i})} \lor \\ \lor_{j = 1}^{m} {η_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \lor η_{A_{2}} (k_{i})} \\ = η_{\underline{H_{C}^{β}} (A_{1})} (k_{i}) \lor η_{\underline{H_{C}^{β}} (A_{2})} (k_{i}) \end{matrix}$

Therefore $\begin{matrix} \underline{H_{C}^{β}} (A_{1} \cap A_{2}) = \underline{H_{C}^{β}} (A_{1}) \cap \underline{H_{C}^{β}} (A_{1}) \end{matrix}$

v: Next to prove $\begin{matrix} \underline{H_{C}^{β}} (A_{1} \cap A_{2}) \supseteq \underline{H_{C}^{β}} (A_{1}) \cup \underline{H_{C}^{β}} (A_{1}) \end{matrix}$ we have to show k_i ∈ U $\begin{matrix} μ_{\underline{H_{C}^{β}} (A_{1} \cap A_{2})} (k_{i}) \geq μ_{\underline{H_{C}^{β}} (A_{1})} (k_{i}) \lor μ_{\underline{H_{C}^{β}} (A_{2})} (k_{i}) \end{matrix}$ $\begin{matrix} η_{\underline{H_{C}^{β}} (A_{1} \cap A_{2})} (k_{i}) \leq η_{\underline{H_{C}^{β}} (A_{1})} (k_{i}) \land η_{\underline{H_{C}^{β}} (A_{2})} (k_{i}) \end{matrix}$

Consider $\begin{matrix} μ_{\underline{H_{C}^{β}} (A_{1} \cap A_{2})} (k_{i}) = \land_{j = 1}^{m} \\ {μ_{H_{C}^{β} (A_{1})} (k_{i}, k_{j}) \land μ_{(A_{1} \cup A_{2})} (k_{i})} \\ = \land_{j = 1}^{m} {μ_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \land {μ_{A_{1}} (k_{i}) \lor μ_{A_{2}} (k_{j})}} \\ \begin{matrix} \geq \land_{j = 1}^{m} {μ_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \land μ_{A_{1}} (k_{i})} \lor \\ \land_{j = 1}^{m} {μ_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \land μ_{A_{2}} (k_{j})} \end{matrix} \\ μ_{\underline{H_{C}^{β}} (A_{1} \cup A_{2})} (k_{i}) \geq μ_{\underline{H_{C}^{β}} (A_{1})} (k_{i}) \lor μ_{\underline{H_{C}^{β}} (A_{2})} (k_{i}) \end{matrix}$

Further $\begin{matrix} η_{\underline{H_{C}^{β}} (A_{1} \cup A_{2})} (k_{i}) = \lor_{j = 1}^{m} \\ {η_{H_{C}^{β} (k_{i})} (k_{i}) \lor η_{H_{C}^{β} (A_{1} \cup A_{2})} (k_{i})} \\ = \lor_{j = 1}^{m} {η_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \lor \\ {η_{H_{C}^{β} (A_{1})} (k_{i}) \land η_{H_{C}^{β} (A_{2})} (k_{i})}} \\ \leq \lor_{j = 1}^{m} {η_{H_{C}^{β} (k_{i})} (k_{i}, k_{j})} \land \lor_{j = 1}^{m} \\ {η_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \lor {η_{(A_{2})} (k_{i})} \\ η_{\underline{H_{C}^{β}} (A_{1} \cup A_{2})} (k_{i}) \leq {η_{\underline{H_{C}^{β}} (A_{1})} (k_{i}) \land η_{\underline{H_{C}^{β}} (A_{2})} (k_{i})} \end{matrix}$

Hence $\underline{H_{C}^{β}} (A_{1} \cap A_{2}) \supseteq \underline{H_{C}^{β}} (A_{1}) \cup \underline{H_{C}^{β}} (A_{2})$

Proofs of vi: and vii: is similar to iv: and v:

4. A new proposal for multi-attribute decision making using q-rung orthopair fuzzy rough sets hybrid with TOPSIS

In this section, a new technique for MADM is proposed. Here concepts of Cq-ROFRS model will be employed, which are stated in Section 3. Major steps for this decision making method and its associated algorithms are presented in the following.

In real life situation MADM has an important role and an intelligent decision approach solve the complex and uncertain decisions under senior experts. The basic concepts of this proposed method for MADM are given as. Let U = {k₁, k₂, . . . , k_n} be any set of n feasible alternatives, $C = {C_{1}, C_{2}, . . ., C_{m}}$ be the finite set of attributes and w = {w₁, w₂, . . . , w_m} ^T be the weight vector of all attributes such that 0 ≤ w_i ≤ 1 and $\sum_{i = 1}^{m} w_{i} = 1$ with i = 1, . . . . m. Decision makers D_mem and D_non-mem put forward the assessment values of all the alternatives k_i (i = 1, . . . , n) corresponding to the set of attribute $C_{j} (j = 1, . . ., m)$ by μ_ij and η_ij respectively. So combining these two values as a q-ROFN we have q-ROF decision matrix $C_{j} (k_{i}) = (μ_{ij}, η_{ij})$ . This means that the decision maker D_mem provides membership grade μ_ij to the alternative k_i according to the attribute $C_{j}$ . Whereas the decision maker D_non-mem provides nonmembership grade η_ij to the alternative k_i according to the attribute $C_{j}$ and their decision matrix is given as: $\begin{matrix} C_{j} (k_{j}) = (\begin{matrix} (μ_{11}, η_{11}) & (μ_{12}, η_{12}) & \dots & (μ_{1 j}, η_{1 j}) \\ (μ_{21}, η_{21}) & (μ_{22}, η_{22}) & \dots & (μ_{2 j}, η_{2 j}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ (μ_{i 1}, η_{i 1}) & (μ_{i 2}, η_{i 2}) & \dots & (μ_{ij}, η_{ij}) \end{matrix}) \end{matrix}$

In order to tackle a MADM problem with the help of the proposed model given in Section 3. We will discuss the first decision model and steps Cq-ROFRS model for the proposed MADM problem, which consist of mainly in three steps. In the first step decision makers D_mem and D_non-mem provide their input to find a q-ROFSs as explained above. By using the q-rung orthopair fuzzy TOPSIS (q-ROF-TOPSIS) approach, we will present q-rung orthopair fuzzy positive ideal solution (q-ROF-PIS) $P^{+} = {C_{j}, max {s (C_{j} (k_{i}))} / j = 1, . . ., m}$ and q-rung orthopair fuzzy negative ideal solution (q-ROF-NIS) $P^{-} = {C_{j}, min {s (C_{j} (k_{i}))} / j = 1, . . ., m}$ , through the score function by Definitions 3.5. With the help of Definitions 3.11, the distance D⁺ and D^- are determined among alternatives k_i and q-ROF-PIS P⁺ and q-ROF-NIS P^-. Therefore, the new q-ROFS D = {< k, μ_D (k) , η_D (k) > _q/k ∈ U} = (μ_D, η_D) = (D⁺, D^-) _q can be constructed. Hence a multi-attribute q-rung orthopair fuzzy decision making information system (MAq-ROFDMIS) $(U, C, P, D)$ has been obtained. Then to find the optimal object or ranking among all the objects, they are arranged according to the preference evaluation. In second phase the lower and upper approximations of the q-ROFSs are calculated with the precision parameter β (0 < β ≤ 1) (where β the precision parameter, which is used on the Cq-ROFRS model explain the consistency consensus threshold by the decision maker). Finally to find the optimal object via ranking or decision making process among all the alternatives and then arranged according to the preference evaluation.

Here the detail of first step is presented and first suggesting the q-ROF-TOPSIS method. In this method the optimal alternative should have the shortest distance (that is the alternative should have higher score value) from q-ROF-PIS P⁺ and the farthest distance (that is the alternative should have least score value) from the q-ROF-NIS P^-. By the use of Definition 3.5, to identify q-ROF-PIS P⁺ and q-ROF-NIS P^- as follows. $\begin{matrix} \begin{matrix} P^{+} = {C_{j}, max {s (C_{j} (k_{i}))} / j = 1, . . ., m} \\ = {< C_{1}, μ_{1}^{+}, η_{1}^{+} >, < C_{2}, μ_{2}^{+}, η_{2}^{+} >, . . ., \\ < C_{2}, μ_{m}^{+}, η_{m}^{+} >} \end{matrix} \end{matrix}$ and $\begin{matrix} \begin{matrix} P^{-} = {C_{j}, min {s (C_{j} (k_{i}))} / j = 1, . . ., m} \\ = {< C_{1}, μ_{1}^{-}, η_{1}^{-} >, < C_{2}, μ_{2}^{-}, η_{2}^{-} >, . . ., \\ < C_{2}, μ_{m}^{-}, η_{m}^{-} >} \end{matrix} \end{matrix}$

Further with the help of Definition 3.11, to compute the weighted distances D⁺ and D^- between the alternative k_i and q-ROF-PIS P⁺ and q-ROF-NIS P^- is defined as the following: $\begin{matrix} D^{+} = \sum_{j = 1}^{m} w_{j} d (C_{j} (k_{j}), C_{j} (P^{+}) \\ = {\frac{1}{2 n} \sum_{j = 1}^{m} w_{j} | μ_{ij} - μ_{j}^{+} |^{p} \\ {+ \frac{1}{2 n} \sum_{j = 1}^{m} w_{j} | η_{ij} - η_{j}^{+} |^{p}}}^{\frac{1}{p}} \end{matrix}$ for (i = 1, . . . , n) and $\begin{matrix} D^{-} = \sum_{j = 1}^{m} w_{j} d (C_{j} (k_{j}), C_{j} (P^{-}) \\ = {\frac{1}{2 n} \sum_{j = 1}^{m} w_{j} | μ_{ij} - μ_{j}^{-} |^{p} \\ {+ \frac{1}{2 n} \sum_{j = 1}^{m} w_{j} | η_{ij} - η_{j}^{-} |^{p}}}^{\frac{1}{p}} . \end{matrix}$ for (i = 1, . . . , n)

Therefore we put together the new q-ROFS $\begin{matrix} D = (μ_{D}, η_{D}) = (D^{+}, D^{-}) . \end{matrix}$

4.1. Definition

A mapping T : [0, 1] × [0, 1] → [0, 1] is known to be a q-rung orthopair fuzzy triangular norm (briefly q-ROF t-norm), if it is commutative, associative and increasing with the boundary condition that is T : (k, 1) = k ∀ k ∈ [0, 1].

At the same time mapping T : [0, 1] × [0, 1] → [0, 1] is known to be a q-rung orthopair fuzzy triangular t-conorm (briefly q-ROF t-conorm), if it is commutative, associative and increasing with the boundary condition that is T : (k, 1) = k ∀ k ∈ [0, 1].

Here in this paper q-ROF t-norm and q-ROF t-conorm are used for MADM problem.

$T_{A} (k_{1}, k_{2}) = \frac{k_{1} k_{2}}{q \sqrt{1 + (1 - k_{1}^{q}) (1 - k_{1}^{q})}}$ and T_A (k₁, k₂) = $q \sqrt{\frac{k_{1}^{q} + k_{2}^{q}}{1 + k_{1}^{q} k_{2}^{q}}}$ .

Further by the use of Definition 3.7, the lower and upper approximations of best and worst q-ROFDM objects are found under the consistency consensus threshold β (0 < β ≤ 1) as the following. $\begin{matrix} μ_{\underline{H_{C}^{β}} (D)} (k_{i}) = \land_{j = 1}^{m} {μ_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \land μ_{D} (k_{j})} \\ η_{\underline{H_{C}^{β}} (D)} (k_{i}) = \land_{j = 1}^{m} {η_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \land η_{D} (k_{j})} \end{matrix}$ and $\begin{matrix} μ_{\bar{H_{C}^{β}} (D)} (k_{i}) = \land_{j = 1}^{m} {μ_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \land μ_{D} (k_{j})} \\ η_{\bar{H_{C}^{β}} (D)} (k_{i}) = \land_{j = 1}^{m} {η_{H_{C}^{β} (k_{i})} (k_{i}, k_{j}) \land η_{D} (k_{j})} \end{matrix}$

Finally based on Definition 4.1, to find the rank of all the alternatives and then arranged according to the preference evaluation under the consistency consensus threshold β (0 < β ≤ 1).

4.2. Definition

Suppose the MAq-ROFDMIS is $(U, C, P, D)$ . For the q-ROFDM object $D = (D^{+}, D^{-}$ ) ∈ q-ROF (U) represented by the preference information of decision maker D and risk preference threshold α (0 < α ≤ 1). Now define the ranking function of alternative k_i (i = 1, . . . , n) as: $\begin{matrix} T_{A} (k_{i}) = α T_{A} (μ_{\underline{H_{C}^{β}} (D)} (k_{i}), η_{\underline{H_{C}^{β}} (D)} (k_{i})) \\ + (1 - α) T_{A} (μ_{\bar{H_{C}^{β}} (D)} (k_{i}), η_{H_{C}^{β} (D)} (k_{i})) \end{matrix}$

From the definition of ranking function it is cleared that 0 ≤ T_A (k_l) ≤1.

With the help of above interpretation, the algorithm of the MADM approach based on Cq-ROFRS consists of the following steps:

input MAq-ROFDMIS ( $U, C, P, D$ );

output The sort ordering for all alternatives;

Step (i): Calculate the q-ROF-NIS P⁺ and q-ROF-NIS P^-,

Step (ii): Calculate the $D^{+} = μ_{D}$ and $D^{+} = η_{D}$ between the alternatives and the q-ROF-PIS P⁺ and q-ROF-NIS P^-,

Step (iii): Next find the lower and upper approximations

$μ_{\underline{H_{C}^{β}} (D)} (k_{i}), η_{\underline{H_{C}^{β}} (D)} (k_{i}), μ_{\bar{H_{C}^{β}} (D)} (k_{i})$ and $η_{\bar{H_{C}^{β}} (D)} (k_{i})$ ,

Step (iv): Determine the ranking function T_A (k_i),

Step (v): On the basis of value order the ranking of all alternatives.

5. Illustrative example

Here in this section we will present the proposed method of MADM which is based on Cq-ROFRS models that relates the assessment and rank of appointment of new faculty position in Universities. Then q-ROF-TOPSIS provides the desired ranking.

For a certain senior position in Universities, the appointment of a new faculty has to face a very complex evaluation and decision making process. The skill and ability of a candidate may be judged with respect to various attributes like as “managerial skills” “ability to work under pressure” “research productivity” etc. In order to take the right decision about the candidate the opinions of professional experts are needed for these criteria.

Consider that U = {k₁, k₂, k₃, k₄, k₅} be set of five candidates who fulfil the requirements for the senior faculty position in Y University. In order to appoint the most qualified and suitable person for the required position, a team of experts is organized and chaired by Prof. Z as a director. The team of experts will judge the candidates upon the criteria in the set of attribute $C = {C_{1}, C_{2}, C_{3}, C_{4}, C_{5}, C_{6}}$ where

$C_{1}$ = Research productivity

$C_{2}$ = Managerial skill

$C_{3}$ = Impact on research community

$C_{4}$ = Ability to work under pressure

$C_{5}$ = Academic leadership qualities

$C_{6}$ = Contribution to Y University

According to the background and expertise the team of experts wants to appoint the candidate who qualifies with the criteria of e to the utmost extent from candidate in U. Suppose that the evaluation values of each alternative with respect to each attribute provided by the decision makers D_mem and D_non-mem are presented in the decision matrix given in Table 3, and the weights of all the attributes set are given as the following:

Table 3
Tabular representation of q-ROFSs for $C$

$U_{C}$	$C_{1}$	$C_{2}$	$C_{3}$	$C_{4}$	$C_{5}$	$C_{6}$
k ₁	(0.93, 0.3)	(0.7, 0.4)	(0.8, 0.2)	(0.9, 0.1)	(0.7, 0.6)	(0.4, 0.3)
k ₂	(0.9, 0.4)	(0.7, 0.8)	(0.7, 0.8)	(0.6, 0.3)	(0.65, 0.87)	(0.6, 0.2)
k ₃	(0.8, 0.7)	(0.7, 0.2)	(0.95, 0.4)	(0.8, 0.4)	(0.5, 0.2)	(0.8, 0.3)
k ₄	(0.8, 0.3)	(0.6, 0.3)	(0.7, 0.4)	(0.9, 0.2)	(0.8, 0.65)	(0.4, 0.2)
k ₅	(0.5, 0.2)	(0.95, 0.4)	(0.8, 0.3)	(0.7, 0.1)	(0.6, 0.3)	(0.94, 0.33)

\begin{matrix} \begin{matrix} w_{1} = 0.2, w_{2} = 0.18, w_{3} = 0.22, w_{4} = 0.12, \\ w_{5} = 0.15, w_{6} = 0.13 \end{matrix} \end{matrix}

For example, the characteristics of a candidate k₁ under attribute $C_{1}$ is (0.98, 0.3). The value 0.98 is the degree of membership and the value 0.3 is the degree of non-membership of candidate k₁ under criteria $C_{1}$ respectively. In other words, candidate k₁ is qualified and suitable on the membership degree 0.98 and disqualified on the non-membership degree of 0.3.

Further the step wise algorithm for the proposed MADM approach based on Cq-ROFRS consists of the following steps:

Step (i). Now to compute the q-ROF-PIS P⁺ and q-ROF-NIS P^- by use of Definition 3.5, when q = 3 as follows. $\begin{matrix} P^{+} = \\ {\begin{matrix} (C_{1}, 0.98, 0.3), (C_{2}, 0.95, 0.4), (C_{3}, 0.95, 0.4) \\ (C_{4}, 0.9, 0.1), (C_{5}, 0.8, 0.65), (C_{6}, 0.94, 0.38) \end{matrix}} \end{matrix}$ $\begin{matrix} P^{-} = \\ {\begin{matrix} (C_{1}, 0.5, 0.2), (C_{2}, 0.7, 0.8), (C_{3}, 0.7, 0.5) \\ (C_{4}, 0.6, 0.3), (C_{5}, 0.65, 0.87), (C_{6}, 0.4, 0.3) \end{matrix}} \end{matrix}$

Step (ii). Furthermore, to compute the distance $D^{+} = μ_{D}$ and $D^{+} = η_{D}$ , between the alternatives and the q-ROF-PIS P⁺ and q-ROF-NIS P^- when p = 3 $\begin{matrix} D^{+} = μ_{D} = \frac{0.12943}{k_{1}}, \frac{0.13642}{k_{2}}, \frac{0.14903}{k_{3}}, \\ \frac{0.14129}{k_{4}}, \frac{0.13849}{k_{5}} \end{matrix}$ $\begin{matrix} D^{-} = η_{D} = \frac{0.15782}{k_{1}}, \frac{0.10907}{k_{2}}, \frac{0.2201}{k_{3}}, \\ \frac{0.11109}{k_{4}}, \frac{0.17627}{k_{5}} \end{matrix}$

Step (iii). Next, to determine the lower and upper approximation that is $μ_{\underline{H_{C}^{β}} (D)} (k_{i})$ , $η_{\underline{H_{C}^{β}} (D)} (k_{i})$ , $μ_{\bar{H_{C}^{β}} (D)} (k_{i})$ and $η_{\bar{H_{C}^{β}} (D)} (k_{i})$ ,

First to compute q-ROF β-neighborhood for each k_i ∈ U (i = 1, 2, . . . , 5), and let the consistency threshold q-ROF β = (0.8, 0.4) Then $\begin{matrix} \begin{matrix} H_{C (k_{1})}^{(0.8, 0.4)} = C_{1} \cap C_{3} \cap C_{4}, H_{C (k_{2})}^{(0.8, 0.4)} = C_{1}, \\ H_{C (k_{3})}^{(0.8, 0.4)} = C_{3} \cap C_{4} \cap C_{6}, H_{C (k_{4})}^{(0.8, 0.4)} = C_{1} \cap C_{4}, \\ H_{C (k_{5})}^{(0.8, 0.4)} = C_{2} \cap C_{3} \cap C_{6} \end{matrix} \end{matrix}$

Further Table 4 for

H C^{(0.8, 0.4)}

Table 4

Tabular representation of $H C^{(0.8, 0.4)}$

$U_{C}$	$C_{1}$	$C_{2}$	$C_{3}$	$C_{4}$	$C_{5}$
k ₁	(0.8, 0.3)	(0.6, 0.5)	(0.8, 0.7)	(0.7, 0.4)	(0.5, 0.3)
k ₂	(0.7, 0.4)	(0.7, 0.8)	(0.4, 0.2)	(0.6, 0.5)	(0.59, 0.4)
k ₃	(0.4, 0.3)	(0.6, 0.5)	(0.8, 0.4)	(0.4, 0.4)	(0.7, 0.38)
k ₄	(0.9, 0.3)	(0.6, 0.4)	(0.8, 0.7)	(0.8, 0.3)	(0.5, 0.2)
k ₅	(0.4, 0.4)	(0.6, 0.8)	(0.7, 0.4)	(0.4, 0.5)	(0.58, 0.4)

Now to compute $\begin{matrix} μ_{\underline{H_{C}^{β}} (D)} (k_{i}), η_{\underline{H_{C}^{β}} (D)} (k_{i}), μ_{\bar{H_{C}^{β}} (D)} (k_{i}) \end{matrix}$ and $\begin{matrix} η_{\bar{H_{C}^{β}} (D)} (k_{i}), \end{matrix}$ $\begin{matrix} μ_{\underline{H_{C}^{β}} (D)} = \frac{0.12943}{k_{1}}, \frac{0.12943}{k_{2}}, \frac{0.12943}{k_{3}}, \\ \frac{0.12943}{k_{4}}, \frac{0.12943}{k_{5}} \end{matrix}$ $\begin{matrix} η_{\underline{H_{C}^{β}} (D)} = \frac{0.7}{k_{1}}, \frac{0.8}{k_{2}}, \frac{0.5}{k_{3}}, \frac{0.7}{k_{4}}, \frac{0.8}{k_{5}} \end{matrix}$ $\begin{matrix} μ_{\bar{H e^{β}} (D)} = \frac{0.7}{k_{1}}, \frac{0.8}{k_{2}}, \frac{0.8}{k_{3}}, \frac{0.9}{k_{4}}, \frac{0.8}{k_{5}} \end{matrix}$ $\begin{matrix} η_{\bar{H_{C}^{β}} (D)} = \frac{0.10907}{k_{1}}, \frac{0.10907}{k_{2}}, \frac{0.10907}{k_{3}}, \\ \frac{0.10907}{k_{4}}, \frac{0.10907}{k_{5}} \end{matrix}$

Step (iv). Now to compute the ranking function T_A (k_i), and for this consider the risk preference threshold α = 0.75, where (0 < α ≤ 1) as follows: $\begin{matrix} T_{A} = \frac{0.076550}{k_{1}}, \frac{0.084171}{k_{2}}, \frac{0.058484}{k_{3}}, \\ \frac{0.080097}{k_{4}}, \frac{0.087149}{k_{5}} \end{matrix}$

Step (v). Finally, we are able to present the most suitable candidate for the senior faculty position according to the values of ranking function. Therefore, k₅ is the most suitable candidate for the senior faculty position. $\begin{matrix} k_{5} > k_{2} > k_{4} > k_{1} > k_{3} \end{matrix}$

Hence through the process of decision making finally we get most suitable candidate for the senior faculty position by the use of Cq-ROFRS model based on MADM method. Therefore, from the numerical calculation it is clear that the 5th candidate is the most suitable candidate for the senior faculty position.

5.1. Comparative analysis

Yager [24], developed the concept to PFSs and presented an important method based on Pythagorean fuzzy weighted averaging operator to solve MCDM problems. On the same concept Zhang and Xu [33], presented TOPSIS to solve MCDM with Pythagorean fuzzy information. These methods fail to handle situations when the membership grade is 0.9 and non-membership grade is 0.8. In this case (0.9) ² + (0.8) ² > 1 and the methods proposed in [24] and [33] fail to tackle the situation. The method proposed in this paper handle such situations very easily, for example (0.9) ² + (0.8) ² > 1 for q ≥ 5. So from the analysis it is clear that the method presented in this paper is more suitable to meet a variety of situations by adjusting values of q. Therefore the proposed method is the more superior than the methods proposed in [24] and [33] because it makes the information process more flexible by a parameter q. By increasing the value of parameter q, the range of decision making process will become wider and avoid the distortion of information. Therefore the method presented in this paper is more suitable because it provides more space to the decision maker in decision making problems.

6. Conclusion

The main focus of this paper is to improve the proposed model to tackle decision making problems where strong point of views are in the favor and against of some projects, entities or plans. The proposed model is very useful where in decision making problems the decision makers have contradictory point of views about certain plan or proposal. Therefore a new approach of Cq-ROFRSs model is adopted, which is a significant and powerful generalization of Pawlak’s rough sets. The proposed Cq-ROFRSs model is a nice tools to cope with complexities and uncertainties. The concept of Cq-ROFRSs model via q-ROF β-neighborhoods has been investigated by hybridizing the prominent concepts of CRSs, q-ROFSs and FRSs. Further it has been observed in Section 3, that the Cq-ROFRS is the significant generalization of cover based intuitionistic fuzzy rough sets by adjusting the parameter q = 1 and cover based Pythagorean fuzzy rough sets by adjusting the parameter q = 2. Then we have presented the q-ROF-TOPSIS technique for the MADM problem which depend on the Cq-ROFRSs model. The comparative analysis of the proposed model with existing literature is given in Table 5 by considering the above Illustrative Example of Section 5.

Table 5

Comparative analysis of different methods

Method	Score values					Ranking
Method	k ₁	k ₂	k ₃	k ₄	k ₅	Ranking
CFRSs [12]	Fail to handle					X
CFRSs [27]	Fail to handle					X
CIFRSs [31]	Fail to handle					X
PFSs [24]	Fail to handle					X
PFSs [33]	Fail to handle					X
CPFRSs	Fail to handle					X
Cq-ROFRS	0.076550	0084171	0038484	0080097	0.0087149	k₁> k₂ > k₃ > k₄ > k₅

From above Table 5, it is clear that the methods proposed in [12] and [27] are fail to handle situation because only handle the fuzzy membership values and having no information about nonmembership values. Similarly the CIFRSs method proposed in [33] also fail to handle due to the limitation on membership and nonmembership that their sum is less than or equal to 1. Analogously the methods proposed in [24, 33] and CPFRSs are also fail to handle due to the limitation on membership and nonmembership grades that their square sum is less than or equal to 1 The main advantages of the proposed method has the ability to cope these situations and provides a huge space and freedom to the decision makers to assign values freely by adjusting the value of q and hence the method proposed in this paper is superior than existing methods.

Footnotes

Acknowledgments

The authors would like to thank the editor in chief Reza Langari, associate editor Jian Wu and the anonymous referees for detailed and valuable comments which helped to improve this manuscript.

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Covering based q-rung orthopair fuzzy rough set model hybrid with TOPSIS for multi-attribute decision making

Abstract

Keywords

1. Introduction

1.1. Problem statement

2. Preliminaries

2.1. Definition [2]

2.2. Definition [23, 24]

2.3. Definition [25, 26]

2.4. Definition [25]

2.5. Definition [34]

2.6. Definition [34]

2.7. Definition [20]

2.8. Definition [12]

2.9. Definition [12]

2.10. Definition [12]

3. Covering based q-rung orthopair fuzzy rough set

3.1. Definition

3.2. Remarks

3.3. Definition

3.4. Remark

3.5. Definition [18]

3.6. Example

3.8. Remark

3.9. Remark

3.10. Example

3.11. Definition

3.12. Theorem

4. A new proposal for multi-attribute decision making using q-rung orthopair fuzzy rough sets hybrid with TOPSIS

4.1. Definition

4.2. Definition

5. Illustrative example

Table 3 Tabular representation of q-ROFSs for C

6. Conclusion

Footnotes

Acknowledgments

References

Table 3
Tabular representation of q-ROFSs for $C$