Another view on tolerance based multigranulation q -rung orthopair fuzzy rough sets with applications

Abstract

The q-rung orthopair fuzzy sets accommodate more uncertainties than the Pythagorean fuzzy sets and hence their applications are much extensive. Under the q-rung orthopair fuzzy set, the objective of this paper is to develop new types of q-rung orthopair fuzzy lower and upper approximations by applying the tolerance degree on the similarity between two objects. After employing tolerance degree based q-rung orthopair fuzzy rough set approach to it any times, we can get only the six different sets at most. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Furthermore, we propose tolerance degree based multi granulation optimistic/pessimistic q-rung orthopair fuzzy rough sets and investigate some of their properties. Another main contribution of this paper is to disclose the ideas of different kinds of approximations called approximate precision, rough degree, approximate quality and their mutual relationship. Finally a technique is devloped to rank the alternatives in a q-rung orthopair fuzzy information system based on similarity relation. We find that the proposed method/technique is more efficient when compared with other existing techniques.

Keywords

fuzzy rough set similarity relation tolerance classes

1 Introduction

Multiple attribute decision making, as an effective framework for comparison, has always been used to find the most desirable one from a finite set of alternatives on predefined attributes. An important problem of the decision process is expressing the attribute value. However, due to the intrinsic complexity of natural objects, there exists much uncertain information in many real world problems. So, it is difficult for experts or decision makers to give their assessments on attributes in crisp numbers.

1.1 A brief review on q-rung orthopair fuzzy sets

The intuitionistic fuzzy set [1] is an effective tool to express complex fuzzy information because it is characterized by three parameters: a membership degree, a nonmembership degree, and an indeterminacy degree. That is, an intuitionistic fuzzy set B in a finite universe of discourse $U$ has such a structure $B = {(x_{i}, μ_{B} (x_{i}), ν_{B} (x_{i})) : x_{i} \in U}$ , where sμ_B (x_i) represents the membership degree and ν_B (x_i) is the nonmembership degree with the condition that 0 ≤ μ_B (x_i) + ν_B (x_i) ≤ 1. The operator theory is an effective and efficient tool for ranking the alternatives while making a decision in intuitionistic fuzzy information. Zhao et al. [67] generalized the operators of Xu [46], and developed the generalized intuitionistic fuzzy weighted average operator, intuitionistic fuzzy ordered weighted average operator and intuitionistic fuzzy hybrid operator. Liu et al. in [21] initiated the linguistic intuitionistic fuzzy partitioned heronian mean operator, the linguistic intuitionistic fuzzy weighted partitioned heronian mean operator, the linguistic intuitionistic fuzzy partitioned geometric heronian mean operator and linguistic intuitionistic fuzzy weighted partitioned geometric heronian mean operator for multi attribute group decision making problems. Xu and Yager [47], proposed intuitionistic fuzzy weighted geometric operator, intuitionistic fuzzy ordered weighted geometric operator and intuitionistic fuzzy hybrid geometric operator. However, the application range of intuitionistic fuzzy sets is narrow because it has the constraint that sum of μ_B (x_i) and ν_B (x_i) is limited to 1. But, in these days for complex decision making problems, the decision makers may provide their preferences in such a way that the above condition violates. For instance, if an expert uses the intuitionistic fuzzy set environment to give his/her preferences with μ_B (x_i) = 0.7 and ν_B (x_i) = 0.6, then clearly their sum is 1.3nleq1. Therefore, this situation is not properly described by the intuitionistic fuzzy sets. To solve this problem, Yager [54] introduced the nonstandard fuzzy sets called Pythagorean fuzzy sets with μ_B (x_i) and ν_B (x_i) satisfying the condition that 0 ≤ (μ_B (x_i)) ² + (ν_B (x_i)) ² ≤ 1. Obviously, the Pythagorean fuzzy sets accommodate more uncertainties than the intuitionistic fuzzy sets, and hence their applications are more extensive than intuitionistic fuzzy sets. Subsequently, Yager [54], introduced Pythagorean fuzzy weighted average operator, Pythagorean fuzzy ordered weighted average operator, Pythagorean fuzzy hybrid average operator, Pythagorean fuzzy weighted geometric operator, Pythagorean fuzzy ordered weighted geometric operator, Pythagorean fuzzy hybrid geometric operator under Pythagorean fuzzy environment. By using the Einstein operation Garg [10] initiated the Pythagorean fuzzy weighted aggregation operator by the help of generalized Pythagorean fuzzy information. Based on Einstein t-norm and t-conorm Garg [11] further introduced the Pythagorean fuzzy geometric aggregation operator by using the generalized Pythagorean fuzzy information and discussed their applications in decision making problems. In [12] Garg proposed the confidence level based Pythagorean fuzzy aggregation operator and its application to decision making process. In [13] Garg put forward the idea of new types of Pythagorean fuzzy probabilistic aggregation operator and applied it to multi criteria decision making. Zeng [64] introduced Pythagorean fuzzy probabilistic ordered weighted averaging operator and its application in decision making problem. In [65] Zeng et al. initiated Pythagorean fuzzy weighted induced generalized weighted averaging operator and its application to R & D projections selection. Zhang and Xu [66], developed Pythagorean fuzzy TOPSIS method for multi criteria group decision-making under Pythagorean fuzzy environment. Peng and Yang [35], discussed the relationship between the Pythagorean fuzzy aggregation operators proposed by Yager, and developed a superiority/inferiority ranking method for multi criteria group decision making problems. Recently, the q-rung orthopair fuzzy sets, which was firstly proposed by Yager [55] have regarded as an efficient tool to describe vagueness of the multi criteria decision making problems. The q-rung orthopair fuzzy sets are also characterized by the membership degree and the nonmembership degree, whose sum of the qth power of the membership degree and the qth power of the degrees of nonmembership is either equal or less than 1. For example, (μ_B (x_i), ν_B (x_i)) = (0.8, 0.1) is an intuitionistic membership degree since 0.8 + 0.1 ≤ 1. If the degree of nonmembership ν_B (x_i) = 0.6, then due to (0.8) ² + (0.6) ² ≤ 1, (μ_B (x_i), ν_B (x_i)) = (0.8, 0.6) is a Pythagorean membership degree (but not an intuitionistic membership grade). However, if the degree of nonmembership ν_B (x_i) = 0.7, then this situation cannot be described by using neither intuitionistic fuzzy sets nor Pythagorean fuzzy sets. By (0.8) ³ + (0.7) ³ ≤ 1, (μ_B (x_i), ν_B (x_i)) = (0.8, 0.7) is a q-rung orthopair membership grade (for q ≥ 3), and thus it is suitable to use the q-rung orthopair fuzzy set to deal such situation. It is easily known that the q-rung orthopair fuzzy sets are general because intuitionistic fuzzy sets and Pythagorean fuzzy sets are all their special cases. It is worth noting that as the rung q increases, the space of acceptable orthopairs increases, and more orthopairs satisfy the bounding constraints. Therefore, we can express a wider range of fuzzy information by using q-rung orthopair fuzzy sets. In other words, we can continue to adjust the value of the parameter q to determine the information expression range, and thus q-rung orthopair fuzzy sets are more flexible and more suitable for the uncertain environment. Due to these unique advantages of the q-rung orthopair fuzzy sets, Yager and Alajlan [56] presented approximate reasoning with q-rung orthopair fuzzy sets by formulation of the ideas of possibility and certainty. Liu and Wang [26] proposed q-rung orthopair fuzzy aggregation operators for aggregating the evaluation information. In the meanwhile, Liu and his coworkers [22 –25] developed some new q-rung orthopair fuzzy aggregation operators based on Bonferroni mean and power Maclaurin symmetric mean for aggregating the decision making information given by experts. Wei et al. [63] and Liu et al. [29] explored some q-rung orthopair fuzzy Heronian mean operators in multicriteria decision making. Peng et al. [36] studied exponential operation and aggregation operator for q-rung orthopair fuzzy sets based on a new score function and applied them to the selection of the teaching management system. Du [6] developed Minkowski type distance measures including Hamming, Euclidean, and Chebyshev distances for q-rung orthopair fuzzy sets and discussed their applications to multi criteria decision making problems. Liu et al. [28] put forward multi criteria decision making method for dealing with heterogeneous relationship among attributes and unknown attribute weight information under q-rung orthopair fuzzy sets environment. Yager et al. [57] investigated the concepts of possibility and certainty as well as plausibility and belief in q-rung orthopair fuzzy set environment. Many aggregation operators in the environment of q-rung orthopair fuzzy sets have been developed [3 , 70].

1.2 A brief review on rough set theory

Pawlak proposed the notion of rough set theory in [34], which is the fundamental method of solving uncertain knowledge and is widely applied to the field of expert system, pattern recognition, image processing, decision analysis, artificial intelligence and so on. In addition, Pawlak provided a systematic classification mechanism for objects by employing an equivalence relation. Using this classification method, regulars hidden in information systems are extracted and decision rules are obtained. However, whether in theory or in practice, classical equivalence relation is very stringent, which can lead to unreasonable classification results, thus it confined the applications of rough sets. Accordingly, to enhance the utilization rate of information in the information systems, several authors have extended the definition of rough set approximation employing the non-equivalence relations [2 , 45]. Similarly the non-equivalence relation-based rough set models have been applied to incomplete information tables. Due to the uncertainty of propositions and concepts reflected in knowledge, a multitude of authors have devoted their attention to rough approximation in fuzzy information systems, where the results are known as rough fuzzy sets and fuzzy rough sets [7, 32]. Then, the upper and lower approximation operators of approximate space in fuzzy environment are widely used [5].

In practical life, we often need to describe the concept via multiple relations over the universe based on user requirement or the target of tackling the problem. Hence, Qian et al. in [38] extended the single granulation rough set model to the multi-granulation rough set model, which emerged as a prominent topic in artificial intelligence, attracting a wide range of researchers to both theoretical and application perspectives. Based on fuzzy information environment, Feng and Mi introduced the idea of generalized multi-granulation fuzzy decision theoretic rough set model and presented its application in three way decision analysis [8]. Li et al. [20], presented the notion of three types of double quantitative multi-granulation decision theoretic rough fuzzy set models and discussed case study related to medical diagnosis problem. Also in the context of decision making, Lin et al. developed the idea of multi-granulation rough sets by using neighborhood approach [19]. She and He [42], discussed the topological structure and the lattice structure on the basis of three different kinds of definable sets obtained from multi-granulation rough set model. Sun et al. [43], established multi-granulation approximation technique for two universes and applied it to group decision making. Xu et al. [48], initiated the notions of support characteristic function and information level for the construction of generalized multi-granulation approximation theory and discussed their application in granular selection. Xu et al. [50], applied tolerance relations for the construction of multi-granulation approximation theory. After that, Xu et al. described the idea of multi-granulation approximations in fuzzy environment and investigated their various properties for the developing of this study [49]. Based on tolerance relations, Xu et al. established the notion of generalized fuzzy rough approximations in fuzzy environment with applications to decision making methodology [51]. Yang et al. [59], established the relationship between three different hierarchical structures and different multiple granulation approximations and discussed some of their properties. You et al. [60], presented the idea of multiple granulation approximations in the environment of neighborhood covering information by applying the notion of neighborhood of objects.

1.3 Motivation of our research

As the uncertainty of decision-making situation increases the crisp number no longer satisfy the circumstance and q-rung orthopair fuzzy set emerges. Sun et al. [44], explored the rough approximation of the uncertainty information with Pythagorean fuzzy information on multi-granularity space over two universes combined with grey relational analysis. Hussain et al. [16], introduced covering based q-rung orthopair fuzzy rough set model and discussed their applications in multi attributes group decision making problems. In literature the researchers employed the idea of fuzzy similarity relation to construct the fuzzy rough approximations. To the best of our knowledge, nowadays, there does not exist any research methodology for q-rung orthopair fuzzy rough set model based on δ-fuzzy similarity relations where δ ∈ [0, 1]. So, in order to accommodate this research space, this paper aims to extend the notion of Pawlak’s rough sets to the so-called q-rung orthopair fuzzy rough sets based on δ-fuzzy similarity relations. Moreover, the approximation defined based on q-rung orthopair fuzzy rough sets applying δ-fuzzy similarity relations play a key role between δ-fuzzy similarity relation and crisp set.

Hereby, we propose new types of q-rung orthopair fuzzy lower and upper approximations by applying a tolerance degree on the similarity between two objects. After employing the tolerance degree based q-rung orthopair fuzzy rough set approach to it any times, we can only get the six different sets at most. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. In addition, the relationships among these six sets are established. Furthermore, we propose tolerance degree based multi granulation optimistic/pessimistic q-rung orthopair fuzzy rough sets and investigate some of their properties. Another main contribution of this paper is to disclose the ideas of different kinds of approximations called approximate precision, rough degree, approximate quality and their mutual relationship.

In addition, it should be pointed out that the obtained results in this work not only can be a supplement of the research topic of q-rung orthopair fuzzy rough set from the view point of mathematics, also, they provide theoretical basis and more possibilities for the applications of q-rung orthopair fuzzy rough set in real application problems, such as in expert systems, machine learning, decision analysis, pattern recognition and so on.

1.4 Organization of the paper

The arrangement of this article is as follows: Section 2 focuses mainly on the basic concepts of q-rung orthopair fuzzy sets and proposes a new type of q-rung orthopair fuzzy lower and upper approximations by applying a tolerance degree on the similarity between two objects. In Section 3 we propose tolerance degree based multigranulation optimistic/pessimistic q-rung orthopair fuzzy rough sets and investigate their properties. In addition, several uncertainty measures, such as approximate precision, rough degree, approximate quality and their mutual relationships are discussed in a q-rung orthopair fuzzy information system. In section 4 we focus our attention on the development of decision making method based on q-rung orthopair fuzzy similarity relation and illustrative example. In the end of this section, we present another idea for the development of attribute reduction in a q-rung orthopair fuzzy environment. This paper is concluded in Section 5.

2 q-rung orthopair fuzzy rough sets

In this section, we recall basic concepts of q-rung orthopair fuzzy sets and propose a new type of q-rung orthopair fuzzy lower and upper approximations by applying a tolerance degree on the similarity between two objects.

Definition 1. [34] Let R be an equivalence relation on $U$ and $(U, R)$ be the a Pawlak’s approximation space. Based on $(U, R),$ the lower and upper approximations of $X \subseteq U$ are defined, respectively, as

$\begin{matrix} R (\underline{X}) & = & {x \in U : {[x]}_{R} \subseteq X} \\ R (\bar{X}) & = & {x \in U : {[x]}_{R} \cap X \neq \emptyset} . \end{matrix}$

Moreover, if the lower approximation $R (\underline{X})$ and upper approximation $R (\bar{X})$ of the set $X$ are equal to $X,$ then $X$ is called definable, otherwise $X$ is considered a rough set. So both the $R (\underline{X})$ and $R (\bar{X})$ of the set $X$ in Pawlak’s rough set model are definable sets.

Definition 2. [55] A q-rung orthopair fuzzy set B in a finite universe of discourse $U$ is given by $B = {(x_{i}, μ_{B} (x_{i}), ν_{B} (x_{i})) : x_{i} \in X}$ where $μ_{B} : U \to [0, 1]$ denotes the degree of membership and $ν_{B} : U \to [0, 1]$ the degree of nonmembership of the element $x \in U$ to the set B, with the condition that $0 \leq μ_{B}^{q} (x_{i}) + ν_{B}^{q} (x_{i}) \leq 1,$ (q ≥ 1). The degree of indeterminacy is given by $π_{B}^{q} (x_{i}) = μ_{B}^{q} (x_{i}) + ν_{B}^{q} (x_{i}) - μ_{B}^{q} (x_{i}) ν_{B}^{q} (x_{i}) .$

Definition 3. A q-rung orthopair fuzzy information system is a quadruple $(U, A, V_{qf}, qf),$ where $U$ is a finite universe of discourse, A is a set of criteria, V_qf consists of all q-rung orthopair fuzzy values and qf is characterized by $qf : U \times A \to V_{qf},$ such that qf (x, A_i) = (x, μ_{A
_i} (x), ν_{A
_i} (x)), for all $x \in U$ and A_i ∈ A.

Definition 4. Consider the q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ The similarity between two actions x_i and x_j with respect to attribute A_i ∈ A is defined by $\begin{matrix} q - {Si}_{A_{i}} (x_{i}, x_{j}) \\ = & 1 - \sqrt{α {(μ_{A_{i}}^{q} (x_{i}) - μ_{A_{i}}^{q} (x_{j}))}^{2} + β {(ν_{A_{i}}^{q} (x_{i}) - ν_{A_{i}}^{q} (x_{j}))}^{2} + γ {(π_{A_{i}}^{q} (x_{i}) - π_{A_{i}}^{q} (x_{j}))}^{2}} \end{matrix}$ where μ_{A
_i} (x_i), ν_{A
_i} (x_i) and π_{A
_i} (x_i) are membership, nonmembership and indeterminacy degrees of the actions with respect to attribute A_i ∈ A, respectively, and α, β and γ are weight factors. In the q-rung orthopair fuzzy information system, the values of these parameters can be selected according to the requirement of different experts along with following conditions:

(i) α≥ β > γ ;

(ii) α+ β + γ = 1 ;

(iii) 0 ≤ α, β, γ ≤ 1.

For a subset of attributes A, the similarity relation between two actions x_i and x_j is defined by $(x_{i}, x_{j}) \in q - {Si}_{A}^{δ} \Leftrightarrow ⋂_{A_{i} \in A} q - {Si}_{A_{i}} (x_{i}, x_{j}) \geq δ,$ where δ is a similarity threshold. The tolerance class of an action x_i based on similarity relation is as follows; $q - {Si}_{A}^{δ} (x_{i}) = {x_{j} \in U ∣ (x_{i}, x_{j}) \in q - {Si}_{A}^{δ}} .$

Example 1. An investor wants to invest in a company. There are five potential companies as alternatives which are x₁, x₂, x₃, x₄, x₅ and six attributes, that are used to evaluate the ability of these companies, including A₁ : the technical ability, A₂ : the expected benefit, A₃ : the competitive power on the market, A₄ : the ability to bear risk, A₅ : the management capability, A₆ : Competencies and recruiting the right talent. The attribute weight is w = (0.20, 0.10, 0.30, 0.15, 0.15, 0.10) ^T . The attribute value of each alternative are evaluated by the q-rung orthopair fuzzy numbers. Consider the q-rung orthopair fuzzy information

(U, A, V_{qf}, qf)

as given in Table 1.

Table 1
q-rung orthopair fuzzy formation

	A ₁	A ₂	A ₃	A ₄	A ₅
x ₁	(0.5, 0.2)	(0.8, 0.3)	(0.8, 0.3)	(0.7, 0.3)	(0.4, 0.2)
x ₂	(0.6, 0.3)	(0.5, 0.8)	(0.6, 0.5)	(0.6, 0.5)	(0.7, 0.4)
x ₃	(0.3, 0.4)	(0.8, 0.5)	(0.7, 0.4)	(0.6, 0.4)	(0.6, 0.2)
x ₄	(0.7, 0.4)	(0.5, 0.6)	(0.7, 0.4)	(0.5, 0.5)	(0.7, 0.6)
x ₅	(0.7, 0.6)	(0.6, 0.4)	(0.4, 0.7)	(0.4, 0.3)	(0.7, 0.7)

Let α = β = 0.4, γ = 0.2 and q = 8. Then the similarity relation with respect to A₁ is given by $\begin{matrix} Similarity relation with respect to A_{1} \\ {Si}_{A_{1}} (x_{i}, x_{j}) & = & (\begin{matrix} 1.0000 & 0.9900 & 0.9972 & 0.9582 & 0.9527 \\ 0.9900 & 1.0000 & 0.9872 & 0.9682 & 0.9623 \\ 0.9972 & 0.9872 & 1.0000 & 0.9554 & 0.9501 \\ 0.9582 & 0.9682 & 0.9554 & 1.0000 & 0.9877 \\ 0.9527 & 0.9623 & 0.9501 & 0.9877 & 1.0000 \end{matrix}) . \end{matrix}$

Similarly

$\begin{matrix} Similarity relation with respect to A_{2} \\ {Si}_{A_{2}} (x_{i}, x_{j}) & = & (\begin{matrix} 1.0000 & 0.8577 & 0.9972 & 0.8768 & 0.8832 \\ 0.8517 & 1.0000 & 0.8581 & 0.8832 & 0.8737 \\ 0.9972 & 0.8381 & 1.0000 & 0.8762 & 0.8824 \\ 0.8768 & 0.8832 & 0.8762 & 1.0000 & 0.9869 \\ 0.8832 & 0.8737 & 0.8824 & 0.9869 & 1.0000 \end{matrix}), \end{matrix}$ $\begin{matrix} Similarity relation with respect to A_{3} \\ {Si}_{A_{3}} (x_{i}, x_{j}) & = & (\begin{matrix} 1.0000 & 0.8840 & 0.9179 & 0.9149 & 0.8780 \\ 0.8840 & 1.0000 & 0.9640 & 0.9691 & 0.9607 \\ 0.9179 & 0.9640 & 1.0000 & 0.9877 & 0.9551 \\ 0.9149 & 0.9691 & 0.9877 & 1.0000 & 0.9490 \\ 0.8780 & 0.9607 & 0.9551 & 0.9490 & 1.0000 \end{matrix}), \end{matrix}$ $\begin{matrix} Similarity relation with respect to A_{4} \\ {Si}_{A_{4}} (x_{i}, x_{j}) & = & (\begin{matrix} 1.0000 & 0.9692 & 0.9685 & 0.9593 & 0.9559 \\ 0.9692 & 1.0000 & 0.9975 & 0.9900 & 0.9862 \\ 0.9685 & 0.9975 & 1.0000 & 0.9906 & 0.9873 \\ 0.9593 & 0.9900 & 0.9906 & 1.0000 & 0.9955 \\ 0.9559 & 0.9862 & 0.9873 & 0.9955 & 1.0000 \end{matrix}), \end{matrix}$ and

$\begin{matrix} Similarity relation with respect to A_{5} \\ {Si}_{A_{5}} (x_{i}, x_{j}) & = & (\begin{matrix} 1.0000 & 0.9557 & 0.9875 & 0.9503 & 0.9285 \\ 0.9557 & 1.0000 & 0.9682 & 0.9877 & 0.9567 \\ 0.9875 & 0.9682 & 1.0000 & 0.9623 & 0.9383 \\ 0.9503 & 0.9877 & 0.9623 & 1.0000 & 0.9690 \\ 0.9285 & 0.9567 & 0.9383 & 0.9690 & 1.0000 \end{matrix}) . \end{matrix}$ Now the similarity relation with respect to A is given by $\begin{matrix} Similarity relation with respect to A \\ {Si}_{A} (x_{i}, x_{j}) & = & (\begin{matrix} 1.0000 & 0.8577 & 0.9179 & 0.8768 & 0.8780 \\ 0.8577 & 1.0000 & 0.8581 & 0.8832 & 0.8737 \\ 0.9179 & 0.8581 & 1.0000 & 0.8762 & 0.8824 \\ 0.8768 & 0.8832 & 0.8762 & 1.0000 & 0.9490 \\ 0.8780 & 0.8737 & 0.8824 & 0.9490 & 1.0000 \end{matrix}) . \end{matrix}$

Theorem 1. Consider the q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ If δ₁ ≥ δ₂, then q- ${Si}_{A}^{δ_{1}} \subseteq q$ - ${Si}_{A}^{δ_{2}} .$

Proof. The proof is straightforward.□

Example 2. (ContinuedformExample1) Let δ₁ = 0.8800 and δ₂ = 0.8600. Then the tolerance classes with δ = 0.8800 are given in Table 2 below as follows:

Table 2

Tolerance classes

$U$	${Si}_{A}^{0.8800} (x_{i})$
x ₁	{x₁, x₃ },
x ₂	{x₂, x₄ },
x ₃	{x₁, x₃, x₅ },
x ₄	{x₂, x₄, x₅ },
x ₅	{x₃, x₄, x₅ }.

Similarly the tolerance classes with δ = 0.8600 are given in Table 3 below as follows:

Table 3

Tolerance classes

$U$	${Si}_{A}^{0.8600} (x_{i})$
x ₁	{x₁, x₃, x₄, x₅ },
x ₂	{x₂, x₄, x₅ },
x ₃	{x₁, x₃, x₄, x₅ },
x ₄	{x₁, x₂, x₃, x₄, x₅ },
x ₅	{x₁, x₂, x₃, x₄, x₅ }.

Based on Tables 3, we obtain q- ${Si}_{A}^{δ_{1}} (x_{i}) \subseteq q$ - ${Si}_{A}^{δ_{2}} (x_{i})$ for each $x_{i} \in U .$

Theorem 2. Consider the q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ If $F_{q - {Si}_{A}^{δ}} = {q - {Si}_{A}^{δ} ∣ δ \in [0, 1]}$ is the collection of all q- ${Si}_{A_{i}}^{δ}$ with respect to different values of δ, then $(F_{q - {Si}_{A}^{δ}}, \cap, \cup)$ is a lattice, where ∩ and ∪ are intersection and union operations defined for classical sets.

Proof. To prove $(F_{q - {Si}_{A}^{δ}}, \cap, \cup)$ is a lattice, we need to prove idempotent, commutative, associative and absorption laws,

(1) Idempotent law: For all q- ${Si}_{A}^{δ} \in F_{q - {Si}_{A}^{δ}},$ clearly

$q - {Si}_{A}^{δ} \cap q - {Si}_{A}^{δ} = q - {Si}_{A}^{δ}$ and $q - {Si}_{A}^{δ} \cup q - {Si}_{A}^{δ} = q - {Si}_{A}^{δ} .$

(2) Commutative law: For all q- ${Si}_{A}^{δ_{1}}, q$ - ${Si}_{A}^{δ_{2}} \in F_{q - {Si}_{A}^{δ}}$ we have $q - {Si}_{A}^{δ_{1}} \cap q - {Si}_{A}^{δ_{2}} = q - {Si}_{A}^{δ_{2}} \cap q - {Si}_{A}^{δ_{1}}$ and $q - {Si}_{A}^{δ_{1}} \cup q - {Si}_{A}^{δ_{2}} = q - {Si}_{A}^{δ_{2}} \cup q - {Si}_{A}^{δ_{1}}$

(3) Associative law: For all q- ${Si}_{A}^{δ_{1}},$ q- ${Si}_{A}^{δ_{2}}$ , q- ${Si}_{A}^{δ_{3}} \in F_{q - {Si}_{A}^{δ}}$ using the set theoretic operations it follows that

$\begin{matrix} q - {Si}_{A}^{δ_{1}} \cap (q - {Si}_{A}^{δ_{2}} \cap q - {Si}_{A}^{δ_{3}}) \\ = (q - {Si}_{A}^{δ_{1}} \cap q - {Si}_{A}^{δ_{2}}) \cap q - {Si}_{A}^{δ_{3}} \end{matrix}$ and $\begin{matrix} q - {Si}_{A}^{δ_{1}} \cup (q - {Si}_{A}^{δ_{2}} \cup q - {Si}_{A}^{δ_{3}}) \\ = (q - {Si}_{A}^{δ_{1}} \cup q - {Si}_{A}^{δ_{2}}) \cup q - {Si}_{A}^{δ_{3}} \end{matrix}$

(4) Absorption law: For all q- ${Si}_{A}^{δ_{1}},$ q- ${Si}_{A}^{δ_{2}} \in F_{q - {Si}_{A}^{δ}}$ using the set theoretic operation $\begin{matrix} q - {Si}_{A}^{δ_{1}} \cap (q - {Si}_{A}^{δ_{1}} \cup q - {Si}_{A}^{δ_{2}}) \\ = (q - {Si}_{A}^{δ_{1}} \cap q - {Si}_{A}^{δ_{1}}) \cup (q - {Si}_{A}^{δ_{1}} \cap q - {Si}_{A}^{δ_{2}}) \\ = q - {Si}_{A}^{δ_{1}} \cup (q - {Si}_{A}^{δ_{1}} \cap q - {Si}_{A}^{δ_{2}}) \\ = q - {Si}_{A}^{δ_{1}} . \end{matrix}$

Similarly it is not difficult to prove that q- ${Si}_{A}^{δ_{1}} \cup (q - {Si}_{A}^{δ_{1}} \cap q - {Si}_{A}^{δ_{2}}) = q$ - ${Si}_{A}^{δ_{1}} .$ Thus $(F_{q - {Si}_{A}^{δ}}, \cap, \cup)$ is a lattice. In the lattice $(F_{q - {Si}_{A}^{δ}}, \cap, \cup)$ the partial order induced by such algebraic lattice is ⊆ (asinTheorem1), which depicts that q- ${Si}_{A}^{δ_{1}}$ is stricter than q- ${Si}_{A}^{δ_{2}}$ iff q- ${Si}_{A}^{δ_{1}} \subseteq q$ - ${Si}_{A}^{δ_{2}} .$ Hence $(F_{P_{D}^{+}}, \subseteq)$ is a poset. Clearly in this lattice, the greatest element is q- ${Si}_{A}^{δ_{0}}$ while the least element is q- ${Si}_{A}^{δ_{1}} .$ Therefore $(F_{q - {Si}_{A}^{δ}}, \cap, \cup)$ is a bounded lattice.□

Theorem 3. Consider the q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ Then $(F_{q - {Si}_{A}^{δ}}, \cap, \cup)$ is a distributive lattice.

Proof. Using the basic set theoretic properties of ∩ and ∪, the proof is straightforward.□

2.1 Similarity based q-rung orthopair fuzzy rough sets

Definition 5. Consider the q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ For any set of attributes A and $X \subseteq U,$ the lower and upper approximations are respectively defined by: $\begin{matrix} \underline{app (X_{A}^{δ})} = \cup {q - {Si}_{A}^{δ} (x_{i}) ∣ q - {Si}_{A}^{δ} (x_{i}) \subseteq X} \end{matrix}$ and $\begin{matrix} \bar{app (X_{A}^{δ})} = - \underline{app (- X_{A}^{δ})} . \end{matrix}$ where $- X$ denote the complementary set $U - X$ of $X .$ The ordered pair $(\underline{app (X_{A}^{δ})}, \bar{app (X_{A}^{δ})})$ is called q-rung orthopair fuzzy tolerance rough set.

Example 3. (Continued form Example 1) Let δ = 0.8800 and $X = {x_{1}, x_{2}, x_{4}} .$ Then $\underline{app (X_{A}^{0.8800})} = {x_{2}, x_{4}}$ and $\bar{app (X_{A}^{0.8800})} = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}} .$

Theorem 4. Consider a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ If δ₁ ≥ δ₂ and $X \subseteq U,$ then

(i) $\underline{app (X_{A}^{δ_{2}})} \subseteq \underline{app (X_{A}^{δ_{1}})},$

(ii) $\bar{app (X_{A}^{δ_{1}})} \subseteq \bar{app (X_{A}^{δ_{2}})} .$

Example 4. (ContinuedformExample1) Let δ₁ = 0.8800, δ₂ = 0.8600 and $X = {x_{1}, x_{2}, x_{4}} .$ Then $\underline{app (X_{A}^{0.8800})} = {x_{2}, x_{4}}$ and $\underline{app (X_{A}^{0.8600})} = \emptyset .$ Thus $\underline{app (X_{A}^{0.8600})} \subseteq \underline{app (X_{A}^{0.8800})} .$ Further, let {x₁, x₃, x₅ }. Then $\bar{app (X_{A}^{0.8800})} = {x_{1}, x_{3}, x_{5}}$ and $\bar{app (X_{A}^{0.8600})} = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}} .$ Thus $\bar{app (X_{A}^{0.8800})} \subseteq \bar{app (X_{A}^{0.8600})} .$

Theorem 5. Consider a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf)$ and $X \subseteq U .$ Then

(i) $\bar{app} (\bar{app (X_{A}^{δ})}) = \bar{app (X_{A}^{δ})},$

(ii) $\underline{app} (\underline{app (X_{A}^{δ})}) = \underline{app (X_{A}^{δ})} .$

In classical rough set theory, both the lower and upper approximations of a subset $X$ are crisp sets that is, $R (\underline{R (\underline{X})}) = R (\underline{X}) = R (\bar{R (\underline{X})})$ and $R (\underline{R (\bar{X})}) = R (\bar{X}) = R (\bar{R (\bar{X})}) .$ But in generalized rough set theory, including binary relation-based rough set theory, covering rough set theory and fuzzy rough set theory, the lower and upper approximations are hardly crisp sets. In general, they’re still rough sets. For q-rung orthopair fuzzy tolerance rough set theory, the lower and upper approximations are hardly definable sets. In general, they are still rough sets. That is, $\underline{app} (\underline{app (X_{A}^{δ})}) = \underline{app (X_{A}^{δ})} \neq \bar{app} (\underline{app (X_{A}^{δ})}),$ $\bar{app} (\bar{app (X_{A}^{δ})}) = \bar{app (X_{A}^{δ})} \neq \underline{app} (\bar{app (X_{A}^{δ})}) .$ In the following we present an example to show this fact.

Example 5. Consider a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ Let $U = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}}$ be a universe of discourse and A ={ A₁, A₂, A₃, A₄, A₅ } be a set of criteria. Consider the q-rung orthopair fuzzy information as given in Table 4.

Table 4
q-rung orthopair fuzzy information

A ₁ A ₂ A ₃ A ₄ A ₅

x ₁ (0.8, 0.3) (0.8, 0.3) (0.7, 0.3) (0.4, 0.2) (0.4, 0.8)

x ₂ (0.5, 0.8) (0.6, 0.5) (0.6, 0.5) (0.7, 0.4) (0.5, 0.6)

x ₃ (0.8, 0.5) (0.7, 0.4) (0.6, 0.4) (0.6, 0.2) (0.4, 0.7)

x ₄ (0.5, 0.6) (0.7, 0.4) (0.5, 0.5) (0.7, 0.6) (0.6, 0.5)

x ₅ (0.6, 0.4) (0.4, 0.7) (0.4, 0.3) (0.7, 0.7) (0., 0.4)

	A ₁	A ₂	A ₃	A ₄	A ₅
x ₁	(0.8, 0.3)	(0.8, 0.3)	(0.7, 0.3)	(0.4, 0.2)	(0.4, 0.8)
x ₂	(0.5, 0.8)	(0.6, 0.5)	(0.6, 0.5)	(0.7, 0.4)	(0.5, 0.6)
x ₃	(0.8, 0.5)	(0.7, 0.4)	(0.6, 0.4)	(0.6, 0.2)	(0.4, 0.7)
x ₄	(0.5, 0.6)	(0.7, 0.4)	(0.5, 0.5)	(0.7, 0.6)	(0.6, 0.5)
x ₅	(0.6, 0.4)	(0.4, 0.7)	(0.4, 0.3)	(0.7, 0.7)	(0., 0.4)

Let α = β = 0.4, γ = 0.2 and q = 8. Then the similarity relation with respect to A is given by $\begin{matrix} Similarity relation with respect to A \\ {Si}_{A} (x_{i}, x_{j}) & = & (\begin{matrix} 1.0000 & 0.8577 & 0.9147 & 0.8767 & 0.8714 \\ 0.8577 & 1.0000 & 0.8581 & 0.8832 & 0.8737 \\ 0.9147 & 0.8581 & 1.0000 & 0.8762 & 0.8824 \\ 0.8767 & 0.8832 & 0.8762 & 1.0000 & 0.9490 \\ 0.8714 & 0.8737 & 0.8824 & 0.9490 & 1.0000 \end{matrix}) . \end{matrix}$ The tolerance classes with δ = 0.8700 are given in Table 5 below as follows:

Table 5

Tolerance classes

$U$	${Si}_{A}^{0.8700} (x_{i})$
x ₁	{x₁, x₃, x₄, x₅ },
x ₂	{x₂, x₄, x₅ },
x ₃	{x₁, x₃, x₄, x₅ },
x ₄	{x₁, x₂, x₃, x₄, x₅ },
x ₅	{x₁, x₂, x₃, x₄, x₅ }.

For $X = {x_{1}, x_{3}, x_{4}, x_{5}},$ the approximations $\underline{app (X_{A}^{0.8700})} = {x_{1}, x_{3}, x_{4}, x_{5}} = \underline{app} (\underline{app (X_{A}^{0.8700})})$ and $\bar{app} (\underline{app (X_{A}^{0.8700})}) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}} .$ Therefore $\underline{app} (\underline{app (X_{A}^{0.8700})}) = \underline{app (X_{A}^{0.8700})} \neq \bar{app} (\underline{app (X_{A}^{0.8700})}) .$

The tolerance classes with δ = 0.8800 are given in Table 6 below as follows:

Table 6

Tolerance classes

$U$	${Si}_{A}^{0.8800} (x_{i})$
x ₁	{x₁, x₃ },
x ₂	{x₂, x₄ },
x ₃	{x₁, x₃, x₅ },
x ₄	{x₂, x₄, x₅ },
x ₅	{x₃, x₄, x₅ }.

For $X = {x_{1}, x_{2}},$ the approximations are $\bar{app (X_{A}^{0.8800})} = {x_{1}, x_{2}} = \bar{app} (\bar{app (X_{A}^{0.8800})})$ and $\underline{app} (\bar{app (X_{A}^{0.8800})}) = \emptyset .$ Therefore $\bar{app} (\bar{app (X_{A}^{0.8800})}) = \bar{app (X_{A}^{0.8800})} \neq \underline{app} (\bar{app (X_{A}^{0.8800})}) .$

From the above example we see that $\underline{app (X_{A}^{δ})}$ and $\bar{app (X_{A}^{δ})}$ are still rough sets. If we apply the lower or upper approximation operations over and over again to a subset $X,$ we obtain six different sets at most. These sets are $\underline{app (X_{A}^{δ})},$ $\bar{app} (\underline{app (X_{A}^{δ})}),$ $\underline{app} (\bar{app} (\underline{app (X_{A}^{δ})})),$ $\bar{app (X_{A}^{δ})},$ $\underline{app} (\bar{app (X_{A}^{δ})}),$ $\bar{app} (\underline{app} (\bar{app (X_{A}^{δ})})) .$

Theorem 6. Consider a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ Then for any $X \subseteq U,$ the following hold:

(i) $\bar{app} (\underline{app} (\bar{app} (\underline{app (X_{A}^{δ})}))) = \bar{app} (\underline{app (X_{A}^{δ})});$

(ii) $\underline{app} (\bar{app} (\underline{app} (\bar{app (X_{A}^{δ})}))) = \underline{app} (\bar{app (X_{A}^{δ})}) .$

Proof. (i) As we know that $\underline{app (X_{A}^{δ})} \subseteq \bar{app} (\underline{app (X_{A}^{δ})}) .$ This implies that

$\underline{app} (\underline{app (X_{A}^{δ})}) \subseteq \underline{app} (\bar{app} (\underline{app (X_{A}^{δ})})) .$ Therefore $\underline{app (X_{A}^{δ})} \subseteq \underline{app} (\bar{app} (\underline{app (X_{A}^{δ})})) .$ Also $\bar{app} (\underline{app (X_{A}^{δ})}) \subseteq \bar{app} (\underline{app} (\bar{app} (\underline{app (X_{A}^{δ})}))) .$ Now $\underline{app} (\bar{app} (\underline{app (X_{A}^{δ})})) \subseteq \bar{app} (\underline{app (X_{A}^{δ})}) .$ This implies that

$\bar{app} (\underline{app} (\bar{app} (\underline{app (X_{A}^{δ})}))) \subseteq \bar{app} (\bar{app} (\underline{app (X_{A}^{δ})})) = \bar{app} (\underline{app (X_{A}^{δ})}) .$ Therefore $\bar{app} (\underline{app} (\bar{app} (\underline{app (X_{A}^{δ})}))) = \bar{app} (\underline{app (X_{A}^{δ})}) .$

(ii) Since $\underline{app} (\bar{app (X_{A}^{δ})}) \subseteq \bar{app (X_{A}^{δ})} .$ So $\bar{app} (\underline{app} (\bar{app (X_{A}^{δ})})) \subseteq \bar{app} (\bar{app (X_{A}^{δ})}) = \bar{app (X_{A}^{δ})} .$ This implies that $\underline{app} (\bar{app} (\underline{app} (\bar{app (X_{A}^{δ})}))) \subseteq \underline{app} \bar{(app (X_{A}^{δ}))} .$ It follows that $\underline{app} \bar{(app (X_{A}^{δ}))} \subseteq \bar{app} (\underline{app} (\bar{app (X_{A}^{δ})})) .$ Also $\underline{app} (\underline{app} (\bar{app (X_{A}^{δ})})) \subseteq \underline{app} (\bar{app} (\underline{app} (\bar{app (X_{A}^{δ})}))) .$ This implies that $\underline{app} (\bar{app (X_{A}^{δ})}) \subseteq \underline{app} (\bar{app} (\underline{app} (\bar{app (X_{A}^{δ})}))) .$ Therefore $\underline{app} (\bar{app} (\underline{app} (\bar{app (X_{A}^{δ})}))) = \underline{app} (\bar{app (X_{A}^{δ})}) .$ □

The following theorem gives the relationship between the aforesaid six sets.

Theorem 7. Consider a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ Then for any $X \subseteq U,$ the following hold:

(i) $\underline{app} (\bar{app} (\underline{app} (\bar{app (X_{A}^{δ})}))) = \underline{app} (\bar{app (X_{A}^{δ})}) = \underline{app} (\underline{app} (\bar{app (X_{A}^{δ})}));$

(ii) $\bar{app} (\underline{app} (\bar{app} (\underline{app (X_{A}^{δ})}))) = \bar{app} (\underline{app (X_{A}^{δ})}) = \bar{app} (\bar{app} (\underline{app (X_{A}^{δ})}));$

(iii) If $\underline{app (X_{A}^{δ})} = \bar{app (X_{A}^{δ})},$ then $\underline{app (X_{A}^{δ})} = X = \bar{app (X_{A}^{δ})} = \underline{app} (\bar{app (X_{A}^{δ})}) = \bar{app} (\underline{app} (\bar{app (X_{A}^{δ})}))$

$= \bar{app} (\underline{app (X_{A}^{δ})}) = \underline{app} (\bar{app} (\underline{app (X_{A}^{δ})}));$

(iv) If $\bar{app (X_{A}^{δ})} = \underline{app} (\bar{app} (\underline{app (X_{A}^{δ})})),$ then $\bar{app (X_{A}^{δ})} = \underline{app} (\bar{app (X_{A}^{δ})})$

$= \bar{app} (\underline{app} (\bar{app (X_{A}^{δ})})) = \bar{app} (\underline{app (X_{A}^{δ})}) = \underline{app} (\bar{app} (\underline{app (X_{A}^{δ})}));;$

(v) If $\underline{app (X_{A}^{δ})} = \bar{app} (\underline{app} (\bar{app (X_{A}^{δ})})),$ then $\underline{app (X_{A}^{δ})} = \bar{app} (\underline{app (X_{A}^{δ})})$

$= \bar{app} (\underline{app} (\bar{app (X_{A}^{δ})})) = \underline{app} (\bar{app (X_{A}^{δ})}) = \underline{app} (\bar{app} (\underline{app (X_{A}^{δ})}));$

(vi) If $\bar{app (X_{A}^{δ})} = \bar{app} (\underline{app (X_{A}^{δ})}),$ then $\bar{app (X_{A}^{δ})} = \bar{app} (\underline{app (X_{A}^{δ})}) = \bar{app} (\underline{app} (\bar{app (X_{A}^{δ})}))$ and $\underline{app} (\bar{app (X_{A}^{δ})}) = \underline{app} (\bar{app} (\underline{app (X_{A}^{δ})}));$

(vii) If $\underline{app (X_{A}^{δ})} = \underline{app} (\bar{app (X_{A}^{δ})}),$ then $\underline{app (X_{A}^{δ})} = \underline{app} (\bar{app (X_{A}^{δ})}) = \underline{app} (\bar{app} (\underline{app (X_{A}^{δ})}))$ and $\bar{app} (\underline{app (X_{A}^{δ})})$

$= \bar{app} (\underline{app} (\bar{app (X_{A}^{δ})}));$

(viii) If $\underline{app} (\bar{app} (\underline{app (X_{A}^{δ})})) = \bar{app} (\underline{app} (\bar{app (X_{A}^{δ})})),$ then $\bar{app} (\underline{app (X_{A}^{δ})})$

$= \bar{app} (\underline{app} (\bar{app (X_{A}^{δ})})) = \underline{app} (\bar{app (X_{A}^{δ})}) = \underline{app} (\bar{app} (\underline{app (X_{A}^{δ})}));$

(ix) If $\bar{app} (\underline{app (X_{A}^{δ})}) = \underline{app} (\bar{app (X_{A}^{δ})}),$ then $\underline{app} (\bar{app (X_{A}^{δ})}) = \bar{app} (\underline{app} (\bar{app (X_{A}^{δ})})) = \underline{app} (\bar{app (X_{A}^{δ})}) = \underline{app} (\bar{app} (\underline{app (X_{A}^{δ})})) .$

3 Multi-granulation q-rung orthopair fuzzy tolerance rough sets

For the optimists, they usually consider the positive side and believe that the candidates are optimal as long as an individual is satisfied finds it appropriate or an attribute for an individual is in line with forecast. From the perspective, this section presents tolerance degree based multi-granulation optimistic q-rung orthopair fuzzy rough sets. Firstly, we give the definition of tolerance degree based multi-granulation optimistic q-rung orthopair fuzzy rough sets and investigate some of their properties.

Definition 6. Consider a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ Let A₁, A₂, A₃, ..., A_n ∈ A and $X \subseteq U .$ Then the optimistic lower and upper approximation of $X$ are respectively defined by: $\begin{matrix} {\underline{app {(X)}^{δ (o)}}}_{\sum_{i = 1}^{n} A_{i}} \\ = {\begin{matrix} x_{i} \in U ∣ q - {Si}_{A_{1}}^{δ} (x_{i}) \subseteq X \lor q - {Si}_{A_{2}}^{δ} (x_{i}) \subseteq X \\ \lor . . . \lor q - {Si}_{A_{n}}^{δ} (x_{i}) \subseteq X \end{matrix}} \end{matrix}$ $\begin{matrix} \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (o)}} \\ = {\begin{matrix} x_{i} \in U ∣ q - {Si}_{A_{1}}^{δ} (x_{i}) \cap X \neq \emptyset \land q - {Si}_{A_{2}}^{δ} (x_{i}) \cap X \neq \emptyset \\ \land . . . \land q - {Si}_{A_{n}}^{δ} (x_{i}) \cap X \neq \emptyset \end{matrix}} . \end{matrix}$

The ordered pair $({\underline{app {(X)}^{δ (o)}}}_{\sum_{i = 1}^{n} A_{i}}, \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (o)}})$ is called optimistic multigranulation q-rung orthopair fuzzy tolerance rough set.

Theorem 8. Consider a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ Let A₁, A₂, A₃, ..., A_n ∈ A and $X_{1},$ $X_{2},$ $. . ., X_{n} \subseteq U .$ Then the optimistic multigranulation q-rung orthopair fuzzy tolerance rough sets satisfy the following properties:

(i) ${\underline{app {(X_{i})}^{δ (o)}}}_{\sum_{i = 1}^{n} A_{i}} \subseteq X_{i} \subseteq \bar{app {(X_{i})}_{\sum_{i = 1}^{n} A_{i}}^{δ (o)}};$

(ii) ${\underline{app {(U)}^{δ (o)}}}_{\sum_{i = 1}^{n} A_{i}} = U = \bar{app {(U)}_{\sum_{i = 1}^{n} A_{i}}^{δ (o)}};$

(iii) ${\underline{app {(\emptyset)}^{δ (o)}}}_{\sum_{i = 1}^{n} A_{i}} = \emptyset = \bar{app {(\emptyset)}_{\sum_{i = 1}^{n} A_{i}}^{δ (o)}};$

(iv) ${\underline{app {(X_{i}^{c})}^{δ (o)}}}_{\sum_{i = 1}^{n} A_{i}} = {(\bar{app {(X_{i})}_{\sum_{i = 1}^{n} A_{i}}^{δ (o)}})}^{c};$

(v) $\bar{app {(X_{i}^{c})}_{\sum_{i = 1}^{n} A_{i}}^{δ (o)}} = {({\underline{app {(X_{i})}^{δ (o)}}}_{\sum_{i = 1}^{n} A_{i}})}^{c};$

(vi) $X_{1} \subseteq X_{2}$ implies ${\underline{app {(X_{1})}^{δ (o)}}}_{\sum_{i = 1}^{n} A_{i}} \subseteq {\underline{app {(X_{2})}^{δ (o)}}}_{\sum_{i = 1}^{n} A_{i}};$

(vii) $X_{1} \subseteq X_{2}$ implies $\bar{app {(X_{1})}_{\sum_{i = 1}^{n} A_{i}}^{δ (o)}} \subseteq \bar{app {(X_{2})}_{\sum_{i = 1}^{n} A_{i}}^{δ (o)}};$

(viii) ${\underline{app {(⋂_{i = 1}^{n} X_{i})}^{δ (o)}}}_{\sum_{i = 1}^{n} A_{i}} \subseteq ⋂_{i = 1}^{n} {\underline{app {(X_{i})}^{δ (o)}}}_{\sum_{i = 1}^{n} A_{i}};$

(ix) $\bar{app {(⋂_{i = 1}^{n} X_{i})}_{\sum_{i = 1}^{n} A_{i}}^{δ (o)}} \subseteq ⋂_{i = 1}^{n} \bar{app {(X_{i})}_{\sum_{i = 1}^{n} A_{i}}^{δ (o)}};$

(x) ${\underline{app {(⋃_{i = 1}^{n} X_{i})}^{δ (o)}}}_{\sum_{i = 1}^{n} A_{i}} \supseteq ⋃_{i = 1}^{n} {\underline{app {(X_{i})}^{δ (o)}}}_{\sum_{i = 1}^{n} A_{i}};$

(xi) $\bar{app {(⋃_{i = 1}^{n} X_{i})}_{\sum_{i = 1}^{n} A_{i}}^{δ (o)}} \supseteq ⋃_{i = 1}^{n} \bar{app {(X_{i})}_{\sum_{i = 1}^{n} A_{i}}^{δ (o)}} .$

3.1 Multi-granulation pessimistic q-rung orthopair fuzzy rough sets

Now we turn our attention to another q-rung orthopair fuzzy system in pessimistic setting. For the pessimists, they hope that the optimal alternatives are satisfied with all the decision makers or an attribute is in line with the expectation of individuals. In this subsection, we will present tolerance degree based multi-granulation pessimistic q-rung orthopair fuzzy rough sets and investigate some of their properties.

Definition 7. Consider a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ Let A₁, A₂, A₃, ..., A_n ∈ A and $X,$ $Y \subseteq U .$ Then the pessimistic lower and upper approximation of $X$ are defined, respectively, as $\begin{matrix} {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} \\ = {\begin{matrix} x_{i} \in U ∣ q - {Si}_{A_{1}}^{δ} (x_{i}) \subseteq X \land q - {Si}_{A_{2}}^{δ} (x_{i}) \subseteq X \\ \land . . . \land q - {Si}_{A_{n}}^{δ} (x_{i}) \subseteq X \end{matrix}} \end{matrix}$ $\begin{matrix} \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} = \\ {\begin{matrix} x_{i} \in U ∣ q - {Si}_{A_{1}}^{δ} (x_{i}) \cap X \neq \emptyset \lor q - {Si}_{A_{2}}^{δ} (x_{i}) \cap X \neq \emptyset \\ \lor . . . \lor q - {Si}_{A_{n}}^{δ} (x_{i}) \cap X \neq \emptyset \end{matrix}} . \end{matrix}$

The ordered pair $({\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}}, \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}})$ is called pessimistic multigranu-lation q-rung orthopair fuzzy tolerance rough set.

Theorem 9. Consider a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ Let A₁, A₂, A₃, ..., A_n ∈ A and $X_{1},$ $X_{2},$ ..., $X_{n} \subseteq U .$ Then the pessimistic multigranulation q-rung orthopair fuzzy tolerance rough set satisfy the following:

(i) ${\underline{app {(X_{i})}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} \subseteq X_{i} \subseteq \bar{app {(X_{i})}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}};$

(ii) ${\underline{app {(U)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} = U = \bar{app {(U)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}};$

(iii) ${\underline{app {(\emptyset)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} = \emptyset = \bar{app {(\emptyset)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}};$

(iv) ${\underline{app {(X_{i}^{c})}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} = {(\bar{app {(X_{i})}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}})}^{c};$

(v) $\bar{app {(X_{i}^{c})}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} = {({\underline{app {(X_{i})}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}})}^{c};$

(vi) $X_{1} \subseteq X_{2}$ implies ${\underline{app {(X_{1})}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} \subseteq {\underline{app {(X_{2})}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}};$

(vii) $X_{1} \subseteq X_{2}$ implies $\bar{app {(X_{1})}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \subseteq \bar{app {(X_{2})}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}};$

(viii) ${\underline{app {(⋂_{i = 1}^{n} X_{i})}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} = ⋂_{i = 1}^{n} {\underline{app {(X_{i})}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}};$

(ix) $\bar{app {(⋂_{i = 1}^{n} X_{i})}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \subseteq ⋂_{i = 1}^{n} \bar{app {(X_{i})}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}};$

(x) ${\underline{app {(⋃_{i = 1}^{n} X_{i})}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} \supseteq ⋃_{i = 1}^{n} {\underline{app {(X_{i})}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}};$

(xi) $\bar{app {(⋃_{i = 1}^{n} X_{i})}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} = ⋃_{i = 1}^{n} \bar{app {(X_{i})}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} .$

Definition 8. Consider a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ For any $X \subseteq U,$ the approximate precision $ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X)$ of $X$ is defined by: $\begin{matrix} ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) = \frac{| {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} |}{| \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} |}, \end{matrix}$ where $X \neq \emptyset,$ and | · | denotes the cardinality of a set.

Let $μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) = 1 - ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X)$ . Then $μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X)$ is called the rough degree of $X .$

By definition, $0 \leq μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) \leq 1$ and $0 \leq ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) \leq 1 .$ It can be seen that $ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) = 1$ if and only if ${\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} = X = \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}}$ .

The following theorem describes the relationship of the precision $ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X)$ and the rough degree $μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X)$ for the intersection and union of subsets $X$ and $Y$ of the universe $U .$

Theorem 10. Consider a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf)$ and $X,$ $Y \subseteq U .$ Then the rough degree and precision of the subsets $X,$ $Y,$ $X \cup Y$ and $X \cap Y$ satisfy the following relations.

$\begin{matrix} (i) μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cup Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cup \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ \leq & μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | + μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (Y) | \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cap Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | . \end{matrix}$ $\begin{matrix} (ii) ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cup Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cup \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ \geq & ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | + ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (Y) | \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cap Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | . \end{matrix}$

Proof. (i) By definition of rough degree, $\begin{matrix} {\sum_{i = 1}^{μ_{n}^{δ (p)}} A_{i}}^{(X \cup Y)} \\ = 1 - \frac{| \underline{a p p {(X \cup Y)}^{δ (p)}} \sum_{i = 1}^{n} A_{i} |}{| \bar{a p p {(X \cup Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} |} \\ =1 - \frac{| \underline{app {(X \cup Y)}^{δ (p)}} \sum_{i = 1}^{n} A_{i} |}{| \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cup \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} |} \\ \leq 1 - \frac{| \underline{app {(X)}^{δ (p)}} \sum_{i = 1}^{n} A_{i} \cup \underline{app {(Y)}^{δ (p)}} \sum_{i = 1}^{n} A_{i} |}{| \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cup \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} |} . \end{matrix}$ This implies that $\begin{matrix} μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cup Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cup \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ \leq & | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cup \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - | {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} \cup {\underline{app {(Y)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} | . \end{matrix}$

Similarly $\begin{matrix} μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cap Y) \\ = 1 - \frac{| {\underline{app {(X \cap Y)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} |}{| \bar{app {(X \cap Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} |} \\ = 1 - \frac{| {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} \cap {\underline{app {(Y)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} |}{| \bar{app {(X \cap Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} |} \\ \leq 1 - \frac{| {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} \cap {\underline{app {(Y)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} |}{| \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} |} . \end{matrix}$ Hence $\begin{matrix} μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cap Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ \leq & | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - | {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} \cap {\underline{app {(Y)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} | . \end{matrix}$

For any sets A and B, . It follows that $\begin{matrix} μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cup Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cup \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ \leq & | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cup \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - | {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} \cup {\underline{app {(Y)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} | \\ = & | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | + | \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - | {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} | - | {\underline{app {(Y)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} | \\ + | {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} \cap {\underline{app {(Y)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} | \\ = & | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | + | \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - | {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} | - | {\underline{app {(Y)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} | \\ - | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ + | {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} \cap {\underline{app {(Y)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} | \\ \leq & | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | + | \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - | {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} | - | {\underline{app {(Y)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} | \\ - μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cap Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | . \end{matrix}$ Now by definition of rough degree $\begin{matrix} μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) = 1 - \frac{| {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} |}{| \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} |} \end{matrix}$ and $\begin{matrix} μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (Y) = 1 - \frac{| {\underline{app {(Y)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} |}{| \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} |} . \end{matrix}$ We obtain $\begin{matrix} μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ = | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | - | {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} | \end{matrix}$ and $\begin{matrix} μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (Y) | \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ = | \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | - | {\underline{app {(Y)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} | . \end{matrix}$ Therefore $\begin{matrix} μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cup Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cup \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ \leq & {| \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | - | {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} |} + \\ {| \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | - | {\underline{app {(Y)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} |} \\ - μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cap Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ = & μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ + μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (Y) | \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cap Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | . \end{matrix}$

(ii) By definition of rough degree $μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) = 1 - ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cup Y) .$ Since $\begin{matrix} μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cup Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cup \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ \leq & μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ + μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (Y) | \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cap Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | . \end{matrix}$

Therefore $\begin{matrix} {1 - ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cup Y)} | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cup \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ \leq & {1 - ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X)} | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ + {1 - ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (Y)} | \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - {1 - ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cap Y)} | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ = & | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | - ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ + | \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | - ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (Y) | \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ + ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cap Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ = & | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cup \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | - ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (Y) | \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ + ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cap Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | . \end{matrix}$

By routine simplifications, we get $\begin{matrix} ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cup Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cup \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ \geq & ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ + ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (Y) | \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ - ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cap Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | . \end{matrix}$ □

Definition 9. Consider a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$ For any $X \subseteq U,$ the approximate quality $γ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X)$ of $X$ is defined by: $\begin{matrix} γ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) = \frac{| {\underline{app {(X)}^{δ (p)}}}_{\sum_{i = 1}^{n} A_{i}} |}{| U |}, \end{matrix}$ where $0 \leq γ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) \leq 1 .$

The following theorem describes the relationship of the approximate quality $γ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X)$ and the rough degree $μ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X)$ for the intersection and union of subsets $X$ and $Y$ of the universe $U .$

Theorem 11. Consider a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf)$ and $X,$ $Y \subseteq U .$ Then the rough degree and approximate quality satisfy the following:

for all subsets $X,$ $Y,$ $X \cup Y$ and $X \cap Y$ of $U .$

Proof. The proof is similar to the proof of Theorem 10.□

The following theorem highlights the relationship between approximate precision and approximate quality for the intersection and union of two sets.

Theorem 12. Consider a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf)$ and $X,$ $Y \subseteq U .$ Then the approximate quality and precision for all subsets $X,$ $Y,$ $X \cup Y$ and $X \cap Y$ of $U$ satisfy the following relation:

$\begin{matrix} ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cup Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cup \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | \\ \geq & | U | {γ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X) + γ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (Y)} \\ - ρ_{\sum_{i = 1}^{n} A_{i}}^{δ (p)} (X \cap Y) | \bar{app {(X)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} \cap \bar{app {(Y)}_{\sum_{i = 1}^{n} A_{i}}^{δ (p)}} | . \end{matrix}$

Proof. The proof is straightforward.□

4 Decision making method based on q-rung orthopair fuzzy similarity relation

For a multi attribute group decision making problem under q-rung orthopair fuzzy environment: let $U = {x_{1}, x_{2}, x_{3}, . . ., x_{m}}$ be a set of alternatives; E ={ E₁, E₂, ..., E_t } be the set of decision makers and w_l be the weight of decision maker E_l (l = 1, 2, ..., t) where w_l ∈ [0, 1], l = 1, 2, ..., t and $\sum_{l = 1}^{t} w_{l} = 1;$ A ={ A₁, A₂, ..., A_n } is the set of attributes and w_j is the weight of attribute A_j (j = 1, 2, ..., n) with w_j ∈ [0, 1], j = 1, 2, ..., n and $\sum_{j = 1}^{n} w_{j} = 1 .$ Suppose that ${\tilde{R}}^{l} = ({\tilde{r}}_{ij}^{l})_{m \times n}$ is the individual’s q-rung orthopair fuzzy decision matrix given by the decision maker E_l, where ${\tilde{r}}_{ij}^{l} = ({\tilde{μ}}_{ij}^{l}, {\tilde{ν}}_{ij}^{l})$ is a q-rung orthopair fuzzy value given by the decision maker E_l for the alternative x_i (i = 1, 2, ..., m) with respect to the attribute A_j (j = 1, 2, ..., n), ${\tilde{μ}}_{ij}^{l},$ ${\tilde{ν}}_{ij}^{l} \in [0, 1]$ and ${({\tilde{μ}}_{ij}^{l} (x))}^{q} + {({\tilde{ν}}_{ij}^{l} (x))}^{q} \leq 1,$ q ≥ 1. To obtain the most desirable alternative(s), the proposed similarity relations are utilized to develop an approach to solve multi attribute group decision making problem under q-rung orthopair fuzzy environment, which consists of the following steps:

Step 1. Transform the q-rung orthopair fuzzy decision matrix ${\tilde{R}}^{l} = ({\tilde{r}}_{ij}^{l})_{m \times n}$ into the normalized q-rung orthopair fuzzy decision matrix $R^{l} = (r_{ij}^{l})_{m \times n}$ by the following $r_{i j}^{l} = (μ_{i j}^{l}, ν_{i j}^{l}) = {\begin{matrix} ({\tilde{μ}}_{i j}^{l}, {\tilde{ν}}_{i j}^{l}) {for the benefit attribute A}_{j}, \\ ({\tilde{ν}}_{i j}^{l}, {\tilde{μ}}_{i j}^{l}) {for the cost attribute A}_{j} . \end{matrix}$

Step 2. Calculate the similarity relation from normalized q-rung orthopair fuzzy decision matrix $R^{l} = (r_{ij}^{l})_{m \times n}$ as follows: $\begin{matrix} q - {Si}_{A}^{l} (x_{i}, x_{j}) = ⋂_{A_{i} \in A} q - {Si}_{A_{i}}^{l} (x_{i}, x_{j}) \end{matrix}$ where l = 1, 2, ..., t, and i, j = 1, 2, ..., m.

Step 3. Calculate the similarity relation from q- ${Si}_{A}^{l} (x_{i}, x_{j})$ as follows: $\begin{matrix} q - {Si}_{A} (x_{i}, x_{j}) = ⋂_{l = 1}^{t} q - {Si}_{A_{i}}^{l} (x_{i}, x_{j}) \end{matrix}$ where l = 1, 2, ..., t, and i, j = 1, 2, ..., m.

Step 4. Calculate the the score value as follows: $\begin{matrix} S (x_{i}) = \sum_{j = 1}^{m} q - {Si}_{A} (x_{i}, x_{j}) \end{matrix}$ where i, j = 1, 2, ... m.

Step 5. Rank all the alternatives x_i (i = 1, 2, ..., m), according to the score values S (x_i) (i = 1, 2, ..., m), the greater x_i, with the highest value of S (x_i), is the best alternative.

The time complexity of proposed algorithm is O (mn²).

4.1 Illustrative Example

Example 6. The dependence on imported sources are very much reduced and the security of supply are provided by renewable energy. Renewable energy addresses our energy needs by replacing foreign energy imports with reliable and clean home-grown electricity. Also, renewable energy added bonus of fantastic local economic opportunities. By using renewable energy instead of fossil fuels, we would significantly decrease the current levels of greenhouse gas emissions, and this would have positive environmental impact for our entire planet. Renewable energy is not all about environment as it can also give strong boost to our economy in the form of new jobs. Renewable energy often referred to as clean energy, comes from natural sources or processes that are constantly replenished. For example, sunlight or wind keep shining and blowing, even if their availability depends on time and weather. Let $U = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}}$ represents five renewable energy sources (alternatives) where x₁ : wind energy, x₂ = biomass energy, x₃ = tidal energy, x₄ = solar energy, x₅ = hydro-power and A ={ A₁, A₂, A₃, A₄, A₅, A₆ } be a set of criteria where A₁ = water pollution, A₂ = need for waste disposal, A₃ = air pollutant emissions, A₄ = land requirement, A₅ = economic risk, A₆ = land disruption. The attribute weight is w = (0.20, 0.10, 0.30, 0.15, 0.15, 0.10) ^T. The attribute value of each alternative are evaluated by the q-rung orthopair fuzzy numbers. Assume the decision matrix R which is presented in Table 7.

Table 7
Decision matrix R

A ₁ A ₂ A ₃ A ₄ A ₅ A ₆

x ₁ (0.5, 0.2) (0.8, 0.3) (0.8, 0.3) (0.7, 0.3) (0.4, 0.2) (0.4, 0.8)

x ₂ (0.6, 0.3) (0.5, 0.8) (0.6, 0.5) (0.6, 0.5) (0.7, 0.4) (0.5, 0.6)

x ₃ (0.3, 0.4) (0.8, 0.5) (0.7, 0.4) (0.6, 0.4) (0.6, 0.2) (0.4, 0.7)

x ₄ (0.7, 0.4) (0.5, 0.6) (0.7, 0.4) (0.5, 0.5) (0.7, 0.6) (0.6, 0.5)

x ₅ (0.7, 0.6) (0.6, 0.4) (0.4, 0.7) (0.4, 0.3) (0.7, 0.7) (0., 0.4)

	A ₁	A ₂	A ₃	A ₄	A ₅	A ₆
x ₁	(0.5, 0.2)	(0.8, 0.3)	(0.8, 0.3)	(0.7, 0.3)	(0.4, 0.2)	(0.4, 0.8)
x ₂	(0.6, 0.3)	(0.5, 0.8)	(0.6, 0.5)	(0.6, 0.5)	(0.7, 0.4)	(0.5, 0.6)
x ₃	(0.3, 0.4)	(0.8, 0.5)	(0.7, 0.4)	(0.6, 0.4)	(0.6, 0.2)	(0.4, 0.7)
x ₄	(0.7, 0.4)	(0.5, 0.6)	(0.7, 0.4)	(0.5, 0.5)	(0.7, 0.6)	(0.6, 0.5)
x ₅	(0.7, 0.6)	(0.6, 0.4)	(0.4, 0.7)	(0.4, 0.3)	(0.7, 0.7)	(0., 0.4)

4.2 Comparative analysis

The method proposed in this paper expresses a wider range of fuzzy information and is valid with the sum of membership degree and nonmembership degree greater than 1. Thus its scope of applications is wider and closer to the real decision making environment. In real life complex decision making environment, we need effectively expressed fuzzy decision making information to avoid a lot of information distortion. Therefore in this way we can get more real and effective decision making results.

To verify the effectiveness of the proposed method based on the similarity relation, we compare our proposed method with other existing methods including the q-ROFWG operator of Liu and Wang [26]. For the collective decision matrix R = (r_ij) _m×n shown in Table 7, if we use the q-ROFWG operator to aggregate the q-rung orthopair fuzzy assessment values r_ij of the action x_i on all attributes A_j (j = 1, 2, …, n) into the overall assessment values r_i of the action x_i (i = 1, 2, …, m), then the score values and ranking results are given in Table 8.

Table 8
Based on q-ROFWG operator proposed by Liu and Wang [26]

q The score function Ranking

q = 2 S (x₁) = 0.2480, S (x₂) = 0.0758, S (x₃) = 0.0352, S (x₄) = 0.1644, S (x₅) = -0.0883 x₁ ≻ x₄ ≻ x₂ ≻ x₃ ≻ x₅

q = 3 S (x₁) = 0.1701, S (x₂) = 0.0499, S (x₃) = 0.0164, S (x₄) = 0.1349, S (x₅) = 0.0860 x₁ ≻ x₄ ≻ x₂ ≻ x₃ ≻ x₅

q = 5 S (x₁) = 0.0627, S (x₂) = 0.0106, S (x₃) - 0.0021, S (x₄) = 0.0694, S (x₅) = -0.0591 x₄ ≻ x₁ ≻ x₃ ≻ x₂ ≻ x₅

q = 8 S (x₁) = 0.0081, S (x₂) = -0.0066, S (x₃) = -0.0043 S (x₄) = 0.0205, S (x₅) = -0.0246 x₄ ≻ x₁ ≻ x₃ ≻ x₂ ≻ x₅

q = 10 S (x₁) - 0.0007, S (x₂) = -0.0071, S (x₃) - 0.0027, S (x₄) = 0.0086, S (x₅) = -0.0126 x₄ ≻ x₁ ≻ x₃ ≻ x₂ ≻ x₅

q = 12 S (x₁) - 0.0028, S (x₂) = -0.0056, S (x₃) = 0.0015, S (x₄) = 0.0036, S (x₅) = -0.0063 x₄ ≻ x₃ ≻ x₁ ≻ x₂ ≻ x₅

q = 15 S (x₁) = -0.0025, S (x₂) = -0.0033, S (x₃) = -0.0005 S (x₄) = 0.0009, S (x₅) = -0.0022 x₄ ≻ x₃ ≻ x₅ ≻ x₁ ≻ x₂

q = 20 S (x₁) = -0.0010, S (x₂) = -0.0011, S (x₃) = -0.0001 S (x₄) = 0.0001, S (x₅) = -0.0004 x₄ ≻ x₃ ≻ x₅ ≻ x₁ ≻ x₂

q	The score function	Ranking
q = 2	S (x₁) = 0.2480, S (x₂) = 0.0758, S (x₃) = 0.0352, S (x₄) = 0.1644, S (x₅) = -0.0883	x₁ ≻ x₄ ≻ x₂ ≻ x₃ ≻ x₅
q = 3	S (x₁) = 0.1701, S (x₂) = 0.0499, S (x₃) = 0.0164, S (x₄) = 0.1349, S (x₅) = 0.0860	x₁ ≻ x₄ ≻ x₂ ≻ x₃ ≻ x₅
q = 5	S (x₁) = 0.0627, S (x₂) = 0.0106, S (x₃) - 0.0021, S (x₄) = 0.0694, S (x₅) = -0.0591	x₄ ≻ x₁ ≻ x₃ ≻ x₂ ≻ x₅
q = 8	S (x₁) = 0.0081, S (x₂) = -0.0066, S (x₃) = -0.0043 S (x₄) = 0.0205, S (x₅) = -0.0246	x₄ ≻ x₁ ≻ x₃ ≻ x₂ ≻ x₅
q = 10	S (x₁) - 0.0007, S (x₂) = -0.0071, S (x₃) - 0.0027, S (x₄) = 0.0086, S (x₅) = -0.0126	x₄ ≻ x₁ ≻ x₃ ≻ x₂ ≻ x₅
q = 12	S (x₁) - 0.0028, S (x₂) = -0.0056, S (x₃) = 0.0015, S (x₄) = 0.0036, S (x₅) = -0.0063	x₄ ≻ x₃ ≻ x₁ ≻ x₂ ≻ x₅
q = 15	S (x₁) = -0.0025, S (x₂) = -0.0033, S (x₃) = -0.0005 S (x₄) = 0.0009, S (x₅) = -0.0022	x₄ ≻ x₃ ≻ x₅ ≻ x₁ ≻ x₂
q = 20	S (x₁) = -0.0010, S (x₂) = -0.0011, S (x₃) = -0.0001 S (x₄) = 0.0001, S (x₅) = -0.0004	x₄ ≻ x₃ ≻ x₅ ≻ x₁ ≻ x₂

From Tables 8 , 9 and 10, we can see that two methods have the same ranking results, which further verifies the validity of the proposed technique. Comparing the q-ROFWG operator of Liu and Wang [26] with proposed technique, we find that our method/technique is more efficient and have less computation cost than that of Liu and Wang technique. Similarly other techniques/Liu and Wang require the weights equal in numbers to the criteria, while the proposed technique is free from the requirement of weights.

Table 9

Proposed method based on similarity relation

q	The score function, where α = β = 0.4, γ = 0.2	Ranking
q = 2	S (x₁) = 3.5869, S (x₂) = 3.6829, S (x₃) = 3.8177	x₄ ≻ x₃ ≻ x₂ ≻ x₅ ≻ x₁
	S (x₄) = 3.8417, S (x₅) = 3.5980
q = 3	S (x₁) = 3.8287, S (x₂) = 3.7208, S (x₃) = 3.9320	x₄ ≻ x₃ ≻ x₅ ≻ x₁ ≻ x₂
	S (x₄) = 3.9653, S (x₅) = 3.8182
q = 5	S (x₁) = 4.1405, S (x₂) = 4.0306, S (x₃) = 4.1956	x₄ ≻ x₃ ≻ x₅ ≻ x₁ ≻ x₂
	S (x₄) = 4.2311, S (x₅) = 4.1873

Table 10

Proposed method based on similarity relation

q	The score function, where α = β = 0.5, γ = 0	Ranking
q = 2	S (x₁) = 3.5276, S (x₂) = 3.5870, S (x₃) = 3.7739	x₄ ≻ x₃ ≻ x₂ ≻ x₅ ≻ x₁
	S (x₄) = 3.8501, S (x₅) = 3.5488
q = 3	S (x₁) = 3.8253, S (x₂) = 3.7035, S (x₃) = 4.0930	x₄ ≻ x₃ ≻ x₅ ≻ x₁ ≻ x₂
	S (x₄) = 4.1036, S (x₅) = 3.8742
q = 5	S (x₁) = 4.0945, S (x₂) = 3.9888, S (x₃) = 4.1893	x₄ ≻ x₃ ≻ x₅ ≻ x₁ ≻ x₂
	S (x₄) = 4.2370, S (x₅) = 4.1848

A major advantage of the proposed technique is that, it has the ability to solve the real life problems by using their parameterization properties. Hence, the developed concept is more useful for solving the decision making problems than other existing operators in q-rung orthopair fuzzy environment.

4.3 Attribute reduction in a q-rung orthopair fuzzy environment

Reduction of knowledge is an essential part of rough set theory. In this subsection we investigate the issue of attribute reduction in the q-rung orthopair fuzzy environment under the framework of similarity relation. The reducts in classical rough set approach, remains an important and inspiring notion as they involve the idea of finding attribute subsets which are minimal with regard to inclusion and guarantee the same quality of approximation as the whole set of attributes. Further we present a new method of reduction, where we reduce the criterion from the q-rung orthopair fuzzy environment with the help of similarity relation based ranking of the alternatives. Some steps are explored to implement the conditional attribute reduction, which aims to discard the unimportant conditional attributes from original q-rung orthopair fuzzy environment. The steps of attribute reduction are:

Step 1. Given a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf);$

Step 2. Calculate the similarity relation from normalized q-rung orthopair fuzzy decision matrix $R^{l} = (r_{ij}^{l})_{m \times n}$ by: $\begin{matrix} q - {Si}_{A}^{l} (x_{i}, x_{j}) = ⋂_{A_{i} \in A} q - {Si}_{A_{i}}^{l} (x_{i}, x_{j}) \end{matrix}$ where l = 1, 2, ..., t, and i, j = 1, 2, ..., m.

Step 3. Calculate the similarity relation from q- ${Si}_{A}^{l} (x_{i}, x_{j})$ by: $\begin{matrix} q - {Si}_{A} (x_{i}, x_{j}) = ⋂_{l = 1}^{t} q - {Si}_{A_{i}}^{l} (x_{i}, x_{j}) \end{matrix}$ where l = 1, 2, ..., t, and i, j = 1, 2, ..., m.

Step 4. Calculate the the score value by: $\begin{matrix} S (x_{i}) = \sum_{j = 1}^{m} q - {Si}_{A} (x_{i}, x_{j}) \end{matrix}$ where i, j = 1, 2, ..., m.

Step 5. Rank all the alternatives x_i (i = 1, 2, ..., m), according to the score values S (x_i) (i = 1, 2, ..., m), the greater x_i, with the highest value of S (x_i), is the best alternative.

Step 6. Original rank of the alternatives x_i (i = 1, 2, ..., m) is obtained.

Step 7 : Discarding a conditional attribute from original conditional attribute set A and go to Step 2. Finally obtained the new rank of alternatives. If the new ranks are similar to the original rank, then discarding conditional attribute goes to dispensable set, otherwise to the indispensable set.

Example 7. (Continued from Example 6) For the rank of alternatives in a q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf),$ with q = 3, α = β = 0.4 and γ = 0.2, using the above steps we get original rank of the alternatives which is $\begin{matrix} x_{4} ≻ x_{3} ≻ x_{5} ≻ x_{1} ≻ x_{2} . \end{matrix}$ By discarding the conditional attribute A₁ from the original set of conditional attribute A, the new information system is $(U, A = {A_{2}, A_{3}, A_{4}, A_{5}, A_{6}}, V_{qf}, qf),$ and by repeating all the above steps yield the new rank of the alternatives as $\begin{matrix} x_{4} ≻ x_{3} ≻ x_{5} ≻ x_{1} ≻ x_{2} . \end{matrix}$ Since the original rank and new rank are same, so A₁ is dispensable. Similarly if A₂ is reduced from the original set of conditional attribute A, then the new information system $(U, A = {A_{1}, A_{3}, A_{4}, A_{5}, A_{6}}, V_{qf}, qf)$ is achieved and one can get the new rank of the feasible alternatives as $\begin{matrix} x_{2} ≻ x_{3} ≻ x_{4} ≻ x_{1} ≻ x_{5} . \end{matrix}$ Clearly the original rank and new ranks are different, where A₂ is indispensable. Similar calculations show that A₄, A₅ and A₆ are also indispensable while A₃ is dispensable. Hence the core cause for the problem discussed is consisting upon the aforementioned indispensable conditional attributes of the q-rung orthopair fuzzy information system $(U, A, V_{qf}, qf) .$

5 Conclusion and future research

The prominent characteristic of q-rung orthopair fuzzy values is that one can choose an appropriate parameter q to solve decision making problems. The restriction on the memberships is that the sum of the q^th powers of the support for and support against is equal to or less than 1. This very relaxed constraint greatly increases the modelers’ ability to capture their judgment of the orthopair memberships. In this paper, we proposed a new type of q-rung orthopair fuzzy lower and upper approximations by applying a tolerance degree on the similarity between two objects. After employing tolerance degree based q-rung orthopair fuzzy rough set approach to it any times, we got the six different sets at most. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Furthermore, we proposed tolerance degree based multi granulation optimistic/pessimistic q-rung orthopair fuzzy rough sets and investigated some of their properties. Another main contribution of this paper is to disclose the ideas of different kinds of approximations called approximate precision, rough degree, approximate quality and their mutual relationship. In future research, we would like to investigate the large-scale multi-criteria q-rung orthopair fuzzy group decision approach based on multi granulation q-rung orthopair fuzzy tolerance rough sets.

Footnotes

Acknowledgment

The work of the author Noor Rehman is partially supported by the Higher Education Commission of Pakistan through project NRPU-15942.

References

Atanassov

, More on intuitionistic fuzzy sets, Fuzzy SetsSyst. 33 (1989), 37–46.

Bonikowski

, Algebraic structures of rough sets, in: W.P. Ziarko (Ed.), Rough Sets, Fuzzy Sets and Knowledge Discovery, Springer-Verlag, Berlin (1995), 242–247.

Chen

K.E.

and Luo

, Generalized orthopair linguistic Muirhead meanoperators and their application in multi-criteria decision making, J. Intell. Fuzzy Syst. 7(1) (2019), 797–809.

Chen

K.E.

and Luo

, Generalized orthopair linguistic Muirhead meanoperators and their application in multi-criteria decision making, J. Intell. Fuzzy Syst. 37(1) (2019), 797–809.

W.S.

and Hu

B.Q.

, Attribute reduction in ordered decision tables via evidence theory, Inform. Sci. (2016), 91–110.

W.S.

, Minkowski-type distance measures for generalized orthopairfuzzy sets., Int. J. Intell. Syst. 33 (2018), 802–817.

Dubois

and Prade

, Rough fuzzy sets and fuzzy rough sets, Int. J. Gen. Syst. 17 (1990), 191–208.

Feng

and Mi

J.S.

, Variable precision multi-granulation fuzzydecision-theoretic rough sets, Knowl. Based Sys. 91(2016), 93–101.

Garg

, A new generalized Pythagorean fuzzy information aggregation using einstein operations and its application to decision making, Int. J. Intell. Syst. (2011), 1–35.

10.

Garg

, A new generalized Pythagorean fuzzy information aggregationusing Einstein operations and its application to decision making, Int. J. Intell. Syst. 31(9) (2016), 886–920.

11.

Garg

, Generalized Pythagorean fuzzy geometric aggregationoperators using Einstein t-norm and t-conorm for multicriteriadecision-making process, Int. J. Intell. Syst. 6(32) (2017), 597–630.

12.

Garg

, Confidence levels based Pythagorean fuzzy aggregationoperators and its application to decision-making process, Comput. Math. Organ. Theory 23 (2017), 546–571.

13.

Garg

, Some methods for strategic decision-making problems withimmediate probabilities in Pythagorean fuzzy environment, Int.J. Intell. Syst. 33(4) (2018), 687–712.

14.

Herrera-Viedma

, Herrera

, Chiclana

and Luque

, some issueson consistency of fuzzy preference relations, Eur J Oper Res 154 (2004), 98–109.

15.

, Yu

and Guo

, Fuzzy preference based rough sets, Inf. Sci. 180 (2010), 2003–2022.

16.

Hussain

, Ali

M.I.

and Mahmood

, Covering based q-rung orthopairfuzzy rough set model hybrid with TOPSIS for multi-attributedecision making, J. Int. Fuzzy Syst. 37 (2019), 981–993.

17.

, Luo

, Ma

and Wang

, A novel multiple-attribute groupdecision-making method based on q-rung orthopair fuzzy generalizedpower weighted aggregation operators, Int. J. Intell. Syst. 34(9) (2019), 2077–2103.

18.

Liang

, Liao

and Liu

, A social ties-based approach for groupdecision-making problems with incomplete additive preferencerelations, Knowl. Based Syst. 119 (2017), 68–86.

19.

Lin

, Qian

and Li

, NMGRS: Neighborhood-basedmulti-granulation rough sets, Int. J. Appro. Reason. 53(7) (2012), 1080–1093.

20.

, Chen

and Xu

, Double quantitative multigranulation decision theoretic rough fuzzy set model, Int. J. Machine Learn.Cyber. 10 (2019), 3225–3244.

21.

Liu

, Liu

and Merigó

J.M.

, Partitioned Heronian means basedon linguistic intuitionistic fuzzy numbers for dealing withmulti-attribute group decision making, Appl. Soft Comput. 62 (2018), 395–422.

22.

Liu

and Liu

, Some q-rung orthopai fuzzy Bonferroni meanoperators and their application to multi-attribute group decisionmaking, Int. J. Intell. Syst. 33 (2018), 315–347.

23.

Liu

, Chen

S.M.

and Wang

, The g-rung orthopair fuzzy power Maclaurin symmetric mean operators, International Conference on Advanced Computational Intelligence. Xiamen, China (2018), 156–161.

24.

Liu

, Chen

S.M.

and Wang

, Multiple-attribute groupdecision-making based on q-rung orthopair fuzzy power Maclaurinsymmetric mean operators, IEEE Trans. Syst. Man Cyb. Syst. 10(50) (2020), 3741–3756.

25.

Liu

and Wang

, Multiple-attribute decision making based onArchimedean Bonferroni operators of q-rung orthopair fuzzy numbers, IEEE Trans. Fuzzy Syst. 27 (2019), 834–848.

26.

Liu

D. P.

and Wang

, Some q-rung orthopair fuzzy aggregationoperators and their applications to multiple attribute decisionmaking, Int. J. Intell. Syst. 33(2) (2018), 259–280.

27.

Liu

L.G.

and Zhu

, The algebraic structures of generalized roughset theory, Inform. Sci. 178(2) (2008), 4105–4113.

28.

Liu

M.Z.

, Liu

and Liang

, Multiple attribute decision-makingmethod for dealing with heterogeneous relationship among attributesand unknown attribute weight information under q-rung orthopairfuzzy environment, Int. J. Intell. Syst. 33 (2018), 1900–1928.

29.

Liu

Z.M.

, Wang

and Liu

P.D.

, Multiple attribute group decisionmaking based on q-rung orthopair fuzzy Heronian mean operators, Int. J. Intell. Syst. 33 (2018), 2341–2363.

30.

Liu

, Chen

S.M.

and Wang

, Multiple-attribute groupdecision-making based on q-rung orthopair fuzzy power Maclaurinsymmetric mean operators, IEEE Trans. Syst. Man Cybernet.Syst. 50(10) (2020), 3741–3756.

31.

Liu

and Liu

, Some q-rung qrthopai fuzzy Bonferroni meanoperators and their application to multi-attribute group decisionmaking, Int. J. Intell. Syst. 33(2) (2018), 315–347.

32.

Moris

N.N.

and Yakout

M.M.

, Axiomatics for fuzzy rough sets, Fuzzy Sets Syst 100 (1998), 327–342.

33.

Pan

, She

and Wei

, Multi-granulation fuzzy preference relation rough set for ordinal decision system, Fuzzy Sets Syst. 312 (2017), 87–108.

34.

Pawlak

, Information systems theoretical foundations, Inform.Sci. 6(3) (1981), 205–218.

35.

Peng

X.D.

and Yang

, Some results for Pythagorean fuzzy sets, Int. J. Intell. Syst. 30 (2015), 1133–1160.

36.

Peng

X.D.

, Dai

G.J.

and Garg

, Exponential operation andaggregation operator for q-rung orthopair fuzzy set and theirdecision-making method with a new score function, Int. J.Intell. Syst. 33 (2018), 2255–2282.

37.

Peng

, Dai

and Garg

, Exponential operation and aggregationoperator for q-rung orthopair fuzzy set and their decision-makingmethod with a new score function, Int. J. Intell. Syst. 33(11) (2018), 2255–2282.

38.

Qian

, Liang

, Yao

and Dang

, MGRS: A multi-granulationrough set, Inform. Sci. 180(6) (2010), 949–970.

39.

Qian

, Li

, Liang

, Shi

and Wang

, Pessimistic rough setbased decisions: a multi-granulation fusion strategy, Inform.Sci. 264 (2014), 196–210.

40.

Rahman

, Abdullah

, Ahmed

and Ullah

, Pythagorean fuzzyEinstein weighted geometric aggregation operator and theirapplication to multiple attribute group decision making, J.Intell. Fuzzy Syst. 33(1) (2017), 635–647.

41.

Rahman

, Abdullah

, Ali

and Amin

, Pythagorean fuzzy Einstein hybrid averaging aggregation operator and its applicationto multiple-attribute group decision making, J. Intell. Syst. 29(1) (2020), 736–752.

42.

She

and He

, On the structure of the multi-granulation roughset model, Knowl. Based Syst. 36 (2012), 81–92.

43.

Sun

, Ma

, Chen

and Zhang

, Multi-granulation vague roughset over two universes and its application to group decision making, Soft Comput. 23 (2019), 8927–8956.

44.

Sun

, Tong

, Ma

, Wang

and Jiang

, An approach to MCGDM based on multi-granulation Pythagorean fuzzy rough set over twouniverses and its application to medical decision problem, Artif. Intell. Rev. 55 (2021), 1887–1913.

45.

Slowinski

and Stefanowski

, Medical information systems-problems with analysis and way of solution, in: S.K. Pal, A. Skowron (Eds.), Rough Fuzzy Hybridization: A New Trend in Decision-Making, Springer-Verlag, Singapore (1999), 301–315.

46.

Z.S.

, Intuitionistic fuzzy aggregation operators, IEEE Trans Fuzzy Syst 15 (2007), 1179–1187.

47.

Z.S.

and Yager

R.R.

, Some geometric aggregation operators basedon intuitionistic fuzzy sets, Int J General Syst 35(2006), 417–433.

48.

, Li

and Zhang

, Generalized multi-granulation rough setsand optimal granularity selection, Granular Comput. 2(4) (2017), 271–288.

49.

, Wang

and Luo

, Multi-granularion fuzzy rough sets, J. Intell. Fuzzy Syst. 26(3) (2014), 1323–1340.

50.

, Wang

and Zhang

, Multi-granulation rough sets based ontolerance relations, Soft Comput. 17 (2013), 1241–1252.

51.

, Wang

and Zhang

, Multi-granulation fuzzy rough sets in afuzzy tolerance approximation space, Int. J. Fuzzy Syst. 13(4) (2011), 246–259.

52.

Yager

R.R.

and Abbasov

A.M.

, Pythagorean membership grades, complexnumbers, and decision making, Int. J. Intell. Syst. 28(2014), 436–452.

53.

Yager

R.R.

, Pythagorean fuzzy subsets. In: Proc Joint IFSA World Congress and NAFIPS Annual Meeting; June 24-28, 2013; Edmonton, Canada: 57–61.

54.

Yager

R.R.

, Pythagorean membership grades in multicriteria decision making, IEEE Trans. Fuzzy Syst. 22 (2014), 958–965.

55.

Yager

R.R.

, Generalized orthopair fuzzy sets, IEEE Trans. Fuzzy Syst. 25 (2017), 1222–1230.

56.

Yager

R.R.

and Alajlan

, Approximate reasoning with generalized orthopair fuzzy sets, Inf. Fusion 38 (2017), 65–73.

57.

Yager

R.R.

, Alajlan

and Bazi

, Aspects of generalized orthopairfuzzy sets, Int. J. Intell. Syst. 33 (2018), 2154–2174.

58.

Yang

X.B.

, Qi

, Yu

D.J.

, Yu

H.L.

and Yang

J.Y.

, α-Dominancerelation and rough sets in interval-valued information systems, Inform. Sci. 294 (2015), 334–347.

59.

Yang

X.B.

, Qian

Y.H.

and Yang

J.Y.

, Hierarchical structures onmulti-granulation spaces, J. Comput. Sci. Tech. 27(6) (2012), 1169–1183.

60.

You

, Li

and Wang

, Relative reduction ofneighborhood-covering pessimistic multi-granulation rough set basedon evidence theory, Information 10(11) (2019), 334.

61.

Wan Mohd

W.R.

and Abdullah

, Pythagorean fuzzy analytic hierarchyprocess to multi-criteria decision making, In: AIP Conference Proceedings 1905(1) (2017), 040020.

62.

Wei

, Garg

, Gao

and Wei

, Interval-valued Pythagorean fuzzy Maclaurin symmetric mean operators in multiple attribute decision making, IEEE Access 6(1) (2018), 67866–67884.

63.

Wei

G.W.

, Gao

and Wei

, Some q-rung orthopair fuzzy Heronianmean operators in multiple attribute decision making, Int. J.Intell. Syst. 33 (2018), 1426–1458.

64.

Zeng

, Pythagorean fuzzy multi attribute group decision making with probabilistic information and OWA approach, Int. J. Intell.Syst. 32 (2017), 1136–1150.

65.

Zeng

, Cao

and Deng

, Pythagorean fuzzy informationaggregation based on weighted induced operator and its applicationto R & D projections selection, Informatica 29(3) (2018), 567–580.

66.

Zhang

X.L.

and Xu

Z.S.

, Extension of TOPSIS to Multiple criteriadecision making with Pythagorean fuzzy sets, Int. J. Intell.Syst. 29 (2014), 1061–1078.

67.

Zhao

, Xu

Z.S.

, Ni

M.F.

and Liu

S.S.

, Generalized aggregation operators for intuitionistic fuzzy sets, Int. J. Intell. Syst. 25 (2010), 1–30.

68.

Zhang

X.L.

and Xu

Z.S.

, Extension of TOPSIS to multiple criteriadecision making with Pythagorean fuzzy sets, Int J Intell Syst. 29 (2014), 1061–1078.

69.

Zhang

, Liao

and Luo

L.I.

, Additive consistency-basedpriority-generating method of q-rung orthopair fuzzy preferencerelation, Int. J. Intell. Syst. 34(9) (2019), 2151–2176.

70.

Zhang

and Chen

, Group decision making with incomplete q-rung orthopair fuzzy preference relations, Inform. Sci. 553 (2021), 376–396.

Another view on tolerance based multigranulation q -rung orthopair fuzzy rough sets with applications

Abstract

Keywords

1 Introduction

1.1 A brief review on q-rung orthopair fuzzy sets

1.2 A brief review on rough set theory

1.3 Motivation of our research

1.4 Organization of the paper

2 q-rung orthopair fuzzy rough sets

Table 1 q-rung orthopair fuzzy formation

3.1 Multi-granulation pessimistic q-rung orthopair fuzzy rough sets

4 Decision making method based on q-rung orthopair fuzzy similarity relation

4.1 Illustrative Example

5 Conclusion and future research

Footnotes

Acknowledgment

References

Table 1
q-rung orthopair fuzzy formation