Rough q -Rung Orthopair Fuzzy Sets and Their Applications in Decision-Making

: Yager recently introduced the q -rung orthopair fuzzy set to accommodate uncertainty in decision-making problems. A binary relation over dual universes has a vital role in mathematics and information sciences. During this work, we deﬁned upper approximations and lower approximations of q -rung orthopair fuzzy sets using crisp binary relations with regard to the aftersets and foresets. We used an accuracy measure of a q -rung orthopair fuzzy set to search out the accuracy of a q -rung orthopair fuzzy set, and we deﬁned two types of q -rung orthopair fuzzy topologies induced by reﬂexive relations. The novel concept of a rough q -rung orthopair fuzzy set over dual universes is more ﬂexible when debating the symmetry between two or more objects that are better than the prevailing notion of a rough Pythagorean fuzzy set, as well as rough intuitionistic fuzzy sets. Furthermore, using the score function of q -rung orthopair fuzzy sets, a practical approach was introduced to research the symmetry of the optimal decision and, therefore, the ranking of feasible alternatives. Multiple criteria decision making (MCDM) methods for q -rung orthopair fuzzy sets cannot solve problems when an individual is faced with the symmetry of a two-sided matching MCDM problem. This new approach solves the matter more accurately. The devised approach is new within the literature. In this method, the main focus is on ranking and selecting the alternative from a collection of feasible alternatives, reckoning for the symmetry of the two-sided matching of alternatives, and providing a solution based on the ranking of alternatives for an issue containing conﬂicting criteria, to assist the decision-maker in a ﬁnal decision.


Introduction
The concept of rough sets (RS) was proposed by Pawlak [1] as a mathematical way to handle vagueness, uncertainty, and imprecision in data. To date, RS theory has been successfully utilized in solving a spread of problems [2], especially within multi-criteria higher cognitive processes and group higher cognitive processes.
In [3], fuzzy set (F z S) proposed by Zadeh could be applied in various fields. Numerous researchers have worked in fuzzy theory. In [4], Zhang et al. provided a method that involved two-sided matching, decided with a F z PR (fuzzy preference relation)-supported logarithmic statistical procedure, and proposed two algorithms. In [3,5], Zhang et al. provided methods to pander to two-sided matching (TSM) under multi-granular hesitant fuzzy linguistic term sets (MGHF z LSs), as well as the consensus approach, in the context of social network group decision-making (GDM).
While solving decision-making problems (DMPs), different evaluation results are produced by different experts. The non-membership degree (NMD) is needed with the membership degree (MD) in F z S in a number of real-life issues. To solve this issue, Atanassov [6] presented the concept of an intuitionistic fuzzy set (IF z S). In an intuitionistic fuzzy set,

Preliminaries
This section consists of fundamental concepts, and notions of BR, RS, IF z S, PF z S, and q-ROF z S are provided.
Throughout the work, M and N will be considered as two non-empty finite universes unless stated. The set T(A) consists of elements whose T-related elements belong to A, and T(A) consists of elements, such as a minimum of one amongst whose T-related elements is in A. The pair (T(A), T(A)) is said to be the generalized RS of A induced by T. Its physical meaning depends on the interpretations of the universe and, therefore, the relation T, specifically in applications. (M, T) is termed as generalized approximation space.
Let M be a non-empty finite universe and T be an E q R on M. Then (M, T) is known to be an approximation space. If A ⊆ M and A can be written as the union of some or all of E q classes of the universe set M, then A is T-definable, [1].
If A is not definable, then A can be approximated by a pair of definable subsets called U AP T

Definition 3 ([6]
). An IF z S A in the universe M is a set given by ) denotes the hesitancy degree or degree of indeterminacy.

Definition 4 ([16]).
A PF z S A in the universe M is a set given by , for any m ∈ M, is named a Pythagorean fuzzy degree (PF z D).
2 denotes the hesitancy degree, also known as degree of indeterminacy.
In [17], Yager proposed the idea of the q-rung orthopair fuzzy set q-ROF z S. This idea has enlarged the range of membership degrees (MDs). Within the following, a quick review of q-ROF z Ss is given. Definition 5 ([17]). A q-ROF z S A in the universe M is given by , for any m ∈ M, is known to be a q-rung orthopair fuzzy degree (q-ROF z D). The collection of all q-ROF z Ss in M is represented by q-ROF z S(M).
We can see that if q = 1, then (Y A (m), Y c A (m)) is an IF z N and if q = 2, then it is a PF z N. From Figure 1, it is evident that q-ROF z S incorporates a big selection for the q-ROF z Ds. Thus, q-ROF z Ss are more general than PF z Ss and IF z Ss.
Then the fundamental operations on q-ROF z S(M) defined by Lie and Wang [18] are as follows: The q-ROF z S 1 M = (1, 0) and q-ROF z S 0 M = (0, 1), where 1(m) = 1 and 0(m) = 0, for all m ∈ M. Definition 7 ([24]). The score value of any q-ROF z D A = (Y A (m), Y c A (m)), m ∈ M, is defined as for q ≥ 1. The greater the worth of score function, the better will be the q-ROF z D.

Rough q-ROF z Ss
In this section, we consider a CBR from M to N, approximate a q-ROF z S over M by using FRs and acquire two q-ROF z Ss over N.
Likewise, we approximate a q-ROF z S of N by using AFs and acquire two q-ROF z Ss over M. We additionally talk about a number of their properties.

Definition 8. Let T be a CBR from M to N and
with respect to AFs, as follows: where mT = {n ∈ N : (m, n) ∈ T}, and is called the afterset (AR) of m for all m ∈ M.
It can be verified that T A , T A are q-ROF z Ss of M. Moreover, the operators T A , T A : q-ROF z S(N) → q-ROF z S(M) are the upper and lower rough q-ROF z approximation operators, respectively.
The pair (T A , T A ) is named the rough q-ROF z S with respect to ARs.

Definition 9. Let T be a CBR from M to N and
with respect to FRs as follows: where Tn = {m ∈ M : (m, n) ∈ T}, and is called the foreset (FR) of n for all n ∈ N.
It can be verified that A T, A T are q-ROF z Ss of N. And A T, T A : q-ROF z S(M) → q-ROF z S(N) are upper and lower rough q-ROF z approximation operators, respectively.
The pair ( A T, A T) is termed the rough q-ROF z S with reference to FRs.
The above defined concepts are elaborated in the next example. , (m 4 , n 3 )} represents the CBR between models and colors available at the shop. Now, let A ∈ q-ROF z S(N) and B ∈ q-ROF z S(M), for q = 5, where the preference of colors is represented by the q-ROF z S A and the q-ROF z S B represents the choice of models, provided by the student, are: Then LAP and U AP of q-ROF z S A with respect to ARs m i T are two q-ROF z Ss on M, given by; Thus, (T A , T A ) is a rough q-ROF z S with respect to ARs.
Similarly, the LAP and U AP of q-ROF z S B with respect to FRs Tn i are two q-ROF z Ss on N, given by; Thus, ( B T, B T) is a rough q-ROF z S with respect to FRs. Theorem 1. Let T be a CBR from M to N, that is, T ∈ P(M × N). For any three q-ROF z Ss Proof.
Theorem 2. Let T be a CBR from M to N; that is, T ∈ P(M × N). For any three q-ROF z Ss Proof. The proof follows directly from the proof of Theorem 1.
Example 2 confirms that the converse is not true in (iv) and (v) parts of Theorem 1.

Example 2.
Revisiting example 1, we define two q-ROF z Ss A 1 , A 2 on N by: Then, . Thus, the LAP of the union of two q-ROF z Ss is not equal to the union of LAPs of two q-ROF z Ss; that is, Similarly, from Table 2, we see that intersection of the U AP of the intersection of two q-ROF z Ss is not equal to U APs of two q-ROF z Ss; that is, Thus, the converse in not true in (iv) and (v) parts of Theorem 1.  Table 2. Intersection of U APs and U AP of intersection of two q-ROF z Ss.
Theorem 4. Let T 1 , T 2 be two BRs from M to N, such that T 1 ⊆ T 2 . Then, for any Proof. The proof follows directly from the proof of Theorem 3.
Theorem 5. Let T 1 , T 2 be two BRs from M to N. Then, for any A ∈ q-ROF z S(N), the following are true: Proof. The proof follows directly from Theorem 3.
Following Theorem 5, we have the following result.
Theorem 6. Let T 1 , T 2 be two BRs from M to N. Then, for any A ∈ q-ROF z S(M), the following hold: Theorem 7. Let T be a BR from M to N and {A i : i ∈ I} be a finite set of q-ROF z Ss defined on N. Then the following hold:

Proof.
(i) Let A i ∈ q-ROF z S(N), for i ∈ I. Then The proof follows directly from the proof of part (i). (iv) The proof follows directly from the proof of part (ii). Theorem 8. Let T be a BR from M to N and {A i : i ∈ I} be a finite set of q-ROF z Ss defined on M. Then the following hold: Proof. The proof follows directly from the proof of Theorem 7.
Theorem 9. Let M be a finite universe and T be a reflexive relation (RR) on M. Then, for any A ∈ q-ROF z S(M), the following properties for LAP and U AP with respect to ARs hold: Theorem 10. Let T be a RR over M. For any A ∈ q-ROF z S(M), the following properties for LAP and U AP with respect to FRs hold: Proof. The proof follows directly from the proof of Theorem 9.

q-ROF z Ts Induced by RR
Cheng [29] proposed the idea of fuzzy topological space and extended some basic terms related to topology. Olgun [30] et al. proposed the idea of a q-rung orthopair fuzzy topological space (q-ROF z TS) and discussed continuity between two q-ROF z TSs.
Here, we give two kinds of q-ROF z Ts based on a RR.

Definition 10 ([30]).
A family A ⊆ q-ROF z S(M) of q-ROF z Ss on M is called a q-ROF z T on M if it satisfies:
From the relations (1) and (2), we get T ( i∈I A i ) = i∈I A i . Hence, T is a q-ROF z T on M.

Theorem 12. If T is a RR on M, then
Proof. The proof follows directly from the proof of Theorem 11.

Similarity Relations (S m Rs) Based on CBR
Here, we discuss some similarity relations (S m Rs) between q-ROF z Ss based on their rough U APs, LAPs and prove some results. Definition 11. Let T be a CBR from M to N. For A 1 , A 2 ∈ q-ROF z S(N), we define the relations S,S and S on N, as follows: Definition 12. Let T be a CBR from M to N. For A 1 , A 2 ∈ q-ROF z S(M), we define the relations s,s and s on M, as follows: The above CBRs are called as the lower q-ROF z similarity relation (q-ROF z S m R), upper q-ROF z similarity relation (q-ROF z S m R), and q-ROF z similarity relation (q-ROF z S m R), respectively.

Proposition 1. The relations S,S, S are E q Rs on q-ROF z S(M).
Proof. Straightforward.

Proposition 2.
The relations s,s, s are E q Rs on q-ROF z S(M).

Proof. Straightforward.
Theorem 13. Let T be a CBR from M to N and A 1 , A 2 , A 3 , A 4 ∈ q-ROF z S(N). Then: Conversely, if A 1S 0 and A 2S 0, then T A 1 = T 0 and T A 2 = T 0 . By Theorem 1, Thus, T 1 = T A 1 and T 1 = T A 2 . Hence, A 1S 1 and A 2S 1.

Theorem 14.
Let T be a CBR from M to N and A 1 , A 2 , A 3 , A 4 ∈ q-ROF z S(M). Then: , then A 1s 1 and A 2s 1.
Proof. The proof follows directly from the proof of Theorem 13.
Theorem 15. Let T be a CBR from M to N and A 1 , A 2 , A 3 , A 4 ∈ q-ROF z S(N). Then the following hold: (v) If A 1 ⊆ A 2 and A 1 S1, then A 2 S1 (vi) If (A 1 ∩ A 2 )S1, then A 1 S1 and A 2 S1.

Proof. Straightforward.
Theorem 16. Let T be a CBR from M to N and A 1 , A 2 , A 3 , A 4 ∈ q-ROF z S(M). Then the following hold:

Proof. Straightforward.
Theorem 17. Let T be a CBR from M to N and A 1 , A 2 ∈ q-ROF z S(N). Then the following hold: if and only if A 1 S0 and A 2 S0 (iv) If (A 1 ∩ A 2 )S1, then A 1 S1 and A 2 S1.
(v) If A 1 ⊆ A 2 and A 1 S1, then A 2 S1 Proof. The proof follows directly from Theorems 13 and 15.
Theorem 18. Let T be a CBR from M to N and A 1 , A 2 ∈ q-ROF z S(M). Then the following hold: Proof. The proof follows directly from Theorems 14 and 16.

Accuracy Measures of q-ROF z Ds
The approximation of q-ROF z Ss gives a new method for checking how much accurate a q-ROMD is, which describe the objects. First we define (α, β)-level cut of a q-ROF z S A. Definition 13. Let A ∈ q-ROF z S(M) and α, β ∈ [0, 1] be such that α q + β q ≤ 1, for q ≥ 1. Then we define (α, β)-level cut set of a q-ROF z S A by

sets of β-level and strong β-level cuts of A.
Thus, we can define the other cuts sets of a q-ROF z S A as: which we call as (α,β)-level cut set of A.
Now by using (i), we get (v) Let m be an element of ( Note that if T is a CBR over M, then T A β α is the LAP of the crisp set A β α and (T A ) β α will be (α, β)-level cut set of T A with regard to the ARs. Thus, we have, with regard to ARs. Similarly, with respect to FRs.

Lemma 2.
Let T be a RR on a finite universe M and A ∈ q-ROF z S(M). Let α, β ∈ [0, 1] be such Proof. The proof follows directly from the proof of Lemma 1.
The accuracy degree (AD) and roughness degree (RD) of a q-ROF z S are defined below.

Definition 14.
Let T be a RR on a finite universe M. The accuracy degree of A ∈ q-ROF z S(M), with regard to the parameters α, β, θ, γ ∈ [0, 1] such that β ≥ θ, α ≤ γ, and γ q + θ q ≤ 1, α q + β q ≤ 1, for q ≥ 1, and with regard to ARs, is given as: The roughness degree for the membership of A ∈ q-ROF z S(M) is given as: Similarly, the accuracy degree for the membership of A ∈ q-ROF z S(M), with regard to FRs, can be given as: Hence, δ (γ,θ) (α,β) (T A ) can be considered as the MD to how much A is accurate, with regard to (γ, θ) and (α, β).
The next example illustrates the above concepts related to degrees.
where S is the score function as given in Definition 7. Thus, the element m i ∈ M is considered the best decision if m i has the greatest choice-value λ i , and the object m i ∈ M is considered the worst decision if m i has the least choice-value λ i for the DMP. If there exists more that one element m i ∈ M with the same greatest (least) choice-values λ i , then we can take any one of them as the optimum decision for the DMP.
Here, two algorithms for the proposed method are presented: one can use the ring product operation ⊗ to perform Algorithms 1 and 2.

Algorithm 1 Selection of Best and Worst alternative based on ARs
Step i Using Definition 8, finds the lower q-ROF z S approximation T A and upper q-ROF z S approximation T A of a q-ROF z S A with respect to the ARs.
Step ii By sum operation ⊕, calculate the choice set; Step iii Compute the choice value using score function given in Definition 7, Step iv The best decision is m t ∈ M if λ t = max i λ i , i = 1, 2, 3, . . . | M |.
Step vi If there is more than one value for t, then take any m t as the best/worst alternative.

Algorithm 2 Selection of Best and Worst alternative based on FRs
Step i Using Definition 9 find the lower q-ROF z S approximation A T and upper q-ROF z S approximation A T of a q-ROF z S A with respect to the FRs.
Step ii By the sum operation ⊕, calculate the choice set; T = T A ⊕ T A Step iii Compute the choice value using the score function given in Definition 7, λ i = S(T (m i )) Step iv The best decision is m t ∈ M if λ t = max i λ i , i = 1, 2, 3, ... | M |.
Step vi If there is more than one value for t, then take any m t as the best/worst alternative.

An Application of the DM Approach
Here, we study emergency DM under the framework of rough q-ROF z S over dual universes. Plans for sound emergency preparedness can guarantee a quick and an efficient emergency response and can keep loss to a minimum. Existing research focuses on qualitative evaluation criteria, including economy, effectiveness, adequacy of protection, etc. The literature presents methodologies on how to determine the corresponding significance of each criterion and indicator and, thus, the weight of each expert opinion, the method to aggregate group opinions and judgments, and other related issues. Meanwhile, the outputs of a quantitative evaluation are provided using the method to choose a plan for emergency preparedness. Thus, this work provides a basis for decision-makers to choose the best emergency plan in practice.

Problem Statement
The criteria and evaluation indicators for an emergency DM are the fundamental characteristics of a plan for an emergency situation. Therefore, we do not depend upon scoring of expert or pairwise comparisons to evaluate the indicators. Instead, to evaluate the indicators, for instance, specificity, quick response to a situation, completeness, and other main characteristics of the plan are considered to be a finite collection or universe, denoted by N. That is, the universe N will stand for characteristics of the plan for an emergency situation, i.e., N = {soundness of personnel and resources allocation (c 1 ), good intersectoral collaboration (c 2 ), . . . , and reasonable cost (c l )}. Generally speaking, N is finite as the indicators describing the basic features of the plan are finite. Meanwhile, we collect all of the plans for an emergency situation into a set, denoted by M, i.e., M = {p 1 , p 2 , p 3 , . . . , p k }, where each p i stands for the ith emergency plan. A subset T of M × N is the relation between the plan set M and the set of characteristics N. That is, for any plan of emergency p i ∈ M, the characteristic is that the AR p i T. Then, the details of emergency DMP are as follows: First, suppose that each plan of emergency (denoted by universe M = {p i : i = 1, 2, . . . , k}) will be linked with several characteristics.
Secondly, the choice of the decision-makers are given with the most characteristics (denote because the q-ROF z S A of universe N = {c j : j = 1, 2, . . . , l}), which are related to an optimal plan for emergency situations, depending on online information and realtime scenarios.
Finally, decision-makers will choose one amongst the plans, p i ∈ M(i = 1, 2 . . . , k), with minimum risk of losing since the criterion for the optimal plan is to implement the plan.