# Novel Concepts ofq-Rung Orthopair Fuzzy Topology and WPM Approach for Multicriteria Decision-Making

Authors: Muhammad Riaz; Hafiz Muhammad Athar Farid; S. A. Alblowi; Yahya Almalki
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Aq-rung orthopair fuzzy set (q-ROFS) is a robust approach for fuzzy modeling, computational intelligence, and multicriteria decision-making (MCDM) problems. The aim of this paper is to study the topological structure on q-ROFSs and define the idea of q-rung orthopair fuzzy topology (q-ROF topology). The characterization of q-ROF α-continuous mappings between q-ROF topological spaces and q-ROF connectedness is investigated. Some relationships of different types of q-rung orthopair fuzzy connectedness are also investigated. Additionally, the “q-rung orthopair fuzzy weighted product model” (q-ROF WPM) is developed for MCDM of a hierarchical healthcare system. Due to limited and insufficient resources, a hierarchical healthcare system (HHS) is very effective to deal with the increasing problems of healthcare. Recognizing the stage of a disease with the symptoms, ranking the critical condition of patients, and referring patients to feasible hospitals are key features of HHS. A HHS will provide healthcare services in three levels, a primary health centers for initial stage of disease, secondary hospitals for secondary stage of disease, and tertiary hospital for the third-order stage. A numerical example is illustrated to demonstrate the efficiency of q-ROF WPM and advantages of HHS.


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## Body

## 1. Introduction

Topological data analysis (TDA) methods are rapidly growing approaches to infer persistent key features for possibly complex data. Topology and big data have inspiration to conventional and information analysis in a variety of computational intelligence fields, learning algorithms [1], analysis techniques [2, 3], and big data [4, 5]. Topology corresponds to the link amongst spatial structures and features, and it can be utilized to explain some specific spatial functions as well as to conceptualize data sets with better reliability and stronger integrity of data. The isomorphisms of the category of topological spaces, known as homeomorphisms, contribute significantly to the theory. The concepts of topology like convergence, continuity, homeomorphisms, and simplicial complexes have robust geometrical interpretation.The classical methods can not deal with uncertain and vague information in data analysis. To address these challenges, Zadeh [6] established the notion of fuzzy set (FS) theory and membership function. The membership of an element in a fuzzy set with a range of 0,1 is represented by a single value. Fuzzy logic is a type of multivalued logic in which the values of variables can range from 0 to 1. The idea behind fuzzy logic is that people take decisions relying on ambiguous and nonnumerical information. Fuzzy sets are mathematical interpretations of uncertainty and incorrect data. They are often referred to as fuzzy models. It is a term used to describe the notion of partial truth, which states that the truth value can range between false and true.Chang [7] developed fuzzy topology and topological space and some fundamental topological concepts such as closed set, open set, compactness, and continuity. Lowen introduced a new concept and definition for fuzzy topological spaces [8, 9]. Fuzzy modeling provide a mathematical framework for uncertain information making judgments based on fuzzy descriptions of data. It is based on the membership function of fuzzy set, which assign a degree of membership or satisfaction. The researchers noticed that the membership grades are not sufficient to analyze data or objects. Atanassov [10] proposed a direct extension of FS, namely, “intuitionistic fuzzy set” (IFS), which depends on both membership grade (MG) and nonmembership grade (NMG).Coker developed the idea of intuitionistic fuzzy topological space and investigated various counterpart versions of traditional topological properties such as compactness and continuity [11, 12]. Additional results on intuitionistic fuzzy topological spaces are explored in [13, 14]. Fuzzy metric, distance function, and fuzzy metric spaces are the generalization of classical metric spaces [15]. Thus, rather than specific fixed boundaries, the degree-theoretic structure of fuzziness has been incorporated in the notion of topology. Singal and Rajvanshi pioneered the notions of open map, fuzzy alpha open and closed sets, and continuous functions [16].Ajmal and Kohli [17] developed the idea of “connectedness in fuzzy topological spaces,” and Chaudhuri and Das [18] initiated the notion of “fuzzy connected sets in fuzzy topological spaces.” Olgun et al. [19] and Öztürk and Yolcu [20] proposed the notion of “Pythagorean fuzzy topology and Pythagorean fuzzy topological spaces.” Turkarslan et al. [21] proposed some results of “q-rung orthopair fuzzy topological spaces”, and Charisma and Ajay [22] gave the idea of “Pythagorean fuzzy α-continuity.” Haydar gave the notion of connectedness for Pythagorean fuzzy topological space [23].MCDM techniques with fuzzy modeling have been increasingly studied and applied to real-world problems and computational intelligence. Numerous MCDM approaches have been developed to evaluate the robustness of fuzzy modeling in evaluating a set of available objects against a set of criteria. Numerous fields rely on information aggregation and synthesis, including machine learning, neural network, decision analysis, and pattern recognition. Aggregation, in a wide sense, is the process of combining several bits of data to obtain a result. Additionally, it was demonstrated that fundamental data handling algorithms based on crisp integers are incapable of describing working conditions in human cognitive systems. Decision-makers (DMs) are left with hazy findings and perplexing judgements as a result of these approaches. As a result, in order to deal with the world’s ambiguous and fuzzy scenarios, DMs seek new philosophies that enable them to interpret ambiguous data values while maintaining their judgement demands under a variety of circumstances.Yager and Abbasov [24] and Yager [25] introduced the notion of “Pythagorean fuzzy set” (PFS), and Yager [26] introduced the generalized membership grading concept named as “q-rung orthopair fuzzy set.” The constraint of q-ROFS is that the sum of qth powers of MG and NMG must be less than or equal to one. Clearly, the higher value of q gives higher the q-rung (q-orbit), the more orthopair’s fulfil the constraining requirement, and hence, a broader space is available for q-rung orthopair’s [27].Xu and Yager [28, 29] established geometric and averaging aggregation operators (AOs) for IFSs. Ashraf and Abdullah [30] proposed mathematical approach for MCDM in COVID-19 by utilizing spherical fuzzy information. Saha et al. [31] suggested a new hybrid hesitant fuzzy weighted AOs based on Archimedean and Dombi operations for MCDM. Wei and Zhang [32] utilized single-valued neutrosophic Bonferroni power AOs to select optimal strategic providers. Wei and Wei [33] developed Dombi prioritized AOs using SVNSs. Garg [34] developed AOs of intuitionistic multiplicative set. Akram et al. [35] proposed some novel Dombi AOs for m-polar fuzzy set with applications to MCDM. Sitara et al. [36] proposed q-rung picture fuzzy (q-RPF) graph structures and numerous significant results for modeling with q-RPF graph. Kanwal et al. [37] proposed the existence of fixed points in fuzzy strong b-metric spaces, and Kanwal and Azam [38] gave the notion of common fixed points of IFS maps for Meir-Keeler type contractions. Many AOs for different extensions of fuzzy sets are proposed in [39, 40]. Mahmood et al. [41] proposed the spherical fuzzy set and T spherical fuzzy set with applications to medical diagnosis. The concept of “linear Diophantine fuzzy set” (LDFS) initiated by Riaz and Hashmi [42] is strong model for fuzzy modeling and MADM. They suggested the idea LDF-topology and LDF-AOs with applications. Farid and Riaz suggested improved operational laws for q-ROF information and “q-ROF Einstein interactive geometric” [43], q-ROF hybrid AOs [44], and TOPSIS and VIKOR approaches for q-ROFSSs for MCDM [45].Çag˘man et al. [46] proposed the idea of soft topology on soft sets. Shabir and Naz [47] introduced the novel concepts of soft topological spaces and characterization of soft topology. Peng and Liu [48] proposed information measures, such as similarity measures, distance measures, and entropy, for q-ROFSs with applications to medical diagnosis, pattern recognition, and clustering analysis. Feng et al. [49] proposed MADM application by using a new score function for ranking of alternatives with generalized orthopair fuzzy membership grades. Akram et al. [50] suggested a hybrid decision-making framework by using aggregation operators under a complex spherical fuzzy prioritization approach.The main goals of this manuscript are given in the following:(i)
To define topological structure onq-rung orthopair fuzzy sets and introduce the notion of q-rung orthopair fuzzy topology(ii)
To discuss characterization of q-ROF topology and q-ROF topological spaces like closure, interior, and frontier(iii)
To investigate some significant results related to images and inverse images of q-ROFSs under q-ROF mapping(iv)
To define q-ROFα-continuity and q-ROF connectedness, some relationships between different types of q-ROF connectedness are also investigated(v)
An extension of “weighted product model” WPM to q-ROFSs is developed. A WPM is a robust approach for MCDM frameworks to analyze a set of feasible alternatives in terms of a set of choice criteria. Other key features of WPM include multiplying a number of ratios, one for each choice criterion, each decision alternative is compared to the others. Each ratio is raised to the power of the related criterion’s relative weight(vi)
A hierarchical healthcare system (HHS) is suggested that will provide healthcare services in three levels, a primary health centers for initial stage of disease, secondary hospitals for secondary stage of disease, and tertiary hospital for the third-order stage(vii)
An application to HHS for COVID-19 is also presented to illustrate the feasibility and reliability of q-ROF WPMThe remainder of the paper is organized as follows. In Section2, we review the rudiments of q-ROFSs and q-ROFNs. In Section 3, the idea of q-ROF topology is studied. In Section 4, we presented main results about q-ROFTSs. Section 5 introduces the concept of q-ROF α-continuity. Section 6 introduces certain principles related to q-ROF connectedness, and Section 7 proposes a framework of WPM under the q-ROFSs with an application related to hierarchical healthcare system for COVID-19. Section 8 summarizes the study paper’s key findings.


## 2. Fundamental Concepts

In this section, we review some fundamental concepts and operational procedures ofq-rung orthopair fuzzy sets (q-ROFSs) and q-rung orthopair fuzzy numbers (q-ROFNs). A detailed study of these concepts can be seen in [26, 27, 48, 51].Definition 1 (see [26]).
A q-ROFS⋎ˇ on X is defined as
(1)⋎ˇ=ϱ¯,μ⋎ˇϱ¯,ν⋎ˇϱ¯:ϱ¯∈X,here, μ⋎ˇ,ν⋎ˇ:X⟶0,1 denotes the MF and NMF of the alternative ϱ¯∈X and ∀ϱ¯; we have
(2)0≤μ⋎ˇqϱ¯+ν⋎ˇqϱ¯≤1.
Furthermore,π⋎ˇϱ¯=1−μ⋎ˇqϱ¯−ν⋎ˇqϱ¯q is called the indeterminacy degree of ϱ¯ to ⋎ˇ.
Liu and Wang [51] suggested some operations on q-ROFNs with the given principles.Definition 2 (see [51]).
Letδ~1⋏=μ1,ν1 and δ~2⋏=μ2,ν2 be q-ROFNs. σ>0, Then,
(1)
δ~1⋏c=ν1,μ1(2)
δ~1⋏∨δ~2⋏=maxμ1,ν1,minμ2,ν2(3)
δ~1⋏∧δ~2⋏=minμ1,ν1,maxμ2,ν2(4)
δ~1⋏⊕δ~2⋏=μ1q+μ2q−μ1qμ2qq,ν1ν2(5)
δ~1⋏⊗δ~2⋏=μ1μ2,ν1q+ν2q−ν1qν2qq(6)
σδ~1⋏=1−1−μ1qσq,ν1σ(7)
δ~1⋏σ=μ1σ,1−1−ν1qσqDefinition 3 (see [51]).
Letδ~⋏=μ,ν be the q-ROFN; then, the “score function” (SF) S⊺ of δ~⋏ is defined as
(3)S⊺δ~⋏=μq−νq,S⊺δ~⋏∈−1,1.Definition 4 (see [51]).
Letδ~⋏=μ,ν be the q-ROFN; then, the accuracy function (AF) H˘ of δ~⋏ is defined as
(4)H˘δ~⋏=μq+νq,H˘δ~⋏∈0,1.Definition 5.
Considerδ~1⋏=μ1,ν1 and δ~2⋏=μ2,ν2 be two q-ROFNs. Then, the subtraction and division of q-ROFNs are defined as
(1)
δ~1⋏⊖δ~2⋏=μ1q−μ2q/1−μ2qq,v1/v2, if μ1≥μ2,v1≤minv2,v2π1/π2(2)
δ~1⋏⊘δ~2⋏=μ1/μ2,v1q−v2q/1−v2qq if v1≥v2,μ1≤minμ2,μ2π1/π2


### 2.1. Superiority of q-ROFNs and Comparison with Other Fuzzy Numbers

The generalized MG and NMG ofq-rung orthopair fuzzy numbers (q-ROFNs) provide a robust approach for computational intelligence, fuzzy modeling, and MCDM problems. A q-ROFN is supeior than other fuzzy numbers such as (FNs), IFNs, and PFNs. Table 1 provides analysis of the merits and limitations of q-ROFNs with other fuzzy numbers.Table 1
Comparative analysis of q-ROFNs.
TheoriesMeritsLimitationsFuzzy sets [6]Assign MG in0,1Can not assign NMGIFSs [10]Assign both MG and NMGFails whenMG+NMG>1PFSs [24]Assign both MG and NMG, superior than the IFNsFails whenMG2+NMG2>1FFSs [52]Assign both MG and NMG, superior than IFNs and PFNsFails whenMG3+NMG3>1q-ROFSs [26]Assign both MG and NMG, superior than IFNs, PFNs, and FFNs, a broader space for MG and NMGCan not deal withMGq+NMGq>1 and MG=NMG=1


## 2.1. Superiority of q-ROFNs and Comparison with Other Fuzzy Numbers

The generalized MG and NMG ofq-rung orthopair fuzzy numbers (q-ROFNs) provide a robust approach for computational intelligence, fuzzy modeling, and MCDM problems. A q-ROFN is supeior than other fuzzy numbers such as (FNs), IFNs, and PFNs. Table 1 provides analysis of the merits and limitations of q-ROFNs with other fuzzy numbers.Table 1
Comparative analysis of q-ROFNs.
TheoriesMeritsLimitationsFuzzy sets [6]Assign MG in0,1Can not assign NMGIFSs [10]Assign both MG and NMGFails whenMG+NMG>1PFSs [24]Assign both MG and NMG, superior than the IFNsFails whenMG2+NMG2>1FFSs [52]Assign both MG and NMG, superior than IFNs and PFNsFails whenMG3+NMG3>1q-ROFSs [26]Assign both MG and NMG, superior than IFNs, PFNs, and FFNs, a broader space for MG and NMGCan not deal withMGq+NMGq>1 and MG=NMG=1


## 3. q-ROF Topology

Turkarslan et al. [21] proposed the idea of q-ROF topological spaces as a generalization of Pythagorean fuzzy topological spaces [19, 20].Definition 6 (see [21]).
LetX≠∅ be a set and £˘τ be a family of q-ROF subsets of X. If
T10X,1X∈£˘τ,
T2 for anyTℸ1,Tℸ2∈£˘τ, we have Tℸ1∩Tℸ2∈£˘τ,
T3 for anyTℸii∈I⊆£˘τ, we have ∪i∈ITℸi∈£˘τ.,
then£˘τ is called a q-ROF topology on X and the pair X,£˘τ is said to be a q-ROFTS. Each member of £˘τ is called a q-ROF open set (q-ROFOS). The complement of a q-ROF open set is called a q-ROF closed set (q-ROFCS).Remark 7.
Because any IFS or PFS may be thought of as a q-ROF set, we can conclude that any IF topological space or PF topological space is also a q-ROFTS. In contrast, it is evident that q-ROFTS does not have to be an IF or PF topological space. Even a q-ROF open set may not be an IFS or a PFS.Example 1.
LetX=α˘1γ,α˘2γ,α˘3γ. Consider the following family of q-ROF subsets:
£˘τ=0X,1X,Tℸ1,⋯,Tℸ4 where
(5)Tℸ1=α˘1γ,0.59,0.79,α˘2γ,0.69,0.59,α˘3γ,0.29,0.19,Tℸ2=α˘1γ,0.61,0.77,α˘2γ,0.73,0.54,α˘3γ,0.31,0.17,Tℸ3=α˘1γ,0.65,0.73,α˘2γ,0.75,0.49,α˘3γ,0.35,0.15,Tℸ4=α˘1γ,0.72,0.68,α˘2γ,0.81,0.44,α˘3γ,0.45,0.11.
One can see thatX,£˘τ is a q-ROFTS.
Moreover, we see that the following collections are all q-ROFTSs:
£˘1τ=0X,1X,Tℸ1, £˘2τ=0X,1X,Tℸ2, £˘3τ=0X,1X,Tℸ3, £˘4τ=0X,1X,Tℸ4,
£˘5τ=0X,1X,Tℸ1,Tℸ2, £˘6τ=0X,1X,Tℸ1,Tℸ3, £˘7τ=0X,1X,Tℸ1,Tℸ4,
£˘8τ=0X,1X,Tℸ2,Tℸ3, £˘9τ=0X,1X,Tℸ2,Tℸ4, £˘10τ0X,1X,Tℸ3,Tℸ4,
£˘11τ0X,1X,Tℸ1,Tℸ2,Tℸ3, £˘12τ=0X,1X,Tℸ1,Tℸ3,Tℸ4.Definition 8 (see [21]).
LetX and Y be two nonempty sets, let J:X⟶Y be a mapping, and let E˘ and W~ be q-ROF subsets of X and Y, respectively. Then, the membership and nonmembership functions of image of E˘ with respect to J that is denoted by JE˘ are defined by
(6)μJE˘y=supz∈J−1yμE˘z,ifJ−1y≠∅,0,otherwise,νJE˘y=infz∈J−1yνE˘z,ifJ−1y≠∅,0,otherwise,respectively. The membership and nonmembership functions of preimage of W~ with respect to J that is denoted by J−1W~ are defined by
(7)μJ−1W~x=μW~Jx,νJ−1W~x=νW~Jx,
respectively.
In [21], they showed that μJE˘q+νJE˘q≤1 q-ROF membership condition is provided for q-ROF image and preimage.Proposition 9 (see [21]).
LetX and Y be two nonempty sets and J:X⟶Y be a q-ROF function. Then, we have
(1)
J−1W~c=J−1W~c for any q-ROF subset W~ of Y(2)
JE˘c⊆JE˘c for any q-ROF subset E˘ of X(3)
IfW~1⊆W~2 then J−1W~1⊆J−1W~2 where W~1 and W~2 are q-ROF subset of Y(4)
IfE˘1⊆E˘2 then JE˘1⊆JE˘2 where E˘1 and E˘2 are q-ROF subset of X(5)
JJ−1W~⊆W~ for any q-ROF subset W~ of Y(6)
E˘⊆J−1JE˘ for any q-ROF subset E˘ of X


## 4. Main Results

In this section, numerous results related toq-rung orthopair fuzzy topology (q-ROFT) are proposed.Definition 10.
LetTℸi:i∈I, where Tℸi=ℵ˘,μTℸiℵ˘,νTℸiℵ˘:ℵ˘∈X, be the indexed family of q-ROFSs over X. Then, the intersection and union of the family is defined as
(8)⋂i∈I‍Tℸi=ℵ˘,infμTℸiℵ˘,supνTℸiℵ˘:ℵ˘∈X,⋃i∈I‍Tℸi=ℵ˘,supμTℸiℵ˘,infνTℸiℵ˘:ℵ˘∈X.Note that⋂i∈I‍Tℸi and ∪i∈ITℸi are q-ROFSs over X.Theorem 11.
LetTℸi:i∈I, where Tℸi=ℵ˘,μTℸiℵ˘,νTℸiℵ˘:ℵ˘∈X, be the indexed family of q-ROFSs over X. Then,
(1)
⋂i∈I‍Tℸi¯=∪i∈ITℸi¯(2)
⋃i∈I‍Tℸi¯=⋂i∈I‍Tℸi¯Proof.
The proof is obvious by Definition10.Definition 12.
LetX,£˘τ be a q-ROFTS and Tℸ=ℵ˘,μTℸℵ˘,νTℸℵ˘:ℵ˘∈X be a q-ROFS over X. Then, the q-ROF interior, q-ROF closure, and q-ROF frontier or boundary of Tℸ are defined as
(1)
IntTℸ=∪G:Gisaq‐ROFOSinXandG⊆Tℸ
i.e., “IntTℸ is the q-ROF union of q-ROF open sets contained in Tℸ”(2)
ClTℸ=∩K:Kisaq‐ROFCSinXandTℸ⊆K
i.e., “ClTℸ is the q-ROF intersection of q-ROF closed supersets of Tý”(3)
FrTℸ=ClTℸ∩ClTℸc(4)
ExtTℸ=IntTℸcRemark 13.
By Definition10, we have the following observations:
(i)
q-ROF interior, q-ROF closure, and q-ROF boundary of a q-ROFS are always q-ROFSs(ii)
IntTℸ is the largest q-ROF open set contained Tℸ(iii)
ClTℸ is the smallest q-ROF closed set containing TℸExample 1.
LetX=α˘1γ,α˘2γ,α˘3γ. Consider the family of q-ROF sets given as
(9)£˘τ=1X,0X,Tℸ1,Tℸ2,Tℸ3,Tℸ4,,Tℸ1=α˘1γ,0.59,0.79,α˘2γ,0.69,0.59,α˘3γ,0.29,0.19,Tℸ2=α˘1γ,0.61,0.77,α˘2γ,0.73,0.54,α˘3γ,0.31,0.17,Tℸ3=α˘1γ,0.65,0.73,α˘2γ,0.75,0.49,α˘3γ,0.35,0.15,Tℸ4=α˘1γ,0.72,0.68,α˘2γ,0.81,0.44,α˘3γ,0.45,0.11.It is clear thatX,£˘τ is a q-ROF topological space. Now, assume that
(10)λ=α˘1γ,0.79,0.49,α˘2γ,0.89,0.29,α˘3γ,0.58,0.09,is a q-ROF subset over X. Then,
(11)Intλ=0X∪Tℸ1∪Tℸ2∪Tℸ3∪Tℸ4=Tℸ4=α˘1γ,0.72,0.68,α˘2γ,0.81,0.44,α˘3γ,0.45,0.11.On the other hand, in order to find the q-ROF closure ofTℸ, it necessary to determine the q-ROF closed sets over X. Then,
(12)Tℸ1c=α˘1γ,0.79,0.59,α˘2γ,0.59,0.69,α˘3γ,0.19,0.29,Tℸ2c=α˘1γ,0.77,0.61,α˘2γ,0.54,0.73,α˘3γ,0.17,0.31,Tℸ3c=α˘1γ,0.73,0.65,α˘2γ,0.49,0.75,α˘3γ,0.15,0.35,Tℸ4c=α˘1γ,0.68,0.72,α˘2γ,0.44,0.81,α˘3γ,0.11,0.45.Hence,(13)Clλ=1X..Similarly to find the q-ROF boundary ofTℸ,
(14)λc=α˘1γ,0.49,0.79,α˘2γ,0.29,0.89,α˘3γ,0.09,0.58,Clλc=1X∩Tℸ1c∩Tℸ2c∩Tℸ3c∩Tℸ4c=Tℸ4c=α˘1γ,0.68,0.72,α˘2γ,0.44,0.81,α˘3γ,0.11,0.45,Frλ=Clλ∩Clλc=1X∩Tℸ4c=α˘1γ,0.68,0.72,α˘2γ,0.44,0.81,α˘3γ,0.11,0.45,Extλ=IntTℸc=0X.Remark 14.
Furthermore, we investigate that some results that hold in crisp topology but do not hold in q-ROFTSX,£˘τ. Table 2 shows the comparison of some results of crisp topology and q-ROF topology.Proposition 15.
LetX,£˘τn be a q-ROFTS and Tℸ,Tℸ1,andTℸ2 be q-ROFSs over X. Then, the following properties hold:
(1)
IntTℸ⊆Tℸ(2)
IntIntTℸ=IntTℸ(3)
Tℸ1⊆Tℸ2⇒IntTℸ1⊆IntTℸ2(4)
IntTℸ1∩Tℸ2=IntTℸ1∩IntTℸ2(5)
Int1X=1X,Int0X=0X(6)
IntTℸ1∪Tℸ2⊆IntTℸ1∩IntTℸ2Proof.
(1), (2), (3), and (5) can be easily obtained from the definition of the q-ROF interior (4).
FromIntTℸ1∩Tℸ2⊆IntTℸ1 and IntTℸ1∩Tℸ2⊆IntTℸ2, we obtain IntTℸ1∩Tℸ2⊆IntTℸ1∩IntTℸ2. On the other hand, from the facts IntTℸ1⊆Tℸ1 and IntTℸ2⊆Tℸ2⇒IntTℸ1∩IntTℸ2⊆Tℸ1∩Tℸ2 and IntTℸ1∩IntTℸ2∈£˘τ, we have IntTℸ1∩IntTℸ2⊆IntTℸ1∩Tℸ2. So, proof of the axioms (4) is obtained from these two inequalities.Theorem 16.
LetJ:q‐ROFSX⟶q‐ROFSX be a mapping. The family £˘τ=Tℸ∈q‐ROFSX:JTℸ=Tℸ is a q-ROF topology over X, if the mapping J satisfies the following conditions:
(1)
JTℸ⊆Tℸ(2)
J1X=1X(3)
JJTℸ=JTℸ(4)
JTℸ1∩Tℸ2=JTℸ1∩JTℸ2
JTℸ=IntTℸ for each q-ROF set Tℸ in this q-ROF topological space.Proposition 17.
LetX,£˘τ be a q-ROFTS and Tℸ,Tℸ1,andTℸ2 be q-ROFSs over X. Then, the following properties hold:
(i)
Tℸ⊆ClTℸ(ii)
ClClTℸ=ClTℸ(iii)
Tℸ1⊆Tℸ2⇒ClTℸ1⊆ClTℸ2(iv)
ClTℸ1∪Tℸ2=ClTℸ1∪ClTℸ2(v)
Cl1X=1X,Cl0X=0XProof.
(1), (2), (3), and (5) can be easily obtained from the definition of the q-ROF closure (4).
FromClTý1⊆ClTý1∪Tý2 and ClTý2⊆ClTý1∪Tý2, we obtain ClTý1∪ClTý2⊆ClTý1∪Tý2. On the other hand, from the facts Tý1⊆ClTý1 and Tý2⊆ClTý2⇒Tý1∪Tý2⊆ClTý1∪ClTý2 and ClTý1∪ClTý2∈Tý, we have ClTý1∪Tý2⊆ClTý1∪ClTý2. Thus, proof of the axioms (4) is obtained from these two inequalities.Theorem 18.
LetC:q‐ROFSX⟶q‐ROFSX be a mapping. The family £˘τ=Tℸ∈q‐ROFSX:CTℸc=Tℸc is a q-ROF topology over X, if the mapping C satisfies the following conditions:
(1)
Tℸ⊆CTℸ(2)
C0X=0X(3)
CCTℸ=CTℸ(4)
CTℸ1∪Tℸ2=CTℸ1∪CTℸ2
Also,CTℸ=ClTℸ for each q-ROF set Tℸ in this q-ROF topological space.Theorem 19.
LetX,£˘τ be a q-ROFTS and Tℸ be a q-ROFS over X. Then,
(a)
ClTℸc=IntTℸc(b)
IntTℸc=ClTℸcProof.
(a)
LetTℸ=ℵ˘,μTℸℵ˘,νTℸℵ˘:ℵ˘∈X and assume that the family of q-ROFSs contained in Tℸ is indexed by the family Tℸi=ℵ˘,μTℸiℵ˘,νTℸiℵ˘:ℵ˘∈Xi∈I. Then, we see that IntTℸ=ℵ˘,supμTℸiℵ˘,infνTℸiℵ˘:ℵ˘∈X and hence IntTℸc=ℵ˘,infνTℸiℵ˘,supμTℸiℵ˘:ℵ˘∈X. Since Tℸc=ℵ˘,νTℸℵ˘,μTℸℵ˘:ℵ˘∈X and μTℸiℵ˘≤μTℸℵ˘,νTℸiℵ˘≥νTℸℵ˘ for each i∈I, we obtain that Tℸi=ℵ˘,μTℸiℵ˘,νTℸiℵ˘:ℵ˘∈Xi∈I is the family of q-ROFSs containing Tℸc, i.e., ClTℸc=ℵ˘,infνTℸiℵ˘,supμTℸiℵ˘:ℵ˘∈X. Therefore, ClTℸc=IntTℸc immediately(b)
This analogous to (a)Definition 20.
A q-ROFN (or q-ROF point)ℵ=μ,ν is said to be contained in q-ROFS ⋎ˇ=ϱ¯,μ⋎ˇϱ¯,ν⋎ˇϱ¯:ϱ¯∈X, written as ℵ∈⋎ˇ, if μ≤μ⋎ˇϱ¯ and if ν≥νË˘ϱ¯,∀ϱ¯∈X.Definition 21.
A q-ROFNℵ=μ,ν in q-ROFS ⋎ˇ is said to be q-ROF interior point of ⋎ˇ if there exist q-ROF open set U such that ℵ∈U⊆⋎ˇ. Then, q-ROFS ⋎ˇ is called q-ROF neighborhood of q-ROFN ℵ=μ,ν.Theorem 22.
ConsiderX,£˘τ be a q-ROFTS, then
(1)
Ifϕ and φ are the neighborhood of q-ROFN ℵ, then ϕ⋂φ and ϕ⋃φ are also neighborhood of q-ROFN ℵ(2)
Ifψ is neighborhood of q-ROFN ℵ then each q-ROF superset δ⊃ψ is also neighborhood of q-ROFN ℵProposition 23.
LetX,£˘1τ and Y,£˘2τ be two q-ROFTSs and J:X⟶Y be a q-ROF mapping. Then, the following statements are equivalent:
(a)
J is a q-ROF continuous mapping(b)
JClTℸ⊆ClJTℸ for each q-ROFS Tℸ in X(c)
ClJ−1K⊆J−1ClK for each q-ROFS K in Y(d)
J−1IntK⊆IntJ−1K for each q-ROFS K in YProof (a ⇒ b).
LetJ:X⟶Y be a q-ROF continuous mapping and Tℸ be a q-ROFS over X. Then, JTℸ⊆ClJTℸ and Tℸ⊆J−1ClJTℸ. Since ClJTℸ is a q-ROF closed set in Y and J is a q-ROF continuous mapping, J−1ClJTℸ is a q-ROF closed set in X. On the other hand, if ClTℸ is the smallest q-ROF closed set containing Tℸ, then ClTℸ⊆J−1ClJTℸ and so JClTℸ⊆ClJTℸ.
(b ⇒ c) Suppose thatTℸ=J−1K. From (b), JClTℸ=JClJ−1K⊆ClJTℸ=ClJJ−1K⊆ClK. Then, ClJ−1K=ClTℸ⊆J−1JClTℸ⊆J−1ClK.
(c⇒ d) Since IntK=ClKcc, then ClJ−1K=ClTℸ⊆J−1JClTℸ⊆J−1ClK.
Assume that,G is a q-ROF open set in Y. Then, IntG=G. From (d), J−1G=J−1IntG⊆IntJ−1G⊆J−1G. Therefore, J is a q-ROF continuous mapping.Definition 24.
LetX,£˘τ be a q-ROFTS.
(1)
A subfamilyΓ of £˘τ is called a q-ROF basis (q-ROFB) for £˘τ, if for each Tℸ∈£˘τ, Tℸ=0X or there exists Γ′⊆Γ such that Tℸ=∪Γ′(2)
A collectionΦ of some q-ROFSs over X is called a q-ROF subbase for some q-ROF topology £˘τ, if the finite intersections of members of Φ form a q-ROF basis for £˘τTheorem 25.
X,£˘1τ and Y,£˘2τ be two q-ROFTSs and J:X⟶Y be a q-ROF mapping. Then,
(1)
J is a q-ROF continuous mapping iff for each W~∈Γ we have J−1W~ is a q-ROF open subset of X such that Γ is a q-ROF basic for £˘2τ(2)
J is a q-ROF continuous mapping iff for each K∈Tℸhi we have J−1K is a q-ROF open subset of X such that Φ is a q-ROF subbase for £˘2τProof.
(i)
LetJ be a q-ROF continuous mapping. Since each W~∈Γ⊆£˘2τ and J is a q-ROF continuous mapping, then J−1W~∈£˘1τ
Conversely, suppose thatΓ is a q-ROF basic for £˘2τ and J−1W~∈£˘1τ for each W~∈Γ, then for arbitrary a q-ROF open set Tℸ∈£˘2τ,
(15)J−1Tℸ=J−1∪W~∈ΓW~=∪W~∈ΓJ−1W~∈£˘1τ.
That is,J is a q-ROF continuous mapping.
(ii)
LetJ be a q-ROF continuous mapping. Since each K∈Φ⊆£˘2τ and J is a q-ROF continuous mapping, then J−1K∈£˘1τ
Conversely, assume thatΦ is a q-ROF subbase for £˘2τ and J−1K∈£˘1τ for each K∈Φ, then for arbitrary a q-ROF open set Tℸ∈£˘2τ,
(16)J−1Tℸ=J−1∪ij∈IKi1∩Ki2∩⋯∩Kin=⋃ij∈I‍J−1Ki1∩J−1Ki2∩⋯∩J−1Kin∈£˘1τ.
That is,J is a q-ROF continuous mapping.Definition 26.
LetX,£˘1τ and Y,£˘2τ be two q-ROFTSs and J:X⟶Y be a q-ROF mapping. Then,
(1)
J is called a q-ROF open mapping if JTℸ is a q-ROF open set over Y for every q-ROF open set Tℸ over X(2)
J is called a q-ROF closed mapping if JK is a q-ROF closed set over Y for every q-ROF closed set K over XExample 2.
LetX=α˘1γ,α˘2γ,α˘3γ and Y=y1,y2,y3. Consider the following families of q-ROF sets £˘1τ=0X,1X,Tℸ1,Tℸ2,Tℸ3,Tℸ4 and £˘2τ=0Y,1Y,S1,S2,S3,S4 where
(17)Tℸ1=α˘1γ,0.29,0.45,α˘2γ,0.59,0.19,α˘3γ,0.59,0.49,Tℸ2=α˘1γ,0.59,0.49,α˘2γ,0.79,0.29,α˘3γ,0.69,0.59,Tℸ3=α˘1γ,0.59,0.40,α˘2γ,0.79,0.19,α˘3γ,0.69,0.49,Tℸ4=α˘1γ,0.29,0.42,α˘2γ,0.59,0.29,α˘3γ,0.59,0.59,S1=y1,0.59,0.19,y2,0.29,0.45,y3,0.59,0.49,S2=y1,0.79,0.29,y2,0.59,0.49,y3,0.69,0.59,S3=y1,0.79,0.19,y2,0.59,0.40,y3,0.69,0.49,S4=y1,0.59,0.29,y2,0.29,0.42,y3,0.59,0.59.
It is clear thatX,£˘1τ and Y,£˘2τ are q-ROF topological spaces. If q-ROF mapping J:X⟶Y is defined as
(18)Jα˘1γ=y2,Jα˘2γ=y1,Jα˘3γ=y3.
Then,J is a q-ROF open mapping. However, J is not q-ROF closed mapping on q-ROF topological spaces X,£˘1τ.Theorem 27.
LetX,£˘1τ and Y,£˘2τ be two q-ROFTSs and J:X⟶Y be a q-ROF mapping. Then,
(1)
J is a q-ROF open mapping if JIntTℸ⊆IntJTℸ for each q-ROF set Tℸ over X(2)
J is a q-ROF closed mapping if ClJTℸ⊆JClTℸ for each q-ROF set Tℸ over XProof.
(1)
LetJ be a q-ROF open mapping and Tℸ be a q-ROFS over X. Then, IntTℸ is a q-ROF open set and IntTℸ⊆Tℸ. Since J is a q-ROF open mapping, JIntTℸ is a q-ROF open set over Y and JIntTℸ⊆JTℸ. Thus, JIntTℸ⊆IntJTℸ is obtained
Conversely, suppose thatTℸ is any q-ROF open set over X. Then, Tℸ=IntTℸ. From the condition of the theorem, we have JIntTℸ⊆IntJTℸ. Then, JTℸ=JIntTℸ⊆IntJTℸ⊆JTℸ. This implies that JTℸ=IntJTℸ. That is, J is a q-ROF open mapping.
(2)
LetJ be a q-ROF closed mapping and Tℸ be a q-ROFS over X. Since J is a q-ROF closed mapping then JClTℸ is a q-ROF closed set over Y and JTℸ⊆JClTℸ. Thus, ClJTℸ⊆JClTℸ is obtained
Conversely, assume thatTý is any q-ROF closed set over X. Then, Tℸ=ClTℸ. From the condition of the theorem, we have ClJTℸ⊆JClTℸ=JTℸ⊆ClJTℸ. This means that ClJTℸ=JTℸ. That is, J is a q-ROF closed mapping.Definition 28.
LetX,£˘1τ and Y,£˘2τ be two q-ROFTSs and J:X⟶Y be a q-ROF mapping. Then, J is a called a q-ROF homeomerphism, if
(1)
J is a bijection(2)
J is a q-ROF continuous mapping(3)
J−1 is a q-ROF continuous mappingTheorem 29.
LetX,£˘1τ and Y,£˘2τ be two q-ROFTSs and J:X⟶Y be a q-ROF mapping. Then, the following conditions are equivalent:
(1)
J is a q-ROF homeomerphism(2)
J is a q-ROF continuous mapping and q-ROF open mapping(3)
J is a q-ROF continuous mapping and q-ROF closed mappingProof.
The proof can be easily obtained by using the previous theorems on continuity, opennes, and closedness are omitted.Table 2
Comparison of some results of crisp topology and q-ROF topology.
Crisp topologyq-rung orthopair fuzzy topologyIntλ⋃Extλ⋃Frλ=XIntλ⋃Extλ⋃Frλ≠1XIntλ⋂Extλ=∅Intλ⋂Extλ≠0XExtλ⋂Frλ=∅Extλ⋂Frλ≠0XIntλ⋂Frλ=∅Intλ⋂Frλ≠0X


## 5. q-ROFα-Continuity

Definition 30.
LetX,£˘τ be a q-ROF topological space. A q-ROFS Tℸ over X is called a q-ROF α open set if Tℸ⊆IntClIntTℸ. A q-ROFS whose complement is a q-ROF α open set (q-ROFαOS) is called a q-ROF α closed set (q-ROFαCS).Proposition 31.
LetX,£˘τ be a q-ROFTS. Then, arbitrary union of q-ROFαOS is a q-ROFαOS and arbitrary intersection of q-ROFαCSs is q-ROFαCS.Proof.
LetTℸi=x,μTℸ,γTℸ>i∈I be a family of q-ROFαOSs. Then, for each i∈I, Tℸi⊆IntClIntTℸi. Thus, ∪Tℸi⊆∪IntClIntTℸi⊆Int∪ClIntTℸi⊆IntCl∪IntTℸi⊆IntClInt∪Tℸi.
Hence,∪Tℸi is a q-ROFαOS set. If we take complement of this part, the consecutive will proved (ie. arbitrary intersection of q-ROFαOS is also a q-ROFαOS).
Every q-ROFOS is a q-ROFαOS, and every q-ROFCS is a q-ROFαCS but the converse is not true.Definition 32.
The q-ROFα closure of a q-ROFS Tℸ in a q-ROFTS X;£˘τ is represented as ClαTℸ and defined by ClαTℸ=⋂Ci∣Ciisaq‐ROFαCsetandTℸ⊆CiProposition 33.
In a q-ROFTSX,£˘τ, a q-ROFS Tℸ is q-ROFαC if and only if Tℸ=ClαTℸ.Proof.
Assume thatTℸ is a q-ROFαC set. Then, Tℸ∈Ci∣Ciisaq‐ROFαCsetandTℸ⊆Ci, so Tℸ=∩Ci∣Ciisaq‐ROFαCandTℸ⊆Ci=ClαTℸ.
Conversely, considerTℸ=ClαTℸ,
(19)Tℸ∈Ci∣Ciisaq‐ROFαCsetandTℸ⊆Ci.
Thus,Tℸ is a q-ROFα-closed set.Proposition 34.
In a q-ROFTSX,£˘τ, the following hold for q-ROF α-closure:
(1)
Clα0¯=0¯(2)
ClαTℸ is a q-ROαC in X,£˘τ for every q-ROFS Tℸ in X(3)
ClαTℸ⊆ClαR whenever Tℸ⊆R for every Tℸ and R in X(4)
ClαClαTℸ=ClαTℸ for every q-ROFS Tℸ in XProof.
(1)
The proof is obvious(2)
By preposition,Tℸ is q-ROFαC iff Tℸ=ClαTℸ we get ClαTℸ is a q-ROFαC for every Tℸ in X(3)
By same preposition, we getTℸ=ClαTℸ and R=ClαR. whenever Tℸ⊆R, we have ClαTℸ⊆ClαR(4)
LetTℸ be a q-ROFFS in X. We know that Tℸ=ClαTℸ
ClαTℸ=ClαClαTℸ. Thus, ClαClαTℸ=ClαTℸ for every Tℸ in X.Definition 35.
LetX,£˘Xτ and Y,£˘Yτ be q-ROFTSs. A mapping J:X⟶Y is named as q-ROFα-continuous (q-ROFαCN) if the inverse image of each q-ROFOS of Y is a q-ROFαO set in X.Theorem 36.
LetJ:X,£˘Xτ⟶Y,£˘Yτ be a mapping from a q-ROFTS X,£˘Xτ to a q-ROFTS Y,£˘Yτ. If J is q-ROFα-continues, then
(1)
JClIntClTℸ⊆ClJTℸ for all q-ROFS Tℸ in X(2)
ClIntJ−1W~⊆J−1ClW~ for all W~ in Y


## 6. q-ROF Connectedness

Here, we generalize the concept of IF-connected topological space, to the case of q-ROFTS.Definition 37.
Let A be a q-ROF subset inX,£˘Xτ.
(a)
If there exist q-ROFOSsUς and Vτ in X satisfying the following properties, then Aζ is called q-ROF Ci-disconnected i=1,2,3,4:
C1Aζ⊆Uς∪Vτ,Uς∩Vτ⊆Aζc,Aζ∩Uς≠0x,Aζ∩Vτ≠0x
C2Aζ⊆Uς∪Vτ,Aζ∩Uς∩Vτ≠0x,Aζ∩Uς≠0x,Aζ∩Vτ≠0x,
C3Aζ⊆Uς∪Vτ,Uς∩Vτ⊆Aζc,UςÚAζc,VτÚAζc,
C4Aζ⊆Uς∪Vτ,Aζ∩Uς∩Vτ≠0x,UςÚAζc,VτÚAζc.
(b)
Aζ is said to be q-ROF Ci-connected i=1,2,3,4, if Aζ is not q-ROF Ci-disconnected i=1,2,3,4
It is clear that in q-ROFTSs, we have the following implications:(20)C1‐connectedness⟶C1‐connectedness↓↓C3‐connectedness⟶C4‐connectedness.Example 1.
LetX=α˘1γ,α˘2γ,α˘3γ. Consider the following family of q-ROF sets:
(21)Tℸ1=α˘1γ,0.50,0.20,α˘2γ,0.50,0.40,α˘3γ,0.40,0.40,Tℸ2=α˘1γ,0.40,0.50,α˘2γ,0.60,0.30,α˘3γ,0.20,0.30,Tℸ3=α˘1γ,0.50,0.20,α˘2γ,0.60,0.30,α˘3γ,0.40,0.30,Tℸ4=α˘1γ,0.40,0.50,α˘2γ,0.50,0.40,α˘3γ,0.20,0.40.
Then,£˘τ=1X,0X,Tℸ1,Tℸ2,Tℸ3,Tℸ4 is a q-ROFTS on X, and consider the q-ROFS E given below:
(22)E=α˘1γ,0.60,0.20,α˘2γ,0.50,0.20,α˘3γ,0.40,0.30,in X. Then, E is q-ROF C1-connected, and E is also q-ROFC2-connected, q-ROFC3-connected, and q-ROFC4-connected.Example 2.
Consider the q-ROFTSX,£˘Xτ given in Example 1 and consider the q-ROFS F given below:
(23)F=α˘1γ,0.20,0.40,α˘2γ,0.30,0.60,α˘3γ,0.20,0.40.
One can check ifF is q-ROFC1-disconnected and hence not q-ROFC1-connected.Definition 38.
LetX,£˘Xτ be a q-ROFTS. If there exists a q-ROFOS Aζ in X such that 0x≠Aζ≠1x, then X is called q-ROF super disconnected. and X is called q-ROF fuzzy superconnected, if X is not q-ROF superdisconnected.Example 3.
LetX=α˘1γ,α˘2γ,α˘3γ. Consider the following family of q-ROF sets:
(24)Tℸ1=α˘1γ,0.40,0.50,α˘2γ,0.60,0.30,α˘3γ,0.20,0.30,Tℸ2=α˘1γ,0.50,0.20,α˘2γ,0.60,0.30,α˘3γ,0.40,0.30,Tℸ3=α˘1γ,0.40,0.50,α˘2γ,0.50,0.40,α˘3γ,0.20,0.4.Then,£˘τ=1X,0X,Tℸ1,Tℸ2,Tℸ3, is a q-ROFTS on X, and X,£˘τ q-ROF superconnected.Proposition 39.
In a q-ROFTSX,£˘Xτ, the following conditions are equivalent:
(i)
X is q-ROF superconnected(ii)
For each q-ROFOSAζ≠0x in X, we have ClAζ=0x(iii)
For each q-ROFCSAζ≠1x in X, we have IntAζ=0x(iv)
There exist no q-ROFOSAζ and Sς in X such that Aζ≠0X≠Sς and Aζ⊆Sςc(v)
There exist no q-ROFOSAζ and Sς in X such that Aζ≠0X≠Sς,Sς=ClAζC,Aζ=ClSςc(vi)
There exist no q-ROFCSAζ and Sς in X such that Aζ≠1x≠Sς(25)Sς=IntlAζc,Aζ=IntSςcProof (i=>ii): assume that there exists a q-ROFOSAζ≠0X such that ClAζ≠1x.
SinceSς=IntClAζ is a q-ROFOS in X, and this is a contradiction with the q-ROF super connectedness of X.
(ii⇒ iii): let Aζ≠1x be a q-ROFCS in X. If we take Sς=Aζc, then Sς=Aζc, then Sς is a q-ROFOS in X and Sς≠0x. Hence, ClSς=1x=>ClSςC=0.
(iii=> iv): let Aζ and Sς be q-ROFSs in X such that Aζ≠0X≠SςX≠Sς and Aζ⊆Sςc. Since Sςc is a q-ROFCS in X and Sς≠0x=>Sςc≠1x, we obtain IntSςc=0x. Now, we have 0x≠Aζ=IntAζ⊆IntSςC=0x, which is a contradiction.
iv=>v: obvious.
i=>v: let Aζ and Sς be q-ROFOSs in X such that Aζ≠0x≠Sς and ≠Sς and Sς=ClAζc,Aζ=ClSςc. No ClAζ=IntSςC=ClSςc=Aζ and Aζ≠0X,Aζ≠1X. This is a contradiction.
v=>i: obvious.
(v=>vi): let Aζ and Sς q-ROFCSs in X such that Aζ≠1x≠Sς, Sς=IntAζC,Aζ=IntSςC. Taking Uς=AζC and Vτ=SςC, Uς and Vτ become q-ROFOSs and Uς≠0x≠Vτ,ClUςc=Vτ and ClVτc=Uς. This is a contradiction.
vi=>v: this is similar to v=>vi.Definition 40.
LetX,£˘Xτ be a q-ROFTS.
(i)
X is said to be q-ROFC5-disconnected if there exists a q-ROFOS and q-ROFCS G such that G≠1X and G≠0X(ii)
X is said to be q-ROFC5-connected if it is not q-ROFC5-disconnectedExample 4.
LetX=1,2 and define the Pythagorean fuzzy subsets Aζ,Sς,C,D as follows.
LetX=α˘1γ,α˘2γ,α˘3γ. Consider the following family of q-ROF sets:
(26)Tℸ1=α˘1γ,0.40,0.30,α˘2γ,0.20,0.70,Tℸ2=α˘1γ,0.30,0.40,α˘2γ,0.70,0.20,Tℸ3=α˘1γ,0.30,0.40,α˘2γ,0.20,0.70,Tℸ4=α˘1γ,0.40,0.30,α˘2γ,0.70,0.20.
Then, family£˘τ=1X,0X,Tℸ1,Tℸ2,Tℸ3,Tℸ4 is a q-ROFTS on X and X,£˘τ is a q-ROFC5-disconnected, since Aζ is a nonzero q-ROFOS and q-ROFCS in X.Definition 41.
LetX,£˘Xτ be a q-ROFTS,
(i)
X is called q-ROF disconnected, if there exist q-ROFOSs Aζ≠0X and Sς≠0X such that AζUςSς=1X and Aζ∩Sς=0X(ii)
X is called q-ROF connected, if X is not q-ROF disconnectedProposition 42.
q-ROFC5-connectedness implies q-ROF connectedness.Proposition 43.
LetX,£˘1τ,Y,£˘2τ be two q-ROFTSs and let f:X⟶Y be a q-ROF continuous surjection. If X,£˘1τ is q-ROF connected,, then so is Y,£˘2τ.Proof.
On the contrary, suppose thatY,£˘2τ is q-ROF disconnected. Then, there exist q-ROFOSs Aζ≠0Y,Sς≠0Y in Y such that AζUςSς=1y,Aζ∩Sς=0Y. Now, we see that Uς=f1Aζ,Vτ=f1Sς are q-ROFOSs in X, since f is q-ROF continuous, From Aζ≠0Y, we get Uς=f1Aζ≠0X. Similarly, Vτ≠0X. Hence, AζUςSς=1y=>f1Aζf1Sς=f11y=1X=>UuςVτ=1X;Aζ∩Sς=0Y⇒f1Aζ∩f1Sς=f10Y=0X⇒Uς∩∩Sς=0Y⇒f1Aζ∩f1Sς=f10Y=0X⇒Uς∩.Corollary 44.
LetX,£˘1τ,Y,£˘2τ be two q-ROFTSs and let f:X⟶Y be a q-ROF continuous surjection. If X,£˘1τ is q-ROF C5-connected, then so is Y,£˘2τ.Definition 45.
A q-ROFTSX,£˘τ is said to be q-ROF strongly connected, if there exists nonzero q-ROFCSs Aζ and Sς such that μAζ+μSς≤1 and ϑAζ+ϑSς≥1.Proposition 46.
LetX,£˘1τ,Y,£˘2τ be two q-ROFTSs and let f:X⟶Y be a q-ROF continuous surjection. If X,£˘1τ is q-ROF strongly connected, then so is Y,£˘2τ.Proof.
This is analogous to proof of Proposition43. It is clear that in q-ROFTSs, q-ROF strongly connectedness does not imply q-ROF C5-connectedness, and the same is true for its converse.


## 7. q-ROF Weighted Product Model

The weighted product model (WPM) is a well-known and often used MCDM framework for analyzing a set of options in terms of a set of choice criteria. By multiplying a number of ratios, one for each choice criterion, each decision alternative is compared to the others. Each ratio is raised to the power of the related criterion’s relative weight.The fundamental task in general MCDM problems is to select one or even more options from a set of available alternatives based on numerous criteria. Assume that there aren alternatives and m criterion with the criteria weight vector under the constraInt, each component of WV is +ev and the sum of WV will be unit, for a specified MCDM problem in the q-ROF domain.Step 1.
Acquire the decision matrix from DMs, DMs represent the evaluation values of the alternativeLℷj in terms of the criterion Ciℸ by Tji=μji,νji. DMs give the decision matrix R=Tjin×m given as
(27)C1ℸC2ℸCmℸμ11,ν11μ12,ν12⋯⋯μ1m,ν1mμ21,ν21μ22,ν22⋯⋯μ2m,ν2m⋮⋮⋱⋱⋮μn1,νn1μn2,νn2⋯⋯μnm,νnm,Lℷ1,Lℷ2,⋮Lℷn.In this matrix, each elementTji=μji,νji is q-ROFNs. This means that if the option Lℷj fulfills the criteria Ciℸ, the grade will be the value μji, and if the alternative Lℷj fails to impress the characteristics Ciℸ, the grade will be the quantity νji.Step 2.
In this step, we normalize the matrixR=Tjin×m with linear approach. We classified the criteria into two types, benefit type (ξB) and cost type (ξC). The “linear normalization” for any i∈ξB is defined as follows:
(28)T¯ji=Tji⊘maxjTji,
wheremaxjTji is defined as maxjTji=maxjμji,minjνji.
And for anyi∈ξB “linear normalization” is defined as
(29)T¯ji=minjTjiTji,
whereminjTji is defined as minjTji=minjμji,maxjνji.
By the definition of SF, we can easily see thatminjTji and maxjTji satisfied the conditions of division operation. The decision matrix R=Tjin×m is transformed into the normalized matrix R¯=T¯jin×m and given as
(30)C1ℸC2ℸCmℸμ¯11,ν¯11μ¯12,ν¯12⋯⋯μ¯1m,ν¯1mμ¯21,ν¯21μ¯22,ν¯22⋯⋯μ¯2m,ν¯2m⋮⋮⋱⋱⋮μ¯n1,ν¯n1μ¯n2,ν¯n2⋯⋯μ¯nm,ν¯nm,Lℷ1Lℷ2⋮LℷnStep 3.
According to q-ROF-WPM, the “total relative importance” of alternativej, denoted as Kj, and defined as
(31)Kj=∏i=1m‍T¯jiwi.
Here, we use definition of taking power of any q-ROFN and product operation of q-ROFNs, as eachKj is also q-ROFN.Step 4.
Find the score of all evaluatedKj, which is defined in previous step.Step 5.
Rank all the alternatives as per SFs ofKj.
The pictorial view of the proposed algorithm is given in Figure1.Figure 1
Procedural steps of q-ROF WPM.


### 7.1. Application Related to Hierarchical Healthcare System Approach for COVID-19

The COVID-19 outbreak, which began in China and has spread rapidly throughout the world, has resulted in a rise in patient and casualty numbers. The countries have suffered huge losses not only in the medical industry but also in a range of other sectors as well. As a result, governments have been tasked with the responsibility of properly implementing a variety of remedies in their own jurisdictions. However, only a few countries benefit in part from the measures implemented, while others fail to benefit at all. In this context, it is vital to choose the most critical course of action that governments should take. To evaluate the various strategies employed by various authorities, a decision problem based on the preferences of several specialists using certain contradictory and disparate parameters should be evaluated. This decision procedure is viewed as an MCDM problem in this article, which also takes into account unpredictability. q-ROFSs are used to accomplish this goal, assisting decision-makers in evaluating across a greater space and dealing more effectively with competing knowledge.COVID-19 and other diseases are a serious threat to public health in Pakistan. Individuals seek treatment in grade III, class A hospitals because the facilities are much better than those found in other nearby hospitals. As a result, major hospitals are frequently overcrowded, with patient capacity much surpassing capacity. On the other side, small hospitals and clinics squander healthcare resources. Additional challenges for Pakistan’s medical system under such circumstances include understanding how to correctly distribute scarce healthcare resources and enhancing the inlet and outlet performances of the health care system. As depicted in Figure2, the primary health facility is responsible for the individual’s initial visit.Figure 2
Procedures for patients to seek for medical services.Implementing a hierarchical medical management program is regarded as a vital and effective technique for addressing the issue of insufficient and imbalanced healthcare assets, in which medical organizations of various sizes receive patients based on the severity and urgency of their diseases. In such a system, common ailments are treated at basic centers, with patients directed to more specialized facilities if their health necessitates it. Severe diseases should be treated in more advanced facilities. At the same time, when victims’ conditions improve, higher-level institutions might transfer them to lower-level facilities. As a result, determining the severity of the illness is an important action in this approach. With the increasing number of sufferers with respiratory illnesses, adopting an appropriate technique to separate patients under diverse circumstances into multiple stages of institutions is an effective way to make full use of restricted healthcare resources and cure more COVID-19 and other epidemics.The hierarchical medical system is introduced as an effective way to relieve the burden on the number of patients in Lahore Hospital, with the purpose of categorizing the difficulty of treatment based on the ailment. The numerous medical institutes’ degrees can then take on the various forms of diseases. The essential challenge in the hierarchical medical system is how to define various degrees of sickness. Patients with different conditions can use the hierarchical healthcare system to go to different levels of hospitals, rather than all patients flocking to grade III, class A institutions. In essence, categorizing the various degrees of disease is an important step in developing the hierarchical structure.The specific statement about the medical diagnosis problem is described as follows.If we consider there are five patient who have +ev COVID-19 test report, namely,Lℷ1, Lℷ2, Lℷ3, Lℷ4, Lℷ5, and Lℷ6. Doctors (in our proposed method, these are DMs) are to be appointed for evaluating the severe condition of the patients under the symptoms (criterion) given in Table 3.Table 3
Symptoms for COVID-19.
CriterionC1ℸChest painC2ℸBreathnessC3ℸSore throatC4ℸFeverC5ℸHeadacheC6ℸTaste & smellC7ℸRed or irritated eyesDoctors can assess the patient’s condition using the diagnostic parameters for the COVID-19. Patients might be assigned to multiple levels and kinds of institutions based on the severity and urgency of COVID-19. Patients with serious diseases should be hospitalized in grade III, class A hospitals, while those with milder symptoms should be treated in grade II facilities. Other frequent ailments are treatable at local hospitals.


#### 7.1.1. MCDM Process

(i)
Step 1Acquire the decision matrixR=Tjin×m from the DMs in the format of q-ROFNs given in Table 4.
(i)
Step 2Table 4
Assessment matrix acquired from DMs.
Lℷ1Lℷ2Lℷ3Lℷ4Lℷ4Lℷ6C1ℸ(0.425, 0.255)(0.352, 0.256)(0.359, 0.215)(0.313, 0.243)(0.241, 0.473)(0.312, 0.283)C2ℸ(0.765, 0.345)(0.236, 0.756)(0.159, 0.715)(0.365, 0.183)(0.453, 0.237)(0.478, 0.317)C3ℸ(0.142, 0.453)(0.189, 0.421)(0.345, 0.426)(0.153, 0.742)(0.135, 0.432)(0.341, 0.532)C4ℸ(0.354, 0.321)(0.331, 0.256)(0.359, 0.215)(0.145, 0.231)(0.431, 0.135)(0.426, 0.351)C5ℸ(0.725, 0.295)(0.652, 0.167)(0.942, 0.233)(0.813, 0.064)(0.653, 0.321)(0.673, 0.134)C6ℸ(0.413, 0.532)(0.152, 0.641)(0.532, 0.542)(0.753, 0.142)(0.341, 0.431)(0.241, 0.321)C7ℸ(0.341, 0.599)(0.572, 0.531)(0.653, 0.159)(0.173, 0.943)(0.378, 0.323)(0.173, 0.493)In this step, we normalize the matrixR=Tjin×m with linear approach. We classified the criteria into two types, benefit type (ξB) and cost type (ξC). Here, C2ℸ and C6ℸ belong to the ξC and other are in ξB. First, we find maxjTji and minjTji and maxjTji and minjTji are given in Table 5. After this, we find normalized matrix by using the Equations (1) and (2) given in Table 6.
(i)
Steps 3 and 4Table 5
maxjTji and minjTji values.
maxjTjiminjTjiC1ℸ(0.425, 0.215)(0.241, 0.473)C2ℸ(0.765, 0.183)(0.159, 0.756)C3ℸ(0.345, 0.421)(0.135, 0.742)C4ℸ(0.431, 0.135)(0.145, 0.351)C5ℸ(0.942, 0.064)(0.652, 0.295)C6ℸ(0.753, 0.142)(0.152, 0.641)C7ℸ(0.653, 0.159)(0.173, 0.943)Table 6
Normalized decision matrix.
Lℷ1Lℷ2Lℷ3Lℷ4Lℷ4Lℷ6C1ℸ(1.000, 0.213)(0.828, 0.216)(0.845, 0.000)(0.737, 0.192)(0.567, 0.468)(0.734, 0.256)C2ℸ(0.208, 0.750)(0.674, 0.000)(1.000, 0.545)(0.436, 0.756)(0.351, 0.755)(0.333, 0.752)C3ℸ(0.412, 0.324)(0.548, 0.000)(1.000, 0.199)(0.444, 0.728)(0.391, 0.245)(0.988, 0.474)C4ℸ(0.821, 0.318)(0.768, 0.251)(0.833, 0.206)(0.336, 0.223)(1.000, 0.000)(0.988, 0.349)C5ℸ(0.769, 0.295)(0.692, 0.166)(1.000, 0.233)(0.863, 0.000)(0.693, 0.321)(0.714, 0.132)C6ℸ(0.368, 0.557)(1.000, 0.000)(0.286, 0.548)(0.201, 0.641)(0.446, 0.611)(0.631, 0.632)C7ℸ(0.522, 0.598)(0.876, 0.530)(1.000, 0.000)(0.265, 0.943)(0.579, 0.318)(0.265, 0.492)According to q-ROF-WPM, the “total relative importance” of alternatives (Kj) by using Equation (3) and their score values have been calculated, given in Table 7.
(i)
Step 5Table 7
Total relative importance and score values.
Alternativesq-ROF WPM valuesScore valuesLℷ1(0.0638991, 0.2917500)-0.0072284Lℷ2(0.3642640, 0.0954248)0.01752330Lℷ3(0.2396050, 0.1870860)0.00207089Lℷ4(0.0219020, 0.5043240)-0.0646900Lℷ5(0.0851057, 0.2975280)-0.0077839Lℷ6(0.1558570, 0.3159770)-0.0093783Rank all the alternatives as per SFs ofKj. Final ranking will be
(32)Lℷ2≻Lℷ3≻Lℷ4≻Lℷ6≻Lℷ5≻Lℷ1.As a result, patientLℷ2 and Lℷ3 status is the most critical, and they must be hospitalized in a “grade III, class A medical center”. Patient Lℷ4 must be served in a local hospital in the meanwhile because his condition is not life-threatening. Patients Lℷ6, Lℷ5, and Lℷ1 may be transferred to various types of institutions based on the capacity of the ward.


## 7.1. Application Related to Hierarchical Healthcare System Approach for COVID-19

The COVID-19 outbreak, which began in China and has spread rapidly throughout the world, has resulted in a rise in patient and casualty numbers. The countries have suffered huge losses not only in the medical industry but also in a range of other sectors as well. As a result, governments have been tasked with the responsibility of properly implementing a variety of remedies in their own jurisdictions. However, only a few countries benefit in part from the measures implemented, while others fail to benefit at all. In this context, it is vital to choose the most critical course of action that governments should take. To evaluate the various strategies employed by various authorities, a decision problem based on the preferences of several specialists using certain contradictory and disparate parameters should be evaluated. This decision procedure is viewed as an MCDM problem in this article, which also takes into account unpredictability. q-ROFSs are used to accomplish this goal, assisting decision-makers in evaluating across a greater space and dealing more effectively with competing knowledge.COVID-19 and other diseases are a serious threat to public health in Pakistan. Individuals seek treatment in grade III, class A hospitals because the facilities are much better than those found in other nearby hospitals. As a result, major hospitals are frequently overcrowded, with patient capacity much surpassing capacity. On the other side, small hospitals and clinics squander healthcare resources. Additional challenges for Pakistan’s medical system under such circumstances include understanding how to correctly distribute scarce healthcare resources and enhancing the inlet and outlet performances of the health care system. As depicted in Figure2, the primary health facility is responsible for the individual’s initial visit.Figure 2
Procedures for patients to seek for medical services.Implementing a hierarchical medical management program is regarded as a vital and effective technique for addressing the issue of insufficient and imbalanced healthcare assets, in which medical organizations of various sizes receive patients based on the severity and urgency of their diseases. In such a system, common ailments are treated at basic centers, with patients directed to more specialized facilities if their health necessitates it. Severe diseases should be treated in more advanced facilities. At the same time, when victims’ conditions improve, higher-level institutions might transfer them to lower-level facilities. As a result, determining the severity of the illness is an important action in this approach. With the increasing number of sufferers with respiratory illnesses, adopting an appropriate technique to separate patients under diverse circumstances into multiple stages of institutions is an effective way to make full use of restricted healthcare resources and cure more COVID-19 and other epidemics.The hierarchical medical system is introduced as an effective way to relieve the burden on the number of patients in Lahore Hospital, with the purpose of categorizing the difficulty of treatment based on the ailment. The numerous medical institutes’ degrees can then take on the various forms of diseases. The essential challenge in the hierarchical medical system is how to define various degrees of sickness. Patients with different conditions can use the hierarchical healthcare system to go to different levels of hospitals, rather than all patients flocking to grade III, class A institutions. In essence, categorizing the various degrees of disease is an important step in developing the hierarchical structure.The specific statement about the medical diagnosis problem is described as follows.If we consider there are five patient who have +ev COVID-19 test report, namely,Lℷ1, Lℷ2, Lℷ3, Lℷ4, Lℷ5, and Lℷ6. Doctors (in our proposed method, these are DMs) are to be appointed for evaluating the severe condition of the patients under the symptoms (criterion) given in Table 3.Table 3
Symptoms for COVID-19.
CriterionC1ℸChest painC2ℸBreathnessC3ℸSore throatC4ℸFeverC5ℸHeadacheC6ℸTaste & smellC7ℸRed or irritated eyesDoctors can assess the patient’s condition using the diagnostic parameters for the COVID-19. Patients might be assigned to multiple levels and kinds of institutions based on the severity and urgency of COVID-19. Patients with serious diseases should be hospitalized in grade III, class A hospitals, while those with milder symptoms should be treated in grade II facilities. Other frequent ailments are treatable at local hospitals.


### 7.1.1. MCDM Process

(i)
Step 1Acquire the decision matrixR=Tjin×m from the DMs in the format of q-ROFNs given in Table 4.
(i)
Step 2Table 4
Assessment matrix acquired from DMs.
Lℷ1Lℷ2Lℷ3Lℷ4Lℷ4Lℷ6C1ℸ(0.425, 0.255)(0.352, 0.256)(0.359, 0.215)(0.313, 0.243)(0.241, 0.473)(0.312, 0.283)C2ℸ(0.765, 0.345)(0.236, 0.756)(0.159, 0.715)(0.365, 0.183)(0.453, 0.237)(0.478, 0.317)C3ℸ(0.142, 0.453)(0.189, 0.421)(0.345, 0.426)(0.153, 0.742)(0.135, 0.432)(0.341, 0.532)C4ℸ(0.354, 0.321)(0.331, 0.256)(0.359, 0.215)(0.145, 0.231)(0.431, 0.135)(0.426, 0.351)C5ℸ(0.725, 0.295)(0.652, 0.167)(0.942, 0.233)(0.813, 0.064)(0.653, 0.321)(0.673, 0.134)C6ℸ(0.413, 0.532)(0.152, 0.641)(0.532, 0.542)(0.753, 0.142)(0.341, 0.431)(0.241, 0.321)C7ℸ(0.341, 0.599)(0.572, 0.531)(0.653, 0.159)(0.173, 0.943)(0.378, 0.323)(0.173, 0.493)In this step, we normalize the matrixR=Tjin×m with linear approach. We classified the criteria into two types, benefit type (ξB) and cost type (ξC). Here, C2ℸ and C6ℸ belong to the ξC and other are in ξB. First, we find maxjTji and minjTji and maxjTji and minjTji are given in Table 5. After this, we find normalized matrix by using the Equations (1) and (2) given in Table 6.
(i)
Steps 3 and 4Table 5
maxjTji and minjTji values.
maxjTjiminjTjiC1ℸ(0.425, 0.215)(0.241, 0.473)C2ℸ(0.765, 0.183)(0.159, 0.756)C3ℸ(0.345, 0.421)(0.135, 0.742)C4ℸ(0.431, 0.135)(0.145, 0.351)C5ℸ(0.942, 0.064)(0.652, 0.295)C6ℸ(0.753, 0.142)(0.152, 0.641)C7ℸ(0.653, 0.159)(0.173, 0.943)Table 6
Normalized decision matrix.
Lℷ1Lℷ2Lℷ3Lℷ4Lℷ4Lℷ6C1ℸ(1.000, 0.213)(0.828, 0.216)(0.845, 0.000)(0.737, 0.192)(0.567, 0.468)(0.734, 0.256)C2ℸ(0.208, 0.750)(0.674, 0.000)(1.000, 0.545)(0.436, 0.756)(0.351, 0.755)(0.333, 0.752)C3ℸ(0.412, 0.324)(0.548, 0.000)(1.000, 0.199)(0.444, 0.728)(0.391, 0.245)(0.988, 0.474)C4ℸ(0.821, 0.318)(0.768, 0.251)(0.833, 0.206)(0.336, 0.223)(1.000, 0.000)(0.988, 0.349)C5ℸ(0.769, 0.295)(0.692, 0.166)(1.000, 0.233)(0.863, 0.000)(0.693, 0.321)(0.714, 0.132)C6ℸ(0.368, 0.557)(1.000, 0.000)(0.286, 0.548)(0.201, 0.641)(0.446, 0.611)(0.631, 0.632)C7ℸ(0.522, 0.598)(0.876, 0.530)(1.000, 0.000)(0.265, 0.943)(0.579, 0.318)(0.265, 0.492)According to q-ROF-WPM, the “total relative importance” of alternatives (Kj) by using Equation (3) and their score values have been calculated, given in Table 7.
(i)
Step 5Table 7
Total relative importance and score values.
Alternativesq-ROF WPM valuesScore valuesLℷ1(0.0638991, 0.2917500)-0.0072284Lℷ2(0.3642640, 0.0954248)0.01752330Lℷ3(0.2396050, 0.1870860)0.00207089Lℷ4(0.0219020, 0.5043240)-0.0646900Lℷ5(0.0851057, 0.2975280)-0.0077839Lℷ6(0.1558570, 0.3159770)-0.0093783Rank all the alternatives as per SFs ofKj. Final ranking will be
(32)Lℷ2≻Lℷ3≻Lℷ4≻Lℷ6≻Lℷ5≻Lℷ1.As a result, patientLℷ2 and Lℷ3 status is the most critical, and they must be hospitalized in a “grade III, class A medical center”. Patient Lℷ4 must be served in a local hospital in the meanwhile because his condition is not life-threatening. Patients Lℷ6, Lℷ5, and Lℷ1 may be transferred to various types of institutions based on the capacity of the ward.


## 7.1.1. MCDM Process

(i)
Step 1Acquire the decision matrixR=Tjin×m from the DMs in the format of q-ROFNs given in Table 4.
(i)
Step 2Table 4
Assessment matrix acquired from DMs.
Lℷ1Lℷ2Lℷ3Lℷ4Lℷ4Lℷ6C1ℸ(0.425, 0.255)(0.352, 0.256)(0.359, 0.215)(0.313, 0.243)(0.241, 0.473)(0.312, 0.283)C2ℸ(0.765, 0.345)(0.236, 0.756)(0.159, 0.715)(0.365, 0.183)(0.453, 0.237)(0.478, 0.317)C3ℸ(0.142, 0.453)(0.189, 0.421)(0.345, 0.426)(0.153, 0.742)(0.135, 0.432)(0.341, 0.532)C4ℸ(0.354, 0.321)(0.331, 0.256)(0.359, 0.215)(0.145, 0.231)(0.431, 0.135)(0.426, 0.351)C5ℸ(0.725, 0.295)(0.652, 0.167)(0.942, 0.233)(0.813, 0.064)(0.653, 0.321)(0.673, 0.134)C6ℸ(0.413, 0.532)(0.152, 0.641)(0.532, 0.542)(0.753, 0.142)(0.341, 0.431)(0.241, 0.321)C7ℸ(0.341, 0.599)(0.572, 0.531)(0.653, 0.159)(0.173, 0.943)(0.378, 0.323)(0.173, 0.493)In this step, we normalize the matrixR=Tjin×m with linear approach. We classified the criteria into two types, benefit type (ξB) and cost type (ξC). Here, C2ℸ and C6ℸ belong to the ξC and other are in ξB. First, we find maxjTji and minjTji and maxjTji and minjTji are given in Table 5. After this, we find normalized matrix by using the Equations (1) and (2) given in Table 6.
(i)
Steps 3 and 4Table 5
maxjTji and minjTji values.
maxjTjiminjTjiC1ℸ(0.425, 0.215)(0.241, 0.473)C2ℸ(0.765, 0.183)(0.159, 0.756)C3ℸ(0.345, 0.421)(0.135, 0.742)C4ℸ(0.431, 0.135)(0.145, 0.351)C5ℸ(0.942, 0.064)(0.652, 0.295)C6ℸ(0.753, 0.142)(0.152, 0.641)C7ℸ(0.653, 0.159)(0.173, 0.943)Table 6
Normalized decision matrix.
Lℷ1Lℷ2Lℷ3Lℷ4Lℷ4Lℷ6C1ℸ(1.000, 0.213)(0.828, 0.216)(0.845, 0.000)(0.737, 0.192)(0.567, 0.468)(0.734, 0.256)C2ℸ(0.208, 0.750)(0.674, 0.000)(1.000, 0.545)(0.436, 0.756)(0.351, 0.755)(0.333, 0.752)C3ℸ(0.412, 0.324)(0.548, 0.000)(1.000, 0.199)(0.444, 0.728)(0.391, 0.245)(0.988, 0.474)C4ℸ(0.821, 0.318)(0.768, 0.251)(0.833, 0.206)(0.336, 0.223)(1.000, 0.000)(0.988, 0.349)C5ℸ(0.769, 0.295)(0.692, 0.166)(1.000, 0.233)(0.863, 0.000)(0.693, 0.321)(0.714, 0.132)C6ℸ(0.368, 0.557)(1.000, 0.000)(0.286, 0.548)(0.201, 0.641)(0.446, 0.611)(0.631, 0.632)C7ℸ(0.522, 0.598)(0.876, 0.530)(1.000, 0.000)(0.265, 0.943)(0.579, 0.318)(0.265, 0.492)According to q-ROF-WPM, the “total relative importance” of alternatives (Kj) by using Equation (3) and their score values have been calculated, given in Table 7.
(i)
Step 5Table 7
Total relative importance and score values.
Alternativesq-ROF WPM valuesScore valuesLℷ1(0.0638991, 0.2917500)-0.0072284Lℷ2(0.3642640, 0.0954248)0.01752330Lℷ3(0.2396050, 0.1870860)0.00207089Lℷ4(0.0219020, 0.5043240)-0.0646900Lℷ5(0.0851057, 0.2975280)-0.0077839Lℷ6(0.1558570, 0.3159770)-0.0093783Rank all the alternatives as per SFs ofKj. Final ranking will be
(32)Lℷ2≻Lℷ3≻Lℷ4≻Lℷ6≻Lℷ5≻Lℷ1.As a result, patientLℷ2 and Lℷ3 status is the most critical, and they must be hospitalized in a “grade III, class A medical center”. Patient Lℷ4 must be served in a local hospital in the meanwhile because his condition is not life-threatening. Patients Lℷ6, Lℷ5, and Lℷ1 may be transferred to various types of institutions based on the capacity of the ward.


## 8. Conclusion

This paper introduces certain properties of q-ROF topology are significant results of q-ROF topological spaces. The novel concepts of q-ROF interior, q-ROF closure, and q-ROF boundary of a q-ROFS are defined and illustrated with the help of examples. The notions of q-ROF base, q-ROF subbase, q-ROF continuous mapping, q-ROF homeomorphism, q-ROF open mapping, and q-ROF closed mapping are introduced, as well as several essential proofs. Additionally, the unique idea of “q-rung orthopair fuzzy α-continuous mapping” is introduced and explored in relation to q-ROFTSs and “q-rung orthopair fuzzy connectedness.” We examine multiple relationships between different types of “q-rung orthopair fuzzy connectedness.” The WPM framework is a well-known and often used MCDM technique for analyzing a set of alternatives in terms of a set of selection criteria. By multiplying a number of ratios, one for each choice criterion, each decision alternative is compared to the others. Each ratio is multiplied by the corresponding criterion’s relative weight. The WPM model was extended to q-ROFSs in this study, and it was applied to a COVID-19 application important to hierarchical healthcare system. Implementing a hierarchical medical management system is viewed as a vital and effective strategy for addressing the issue of limited and uneven healthcare resources.



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*Source: 2094593-2022-04-13.xml*