Soft ζ-Rough Set and Its Applications in Decision Making of Coronavirus

M.A.El Safty,Samirah Al Zahrani,M.K.El-Bably and M.El Sayed

1Department of Mathematics and Statistics,College of Science,Taif University,Taif,21944,Saudi Arabia

2Department of Mathematics,Faculty of Science,Tanta University,Tanta,Egypt

3Department of Mathematics,College of Science and Arts,Najran University,Najran,66445,Saudi Arabia

Abstract: In this paper,we present a proposed method for generating a soft rough approximation as a modification and generalization of Zhaowen et al.approach.Comparisons were obtained between our approach and the previous study and also.Eventually,an application on Coronavirus (COVID-19)has been presented,illustrated using our proposed concept,and some influencing results for symptoms of Coronavirus patients have been deduced.Moreover,following these concepts,we construct an algorithm and apply it to a decision-making problem to demonstrate the applicability of our proposed approach.Finally,a proposed approach that competes with others has been obtained,as well as realistic results for patients with Coronavirus.Moreover,we used MATLAB programming to obtain the results;these results are consistent with those of the World Health Organization and an accurate proposal competing with the method of Zhaowen et al.has been studied.Therefore,it is recommended that our proposed concept be used in future decision making.

Keywords: Soft set;soft rough set;soft ζ rough set;COVID-19;intelligence discovery;decision making

1 Introduction

In 1999 Molodtsov [1] have introduced the soft set notion and progressing basics of this theory as a new diverse for modeling roughness and uncertainties.Diverse fields of applications of his approach were used in solving many practical problems in economics,engineering,social science,medical science...etc.Researchers have implementing various kinds of soft,rough and fuzzy sets (see [1-15]).

Often,the right decision making for many real-life issues is very difficult in our daily lives,which is highly essential for choosing the best solution to our discussions.Therefore,we have to consider various features in order to produce the highest practical solution to these problems.For this cause,we use the chosen mathematical instrument in the current article,namely soft rough set theory,in decision making.Decision making application was applied by Maji et al.[10,11].Using soft set approach and accordingly they expand this approach to fuzzy soft set theory in [13].Soft rough model was defined by [15].

Coronavirus emerged in 2019,in Wuhan,China.This virus is a new strain that has not been previously identified in humans.It was believed that Coronaviruses spread from dirty,dry surfaces,like automatic mucous membrane pollination in the nose,eyes,or mouth,reinforcing the importance of a clear understanding of the persistence of coronaviruses on inanimate surfaces [16].Therefore,two factors which are in contact with infected surfaces and encounters with infected viruses,affect the transmission.As a result,many scientific papers have been published and many researchers have studied this virus,such as ([16-22]).

As a generalization to Pawlak’s rough models [23].Based on this structure,they defined soft rough approximations,soft rough sets and some related concepts,such as ([23-27]).

The main objective of our belief is to have a certain influence on the continuous approximation of such basic mathematical principles and to provide a modern method for computational mathematics of real-life problems.In fact,it considers latest generalized soft,rough approximations,called softζ-rough approximations,are defined as a generalization to Zhaowen et al.[15]approximations and their properties are studied.We will prove that our approaches are more accurate and general from Zhaowen et al.approaches.The importance of the current approximations is not only that it is reducing or deleting the boundary regions,but also,it’s satisfying all properties of Pawlak’s rough sets without any restrictions.Comparisons between our method and the method of Zhaowen et al.are obtained.

Several examples are provided to illustrate the links between topologies and relationships of the soft set.Finally,we are added three applications.in making decisions regarding our strategy.One of them represents a beginning point for apply soft rough approach to solve the problem of Coronavirus contagion.At the end of the paper,we give two an algorithm which can be used to have a decision making for information system in terms of softζ-rough approximations.

The main programming for this paper is as follows:

Step 2: Compute the rough neighborhood from the information table.

Step 3: Compute the softζ-upper approximation,ζ-lower approximation andζ-boundary for the decision setM⊆.

Step 4: Remove a featurea1from the condition’s features (A) and then find the rough neighborhoodA-{a1}.

Step 5: Comparingζ-boundary for the decision setM⊆onA-{ai}with Step 3.

Step 6: Repeat Steps 4 and 5 for all attributes inA.

Step 7: Those attributes inAfor whichBNDεA-{ai}(M)/=BNDεA(M)forms the Core

Finally,we explain the importance of the proposed method in the medical sciences for application in decision-making problems.In fact,a medical application has been introduced in the decision-making process of COVID-19 Medical Diagnostic Information System with the algorithm.This application may help the world to reduce the spread of Coronavirus.

The paper is structured as follows: The basic concepts of the rough set and soft set were explored in section two and three.The implementation of COVID-19 for each subclass of attributes in the information systems and comparative analysis was presented in section four and five.Section six concludes and highlights future scope.

2 Preliminaries

In this section,we give some basic definitions and results that used in sequel are mentioned.

2.1 Pawlak Rough Set Theory

In 1982,Pawlak [23] introduced the theory of rough set as a new mathematical methodology or easy tools in order to deal with the vagueness in knowledge-based systems,information systems and data dissection.This theory has many applications in many fields that are used to process control,economics,such as medical diagnosis,chemistry,psychology,finance,marketing,biochemistry,environmental science,intelligent agents,image analysis,biology,conflict analysis,telecommunication,and other fields (See: [23-27],and the bibliography in these papers).

Definition 2.1[23] Assuming thatbe a set,and ℜ be an equivalence relation on,we usethe a collections of all equivalence classes of ℜ and [x]ℜ.It is indicated an equivalence class in ℜ containing an elementx∈Then,the pairit’s called an approximation s pace and for everyM⊆,we can define the lower and upper approximation ofMby(M)=respectively.

According to Pawlak’s definition,Mit’s called a rough set if

Proposition 2.1[23] Letφbe the empty set andMcbe the complement ofM⊆Pawlak’s rough sets have the next characteristic:

2.2 Soft Set Theory and Soft Rough Set

Let us recall now the soft set notion,which is a newly-emerging mathematical approach to vagueness.Letbe an initial universe of objects andEW(simply denoted byE) the set of certain parameters in relation to the objects in.Parameters are often attributing,characteristics,or properties of the objects in.Letdenote the power set of.Following the Definition 2.1 gives the concept of soft sets as follows.

Definition 2.3[12] Let=(F,A)be soft set over,then we define a binary relation onby

i.xℜf y⇔∃e∈E,{x,y}⊆f (a)for eachx,y∈,then ℜfis called the binary relation induced by(F,A)on.

ii.For eachx∈define successor neighborhood

Definition 2.4[15] Let=(F,A)be a soft set over.Then the pairis called a soft approximation space,we define the soft-lower and soft-upper approximations of any subsetM⊆respectively by the following two operations:

Proposition 2.2[15] Assuming that=(F,A)be a soft set uponanda soft approximation space.Then the soft-lower and-upper approximations ofM⊆:

Proposition 2.3[15] Let=(F,A)be a soft set overanda soft approximation space.Then:

Proposition 2.4[15] Let=(F,A)be a soft set overanda soft approximation space.Then for eachM⊆N:

Definition 2.5[2] Let=(F,A)be a soft set overanda soft approximation space.Then,is said to be a full soft set if=∪e∈AF(e).

It is clear that ifis a full soft set,then ∀x∈,∃e∈Asuch thatx∈F(e).

Proposition 2.5[15] Let=(F,A)be a full soft set overanda soft approximation space.Then,the following conditions are true:

3 Generalized Soft Rough Approximations

In this section,we define new generalized soft,rough approximations so-called softξ-rough approximations.The properties of the suggested approaches are superimposed.Relationship among our approaches and the previous one in Li et al.[15] are obtained.Many examples and counter examples are introduced.We will prove that our approach is a generalization to Pawlak [23] and Feng et al.[2] approaches.

Definition 3.1Let=(F,A)be a soft set overanda soft approximation space.Then,the softξ-lower,ξ-upper approximations of any subsetM⊆respectively by:In general,we refer toas softξ-rough approximations ofM⊆with respect to.

Defniition 3.2Letbe a soft approximation space andM⊆.Then,the softξ-positive,ξ-negative,ξ-boundary regions and theξ-accuracy of the softξ-approximations are defined respectively by:

The main goal of the following results is to introduce and studied the basic properties of softξ-rough approximations

Proposition 3.1Let=(F,A)be a soft set overanda soft approximation space.Then,the softξ-lower andξ-upper approximations ofM⊆satisfy the following properties:

Remark 3.1The inclusion in the above Proposition part (iv) is not instead of to equal the following example shows this remark.

Proof

Remark 3.2Note that the inclusion relations in Proposition 3.3 may be strict,as shown in Examples 3.1 and 3.2.

Table 1:Comparison between our approach and Zhaowen approach for some soft sets

From the above Tab.1,we deduce our method is better than Zhaowen method [15].Also,from the above Tab.1,we get the following Tab.2,

Table 2:Comparison between boundaries of Zhaowen method and our method

4 Relationship Between Our Method and the Pawlak Approximation

In this section,we shall compare between current method and the method of Pawlak.

Remark 4.1Propositions 3.1,3.2 and 3.4 represent one of the deviations between our approach and in [15] approach.By this proposition,our approximations satisfied most of Pawlak’s properties and then Tab.3,summarize these properties and give first comparison among our method and [15] method.We then list codes in Tab.3 to show whether these approximations satisfy the properties (L1) to (U9).In Tab.3,the number 1 denotes yes and 0 denotes not.

Table 3:Properties of soft rough and soft ξ-rough approximations

The main goal of the following results is to illustrate the relationship between soft rough approximations (that given by Wang et al.[16]) and soft pre-rough approximations (that given by our approach in the present paper).

Definition 4.2Assuming that=(F,A)be a full soft set upona soft approximation space andM⊆Then,we define the next four main types of softξ-rough sets:

The intuitive meaning of this classification is as follows:

—IfMis roughly softξ-definable,this suggests that we are able to decide about some elements ofthat they belong toM,and for someUelements,while,we can decide that they belong toMc,by using the knowledge available of the soft approximation space.

—IfMis internally softξ-indefinable,this suggests that we are able to decide about some elements ofthat they belong toMc,but we are incapable to decide for any element ofthat it belongs toM,by employing.

—IfMis externally softξ-indefinable,this suggests that we are able to decide about some elements ofwhich they belong toM,but we are incapable to decide,for any element ofthat it belongs toMc,by employing.

—IfMis totally softξ-indefinable,we are incapable to decide for any element of,whether it belongs toMorMc,by employing.

Theorem 4.2Letbe a softξ-approximation space andM⊆.Then:

i.IfMis roughly softξ-definable thenMis roughly soft-definable.

ii.IfMis internally softξ-definable thenMis internally soft-indefinable.

iii.IfMis externally softξ-definable thenMis externally soft-indefinable.

iv.IfMis totally softξ-indefinable thenMis totally soft-indefinable.

Proof: By Proposition 3.5,the proof is obvious.

Remark 4.2Theorem 4.2 represents a one of differences between soft rough approximations(that given by [15]) and softξ-rough approximations (that given by the present paper).Moreover,it illustrates the importance of our approaches in defining the sets,for example: ifMis totally soft-indefinable which impliesthat is,we are incapable to decide for any element ofwhether it belongs toMorMc.But,by using softξ-rough approximations,and thenMcan be roughly softξ-definable Which implies that we can decide on certain elements ofwhich they belong toM,and this meant while for some elements of,we able should decide that they are belong ofMc,Through using the information obtainable from the soft approximation space.

5 Medical Application via in Decision Making of Covid-19

In this section,we introduce a practical example as an application of our approaches in decision making for information system about infections of Coronavirus (COVID-19).In fact,we identify deciding factors of infections for COVID-19 in humans.In this model,we find gatherings,contact with injured people,and work in hospitals is the only deciding factors for infection transmission.We conclude that staying at home and not being in contact with humans protect and against viral infection with Coronavirus.According to [18] (Human-to-Human transmissions have been described with incubation times between 2-10 days,facilitating its spread via droplets,contaminated hands or surfaces).

Now,we introduce the proposed method;the application can be described as follows,where the objects as in [18]:denotes 10 listed patients,the features asA={a1,a2,...,a6}={Difficulty breathing,Chest pain,Temperature,Dry cough,Headache,Loss of taste or smell} and Decision Coronavirus {d},as follows in information was collected by the World Health Organization as well as through medical groups specializing in Coronavirus (COVID-19).Considering the following information system.

Table 4:Information’s decisions data set

Table 5:Consistent part of Tab.4

We note that,IND(A)/=IND(A-{a1}),...,thena1,a3,a4and a6are indispensable.Also,we geta2removed then we obtain IND(A)=IND(A-{a2}),and superfluous area2,a5.

Algorithm-i: Core attributes one removal based on the rough set with MATLAB program function [core]=core_attributes_one_removal(M);[pos]=object_reduction(M);s=find(pos==0);pos(s)=[];M=M(pos,:);core=[];M1=M;[nl,nc]=size(M1);for i=1:nc M1(:,i)=[];[pos]=object_reduction(M1);if isempty(find(pos==0))==0 core=[core;[i,length(find(pos==0))]];

Then,we get the removal of attributes as the next Tab.6,

Table 6:Consistent part of Tab.5

From Tab.6,we obtain the symptoms of every patient are:

Now,we can generate the following relation:

Algorithm-ii

Step 1:Input the soft set(F,E).

Step 2:Compute the right neighborhood for all elements of.

Step 3:Investigate the softζ-upper approximation,say,(M)and softζ-lower approximation,say,(M),for everyM⊆.According to Definition 3.1.

Step 4: Determine a boundary region,say,BNDζ(M)from Step 2,for everyM⊆.According to Definition 3.2.

Step 5: Calculate the accuracy of the approximation,say,μζ(M)by Step 2,for everyM⊆.According to Definition 3.1.

Step 6: Decide,exactly,rough sets and exact sets using Definition 3.2

We apply this relation for all features in the table to induce the successor neighborhoods as follows

Step 2:When the featuresa2-Chest pain is removed: Thus,the successor neighborhoods of each element inUof this relation are:

Step 4.When the attributea4-Dry cough is removed: Thus,the successor neighborhoods of each element inUof this relation are:

Step 5: When the featuresa5-Headache is removed: Therefore,the symptoms of every patient are:

Step 6:When the featuresa6-Loss of taste or smell is removed: The symptoms of every patient are:

Hence,the CORE is: {a3-Temperature},that is the impact factor for COVID-19 infection.

Case 2: (Patients are not infected with COVID-19)

The set of infected patients withN={x3,x4,x5,x9}.By made the same steps like as Case(1),we obtain the same results.From the CORE,we observed that Temperature is the key factor for COVID-19 infection.

6 Conclusion

In this paper,we introduced a modification as a generalization to soft rough set models that given by Zhaowen et al.,namely,Softζ-rough approximation and study of its properties.A comparison is made between our approach and other works such as Zhaowen et al.and Pawlak.Moreover,according to our results Theorem 3.1 and its corollaries,our approach is more accurate than Zhaowen et al.approach.In addition,we used our approach in applications to Coronavirus symptoms to illustrate the relevance of our proposal in this paper,in decision making to illustrate and also to compare our method with that Zhaowen et al.approach.Also,an algorithm was obtained for our method.According to the results in Section 3 (Theorem 3.1 and Corollary 3.1,3.2),we can say that our method is more accurate than Zhaowen et al.approach to decision making and hence this method is very useful in real-life applications.The importance of the current paper is not only that it introduces a new type of generalized soft rough set approximations,which increases the accuracy measure and reduces the boundary region of the sets which is the main aim of soft rough set,but also our approaches achieved the approximate Pawlak’rough set properties that never hold in Zhaowen et al.Finally,we provided an applied example in real-life problems to an illustrate the importance of our approaches to decision making.In fact,our proposal is helpful in deciding any future real-life problem.

Funding Statement: This research received funding from Taif University,Researchers Supporting and Project Number (TURSP-2020/207),Taif University,Taif,Saudi Arabia.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.