# Covering Fuzzy Rough Sets via Variable Precision

Authors: Mohammed Atef; A. A. Azzam
Journal: Journal of Mathematics
 (2021)

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## Abstract

Lately, covering fuzzy rough sets via variable precision according to a fuzzyγ-neighborhood were established by Zhan et al. model. Also, Ma et al. gave the definition of complementary fuzzy γ-neighborhood with reflexivity. In a related context, we used the concepts by Ma et al. to construct three new kinds of covering-based variable precision fuzzy rough sets. Furthermore, we establish the relevant characteristics. Also, we study the relationships between Zhan’s model and our three models. Finally, we introduce a MADM approach to make a decision on a real problem.


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## Body

## 1. Introduction

Pawlak [1, 2] presented the classical definition of rough sets as a valuable mathematical method to deal with the vagueness and granularity of information systems and data processing. His theory and its generalizations since then have produced applications in different areas [3–13].One of the most elaborated generalizations of rough sets is potentially covering-based rough sets (CRS). There are several scholars working on CRS with various views in previous years, see, for more information, [14–22]. After that, the definition of a fuzzy β-neighborhood was seen by Ma [23] and the fuzzy complementary β-neighborhood by Yang and Hu [24]. Also, Yang and Hu [25, 26] introduced the concepts of fuzzy β-minimal description and fuzzy β-maximal description. They used these definitions to construct a fuzzy β covering approximation space (FβCAS). D’eer et al. [27] studied fuzzy neighborhoods based on fuzzy coverings.The definition of rough fuzzy sets and fuzzy rough sets was found by Dubois and Prade [28]. Different research studies on covering-based rough set and fuzzy rough set have recently been investigated [29–33].Variable precision rough sets’ (VPRSs) notion was obtained by Ziarko [34] and variable precision fuzzy rough sets (VPFRSs) were built by Zhao et al. [35]. In addition, the PROMETHEE II approach based on variable precision fuzzy rough sets was also proposed by Jiang et al. [36]. Different kinds of variable precision were further applied in various areas [37–40].One of the standard decision-making processes is TOPSIS (technique for order preference by similarity to an ideal solution). Yoon and Hwang [41] indicated that TOPSIS will solve the problem of multiattribute decision-making (MADM), where the aim is to obtain an object with the highest effect value (PIS) and the lowest effect value (NIS). There are several papers concerning TOPSIS published in different fields [42–51].Zhan et al. [52] put the definition of fuzzy γ-neighborhoods and also studied the covering-based variable precision fuzzy rough sets (CVPFRSs). Furthermore, Ma et al. [53] defined the complementary fuzzy γ-neighborhoods and presented another two types of neighborhoods by merging the fuzzy γ-neighborhoods and the complementary fuzzy γ-neighborhoods. Based on these kinds of fuzzy γ-neighborhoods, this paper proposes to introduce three new kinds of CVPFRSs models as a generalization of the Zhan et al. [52] method. Thus, we discuss some of their properties. The relationships between these methods are also established. Then, we present and explain the methodology to solve MADM problems. The paper structure is as follows. Section 2 gives the basic notions. Section 3 establishes three novel types of CVPFRSs. A decision-making process to explain the theoretical study is advanced in Section 4. We deduced in Section 5.


## 2. Preliminaries

We extend a short scanning of some concepts utilized over the paper in this section. In this article, we work onℛ-implication operator, in particular, ℐ=ℐℒ, i.e., ℐℒa,b=1∧1−a+b, and T=Tℒ, i.e., Tℒa,b=0∨a+b−1. To get more information, see [54].Definition 1.
(see [32, 55]). Suppose that Ω is the universal arbitrary set and ℱΩ is the fuzzy power set of Ω. We mean C^=C1^,C2^,…,Cm^, for Ci^∈ℱΩi=1,2,…,m, a fuzzy covering of Ω if ∪i=1mCi^a=1,∀a∈Ω.The notion of a fuzzyβ-covering was considered by Ma [23] via substituting 1 for the threshold β0<β≤1, i.e., we mean C^=C1^,C2^,…,Cm^, for Ci^∈ℱΩi=1,2,…,m, a fuzzy β-covering of Ω if ∪i=1mCi^a=1,∀a∈Ω,. In addition, Ω,C^ is referred to as the a fuzzy β-covering approximation space (briefly, FβCAS).Definition 2.
(see [23–26]). Assume that Ω,C^ is a FβCAS for some β∈0,1. For each a∈Ω, the fuzzy β-neighborhood (resp., the fuzzy complementary β-neighborhood and the fuzzy β-minimal description) of a is defined by(1)Naβ=∩Ci^∈C^:Ci^≥β,ℳaβb=Nbβa,ℳdC^βa=C^∈C^:C^a≥β∧∀D^∈C^∧D^a≥β∧D^⊆C^⟶D^=C^.Zhan et al. [52] presented a new definition called fuzzy γ-neighborhood with reflexivity. Using these definitions, they describe the notion of a CVPFRSs based on this definition and solve problems in MADM. The Ω,C^ pair produced by this neighborhood is called a fuzzy γ-covering approximation space (FγCAS for short) and C^ is called a fuzzy γ-covering [51].Definition 3.
(see [52]). Suppose that Ω,C^ is a FγCAS and C^=C1^,C2^,…,Cm^. For every a,b∈Ω, the fuzzy γ-neighborhood of a is as follows:(2)ℕ1γab=∧C^∈ℳdγaℐC^a,C^b.According to the above definition, we have the following result.Assume thatΩ,C^ is a FγCAS and the variable precision parameter is ξ∈0,1. For every a∈Ω and A^∈ℱΩ, the first model of a covering-based variable precision fuzzy rough lower and upper approximation which are denoted by 1-CVPFRLA and 1-CVPFRUA, respectively, are given as follows.Model 1:(3)O−1A^a=∧b∈Ωℐℕ1γab,ξ∨A^b,O+1A^a=∧b∈ΩTℕ1γab,ξ∧A^b.IfO−1A^≠O+1A^, then A^ is said to be a covering-based variable precision fuzzy rough set (briefly, 1-CVPFRS); otherwise it is definable [52].Ma et al. [53] generalizes Zhan’s model by introducing three kinds of neighborhoods as follows.Definition 4.
Assume thatΩ,C^ is a FγCAS. For any a,b∈Ω, three types of the fuzzy γ-neighborhoods of x are as follows:(1)
ℕ2γab=ℕ1γba(2)
ℕ3γab=ℕ1γab∧ℕ2γab(3)
ℕ4γab=ℕ1γab∨ℕ2γabTo explain the comparisons between these four kinds of neighborhoods, we give the next example.Example 1.
IfΩ,C^ is a FγCAS, Ω=a1,a2,a3,a4 and C^=C1^,C2^,C3^ is a three fuzzy γ covering on Ω set as follows:(4)C1^=0.89a1+0.88a2+0.79a3+0.56a4,C2^=0.77a1+0.85a2+0.67a3+0.84a4,C3^=0.69a1+0.78a2+0.93a3+0.63a4.
Letγ=0.74 and ℐ=ℐℒ. Then, the following values hold for each point on Ω for the three types of neighborhoods which are set in Definition 4.
Firstly, we compute the results forℕ2γar∀i∈1,2,3,4:(5)ℕ2γa1=1a1+0.91a2+0.76a3+0.93a4,ℕ2γa2=0.99a1+1a2+0.85a3+1a4,ℕ2γa3=0.9a1+0.82a2+1a3+0.83a4,ℕ2γa4=0.67a1+0.68a2+0.7a3+1a4.
Secondly, we compute the results forℕ3γar∀i∈1,2,3,4:(6)ℕ3γa1=1a1+0.91a2+0.76a3+0.67a4,ℕ3γa2=0.91a1+1a2+0.82x3+0.68a4,ℕ3γa3=0.76a1+0.82a2+1a3+0.7a4,ℕ3γa4=0.67a1+0.68a2+0.7a3+1a4.
Finally, we compute the results forℕ4γar∀i∈1,2,3,4:(7)ℕ4γa1=1a1+0.99a2+0.9a3+0.93a4,ℕ4γa2=0.99a1+1a2+0.85a3+1a4,ℕ4γa3=0.9a1+0.85a2+1a3+0.83a4,ℕ4γa4=0.93a1+1a2+0.83a3+1a4.From the above example, you can see the differences between these kinds of neighborhoods. Also, you can conclude thatℕ3γar is considered as the union between ℕ1γar and ℕ2γar. Furthermore, ℕ4γar is considered as the intersection between ℕ1γar and ℕ2γar. Therefore, it is easy to say that the third neighborhood ℕ3γar is better than others.


## 3. Three New Models of Covering Fuzzy Rough Sets via Variable Precision

Now, we are implementing three CVPFRSs’ models based on different kinds of a reflexive fuzzyγ-neighborhood.Assume thatΩ,C^ is a FγCAS and the parameter ξ∈0,1. For every a∈Ω and A^∈ℱΩ, three models of CVPFRSs are defined as follows.Model 2:(8)O−2A^a=∧b∈Ωℐℕ2γab,ξ∨A^b,O+2A^a=∨b∈ΩTℕ2γab,ξ∧A^b.Model 3:(9)O−3A^a=∧b∈Ωℐℕ3γab,ξ∨A^b,O+3A^a=∨b∈ΩTℕ3γab,ξ∧A^b.Model 4:(10)O−4A^a=∧b∈Ωℐℕ4γab,ξ∨A^b,O+4A^a=∨b∈ΩTℕ4γab,ξ∧A^b,where the three models are called the 2-CVPFRLA (resp., 3-CVPFRLA and 4-CVPFRLA) and the 2-CVPFRUA (resp., 3-CVPFRUA and 4-CVPFRUA), respectively.IfO−2A^ (resp., O−3A^, O−4A^) ≠O+2A^ (resp., O+3A^, O+4A^), then A^ is called a 2-CVPFRS (resp., 3-CVPFRS, 4-CVPFRS)), otherwise it is definable.The next example clarifies the above.Example 2.
(continued from Example1). Suppose that A^=0.58/a1+0.65/a2+0.77/a3+0.76/a4. Then, we have the following results for the above four models (i.e., 1-CVPFRS, 2-CVPFRS, 3-CVPFRS, and 4-CVPFRS).Model 1:(11)O−1A^=0.58a1+0.65a2+0.77a3+0.65a4,O+1A^=0.67a1+0.65a2+0.77a3+0.76a4.Model 2:(12)O−2A^=0.58a1+0.59a2+0.68a3+0.76a4,O+2A^=0.69a1+0.76a2+0.77a3+0.76a4.Model 3:(13)O−3A^=0.58a1+0.65a2+0.77a3+0.76a4,O+3A^=0.67a1+0.65a2+0.77a3+0.76a4.Model 4:(14)O−4A^=0.58a1+0.59a2+0.68a3+0.65a4,O+4A^=0.69a1+0.76a2+0.77a3+0.76a4.Remark 1.
From Example2, it is easy to see that(1)
O−2A^⊈O−1A^ and O−1A^⊈O−2A^(2)
O+2A^⊆O+1A^ and O+1A^⊈O+2A^The 1-CVPFRS model and the 2-CVPFRS model are clearly not capable of containing each other.Next, ifr = 1, we propose Theorem 1, and also, it meets in case r = 2, 3, 4.Theorem 1.
Assume thatΩ,C^ is a FγCAS and the parameter is ξ∈0,1. For any A^,ℬ^∈ℱΩ and ξ,ε∈0,1 (∀r∈1,2,3,4), the following properties hold:(1)
O−rA^c=O+rA^c(2)
O+rA^c=O−rA^c(3)
O−rΩ=Ω(4)
O+r∅=∅(5)
IfA^≤ℬ^, then O−rA^≤O−rℬ^(6)
IfA^≤ℬ^, then O+rA^≤O+rℬ^(7)
O−rA^∧ℬ^=O−rA^∧O−rℬ^(8)
O+rA^∧ℬ^≤O+rA^∧O+rℬ^(9)
O−rA^∨ℬ^≥O−rA^∨O−rℬ^(10)
O+rA^∨ℬ^=O+rA^∨O+rℬ^(11)
Ifξ≤ε, then Oξ−rA^≤Oε−rℬ^(12)
Ifξ≤ε, then Oε+rA^≤Oξ+rℬ^Proof.
We shall only prove (1), (3), (5), (7), (9), and (11).(1)
O−1A^c=∧b∈Ωℐℕ1γab,ξ∨NA^b=∧b∈ΩNTℕ1γab,Nξ∧A^b=N∧b∈ΩTℕ1γab,Nξ∧A^b=O+1A^c.(3)
Asℐ is left monotonic and Ωa=1, for every a∈Ω. Then, we have O−1Ω=∧b∈Ωℐℕ1γab,ξ∨Ωb=∧b∈Ωℐℕ1γab,Ωb=∧b∈Ωℐℕ1γab,1=1=Ωa.(5)
ℐ is right monotonic and for every a∈Ω. If A^≤ℬ^, then we get the following result. O−1A^a=∧b∈Ωℐℕ1γab,ξ∨A^b≤∧b∈Ωℐℕ1γab,ξ∨ℬ^b=O−1ℬ^a.(7)
ℐ is right monotonic and for all a∈Ω. Then, we have O−1A^∧ℬ^a=∧b∈Ωℐℕ1γab,ξ∨A^∧ℬ^b=∧b∈Ωℐℕ1γab,ξ∨A^b∧∧b∈Ωℐℕ1γab,ξ∨ℬ^b=O−1A^a∧O−1ℬ^a.(9)
Asℐ is right monotonic, A^≤A^∨ℬ^ and ℬ^≤A^∨ℬ^. Then, by (3), we obtain the following O−1A^≤O−1A^∨ℬ^ and O−1ℬ^≤O−1A^∨ℬ^. Thus, O−1A^∨ℬ^≥O−1A^∨O−1ℬ^.(11)
It is obtained directly from Definition of Model 1.The relationships between our models and the Zhan model in [52] are defined as follows. The following characteristics are clear and will be seen without proof.Proposition 1.
Assume thatΩ,C^ is a FγCAS of Ω. For every A^∈ℱΩ and ∀r∈1,2,3,4, we have the following properties:(1)
O−rA^⊆A^⊆O+rA^(2)
O−3A^=O−1A^∨O−2A^ and O−3A^=O+1A^∨O+2A^(3)
O−4A^=O−1A^∨O−2A^ and O+4A^=O+1A^∨O+2A^(4)
O−4A^=O−1A^∨O−3A^ and O+3A^=O+1A^∨O+4A^(5)
O−4A^=O−2A^∨O−3A^ and O+3A^=O+2A^∨O+4A^Proposition 2.
Suppose thatΩ,C^ is a FγCAS of Ω. For any A^∈ℱΩ and ∀a∈Ω.
Then,ℕ1γa=ℕ2γa⇔ either O−1A^=O−2A^=O−3A^=O−4A^ or O+1A^=O+2A^=O+3A^=O+4A^.


## 4. Decision-Making Approach to MADM Based on CVPFRS

This section introduces a new decision-making method to solve MADM problems by using CVPFRSs’ models.


### 4.1. Description and Process

In medicine, some types of drugs exist for the treatment of a disease, such as viral fever, dysentery, and chest problems. Assume thatΩ=a1,a2,…,an is n kinds of drugs (alternatives) and C^=C1^,C2^,…,Cm^ is m symptoms (attributes). According to the decision assessment, maker’s efficacy effect of the drug xi on the symptoms Cr^ (∀r=1,2,…,m and i=1,2,…,n) has been determined. Hence, Ω,C^ establishes an FγCAS. According to the presented work, in the next steps, we introduce a decision-making algorithm that finds the most effective drug.Step 1
: fuzzy decision matrixℱ of medicine evaluations set as below:(15)ℱ=ΩC^C1^C2^…Cm^a1d11d12…d1ma2d21d22…d2m⋮⋮⋮⋱⋮andn1dn2…dnm.Step 2
: calculate the lower and upper approximations ofCr^ and evaluate the lower and upper fuzzy decision-making matrix of medicine evaluations:(16)O−kCr^a=∧b∈Ωℐℕqγab,ξ∨Cr^b,O+kCr^a=∨b∈ΩTℕqγab,ξ∧Cr^b,∀k,q∈1,2,3,4.Step 3
: three deflections among the estimations of any two alternatives are called the deflections among drugsD^r, the lower deflections among drugs D^r⊖, and the upper deflections among drugs D^r⊕, respectively. These three deviations are computed as follows:(17)D^rai,aj=Cr^ai−Cr^aj=dir−djr,D^r⊖ai,aj=O−kCr^ai−O−kCr^aj,D^r⊕ai,aj=O+kCr^ai−O+kCr^aj,where k∈1,2,3,4.Step 4
: according to the three deviations, three drug preference values are referred to as drug preference valuesP^r, lower drug preference values P^r⊖, and upper drug preference values P^r⊕. These three values of choice among alternatives are therefore computed as follows:(18)P^rai,aj=0ifD^rai,aj<0,D^rai,ajαif0≤D^rai,aj≤α,1ifD^rai,aj>α,P^r⊖ai,aj=0ifD^r⊖ai,aj<0,D^r⊖ai,ajαif0≤D^r⊖ai,aj≤α,1ifD^r⊖ai,aj>α,P^r⊕ai,aj=0ifD^r⊕ai,aj<0,D^r⊕ai,ajαif0≤D^r⊕ai,aj≤α,1ifD^r⊕ai,aj>α.where α denotes the value of preference threshold.Step 5
: calculate three general drug preference indices, referred to as the overall drug preference indices for alternativesO^, the overall lower drug preference indices for alternatives O^⊖, and the overall upper drug preference indices for alternatives O^⊕, as follows:(19)O^ai,aj=∑r=1mWr^P^rai,aj,O^⊖ai,aj=∑r=1mWr^P^r⊖ai,aj,O^⊕ai,aj=∑r=1mWr^P^r⊕ai,aj,where W^=W1^,W2^,…,Wm^ is the vector of the weight of attributes such that ∑r=1mWr^=1 and Wr^∈0,1.Step 6
: three outflows of medicines are referred to as the outflows of alternativesℒ•, the lower outflows of alternatives ℒ•⊖, and the upper outflows of medicines ℒ•⊕. These flows are thus constructed as follows:(20)ℒ•ai,aj=∑r=1mO^ai,aj,ℒ•⊖ai,aj=∑r=1mO^⊖ai,aj,ℒ•⊕ai,aj=∑r=1mO^⊕ai,aj.We also create three input flows of drugs called the input flows of drugsℒ°, the lower input flows of drugs ℒ°⊖, and the upper input flows of drugs ℒ°⊕, respectively, as follows:(21)ℒ°ai,aj=∑r=1mO^ai,aj,ℒ°⊖ai,aj=∑r=1mO^⊖ai,aj,ℒ°⊕ai,aj=∑r=1mO^⊕ai,aj.Step 7
: the next formula computes the net flow of alternatives:(22)ℒai,aj=ℒ•+ℒ•⊖+ℒ•⊕−ℒ°+ℒ°⊖+ℒ°⊕,hence ranking the alternatives.In accordance with these steps, we include an algorithm based on Model 3 (3-CVPFRS) to solve decision-making issues. Algorithm1 summarizes the measures leading to it.Algorithm 1:Algorithm for the presented drug selections.
Input: theℱ fuzzy decision matrix, the α choice threshold, and the ξ parameter.Output: decision-aking.(1)Compute the lower and upper approximations by using Model 3(2)Compute three deflections among drugs (i.e.,D^r,D^r⊖, and D^r⊕)(3)Compute three preference values among drugs (i.e.,P^r,P^r⊖, and P^r⊕)(4)Compute three overall preference indices among drugs (i.e.,O^,O^⊖, and O^⊕)(5)Compute three leaving flows of drugs (i.e.,ℒ•,ℒ•⊖, and ℒ•⊕)(6)Compute three entering flows of drugs (i.e.,ℒ°,ℒ°⊖, and ℒ°⊕)(7)Compute the net flow of alternativesℒ(8)
Ranking the alternatives and obtain the decision


### 4.2. A Numerical Example

The steps aforementioned have been illustrated as follows with a check instance.Example 3.
Alternatives (medicines) construct a setΩ=a1,a2,…,a6 which are treated a diseases A^, and their symptoms are gathered by the attribute set C^= fever (C1^), cough (C2^), headache (C3^), stomachaches (C4^), dizzy giddy (C5^) . Here, the following steps of the algorithm mentioned are implemented.Step 1: over the set of symptoms, experts analyze each medication and present its conclusions with acceptable values set out in Table1.Step 2: let us fixℐℒ and Tℒ. Then, by 2-CVPFRS, we have the following:(23)ℕ3γa1=1a1+0.89a2+0.82a3+0.77a4+0.54a5+0.49a6,ℕ3γa2=0.89a1+1a2+0.83a3+0.74a4+0.65a5+0.55a6,ℕ3γa3=0.82a1+0.83a2+1a3+0.87a4+0.48a5+0.64a6,ℕ3γa4=0.77a1+0.74a2+0.87a3+1a4+0.73a5+0.51a6,ℕ3γa5=0.54a1+0.65a2+0.48a3+0.73a4+1a5+0.51a6,ℕ3γa6=0.49a1+0.55a2+0.64a3+0.51a4+0.51a5+1a6.Thus, the 3-CVPFRLA and 3-CVPFRUA are obtained as follows:(24)O−3C1^=0.92a1+0.91a2+0.86a3+0.73a4+0.56a5+1a6,O+3C1^=0.9a1+0.9a2+0.9a3+0.77a4+0.56a5+1a6,O−3C2^=0.54a1+0.65a2+0.48a3+0.61a4+1a5+0.51a6,O+3C2^=0.61a1+0.65a2+0.71a3+0.84a4+0.9a5+0.51a6,O−3C3^=0.95a1+0.94a2+0.82a3+0.95a4+0.82a5+0.49a6,O+3C3^=0.9a1+0.9a2+0.82a3+0.9a4+0.82a5+0.49a6,O−3C4^=0.48a1+0.52a2+0.57a3+0.44a4+0.52a5+0.93a6,O+3C4^=0.48a1+0.52a2+0.57a3+0.44a4+0.52a5+0.9a6,O−3C5^=0.28a1+0.25a2+0.42a3+0.51a4+0.24a5+0.28a6,O+3C5^=0.28a1+0.29a2+0.46a3+0.51a4+0.24a5+0.28a6.Steps 3 and 4: by using the previous data, it is easy to compute the three deflections among the estimations of any two alternatives and the three preference values among drugs.Step 5: from this information, we construct the values for three overall preference indices among drugs as set in Tables2–4.Step 6: the three leaving flows of drugs are calculated as follows:(25)ℒ•ai,aj=0.66895a1+0.9367a2+0.6456a3+1.411a4+0.8467a5+0.58875a6,ℒ•⊖ai,aj=0.61255a1+0.70305a2+0.6116a3+0.9596a4+0.9227a5+0.64875a6,ℒ•⊕ai,aj=0.45795a1+0.5091a2+0.7696a3+1.04425a4+0.5787a5+0.58775a6,ℒ°ai,aj=0.6219a1+0.3413a2+0.975a3+0.5367a4+1.0578a5+1.565a6,ℒ°⊖ai,aj=0.4894a1+0.456a2+0.65545a3+0.4811a4+0.9353a5+1.441a6,ℒ°⊕ai,aj=0.51755a1+0.4273a2+0.32625a3+0.3077a4+0.86605a5+1.5025a6.Step 7: the values of the net flow of alternatives are computed as follows:(26)ℒa1=0.1106,ℒa2=0.92425,ℒa3=0.0701,ℒa4=2.08935,ℒa5=−0.51105,ℒa6=−2.68325.Thus, the drugs’ ranking is as follows:(27)a4≥a2≥a1≥a3≥a5≥a6.Table 1
Decision-making matrixℱ with fuzzy information.
a1a2a3a4a5a6C1^0.92110.730.561C2^0.540.650.480.8410.51C3^10.940.8210.820.49C4^0.480.520.570.440.520.93C5^0.280.250.460.510.240.28Table 2
The overall preference indices among drugs.
O^ai,aja1a2a3a4a5a6a100.06220.10950.1950.16070.0945a20.042500.00830.0940.1160.0805a30.110.203500.41750.1740.07a40.05470.1880.077500.0670.1495a50.196750.2120.21030.244500.19425a60.2650.2710.240.460.3290Table 3
The overall lower preference indices among drugs.
O^⊖ai,aja1a2a3a4a5a6a100.04270.07250.1190.16070.0945a20.043100.07830.11260.1160.106a30.1010.1284500.14750.1740.1045a40.05470.07140.062500.1430.1495a50.172750.18950.15830.220500.19425a60.2410.2710.240.360.3290Table 4
The overall upper preference indices among drugs.
O^⊕ai,aja1a2a3a4a5a6a100.024350.1240.1710.10270.0955a2000.09830.15650.0840.0885a30.0390.03900.103750.0640.0805a40.03970.0460.05500.0330.134a50.141250.141250.18630.20400.18925a60.2380.25450.3060.4090.2950


### 4.3. Comparative Analysis

Here, we give the differences between the proposed method (i.e., 2-CVPFRS, 3-CVPFRS, and 4-CVPFRS) and the previous methods (i.e., Jiang’s method [36], PROMETHEE II [56], TOPSIS [57], WAA [58], OWA [59], and VIKOR [60]). Based on the sorting values of various decision-making approaches summarized in Table 5, our approach is therefore rational and effective.Table 5
Table for the ranking results for different methods.
Different modelsObtain a decisionOur modela4≥a2≥a1≥a3≥a5≥a6Jiang model [36]a4≥a2≥a3≥a1≥a5≥a6PROMETHEE II [56]a4≥a2≥a1=a3≥a5≥a6TOPSIS [57]a4≥a2≥a1≥a5≥a3≥a6WAA [58]a4≥a2≥a1=a3≥a5≥a6OWA [59]a4≥a3≥a2≥a1≥a6≥a5VIKOR [60]a4≥a3≥a1≥a2≥a5≥a6According to Table5, (1) the best position of our presented method, Jiang’s method [36], PROMETHEE II [56], TOPSIS [57], WAA [58], OWA [59], and VIKOR [60], is still consistent, that is, a4 is the best drug. Thus, our suggested approach is rational and efficient from the point of view of the decision outcome (the best option in the decision-making process). (2) Five drug classifications based on various methods are not precisely the same in [36], meaning that the best drug is equal (i.e., the drug a4). However, operating on the fuzzy γ-neighborhoods without reflexivity in Jiang’s [36] process, our methodology relies on the fuzzy γ-neighborhoods with reflexivity, which makes our approach proposed more rational and effective.The best way to clarify these results, you can see Figures1 and 2 which simplify the comparisons between the presented method and others.Figure 1
The presentation of lower approximations by using our models and the previous model.Figure 2
The presentation of upper approximations by using our models and the previous model.Figure1 explains the comparisons between the lower approximation for the four models (i.e., 1-CVPFRLA, 2-CVPFRLA, 3-CVPFRLA, and 4-CVPFRLA). This figure clarifies that the 3-CVPFRLA is larger than the others.Figure2 clarifies the comparisons between the upper approximation for the four models (i.e., 1-CVPFRUA, 2-CVPFRUA, 3-CVPFRUA, and 4-CVPFRUA). This figure shows that 3-CVPFRUA is smaller than the others.(1)
Two documented issues with fuzzyγ-neighborhoods are conquered by our presented methods. However, not all techniques can escape the obstacles that are not reflexive operators in fuzzy γ-neighborhoods and that the lower approximations they describe are not usually included in the corresponding upper approximation. For this reason, our approach for solving MADM issues is based on the CVPFRS models (i.e., 1-CVPFRS, 2-CVPFRS, and 3-CVPFRS). Moreover, by a comparative study in Section 4.3, by using fuzzy γ-neighborhoods, the proposed models are more freely used than the classical models.(2)
We can see in Section4 that our presented models (i.e., Algorithm 1) are elastic and scalable, whereby decision makers can use fuzzy γ-neighborhoods to pick various logical operators and parameters according to current status.(3)
We can easily observe from a comparative study that our models presented are superior to Jiang’s method [36], PROMETHEE II [56], TOPSIS [57], WAA [58], OWA [59], and VIKOR [60]. This implies that the innovative decision-making approaches suggested are rational and feasible.


## 4.1. Description and Process

In medicine, some types of drugs exist for the treatment of a disease, such as viral fever, dysentery, and chest problems. Assume thatΩ=a1,a2,…,an is n kinds of drugs (alternatives) and C^=C1^,C2^,…,Cm^ is m symptoms (attributes). According to the decision assessment, maker’s efficacy effect of the drug xi on the symptoms Cr^ (∀r=1,2,…,m and i=1,2,…,n) has been determined. Hence, Ω,C^ establishes an FγCAS. According to the presented work, in the next steps, we introduce a decision-making algorithm that finds the most effective drug.Step 1
: fuzzy decision matrixℱ of medicine evaluations set as below:(15)ℱ=ΩC^C1^C2^…Cm^a1d11d12…d1ma2d21d22…d2m⋮⋮⋮⋱⋮andn1dn2…dnm.Step 2
: calculate the lower and upper approximations ofCr^ and evaluate the lower and upper fuzzy decision-making matrix of medicine evaluations:(16)O−kCr^a=∧b∈Ωℐℕqγab,ξ∨Cr^b,O+kCr^a=∨b∈ΩTℕqγab,ξ∧Cr^b,∀k,q∈1,2,3,4.Step 3
: three deflections among the estimations of any two alternatives are called the deflections among drugsD^r, the lower deflections among drugs D^r⊖, and the upper deflections among drugs D^r⊕, respectively. These three deviations are computed as follows:(17)D^rai,aj=Cr^ai−Cr^aj=dir−djr,D^r⊖ai,aj=O−kCr^ai−O−kCr^aj,D^r⊕ai,aj=O+kCr^ai−O+kCr^aj,where k∈1,2,3,4.Step 4
: according to the three deviations, three drug preference values are referred to as drug preference valuesP^r, lower drug preference values P^r⊖, and upper drug preference values P^r⊕. These three values of choice among alternatives are therefore computed as follows:(18)P^rai,aj=0ifD^rai,aj<0,D^rai,ajαif0≤D^rai,aj≤α,1ifD^rai,aj>α,P^r⊖ai,aj=0ifD^r⊖ai,aj<0,D^r⊖ai,ajαif0≤D^r⊖ai,aj≤α,1ifD^r⊖ai,aj>α,P^r⊕ai,aj=0ifD^r⊕ai,aj<0,D^r⊕ai,ajαif0≤D^r⊕ai,aj≤α,1ifD^r⊕ai,aj>α.where α denotes the value of preference threshold.Step 5
: calculate three general drug preference indices, referred to as the overall drug preference indices for alternativesO^, the overall lower drug preference indices for alternatives O^⊖, and the overall upper drug preference indices for alternatives O^⊕, as follows:(19)O^ai,aj=∑r=1mWr^P^rai,aj,O^⊖ai,aj=∑r=1mWr^P^r⊖ai,aj,O^⊕ai,aj=∑r=1mWr^P^r⊕ai,aj,where W^=W1^,W2^,…,Wm^ is the vector of the weight of attributes such that ∑r=1mWr^=1 and Wr^∈0,1.Step 6
: three outflows of medicines are referred to as the outflows of alternativesℒ•, the lower outflows of alternatives ℒ•⊖, and the upper outflows of medicines ℒ•⊕. These flows are thus constructed as follows:(20)ℒ•ai,aj=∑r=1mO^ai,aj,ℒ•⊖ai,aj=∑r=1mO^⊖ai,aj,ℒ•⊕ai,aj=∑r=1mO^⊕ai,aj.We also create three input flows of drugs called the input flows of drugsℒ°, the lower input flows of drugs ℒ°⊖, and the upper input flows of drugs ℒ°⊕, respectively, as follows:(21)ℒ°ai,aj=∑r=1mO^ai,aj,ℒ°⊖ai,aj=∑r=1mO^⊖ai,aj,ℒ°⊕ai,aj=∑r=1mO^⊕ai,aj.Step 7
: the next formula computes the net flow of alternatives:(22)ℒai,aj=ℒ•+ℒ•⊖+ℒ•⊕−ℒ°+ℒ°⊖+ℒ°⊕,hence ranking the alternatives.In accordance with these steps, we include an algorithm based on Model 3 (3-CVPFRS) to solve decision-making issues. Algorithm1 summarizes the measures leading to it.Algorithm 1:Algorithm for the presented drug selections.
Input: theℱ fuzzy decision matrix, the α choice threshold, and the ξ parameter.Output: decision-aking.(1)Compute the lower and upper approximations by using Model 3(2)Compute three deflections among drugs (i.e.,D^r,D^r⊖, and D^r⊕)(3)Compute three preference values among drugs (i.e.,P^r,P^r⊖, and P^r⊕)(4)Compute three overall preference indices among drugs (i.e.,O^,O^⊖, and O^⊕)(5)Compute three leaving flows of drugs (i.e.,ℒ•,ℒ•⊖, and ℒ•⊕)(6)Compute three entering flows of drugs (i.e.,ℒ°,ℒ°⊖, and ℒ°⊕)(7)Compute the net flow of alternativesℒ(8)
Ranking the alternatives and obtain the decision


## 4.2. A Numerical Example

The steps aforementioned have been illustrated as follows with a check instance.Example 3.
Alternatives (medicines) construct a setΩ=a1,a2,…,a6 which are treated a diseases A^, and their symptoms are gathered by the attribute set C^= fever (C1^), cough (C2^), headache (C3^), stomachaches (C4^), dizzy giddy (C5^) . Here, the following steps of the algorithm mentioned are implemented.Step 1: over the set of symptoms, experts analyze each medication and present its conclusions with acceptable values set out in Table1.Step 2: let us fixℐℒ and Tℒ. Then, by 2-CVPFRS, we have the following:(23)ℕ3γa1=1a1+0.89a2+0.82a3+0.77a4+0.54a5+0.49a6,ℕ3γa2=0.89a1+1a2+0.83a3+0.74a4+0.65a5+0.55a6,ℕ3γa3=0.82a1+0.83a2+1a3+0.87a4+0.48a5+0.64a6,ℕ3γa4=0.77a1+0.74a2+0.87a3+1a4+0.73a5+0.51a6,ℕ3γa5=0.54a1+0.65a2+0.48a3+0.73a4+1a5+0.51a6,ℕ3γa6=0.49a1+0.55a2+0.64a3+0.51a4+0.51a5+1a6.Thus, the 3-CVPFRLA and 3-CVPFRUA are obtained as follows:(24)O−3C1^=0.92a1+0.91a2+0.86a3+0.73a4+0.56a5+1a6,O+3C1^=0.9a1+0.9a2+0.9a3+0.77a4+0.56a5+1a6,O−3C2^=0.54a1+0.65a2+0.48a3+0.61a4+1a5+0.51a6,O+3C2^=0.61a1+0.65a2+0.71a3+0.84a4+0.9a5+0.51a6,O−3C3^=0.95a1+0.94a2+0.82a3+0.95a4+0.82a5+0.49a6,O+3C3^=0.9a1+0.9a2+0.82a3+0.9a4+0.82a5+0.49a6,O−3C4^=0.48a1+0.52a2+0.57a3+0.44a4+0.52a5+0.93a6,O+3C4^=0.48a1+0.52a2+0.57a3+0.44a4+0.52a5+0.9a6,O−3C5^=0.28a1+0.25a2+0.42a3+0.51a4+0.24a5+0.28a6,O+3C5^=0.28a1+0.29a2+0.46a3+0.51a4+0.24a5+0.28a6.Steps 3 and 4: by using the previous data, it is easy to compute the three deflections among the estimations of any two alternatives and the three preference values among drugs.Step 5: from this information, we construct the values for three overall preference indices among drugs as set in Tables2–4.Step 6: the three leaving flows of drugs are calculated as follows:(25)ℒ•ai,aj=0.66895a1+0.9367a2+0.6456a3+1.411a4+0.8467a5+0.58875a6,ℒ•⊖ai,aj=0.61255a1+0.70305a2+0.6116a3+0.9596a4+0.9227a5+0.64875a6,ℒ•⊕ai,aj=0.45795a1+0.5091a2+0.7696a3+1.04425a4+0.5787a5+0.58775a6,ℒ°ai,aj=0.6219a1+0.3413a2+0.975a3+0.5367a4+1.0578a5+1.565a6,ℒ°⊖ai,aj=0.4894a1+0.456a2+0.65545a3+0.4811a4+0.9353a5+1.441a6,ℒ°⊕ai,aj=0.51755a1+0.4273a2+0.32625a3+0.3077a4+0.86605a5+1.5025a6.Step 7: the values of the net flow of alternatives are computed as follows:(26)ℒa1=0.1106,ℒa2=0.92425,ℒa3=0.0701,ℒa4=2.08935,ℒa5=−0.51105,ℒa6=−2.68325.Thus, the drugs’ ranking is as follows:(27)a4≥a2≥a1≥a3≥a5≥a6.Table 1
Decision-making matrixℱ with fuzzy information.
a1a2a3a4a5a6C1^0.92110.730.561C2^0.540.650.480.8410.51C3^10.940.8210.820.49C4^0.480.520.570.440.520.93C5^0.280.250.460.510.240.28Table 2
The overall preference indices among drugs.
O^ai,aja1a2a3a4a5a6a100.06220.10950.1950.16070.0945a20.042500.00830.0940.1160.0805a30.110.203500.41750.1740.07a40.05470.1880.077500.0670.1495a50.196750.2120.21030.244500.19425a60.2650.2710.240.460.3290Table 3
The overall lower preference indices among drugs.
O^⊖ai,aja1a2a3a4a5a6a100.04270.07250.1190.16070.0945a20.043100.07830.11260.1160.106a30.1010.1284500.14750.1740.1045a40.05470.07140.062500.1430.1495a50.172750.18950.15830.220500.19425a60.2410.2710.240.360.3290Table 4
The overall upper preference indices among drugs.
O^⊕ai,aja1a2a3a4a5a6a100.024350.1240.1710.10270.0955a2000.09830.15650.0840.0885a30.0390.03900.103750.0640.0805a40.03970.0460.05500.0330.134a50.141250.141250.18630.20400.18925a60.2380.25450.3060.4090.2950


## 4.3. Comparative Analysis

Here, we give the differences between the proposed method (i.e., 2-CVPFRS, 3-CVPFRS, and 4-CVPFRS) and the previous methods (i.e., Jiang’s method [36], PROMETHEE II [56], TOPSIS [57], WAA [58], OWA [59], and VIKOR [60]). Based on the sorting values of various decision-making approaches summarized in Table 5, our approach is therefore rational and effective.Table 5
Table for the ranking results for different methods.
Different modelsObtain a decisionOur modela4≥a2≥a1≥a3≥a5≥a6Jiang model [36]a4≥a2≥a3≥a1≥a5≥a6PROMETHEE II [56]a4≥a2≥a1=a3≥a5≥a6TOPSIS [57]a4≥a2≥a1≥a5≥a3≥a6WAA [58]a4≥a2≥a1=a3≥a5≥a6OWA [59]a4≥a3≥a2≥a1≥a6≥a5VIKOR [60]a4≥a3≥a1≥a2≥a5≥a6According to Table5, (1) the best position of our presented method, Jiang’s method [36], PROMETHEE II [56], TOPSIS [57], WAA [58], OWA [59], and VIKOR [60], is still consistent, that is, a4 is the best drug. Thus, our suggested approach is rational and efficient from the point of view of the decision outcome (the best option in the decision-making process). (2) Five drug classifications based on various methods are not precisely the same in [36], meaning that the best drug is equal (i.e., the drug a4). However, operating on the fuzzy γ-neighborhoods without reflexivity in Jiang’s [36] process, our methodology relies on the fuzzy γ-neighborhoods with reflexivity, which makes our approach proposed more rational and effective.The best way to clarify these results, you can see Figures1 and 2 which simplify the comparisons between the presented method and others.Figure 1
The presentation of lower approximations by using our models and the previous model.Figure 2
The presentation of upper approximations by using our models and the previous model.Figure1 explains the comparisons between the lower approximation for the four models (i.e., 1-CVPFRLA, 2-CVPFRLA, 3-CVPFRLA, and 4-CVPFRLA). This figure clarifies that the 3-CVPFRLA is larger than the others.Figure2 clarifies the comparisons between the upper approximation for the four models (i.e., 1-CVPFRUA, 2-CVPFRUA, 3-CVPFRUA, and 4-CVPFRUA). This figure shows that 3-CVPFRUA is smaller than the others.(1)
Two documented issues with fuzzyγ-neighborhoods are conquered by our presented methods. However, not all techniques can escape the obstacles that are not reflexive operators in fuzzy γ-neighborhoods and that the lower approximations they describe are not usually included in the corresponding upper approximation. For this reason, our approach for solving MADM issues is based on the CVPFRS models (i.e., 1-CVPFRS, 2-CVPFRS, and 3-CVPFRS). Moreover, by a comparative study in Section 4.3, by using fuzzy γ-neighborhoods, the proposed models are more freely used than the classical models.(2)
We can see in Section4 that our presented models (i.e., Algorithm 1) are elastic and scalable, whereby decision makers can use fuzzy γ-neighborhoods to pick various logical operators and parameters according to current status.(3)
We can easily observe from a comparative study that our models presented are superior to Jiang’s method [36], PROMETHEE II [56], TOPSIS [57], WAA [58], OWA [59], and VIKOR [60]. This implies that the innovative decision-making approaches suggested are rational and feasible.


## 5. Conclusion

As an improvement of the Zhan et al. method [52] and by using the concepts of neighborhoods by Ma et al. in [53], we then established new three kinds of covering-based variable precision fuzzy rough sets (i.e., 2-CVPFRS, 3-CVPFRS, and 4-CVPFRS). Relationship between these three paradigms and the paradigm of Zhan is also dealt with. This correlation indicates that the 3-CVPFRS is better than other models (i.e., the lower approximation is greater than others and the upper approximation is lower than others, as can be seen in Figures 1 and 2 based on Example 3). Finally, we set up an application for MADM to solve a problem. In the existing decision-making principles of interval-valued q-rung orthopair fuzzy sets [61] and linguistic interval-valued Pythagorean fuzzy sets [62], we hope this fuzzy rough concept can be incorporated.



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