# On Some Types of Covering-Basedℐ,T-Fuzzy Rough Sets and Their Applications

Authors: Mohammed Atef; José Carlos R. Alcantud; Hussain AlSalman; Abdu Gumaei
Journal: Journal of Mathematics
 (2021)

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## Abstract

The notions of the fuzzyβ-minimal and maximal descriptions were established by Yang et al. (Yang and Hu, 2016 and 2019). Recently, Zhang et al. (Zhang et al. 2019) presented the fuzzy covering via ℐ,T-fuzzy rough set model (FCℐTFRS), and Jiang et al. (Jiang et al., in 2019) introduced the covering through variable precision ℐ,T-fuzzy rough sets (CVPℐTFRS). To generalize these models in (Jiang et al., 2019 and Zhang et al. 2019), that is, to improve the lower approximation and reduce the upper approximation, the present paper constructs eight novel models of an FCℐTFRS based on fuzzy β-minimal (maximal) descriptions. Characterizations of these models are discussed. Further, eight types of CVPℐTFRS are introduced, and we investigate the related properties. Relationships among these models are also proposed. Finally, we illustrate the above study with a numerical example that also describes its practical application.


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## Body

## 1. Introduction

In the 8th decade of the 20th century, the researchers found a new style of sets called rough set theory under the scientist Pawlak [1, 2]. This new theory helped many more authors to generalize and improve the studies in many areas such as engineering, economics, medicine, biology, speech recognition, chemistry, market research, data analysis, banking, materials science, networking, data mining, linguistics, variable precision rough sets, and other areas [3–15].A widely studied extension of RST is a covering through rough sets (CRSs).Pomykala [16, 17] generated a dual approximation for two pairs of operators. The definitions of granularity and neighborhood presented further insights in the field of approximation operators by Yao [18]. Using the assumption of incomplete knowledge, Bonikowski et al. [19] introduced a model of CRS, which depends on the of minimal description concept. Couso and Dubois [20] deliberated both pairs of operators as well. Other CRS models and relationships among them are the subjects of Zhu [21] and of Zhu and Wang [22–24]. The additional models of CRS were developed by Tsang et al. [25] and Xu and Zhang [26]. Moreover, Liu and Sai [27] compared the CRS model of Zhu and the CRS models of Xu and Zhang. Based on the concepts of neighborhood and complementary neighborhood, Ma [28] established some neighborhood-related forms covering rough sets in 2012. On the other direction, Dubois et al. [29] defined the fuzzy rough set (FRS) and rough fuzzy set (RFS) which assist the authors to a built a new way of CRS called fuzzy covering rough sets (FCRSs). There are many scholars working on this idea such as Atef and El Atik [30] presented some types of covering-based multigranulation fuzzy rough sets, Ma et al. [31] proposed new fuzzy rough coverings models using fuzzy α-neighborhood and presented some of its application, and the multigranulation fuzzy rough sets notion over two universes with its decision-making application studied by [32] and Zhan et al. [33] established the concept of a covering-based multigranulation fuzzy rough sets. In this direction, the meaning of fuzzy β-neighborhood was discovered by Ma [34] to create a fuzzy β-covering approximation space (FCAS). Then, Yang and Hu [35–37] investigated the notions of a fuzzy β-complementary neighborhood, a fuzzy β-minimal description, and a fuzzy β-maximal description to generate new models of FCAS. Deer et al. [38] studied fuzzy neighborhoods based on fuzzy coverings. Further, Atef and Azzam [39] and Jiang et al. [40] introduced covering fuzzy rough sets through variable precision. In [41–43], Hu and Wong have constructed generalized interval-valued fuzzy variable precision rough sets and generalized interval-valued fuzzy rough sets by using the operators of fuzzy logical. Meanwhile, L-fuzzy rough sets have been studied by the constructive method and axiomatic method [44–46]. Yeung et al. [47] have proposed a number of definitions for upper and lower fuzzy sets approximation operators by the means of arbitrary fuzzy relations and studied the relationship among them from the viewpoint of a constructive approach. In an axiomatic approach, they have characterized different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms. After that, Zhang et al. [48] set up the FCℐTFRS paradigms in 2019. In addition, Jiang et al. [49] established CVPℐTFRS with its applications to multiattribute-based decision-making. Furthermore, between 2019 and 2021, Zhan et al. introduced some types of decision-making problems [33, 50]. The later extension of the former model takes advantage of the generality provided by a fuzzy logical implication ℐ and a t-norm T. Both works provided applications of the models to multiattribute group decision-making.The emphasis of theFCℐTFRS models is on granularity, which here is understood in a fuzzy sense. Other settings require alternative models. For example, the Takagi–Sugeno fuzzy models [51] show a higher ability to capture nonlinear behavior. While the former insists on approximation operators, be they defined by equivalence relations, (fuzzy) coverings, or otherwise, and the latter relies on fuzzy rules. For this reason, the Takagi–Sugeno fuzzy models have also become popular in many other applied fields [52–56].Based on these recent developments and to extend and generalize the last studies by [48, 49] (i.e., to raise the lower approximation and lowering the previous works’ upper approximation), in this paper, we contribute to generating eight types of FCℐTFRS models through the descriptions of fuzzy β-minimal and fuzzy β-maximal notions and their complementary.Contrary to the inspiring cases of Zhan et al. [33, 50] and Ma et al. [57], our attention is restricted to one single source of granularity although we keep the general setting that allows for a t-norm and a fuzzy logical implication. The properties of these models as well as their relationships are introduced. Further, eight types of FCℐTFRSs and their related properties are considered. An application to a real issue clarifies the capacity to assist the practitioner for making the decisions.The structure and body of the paper are as follows. Our next section contains various technical preliminary concepts. Then, Section3 describes the eight paradigms of FCℐTFRSs that we propose for investigation. Section 4 introduces the corresponding eight types of CVPℐTFRSs. The relations between these different models are set in Section 5. Section 6 gives the applicable example of this theoretical study. Section 7 puts an end to this paper.


## 2. Preliminary Concepts

Throughout this section, we introduce several fundamental notions of fuzzy logical operators, CFRSs,FCℐTFRSs, and CVPℐTFRSs. In this paper, ℱΩ denotes the set of all fuzzy sets on a crisp set Ω. Also, we use Tℳx,y=x∧y and ℐMx,y=1−x∨y. Further, we say that N is involutive when NNx=x for every x∈0,1. The standard negator operator is defined as Nx=1−x, for any x∈0,1. For more data, see [58].Definition 1.
(see [59, 60]). Presume that Ω be a universe set. Then, Γ˜=C1˜,C2˜,…,Cm˜, where C˜i∈ℱΩi=1,2,…,m, is called a fuzzy covering (FC) of Ω if ∪i=1mC˜ix=1, ∀x∈Ω.As a generalization of FC, Ma [34] defined a fuzzy β-covering by replacing 1 with a threshold β0<β≤1. So, Γ˜ is said to be a fuzzy β-covering of Ω if ∪i=1mC˜ix≥β, ∀x∈Ω. Besides, Ω,Γ˜ is an FCAS.FCASs produce a number of theoretical elements for discussion.Definition 2.
(see [34–37]). Presume that Ω,Γ˜ be an FCAS for β∈0,1. For every x∈Ω, the fuzzy β-neighborhood consecutively, fuzzy complementary β-neighborhood, fuzzy β-minimal description, and fuzzy β-maximal description of x are defined as follows:(1)Nxβ˜=∩C˜i∈Γ˜:C˜i≥β,Mxβ˜y=Nyβ˜x,ℳd˜xβ=C∈Γ˜:Cx≥β∧∀D∈C∧Dx≥β∧D⊆C⟶D=C,ℳD˜xβ=C∈Γ˜:Cx≥β∧∀D∈C∧Dx≥β∧D⊇C⟶D=C.Definition 3.
(see [48]). Presume that Ω,Γ˜ be an FCAS for β∈0,1. For any x∈Ω and X^∈ℱΩ, define the first type of ℐ,T-fuzzy lower approximation (for short 1-ITFLA) (resp., 2-ITFLA, 3-ITFLA, and 4-ITFLA) and the first type of ℐ,T-fuzzy upper approximation (for short 1-ITFUA) (resp., 2-ITFUA, 3-ITFUA, and 4-ITFUA) of x as follows:(2)C˜1−X^x=∧y∈ΩℐNxβ˜y,X^y,C˜1+X^x=∨y∈ΩTNxβ˜y,X^y,C˜2−X^x=∧y∈ΩℐMxβ˜y,X^y,C˜2+X^x=∨y∈ΩTMxβ˜y,X^y,C˜3−X^x=∧y∈ΩℐNxβ˜y∧Mxβ˜y,X^y,C˜3+X^x=∨y∈ΩTNxβ˜y∧Mxβ˜y,X^y,C˜4−X^x=∧y∈ΩℐNxβ˜y∨Mxβ˜y,X^y,C˜4+X^x=∨y∈ΩTNxβ˜y∨Mxβ˜y,X^y.
IfC˜1−X^ (resp., C˜2−X^, C˜3−X^, and C˜4−X^≠C˜1+X^ (resp., C˜2+X^, C˜3+X^, and C˜4+X^, then we call X^ a 1-FCℐTFRS (resp., 2-FCℐTFRS, 3-FCℐTFRS, and 4-FCℐTFRS).Definition 4.
(see [49]). Presume that Ω,Γ˜ be an FCAS for β∈0,1. Then, for every X^∈ℱΩ, a parameter γ∈0,1. For any x∈Ω, define the first type of the variable precision ℐ,T-fuzzy lower approximation (for short I-VPITFLA) (resp., II-VPITFLA, III-VPITFLA, and IV-VPITFLA) and the first type of the variable precision ℐ,T-fuzzy upper approximation (for short I-VPITFUA) (resp., II-VPITFUA, III-VPITFUA, and IV-VPITFUA) as follows:(3)ℒIvpℐX^x=∧X^y≤γℐN˜xβy,γ∧∧X^y>γℐN˜xβy,X^y,UIvpTX^x=∨X^y≥1−γTN˜xβy,1−γ∨∨X^y<1−γTN˜xβy,X^y,ℒIIvpℐX^x=∧X^y≤γℐM˜xβy,γ∧∧X^y>γℐM˜xβy,X^y,UIIvpTX^x=∨X^y≥1−γTM˜xβy,1−γ∨∨X^y<1−γTM˜xβy,X^y,ℒIIIvpℐX^x=∧X^y≤γℐN˜xβy∧M˜xβy,γ∧∧X^y>γℐN˜xβy∧M˜xβy,X^y,UIIIvpTX^x=∨X^y≥1−γTN˜xβy∧M˜xβy,1−γ∨∨X^y<1−γTN˜xβy∧M˜xβy,X^y,ℒIVvpℐX^x=∧X^y≤γℐN˜xβy∨M˜xβy,γ∧∧X^y>γℐN˜xβy∨M˜xβy,X^y,UIVvpTX^x=∨X^y≥1−γTN˜xβy∨M˜xβy,1−γ∨∨X^y<1−γTN˜xβy∨M˜xβy,X^y.
IfℒIvpℐX^ (resp., ℒIIvpℐX^, ℒIIIvpℐX^, and ℒIVvpℐX^≠UIvpTX^ (resp., UIIvpTX^, UIIIvpTX^, and UIVvpTX^), then X^ is called a I-CVPℐTFRS (resp., II-CVPℐTFRS, III-CVPℐTFRS, and IV-CVPℐTFRS)); otherwise, it is variable precision fuzzy definable.


## 3. Eight Types of Covering-Basedℐ,T-Fuzzy Rough Sets

This section contains the definitions of eight types ofFCℐTFRS models. Also, the related characteristics are discussed. The study by Zhan et al. [50] is a direct inspiration by their utilization of a fuzzy logical implication and a t-norm in a similar setting. However, we refrain from using various sources of granularity, which opens the door for additional modelizations.


### 3.1. Four Kinds ofFCℐTFRS via Fuzzy β-Minimal Description

Definition 5.
Presume thatΩ,Γ˜ be an FCAS for a fixed β∈0,1. Select X^∈ℱΩ. For any x∈Ω, the 1ITFLA (resp., 2ITFLA, 3ITFLA, and 4ITFLA) and the 1ITFUA (resp., 2ITFUA, 3ITFUA, and 4ITFUA) are defined as in the following way:(4)ℒ1ℐX^x=∧y∈Ωℐ∨ℳd˜xβy,X^y,U1TX^x=∨y∈ΩT∨ℳd˜xβy,X^y,ℒ2ℐX^x=∧y∈Ωℐℳd˜xβy,X^y,U2TX^x=∨y∈ΩTℳd˜xβy,X^y,ℒ3ℐX^x=∧y∈Ωℐ∨ℳd˜xβy∩ℳd˜xβy,X^y,U3TX^x=∨y∈ΩT∨ℳd˜xβy∩ℳd˜xβy,X^y,ℒ4ℐX^x=∧y∈Ωℐ∨ℳd˜xβy∪ℳd˜xβy,X^y,U4TX^x=∨y∈ΩT∨ℳd˜xβy∪ℳd˜xβy,X^y.
Ifℒ1ℐX^ (resp., ℒ2ℐX^, ℒ3ℐX^, and ℒ1ℐX^≠U1TX^ (resp., U2TX^, U3TX^, and U4TX^), then X^ is a F1CℐTFRS (resp., F2CℐTFRS, F3CℐTFRS, and F4CℐTFRS). In other cases, we say that it is definable.Example 1.
LetΩ,Γ˜ be an FCAS, where β=0.5 and Ω=x1,…,x6 as in Table 1.
Let us fixℐm and Tℳ. Then, we have the following.
Then, we can calculate theℳd˜xiβ and ℳD˜xiβi=1,2,3,4,5,6, respectively. See Table 2 for the results of the corresponding computations.
Now, ifX^=0.6/x1+0.8/x2+0.5/x3+0.7/x4+0.8/x5+0.9/x6 then we introduce fuzzy β-neighborhood and the union of the fuzzy β-minimal description for Γ˜=C1˜ as n Tables 3 and 4.
So, we have the following results:(5)ℒ1ℐX^=0.5x1+0.6x2+0.5x3+0.5x4+0.5x5+0.5x6,U1TX^=0.7x1+0.8x2+0.7x3+0.8x4+0.8x5+0.7x6.
We proceed to calculate the fuzzyβ-complementary for ∨dℳ˜xiβ which is denoted by ℳd˜xiβ and is set in Table 5.
So, we obtain the following values:(6)ℒ2ℐX^=0.5x1+0.7x2+0.5x3+0.5x4+0.6x5+0.5x6,U2TX^=0.6x1+0.8x2+0.8x3+0.9x4+0.8x5+0.7x6.
Then, we calculate∨ℳd˜xβ∩ℳd˜xβ as shown in Table 6.
Thus, we have(7)ℒ3ℐX^=0.5x1+0.7x2+0.5x3+0.5x4+0.6x5+0.5x6,U3TX^=0.6x1+0.8x2+0.7x3+0.8x4+0.8x5+0.7x6.
Let us now calculate∨ℳd˜xβ∪ℳd˜xβ, which produces Table 7.
Then, we get the following outcomes:(8)ℒ4ℐX^=0.5x1+0.6x2+0.5x3+0.5x4+0.5x5+0.5x6,U4TX^=0.7x1+0.8x2+0.8x3+0.9x4+0.8x5+0.7x6.Table 1
A fuzzyβ-covering approximation space.
x1x2x3x4x5x6Γ˜10.60.10.20.50.40.6Γ˜20.50.10.80.60.30.7Γ˜30.10.80.30.40.20.1Γ˜40.30.60.40.80.80.3Γ˜50.50.30.80.90.50.7Table 2
N˜C˜0.5xi, ℳd˜xiβ, and ℳD˜xiβ for i=1,2,3,4,5,6.
x1x2x3x4x5x6N˜Γ˜0.5xiΓ1,Γ2,Γ5Γ3,Γ4Γ2,Γ5Γ1,Γ2,Γ4,Γ5Γ4,Γ5Γ1,Γ2,Γ5ℳd˜xi0.5Γ˜1,Γ˜2Γ˜3,Γ˜4Γ˜2Γ˜1,Γ˜2,Γ˜4,Γ˜5Γ˜4,Γ˜5Γ˜1,Γ˜2ℳd˜xi0.5Γ˜1,Γ˜5Γ˜3,Γ˜4Γ˜5Γ˜1,Γ˜2,Γ˜4,Γ˜5Γ˜4,Γ˜5Γ˜1,Γ˜5Table 3
Fuzzyβ-neighborhood of Γ˜.
x1x2x3x4x5x6N˜x10.50.50.10.20.50.30.6N˜x20.50.10.60.30.40.20.1N˜x30.50.50.10.80.60.30.7N˜x40.50.30.10.20.50.30.3N˜x50.50.30.30.40.80.50.3N˜x60.50.50.10.20.50.30.6Table 4
Values of∨ℳd˜xβy.
x1x2x3x4x5x6∨ℳd˜x10.50.60.10.80.60.40.7∨ℳd˜x20.50.30.80.40.80.80.3∨ℳd˜x30.50.50.10.80.60.30.7∨ℳd˜x40.50.60.60.80.90.80.7∨ℳd˜x50.50.50.60.80.90.80.7∨ℳd˜x60.50.60.10.80.90.40.7Table 5
Values ofℳd˜xβy.
x1x2x3x4x5x6ℳd˜x10.50.60.30.50.60.50.6ℳd˜x20.50.10.80.10.60.60.1ℳd˜x30.50.80.40.80.80.80.8ℳd˜x40.50.60.80.60.90.90.9ℳd˜x50.50.40.80.30.80.80.4ℳd˜x60.50.70.30.70.70.70.7Table 6
The result for∨ℳd˜xβ∩ℳd˜xβ.
x1x2x3x4x5x6∨ℳd˜x10.5∩ℳd˜x10.50.60.10.50.60.40.7∨ℳd˜x20.5∩ℳd˜x20.50.10.80.10.60.60.1∨ℳd˜x30.5∩ℳd˜x30.50.50.10.80.60.30.7∨ℳd˜x40.5∩ℳd˜x40.50.60.60.60.90.80.7∨ℳd˜x50.5∩ℳd˜x50.50.40.60.30.80.80.4∨ℳd˜x60.5∩ℳd˜x60.50.60.10.70.70.40.7Table 7
The result for∨ℳd˜xβ∪ℳd˜xβ.
x1x2x3x4x5x6∨ℳd˜x10.5∪ℳd˜x10.50.60.30.80.60.50.7∨ℳd˜x20.5∪ℳd˜x20.50.30.80.40.80.80.3∨ℳd˜x30.5∪ℳd˜x30.50.80.40.80.80.80.8∨ℳd˜x40.5∪ℳd˜x40.50.60.80.80.90.90.9∨ℳd˜x50.5∪ℳd˜x50.50.50.80.80.90.80.7∨ℳd˜x60.5∪ℳd˜x60.50.70.30.80.90.70.7Theorem1 satisfies all kinds of operators in Definition 5. Here, we prove it for the first kind only and the other kinds similarly.Theorem 1.
Presume thatΩ,Γ˜ be an FCAS for a given β∈0,1. Select X^∈ℱΩ and ∀x∈Ω. Then, the following affirmations hold true:(1)
ℒ1ℐX^=U1TX^cNcN and U1TX^=ℒ1ℐX^cNcN when ℐ is an S-implicator(2)
ℒ1ℐΩ=Ω and U1T∅=∅ when ℐ is a left monotonicity(3)
IfX^⊆Y^, then ℒ1ℐX^⊆ℒ1ℐY^ and U1TX^⊆U1TY^ when ℐ is a right monotonicity(4)
ℒ1ℐX^∩Y^=ℒ1ℐX^∩ℒ1ℐY^ and U1TX^∩Y^⊆U1TX^∩U1TY^ when ℐ is a right monotonicity(5)
ℒ1ℐX^∪Y^⊇ℒ1ℐX^∪ℒ1ℐY^ and U1TX^∪Y^=U1TX^∪U1TY^ when ℐ is a right monotonicityProof.
(1)
U1TX^cNcN=N∨y∈ΩT∨ℳd˜xβy,NX^y=N∨y∈ΩNℐ∨ℳd˜xβy,X^y=NN∧y∈Ωℐ∨ℳd˜xβy,X^y=∧y∈Ωℐ∨ℳd˜xβy,X^y=ℒ1ℐX^.By the same manner, we also obtainU1TX^=ℒ1ℐX^cNcN.(2)
Asℐ is a left monotonic, then we have(9)ℒ1ℐ=∧y∈Ωℐ∨ℳd˜xβy,Ωy=∧y∈Ωℐ∨ℳd˜xβy,1=1=ΩxAlso, we have(10)U1T∅=∧y∈ΩT∨ℳd˜xβy,∅y=∧y∈ΩT∨ℳd˜xβy,0=0=∅x.(3)
Asℐ is a right monotonic, if X^⊆Y^, then we have(11)ℒ1ℐX^=∧y∈Ωℐ∨ℳd˜xβy,X^y≤∧y∈Ωℐ∨ℳd˜xβy,Y^y=ℒ1ℐY^.Thus,ℒ1ℐX^⊆ℒ1ℐY^ holds, and similarly U1ℐX^⊆U1ℐY^ holds.(4)
Asℐ is a right monotonic, then we have(12)ℒ1ℐX^∩Y^x=∧y∈Ωℐ∨ℳd˜xβy,X^∧Y^y=∧y∈Ωℐ∨ℳd˜xβy,X^y∧∧y∈Ωℐ∨ℳd˜xβy,Y^y=ℒ1ℐX^x∧ℒ1ℐY^x=ℒ1ℐX^∩ℒ1ℐY^x.BecauseX^∩Y^⊆X^ and X^∩Y^⊆Y^, we deduce from (3), U1TX^∩Y^⊆U1TX^ and U1TX^∩Y^⊆U1TY^. Thus, U1TX^∩Y^⊆U1TX^∩U1TY^.(5)
Asℐ is a right monotonic, X^⊆X^∪Y^ and Y^⊆X^∪Y^; from (3), we have ℒ1ℐX^⊆ℒ1ℐX^∪Y^ and ℒ1ℐY^⊆ℒ1ℐX^∪Y^. Thus, ℒ1ℐX^∪Y^⊇ℒ1ℐX^∪ℒ1ℐY^.
Also, we obtain(13)U1ℐX^∪Y^x=∧y∈Ωℐ∨ℳd˜xβy,X^∪Y^y=∧y∈Ωℐ∨ℳd˜xβy,X^y∨∧y∈Ωℐ∨ℳd˜xβy,Y^y=U1ℐX^x∨U1ℐY^x=U1ℐX^∪U1ℐY^x.


### 3.2. Another FourFCℐTFRS Models through Fuzzy β-Maximal Description

Definition 6.
Presume thatΩ,Γ˜ be an FCAS for a fixed β∈0,1. Select X^∈ℱΩ. For each x∈Ω, define 5ITFLA (resp., 6ITFLA, 7ITFLA, and 8ITFLA) and 5ITFUA (resp., 6ITFUA, 7ITFUA, and 8ITFUA) as in the following way:(14)ℒ5ℐX^x=∧y∈Ωℐ∧ℳD˜xβy,X^y,U5TX^x=∨y∈ΩT∧ℳD˜xβy,X^y,ℒ6ℐX^x=∨i=1n∧y∈ΩℐℳD˜xβy,X^y,U6TX^x=∧i=1n∨y∈ΩTℳD˜xβy,X^y,ℒ7ℐX^x=∧y∈Ωℐ∧ℳD˜xβy∩ℳD˜xβy,X^y,U7TX^x=∨y∈ΩT∧ℳD˜xβy∩ℳD˜xβy,X^y,ℒ8ℐX^x=∧y∈Ωℐ∧ℳD˜xβy∪ℳD˜xβy,X^y,U8TX^x=∨y∈ΩT∧ℳD˜xβy∪ℳD˜xβy,X^y.
Ifℒ5ℐX^ (resp., ℒ6ℐX^, ℒ7ℐX^ and ℒ8ℐX^≠U5TX^ (resp., U6TX^, U7TX^ and U8TX^, then X^ is a F5CℐTFRS (resp., F6CℐTFRS, F7CℐTFRS and F8CℐTFRS)); otherwise, it is definable.Example 2.
(in continuation of Example1).
IfX^=0.6/x1+0.8/x2+0.5/x3+0.7/x4+0.8/x5+0.9/x6, then we build ∧ℳd˜xiβ∀i=1,2,3,4,5,6 in Table 8.
Hence, we have(15)ℒ5ℐX^=0.6x1+0.7x2+0.5x3+0.7x4+0.6x5+0.6x6,U5TX^=0.6x1+0.6x2+0.7x3+0.5x4+0.7x5+0.6x6.
Then, we compute the fuzzyβ-complementary for the intersection of the fuzzy β-maximal description, which is denoted by ℳD˜xiβ and is shown in Table 9.
Thus, we obtain the following:(16)ℒ6ℐX^=0.5x1+0.7x2+0.5x3+0.5x4+0.5x5+0.5x6,U6TX^=0.5x1+0.6x2+0.5x3+0.8x4+0.5x5+0.6x6.
Then, we calculate∧ℳd˜xβ∩ℳD˜xβ. The results are listed in Table 10.
So, the following results are obtained:(17)ℒ7ℐX^=0.6x1+0.7x2+0.5x3+0.7x4+0.6x5+0.6x6,U7TX^=0.5x1+0.6x2+0.5x3+0.5x4+0.5x5+0.6x6.
Then, we calculate∧ℳd˜xβ∪ℳD˜xβ listed in Table 11.
Therefore, we get the following outcomes:(18)ℒ8ℐX^=0.5x1+0.7x2+0.5x3+0.5x4+0.5x5+0.5x6,U8TX^=0.6x1+0.6x2+0.7x3+0.8x4+0.7x5+0.6x6.Table 8
Values of∧ℳD˜xβ.
x1x2x3x4x5x6∧ℳD˜x10.50.50.10.20.50.40.6∧ℳD˜x20.50.10.60.30.40.20.1∧ℳD˜x30.50.50.30.80.90.50.7∧ℳD˜x40.50.30.10.20.50.30.3∧ℳD˜x50.50.30.30.40.80.50.3∧ℳD˜x60.50.50.10.20.50.40.6Table 9
Values ofℳD˜xβy.
x1x2x3x4x5x6ℳD˜x10.50.50.10.50.30.30.5ℳD˜x20.50.10.60.30.10.30.1ℳD˜x30.50.20.30.80.20.40.2ℳD˜x40.50.50.40.90.50.80.5ℳD˜x50.50.40.20.50.30.50.4ℳD˜x60.50.60.10.70.30.30.6Table 10
The result for∧ℳD˜xβ∩ℳD˜xβ.
x1x2x3x4x5x6∧ℳD˜x10.5∩ℳD˜x10.50.50.10.20.30.30.5∧ℳD˜x20.5∩ℳD˜x20.50.10.60.30.10.20.1∧ℳD˜x30.5∩ℳD˜x30.50.20.30.80.20.40.2∧ℳD˜x40.5∩ℳD˜x40.50.30.10.20.50.30.3∧ℳD˜x50.5∩ℳD˜x50.50.30.20.40.30.50.3∧ℳD˜x60.5∩ℳD˜x60.50.50.10.20.30.30.6Table 11
The result for∧ℳD˜xβ∪ℳD˜xβ.
x1x2x3x4x5x6∧ℳD˜x10.5∪ℳD˜x10.50.50.10.50.50.40.6∧ℳD˜x20.5∪ℳD˜x20.50.10.60.30.40.30.1∧ℳD˜x30.5∪ℳD˜x30.50.50.30.80.90.50.7∧ℳD˜x40.5∪ℳD˜x40.50.50.40.90.50.80.5∧ℳD˜x50.5∪ℳD˜x50.50.40.30.50.80.50.4∧ℳD˜x60.5∪ℳD˜x60.50.60.10.70.50.40.6Theorem2 satisfies all types of operators in Definition 6. Next, we prove it for the fifth model only and the other models similarly.Theorem 2.
Presume thatΩ,Γ˜ be an FCAS for a given β∈0,1. Select X^∈ℱΩ and ∀x∈Ω. Then, the following affirmations hold true:(1)
ℒ5ℐX^=U5TX^cNcN and U5TX^=ℒ5ℐX^cNcN when ℐ is an S-implicator(2)
ℒ5ℐΩ=Ω and U5T∅=∅ when ℐ is a left monotonicity(3)
IfX^⊆Y^, then ℒ5ℐX^⊆ℒ5ℐY^ and U5TX^⊆U5TY^ when ℐ is a right monotonicity(4)
ℒ5ℐX^∩Y^=ℒ5ℐX^∩ℒ5ℐY^ and U5TX^∩Y^⊆U5TX^∩U5TY^ when ℐ is a right monotonicity(5)
ℒ5ℐX^∪Y^⊇ℒ5ℐX^∪ℒ5ℐY^ and U5TX^∪Y^=U5TX^∪U5TY^ when ℐ is a right monotonicityProof.
(1)
U5TX^cNcN=N∨y∈ΩT∧ℳd˜xβy,NX^y=N∨y∈ΩNℐ∧ℳd˜xβy,X^y=NN∧y∈Ωℐ∧ℳd˜xβy,X^y=∧y∈Ωℐ∧ℳd˜xβy,X^y=ℒ5ℐX^.By the same manner, we also obtainU5TX^=ℒ5ℐX^cNcN.(2)
Asℐ is a left monotonic, then we have(19)ℒ5ℐΩ=∧y∈Ωℐ∧ℳd˜xβy,Ωy=∧y∈Ωℐ∧ℳd˜xβy,1=1=Ωx.Also, we have(20)ℒ5ℐ∅=∧y∈Ωℐ∧ℳd˜xβy,∅y=∧y∈Ωℐ∧ℳd˜xβy,0=0=∅x.(3)
Asℐ is a right monotonic, if X^⊆Y^, then we have(21)ℒ5ℐX^=∧y∈Ωℐ∧ℳd˜xβy,X^y≤∧y∈Ωℐ∧ℳd˜xβy,Y^y=ℒ5ℐY^.Thus,ℒ5ℐX^⊆ℒ5ℐY^ holds, and similarly U5ℐX^⊆U5ℐY^ holds.(4)
Asℐ is a right monotonic. Then, we have(22)ℒ5ℐX^∩Y^x=∧y∈Ωℐ∧ℳd˜xβy,X^∧Y^y=∧y∈Ωℐ∧ℳd˜xβy,X^y∧∧y∈Ωℐ∧ℳd˜xβy,Y^y=ℒ5ℐX^x∧ℒ5ℐY^x=ℒ5ℐX^∩ℒ5ℐY^x.AsX^∩Y^⊆X^ and X^∩Y^⊆Y^, from (3), we obtain U5TX^∩Y^⊆U5TX^ and U5TX^∩Y^⊆U5TY^. Thus, U5TX^∩Y^⊆U5TX^∩U5TY^.(5)
Asℐ is a right monotonic, X^⊆X^∪Y^ and Y^⊆X^∪Y^, from (3), we have ℒ5ℐX^⊆ℒ5ℐX^∪Y^ and ℒ5ℐY^⊆ℒ5ℐX^∪Y^. Thus, ℒ5ℐX^∪Y^⊇ℒ5ℐX^∪ℒ5ℐY^.
Also, we obtain(23)U5ℐX^∪Y^x=∧y∈Ωℐ∧ℳd˜xβy,X^∪Y^y=∧y∈Ωℐ∧ℳd˜xβy,X^y∨∧y∈Ωℐ∧ℳd˜xβy,Y^y=U5ℐX^x∨U5ℐY^x=U5ℐX^∪U5ℐY^x.


## 3.1. Four Kinds ofFCℐTFRS via Fuzzy β-Minimal Description

Definition 5.
Presume thatΩ,Γ˜ be an FCAS for a fixed β∈0,1. Select X^∈ℱΩ. For any x∈Ω, the 1ITFLA (resp., 2ITFLA, 3ITFLA, and 4ITFLA) and the 1ITFUA (resp., 2ITFUA, 3ITFUA, and 4ITFUA) are defined as in the following way:(4)ℒ1ℐX^x=∧y∈Ωℐ∨ℳd˜xβy,X^y,U1TX^x=∨y∈ΩT∨ℳd˜xβy,X^y,ℒ2ℐX^x=∧y∈Ωℐℳd˜xβy,X^y,U2TX^x=∨y∈ΩTℳd˜xβy,X^y,ℒ3ℐX^x=∧y∈Ωℐ∨ℳd˜xβy∩ℳd˜xβy,X^y,U3TX^x=∨y∈ΩT∨ℳd˜xβy∩ℳd˜xβy,X^y,ℒ4ℐX^x=∧y∈Ωℐ∨ℳd˜xβy∪ℳd˜xβy,X^y,U4TX^x=∨y∈ΩT∨ℳd˜xβy∪ℳd˜xβy,X^y.
Ifℒ1ℐX^ (resp., ℒ2ℐX^, ℒ3ℐX^, and ℒ1ℐX^≠U1TX^ (resp., U2TX^, U3TX^, and U4TX^), then X^ is a F1CℐTFRS (resp., F2CℐTFRS, F3CℐTFRS, and F4CℐTFRS). In other cases, we say that it is definable.Example 1.
LetΩ,Γ˜ be an FCAS, where β=0.5 and Ω=x1,…,x6 as in Table 1.
Let us fixℐm and Tℳ. Then, we have the following.
Then, we can calculate theℳd˜xiβ and ℳD˜xiβi=1,2,3,4,5,6, respectively. See Table 2 for the results of the corresponding computations.
Now, ifX^=0.6/x1+0.8/x2+0.5/x3+0.7/x4+0.8/x5+0.9/x6 then we introduce fuzzy β-neighborhood and the union of the fuzzy β-minimal description for Γ˜=C1˜ as n Tables 3 and 4.
So, we have the following results:(5)ℒ1ℐX^=0.5x1+0.6x2+0.5x3+0.5x4+0.5x5+0.5x6,U1TX^=0.7x1+0.8x2+0.7x3+0.8x4+0.8x5+0.7x6.
We proceed to calculate the fuzzyβ-complementary for ∨dℳ˜xiβ which is denoted by ℳd˜xiβ and is set in Table 5.
So, we obtain the following values:(6)ℒ2ℐX^=0.5x1+0.7x2+0.5x3+0.5x4+0.6x5+0.5x6,U2TX^=0.6x1+0.8x2+0.8x3+0.9x4+0.8x5+0.7x6.
Then, we calculate∨ℳd˜xβ∩ℳd˜xβ as shown in Table 6.
Thus, we have(7)ℒ3ℐX^=0.5x1+0.7x2+0.5x3+0.5x4+0.6x5+0.5x6,U3TX^=0.6x1+0.8x2+0.7x3+0.8x4+0.8x5+0.7x6.
Let us now calculate∨ℳd˜xβ∪ℳd˜xβ, which produces Table 7.
Then, we get the following outcomes:(8)ℒ4ℐX^=0.5x1+0.6x2+0.5x3+0.5x4+0.5x5+0.5x6,U4TX^=0.7x1+0.8x2+0.8x3+0.9x4+0.8x5+0.7x6.Table 1
A fuzzyβ-covering approximation space.
x1x2x3x4x5x6Γ˜10.60.10.20.50.40.6Γ˜20.50.10.80.60.30.7Γ˜30.10.80.30.40.20.1Γ˜40.30.60.40.80.80.3Γ˜50.50.30.80.90.50.7Table 2
N˜C˜0.5xi, ℳd˜xiβ, and ℳD˜xiβ for i=1,2,3,4,5,6.
x1x2x3x4x5x6N˜Γ˜0.5xiΓ1,Γ2,Γ5Γ3,Γ4Γ2,Γ5Γ1,Γ2,Γ4,Γ5Γ4,Γ5Γ1,Γ2,Γ5ℳd˜xi0.5Γ˜1,Γ˜2Γ˜3,Γ˜4Γ˜2Γ˜1,Γ˜2,Γ˜4,Γ˜5Γ˜4,Γ˜5Γ˜1,Γ˜2ℳd˜xi0.5Γ˜1,Γ˜5Γ˜3,Γ˜4Γ˜5Γ˜1,Γ˜2,Γ˜4,Γ˜5Γ˜4,Γ˜5Γ˜1,Γ˜5Table 3
Fuzzyβ-neighborhood of Γ˜.
x1x2x3x4x5x6N˜x10.50.50.10.20.50.30.6N˜x20.50.10.60.30.40.20.1N˜x30.50.50.10.80.60.30.7N˜x40.50.30.10.20.50.30.3N˜x50.50.30.30.40.80.50.3N˜x60.50.50.10.20.50.30.6Table 4
Values of∨ℳd˜xβy.
x1x2x3x4x5x6∨ℳd˜x10.50.60.10.80.60.40.7∨ℳd˜x20.50.30.80.40.80.80.3∨ℳd˜x30.50.50.10.80.60.30.7∨ℳd˜x40.50.60.60.80.90.80.7∨ℳd˜x50.50.50.60.80.90.80.7∨ℳd˜x60.50.60.10.80.90.40.7Table 5
Values ofℳd˜xβy.
x1x2x3x4x5x6ℳd˜x10.50.60.30.50.60.50.6ℳd˜x20.50.10.80.10.60.60.1ℳd˜x30.50.80.40.80.80.80.8ℳd˜x40.50.60.80.60.90.90.9ℳd˜x50.50.40.80.30.80.80.4ℳd˜x60.50.70.30.70.70.70.7Table 6
The result for∨ℳd˜xβ∩ℳd˜xβ.
x1x2x3x4x5x6∨ℳd˜x10.5∩ℳd˜x10.50.60.10.50.60.40.7∨ℳd˜x20.5∩ℳd˜x20.50.10.80.10.60.60.1∨ℳd˜x30.5∩ℳd˜x30.50.50.10.80.60.30.7∨ℳd˜x40.5∩ℳd˜x40.50.60.60.60.90.80.7∨ℳd˜x50.5∩ℳd˜x50.50.40.60.30.80.80.4∨ℳd˜x60.5∩ℳd˜x60.50.60.10.70.70.40.7Table 7
The result for∨ℳd˜xβ∪ℳd˜xβ.
x1x2x3x4x5x6∨ℳd˜x10.5∪ℳd˜x10.50.60.30.80.60.50.7∨ℳd˜x20.5∪ℳd˜x20.50.30.80.40.80.80.3∨ℳd˜x30.5∪ℳd˜x30.50.80.40.80.80.80.8∨ℳd˜x40.5∪ℳd˜x40.50.60.80.80.90.90.9∨ℳd˜x50.5∪ℳd˜x50.50.50.80.80.90.80.7∨ℳd˜x60.5∪ℳd˜x60.50.70.30.80.90.70.7Theorem1 satisfies all kinds of operators in Definition 5. Here, we prove it for the first kind only and the other kinds similarly.Theorem 1.
Presume thatΩ,Γ˜ be an FCAS for a given β∈0,1. Select X^∈ℱΩ and ∀x∈Ω. Then, the following affirmations hold true:(1)
ℒ1ℐX^=U1TX^cNcN and U1TX^=ℒ1ℐX^cNcN when ℐ is an S-implicator(2)
ℒ1ℐΩ=Ω and U1T∅=∅ when ℐ is a left monotonicity(3)
IfX^⊆Y^, then ℒ1ℐX^⊆ℒ1ℐY^ and U1TX^⊆U1TY^ when ℐ is a right monotonicity(4)
ℒ1ℐX^∩Y^=ℒ1ℐX^∩ℒ1ℐY^ and U1TX^∩Y^⊆U1TX^∩U1TY^ when ℐ is a right monotonicity(5)
ℒ1ℐX^∪Y^⊇ℒ1ℐX^∪ℒ1ℐY^ and U1TX^∪Y^=U1TX^∪U1TY^ when ℐ is a right monotonicityProof.
(1)
U1TX^cNcN=N∨y∈ΩT∨ℳd˜xβy,NX^y=N∨y∈ΩNℐ∨ℳd˜xβy,X^y=NN∧y∈Ωℐ∨ℳd˜xβy,X^y=∧y∈Ωℐ∨ℳd˜xβy,X^y=ℒ1ℐX^.By the same manner, we also obtainU1TX^=ℒ1ℐX^cNcN.(2)
Asℐ is a left monotonic, then we have(9)ℒ1ℐ=∧y∈Ωℐ∨ℳd˜xβy,Ωy=∧y∈Ωℐ∨ℳd˜xβy,1=1=ΩxAlso, we have(10)U1T∅=∧y∈ΩT∨ℳd˜xβy,∅y=∧y∈ΩT∨ℳd˜xβy,0=0=∅x.(3)
Asℐ is a right monotonic, if X^⊆Y^, then we have(11)ℒ1ℐX^=∧y∈Ωℐ∨ℳd˜xβy,X^y≤∧y∈Ωℐ∨ℳd˜xβy,Y^y=ℒ1ℐY^.Thus,ℒ1ℐX^⊆ℒ1ℐY^ holds, and similarly U1ℐX^⊆U1ℐY^ holds.(4)
Asℐ is a right monotonic, then we have(12)ℒ1ℐX^∩Y^x=∧y∈Ωℐ∨ℳd˜xβy,X^∧Y^y=∧y∈Ωℐ∨ℳd˜xβy,X^y∧∧y∈Ωℐ∨ℳd˜xβy,Y^y=ℒ1ℐX^x∧ℒ1ℐY^x=ℒ1ℐX^∩ℒ1ℐY^x.BecauseX^∩Y^⊆X^ and X^∩Y^⊆Y^, we deduce from (3), U1TX^∩Y^⊆U1TX^ and U1TX^∩Y^⊆U1TY^. Thus, U1TX^∩Y^⊆U1TX^∩U1TY^.(5)
Asℐ is a right monotonic, X^⊆X^∪Y^ and Y^⊆X^∪Y^; from (3), we have ℒ1ℐX^⊆ℒ1ℐX^∪Y^ and ℒ1ℐY^⊆ℒ1ℐX^∪Y^. Thus, ℒ1ℐX^∪Y^⊇ℒ1ℐX^∪ℒ1ℐY^.
Also, we obtain(13)U1ℐX^∪Y^x=∧y∈Ωℐ∨ℳd˜xβy,X^∪Y^y=∧y∈Ωℐ∨ℳd˜xβy,X^y∨∧y∈Ωℐ∨ℳd˜xβy,Y^y=U1ℐX^x∨U1ℐY^x=U1ℐX^∪U1ℐY^x.


## 3.2. Another FourFCℐTFRS Models through Fuzzy β-Maximal Description

Definition 6.
Presume thatΩ,Γ˜ be an FCAS for a fixed β∈0,1. Select X^∈ℱΩ. For each x∈Ω, define 5ITFLA (resp., 6ITFLA, 7ITFLA, and 8ITFLA) and 5ITFUA (resp., 6ITFUA, 7ITFUA, and 8ITFUA) as in the following way:(14)ℒ5ℐX^x=∧y∈Ωℐ∧ℳD˜xβy,X^y,U5TX^x=∨y∈ΩT∧ℳD˜xβy,X^y,ℒ6ℐX^x=∨i=1n∧y∈ΩℐℳD˜xβy,X^y,U6TX^x=∧i=1n∨y∈ΩTℳD˜xβy,X^y,ℒ7ℐX^x=∧y∈Ωℐ∧ℳD˜xβy∩ℳD˜xβy,X^y,U7TX^x=∨y∈ΩT∧ℳD˜xβy∩ℳD˜xβy,X^y,ℒ8ℐX^x=∧y∈Ωℐ∧ℳD˜xβy∪ℳD˜xβy,X^y,U8TX^x=∨y∈ΩT∧ℳD˜xβy∪ℳD˜xβy,X^y.
Ifℒ5ℐX^ (resp., ℒ6ℐX^, ℒ7ℐX^ and ℒ8ℐX^≠U5TX^ (resp., U6TX^, U7TX^ and U8TX^, then X^ is a F5CℐTFRS (resp., F6CℐTFRS, F7CℐTFRS and F8CℐTFRS)); otherwise, it is definable.Example 2.
(in continuation of Example1).
IfX^=0.6/x1+0.8/x2+0.5/x3+0.7/x4+0.8/x5+0.9/x6, then we build ∧ℳd˜xiβ∀i=1,2,3,4,5,6 in Table 8.
Hence, we have(15)ℒ5ℐX^=0.6x1+0.7x2+0.5x3+0.7x4+0.6x5+0.6x6,U5TX^=0.6x1+0.6x2+0.7x3+0.5x4+0.7x5+0.6x6.
Then, we compute the fuzzyβ-complementary for the intersection of the fuzzy β-maximal description, which is denoted by ℳD˜xiβ and is shown in Table 9.
Thus, we obtain the following:(16)ℒ6ℐX^=0.5x1+0.7x2+0.5x3+0.5x4+0.5x5+0.5x6,U6TX^=0.5x1+0.6x2+0.5x3+0.8x4+0.5x5+0.6x6.
Then, we calculate∧ℳd˜xβ∩ℳD˜xβ. The results are listed in Table 10.
So, the following results are obtained:(17)ℒ7ℐX^=0.6x1+0.7x2+0.5x3+0.7x4+0.6x5+0.6x6,U7TX^=0.5x1+0.6x2+0.5x3+0.5x4+0.5x5+0.6x6.
Then, we calculate∧ℳd˜xβ∪ℳD˜xβ listed in Table 11.
Therefore, we get the following outcomes:(18)ℒ8ℐX^=0.5x1+0.7x2+0.5x3+0.5x4+0.5x5+0.5x6,U8TX^=0.6x1+0.6x2+0.7x3+0.8x4+0.7x5+0.6x6.Table 8
Values of∧ℳD˜xβ.
x1x2x3x4x5x6∧ℳD˜x10.50.50.10.20.50.40.6∧ℳD˜x20.50.10.60.30.40.20.1∧ℳD˜x30.50.50.30.80.90.50.7∧ℳD˜x40.50.30.10.20.50.30.3∧ℳD˜x50.50.30.30.40.80.50.3∧ℳD˜x60.50.50.10.20.50.40.6Table 9
Values ofℳD˜xβy.
x1x2x3x4x5x6ℳD˜x10.50.50.10.50.30.30.5ℳD˜x20.50.10.60.30.10.30.1ℳD˜x30.50.20.30.80.20.40.2ℳD˜x40.50.50.40.90.50.80.5ℳD˜x50.50.40.20.50.30.50.4ℳD˜x60.50.60.10.70.30.30.6Table 10
The result for∧ℳD˜xβ∩ℳD˜xβ.
x1x2x3x4x5x6∧ℳD˜x10.5∩ℳD˜x10.50.50.10.20.30.30.5∧ℳD˜x20.5∩ℳD˜x20.50.10.60.30.10.20.1∧ℳD˜x30.5∩ℳD˜x30.50.20.30.80.20.40.2∧ℳD˜x40.5∩ℳD˜x40.50.30.10.20.50.30.3∧ℳD˜x50.5∩ℳD˜x50.50.30.20.40.30.50.3∧ℳD˜x60.5∩ℳD˜x60.50.50.10.20.30.30.6Table 11
The result for∧ℳD˜xβ∪ℳD˜xβ.
x1x2x3x4x5x6∧ℳD˜x10.5∪ℳD˜x10.50.50.10.50.50.40.6∧ℳD˜x20.5∪ℳD˜x20.50.10.60.30.40.30.1∧ℳD˜x30.5∪ℳD˜x30.50.50.30.80.90.50.7∧ℳD˜x40.5∪ℳD˜x40.50.50.40.90.50.80.5∧ℳD˜x50.5∪ℳD˜x50.50.40.30.50.80.50.4∧ℳD˜x60.5∪ℳD˜x60.50.60.10.70.50.40.6Theorem2 satisfies all types of operators in Definition 6. Next, we prove it for the fifth model only and the other models similarly.Theorem 2.
Presume thatΩ,Γ˜ be an FCAS for a given β∈0,1. Select X^∈ℱΩ and ∀x∈Ω. Then, the following affirmations hold true:(1)
ℒ5ℐX^=U5TX^cNcN and U5TX^=ℒ5ℐX^cNcN when ℐ is an S-implicator(2)
ℒ5ℐΩ=Ω and U5T∅=∅ when ℐ is a left monotonicity(3)
IfX^⊆Y^, then ℒ5ℐX^⊆ℒ5ℐY^ and U5TX^⊆U5TY^ when ℐ is a right monotonicity(4)
ℒ5ℐX^∩Y^=ℒ5ℐX^∩ℒ5ℐY^ and U5TX^∩Y^⊆U5TX^∩U5TY^ when ℐ is a right monotonicity(5)
ℒ5ℐX^∪Y^⊇ℒ5ℐX^∪ℒ5ℐY^ and U5TX^∪Y^=U5TX^∪U5TY^ when ℐ is a right monotonicityProof.
(1)
U5TX^cNcN=N∨y∈ΩT∧ℳd˜xβy,NX^y=N∨y∈ΩNℐ∧ℳd˜xβy,X^y=NN∧y∈Ωℐ∧ℳd˜xβy,X^y=∧y∈Ωℐ∧ℳd˜xβy,X^y=ℒ5ℐX^.By the same manner, we also obtainU5TX^=ℒ5ℐX^cNcN.(2)
Asℐ is a left monotonic, then we have(19)ℒ5ℐΩ=∧y∈Ωℐ∧ℳd˜xβy,Ωy=∧y∈Ωℐ∧ℳd˜xβy,1=1=Ωx.Also, we have(20)ℒ5ℐ∅=∧y∈Ωℐ∧ℳd˜xβy,∅y=∧y∈Ωℐ∧ℳd˜xβy,0=0=∅x.(3)
Asℐ is a right monotonic, if X^⊆Y^, then we have(21)ℒ5ℐX^=∧y∈Ωℐ∧ℳd˜xβy,X^y≤∧y∈Ωℐ∧ℳd˜xβy,Y^y=ℒ5ℐY^.Thus,ℒ5ℐX^⊆ℒ5ℐY^ holds, and similarly U5ℐX^⊆U5ℐY^ holds.(4)
Asℐ is a right monotonic. Then, we have(22)ℒ5ℐX^∩Y^x=∧y∈Ωℐ∧ℳd˜xβy,X^∧Y^y=∧y∈Ωℐ∧ℳd˜xβy,X^y∧∧y∈Ωℐ∧ℳd˜xβy,Y^y=ℒ5ℐX^x∧ℒ5ℐY^x=ℒ5ℐX^∩ℒ5ℐY^x.AsX^∩Y^⊆X^ and X^∩Y^⊆Y^, from (3), we obtain U5TX^∩Y^⊆U5TX^ and U5TX^∩Y^⊆U5TY^. Thus, U5TX^∩Y^⊆U5TX^∩U5TY^.(5)
Asℐ is a right monotonic, X^⊆X^∪Y^ and Y^⊆X^∪Y^, from (3), we have ℒ5ℐX^⊆ℒ5ℐX^∪Y^ and ℒ5ℐY^⊆ℒ5ℐX^∪Y^. Thus, ℒ5ℐX^∪Y^⊇ℒ5ℐX^∪ℒ5ℐY^.
Also, we obtain(23)U5ℐX^∪Y^x=∧y∈Ωℐ∧ℳd˜xβy,X^∪Y^y=∧y∈Ωℐ∧ℳd˜xβy,X^y∨∧y∈Ωℐ∧ℳd˜xβy,Y^y=U5ℐX^x∨U5ℐY^x=U5ℐX^∪U5ℐY^x.


## 4. Eight Types of Covering-Based Variable Precisionℐ,T-Fuzzy Rough Sets

Based on Section3, this section establishes eight new CVPℐTFRS models. We discuss some of their features too.Definition 7.
LetΩ,Γ˜ be an FCAS and for a given β∈0,1. Let us fix a parameter γ∈0,1. Define 1-VPITFLA (resp., 2-VPITFLA, 3-VPITFLA, 4-VPITFLA, 5-VPITFLA, 6-VPITFLA, 7-VPITFLA, and 8-VPITFLA) and 1-VPITFUA (resp., 2-VPITFUA, 3-VPITFUA, 4-VPITFUA, 5-VPITFUA, 6-VPITFUA, 7-VPITFUA, and 8-VPITFUA), for every X^∈ℱΩ and x∈Ω as follows:(24)L1vpℐX^x=∧X^y≤γℐ∨ℳd˜xβy,γ∧∧X^y>γℐ∨ℳd˜xβy,X^y,U1vpTX^x=∨X^y≥1−γT∨ℳd˜xβy,1−γ∨∨X^y<1−γT∨ℳd˜xβy,X^y,L2vpℐX^x=∧X^y≤γℐℳd˜xβy,γ∧∧X^y>γℐℳd˜xβy,X^y,U2vpTX^x=∨X^y≥1−γTℳd˜xβy,1−γ∨∨X^y<1−γTℳd˜xβy,X^y,L3vpℐX^x=∧X^y≤γℐ∨ℳd˜xβy∩ℳd˜xβy,γ∧∧X^y>γℐ∨ℳd˜xβy∩ℳd˜xβy,X^y,U3vpTX^x=∨X^y≥1−γT∨ℳd˜xβy∩ℳd˜xβy,1−γ∨∨X^y<1−γT∨ℳd˜xβy∩ℳd˜xβy,X^y,L4vpℐX^x=∧X^y≤γℐ∨ℳd˜xβy∪ℳd˜xβy,γ∧∧X^y>γℐ∨ℳd˜xβy∪ℳd˜xβy,X^y,U4vpTX^x=∨X^y≥1−γT∨ℳd˜xβy∪ℳd˜xβy,1−γ∨∨X^y<1−γT∨ℳd˜xβy∪ℳd˜xβy,X^y,L5vpℐX^x=∧X^y≤γℐ∨ℳD˜xβy,γ∧∧X^y>γℐ∨ℳD˜xβy,X^y,U5vpTX^x=∨X^y≥1−γT∨ℳD˜xβy,1−γ∨∨X^y<1−γT∨ℳD˜xβy,X^y,L6vpℐX^x=∧X^y≤γℐℳD˜xβy,γ∧∧X^y>γℐℳD˜xβy,X^y,U6vpTX^x=∨X^y≥1−γTℳD˜xβy,1−γ∨∨X^y<1−γTℳD˜xβy,X^y,L7vpℐX^x=∧X^y≤γℐ∨ℳD˜xβy∩ℳD˜xβy,γ∧∧X^y>γℐ∨ℳD˜xβy∩ℳD˜xβy,X^y,U7vpTX^x=∨X^y≥1−γT∨ℳD˜xβy∩ℳD˜xβy,1−γ∨∨X^y<1−γT∨ℳD˜xβy∩ℳD˜xβy,X^y,L8vpℐX^x=∧X^y≤γℐ∨ℳD˜xβy∪ℳD˜xβy,γ∧∧X^y>γℐ∨ℳD˜xβy∪ℳD˜xβy,X^y,U8vpTX^x=∨X^y≥1−γT∨ℳD˜xβy∪ℳD˜xβy,1−γ∨∨X^y<1−γT∨ℳD˜xβy∪ℳD˜xβy,X^y.
IfL1vpℐX^ (resp., L2vpℐX^, L3vpℐX^, L4vpℐX^, L5vpℐX^, L6vpℐX^, L7vpℐX^, and L8vpℐX^≠U1vpTX^ (resp., U2vpℐX^, U3vpℐX^, U4vpℐX^, U5vpℐX^, U6vpℐX^, U7vpℐX^, and U8vpℐX^), then X^ is a 1-VPℐTFRS (resp., 2-VPℐTFRS, 3-VPℐTFRS, 4-VPℐTFRS, 5-VPℐTFRS, 6-VPℐTFRS, 7-VPℐTFRS, and 8-VPℐTFRS). In other cases, it is definable.Example 3 (in continuation of Examples1 and 2).
Letγ=0.6. We now have(25)ℒ1vpℐX^=0.6x1+0.6x2+0.6x3+0.6x4+0.6x5+0.6x6,U1vpTX^=0.1x1+0.3x2+0.1x3+0.4x4+0.4x5+0.1x6,ℒ2vpℐX^=0.6x1+0.7x2+0.6x3+0.6x4+0.6x5+0.6x6,U2vpTX^=0.3x1+0.1x2+0.4x3+0.4x4+0.3x5+0.3x6,ℒ3vpℐX^=0.6x1+0.7x2+0.6x3+0.6x4+0.6x5+0.6x6,U3vpTX^=0.1x1+0.1x2+0.1x3+0.4x4+0.3x5+0.1x6,ℒ4vpℐX^=0.6x1+0.6x2+0.6x3+0.6x4+0.6x5+0.6x6,U4vpTX^=0.3x1+0.3x2+0.4x3+0.4x4+0.4x5+0.3x6,ℒ5vpℐX^=0.6x1+0.7x2+0.6x3+0.7x4+0.6x5+0.6x6,U5vpTX^=0.1x1+0.1x2+0.3x3+0.1x4+0.3x5+0.1x6,ℒ6vpℐX^=0.6x1+0.7x2+0.6x3+0.6x4+0.6x5+0.6x6,U6vpTX^=0.1x1+0.1x2+0.2x3+0.4x4+0.2x5+0.1x6,ℒ7vpℐX^=0.6x1+0.7x2+0.6x3+0.7x4+0.6x5+0.6x6,U7vpTX^=0.1x1+0.1x2+0.2x3+0.1x4+0.2x5+0.1x6,ℒ8vpℐX^=0.6x1+0.7x2+0.6x3+0.6x4+0.6x5+0.6x6,U8vpTX^=0.1x1+0.1x2+0.3x3+0.4x4+0.3x5+0.1x6.Next, we have the relevant properties of Definition7. The proof of each point in Theorem 3 is easily obtained, so we omit it. Let us select j∈1,2,3,…,8.Theorem 3.
Presume thatΩ,Γ˜ be an FCAS for a given β∈0,1. Select X^∈ℱΩ and ∀x∈Ω. Then, the following affirmations hold true:(1)
ℒjvpℐX^=UjvpTX^cNcN and UjvpTX^=ℒjvpℐX^cNcN when ℐ is an S-implicator(2)
ℒjvpℐΩ=Ω and UjvpT∅=∅ when ℐ is a left monotonicity(3)
IfX^⊆Y^, then ℒjvpℐX^⊆ℒjvpℐY^ and UjvpTX^⊆UjvpTY^ when ℐ is a right monotonicity(4)
ℒjvpℐX^∩Y^=ℒjvpℐX^∩ℒjvpℐY^ and UjvpTX^∩Y^⊆UjvpTX^∩UjvpTY^ when ℐ is a right monotonicity(5)
ℒjvpℐX^∪Y^⊇ℒjvpℐX^∪ℒjvpℐY^ and UjvpTX^∪Y^=UjvpTX^∪UjvpTY^ when ℐ is a right monotonicity


## 5. The Relationships among Our Models

Now, we explain some relationships among the paradigms proposed previously.Remark 1.
Presume thatΩ,Γ˜ be an FCAS for a fixed β∈0,1. Select X^∈ℱΩ. Then, the following affirmation holds true:(1)
∧ℳD˜xβ=N˜xβ(2)
∨ℳD˜xβ=∨C˜i∈Γ˜:C˜i≥β(3)
∨ℳD˜xβ∩ℳD˜xβ≤∨ℳD˜xβ,∨ℳD˜xβ∩ℳD˜xβ≤ℳD˜xβ(4)
∨ℳD˜xβ∪ℳD˜xβ≥∨ℳD˜xβ,∨ℳD˜xβ∪ℳD˜xβ≥ℳD˜xβ(5)
∨ℳD˜xβ∩ℳD˜xβ≤∨ℳD˜xβ,∨ℳD˜xβ∩ℳD˜xβ≤ℳD˜xβ(6)
∨ℳD˜xβ∪ℳD˜xβ≥∨ℳD˜xβ,∨ℳD˜xβ∪ℳD˜xβ≥ℳD˜xβProposition 1.
Presume thatΩ,Γ˜ be an FCAS for a fixed β∈0,1. Select X^∈ℱΩ. Then, the following affirmation holds true:(1)
ℒ4ℐX^⊆ℒ1ℐX^⊆ℒ3ℐX^(2)
ℒ4ℐX^⊆ℒ2ℐX^⊆ℒ3ℐX^(3)
U3TX^⊆U1TX^⊆U4TX^(4)
U3TX^⊆U2TX^⊆U4TX^(5)
ℒ3ℐX^=ℒ1ℐX^∪ℒ2ℐX^(6)
ℒ4ℐX^=ℒ1ℐX^∩ℒ2ℐX^(7)
U3TX^=U1TX^∩U2TX^(8)
U4TX^=U1TX^∪U2TX^Proof.
It is clear by using Definition5.Proposition 2.
Presume thatΩ,Γ˜ be an FCAS for a fixed β∈0,1. Select X^∈ℱΩ. Then, the following affirmation holds true:(1)
ℒ8ℐX^⊆ℒ5ℐX^⊆ℒ7ℐX^(2)
ℒ8ℐX^⊆ℒ6ℐX^⊆ℒ7ℐX^(3)
U7TX^⊆U5TX^⊆U8TX^(4)
U7TX^⊆U6TX^⊆U8TX^(5)
ℒ7ℐX^=ℒ5ℐX^∪ℒ6ℐX^(6)
ℒ8ℐX^=ℒ5ℐX^∩ℒ6ℐX^(7)
U7TX^=U5TX^∩U6TX^(8)
U8TX^=U5TX^∪U6TX^Proof.
It is clear by using Definition6.Proposition 3.
Presume thatΩ,Γ˜ be an FCAS for a fixed β∈0,1. Select X^∈ℱΩ. Then, ∨ℳd˜xβ=ℳd˜xβ, for every x∈Ω⇔ for each X^∈ℱΩ, either ℒ1ℐX^=ℒ2ℐX^=ℒ3ℐX^=ℒ4ℐX^ or U1ℐX^=U2ℐX^=U3ℐX^=U4ℐX^.Proof.
The proof is obtained by using Definition5 and Proposition 1.Proposition 4.
Presume thatΩ,Γ˜ be an FCAS for a fixed β∈0,1. Select X^∈ℱΩ. Then, ∧ℳd˜xβ=ℳD˜xβ, for every x∈Ω⇔ for each X^∈ℱΩ, either ℒ5ℐX^=ℒ6ℐX^=ℒ7ℐX^=ℒ8ℐX^ or U5ℐX^=U6ℐX^=U7ℐX^=U8ℐX^.Proof.
The proof is obtained by using Definition6 and Proposition 2.Proposition 5.
Presume thatΩ,Γ˜ be an FCAS for a fixed β∈0,1. Select X^∈ℱΩ. Then, ∨ℳd˜xβ=N˜xβ, ∀x∈Ω⇔(1)
ℒ1ℐX^=C˜1−X^ or U1ℐX^=C˜1+X^(2)
ℒ2ℐX^=C˜2−X^ or U2ℐX^=C˜2+X^(3)
ℒ3ℐX^=C˜3−X^ or U3ℐX^=C˜3+X^(4)
ℒ4ℐX^=C˜4−X^ or U4ℐX^=C˜4+X^Proof.
The proof is obtained by using Definitions3 and 5.Proposition 6.
Presume thatΩ,Γ˜ be an FCAS for a fixed β∈0,1. Select X^∈ℱΩ. ∧ℳd˜xβ=N˜xβ, ∀x∈Ω⇔(1)
ℒ5ℐX^=C˜1−X^ or U5ℐX^=C˜1+X^(2)
ℒ6ℐX^=C˜2−X^ or U6ℐX^=C˜2+X^(3)
ℒ7ℐX^=C˜3−X^ or U7ℐX^=C˜3+X^(4)
ℒ8ℐX^=C˜4−X^ or U8ℐX^=C˜4+X^Proof.
The proof is obtained by using Definitions3 and 6.Proposition 7.
Presume thatΩ,Γ˜ be an FCAS for a fixed β∈0,1. Select X^∈ℱΩ. Then, the following affirmation holds true:(1)
L4vpℐX^⊆L1vpℐX^⊆L3vpℐX^(2)
L4vpℐX^⊆L2vpℐX^⊆L3vpℐX^(3)
U3vpTX^⊆U1vpTX^⊆U4vpTX^(4)
U3vpTX^⊆U2vpTX^⊆U4vpTX^Proof.
It is obvious from Definition7.Proposition 8.
Presume thatΩ,Γ˜ be an FCAS for a fixed β∈0,1. Select X^∈ℱΩ. Then, the following affirmation holds true:(1)
L8vpℐX^⊆L5vpℐX^⊆L7vpℐX^(2)
L8vpℐX^⊆L6vpℐX^⊆L7vpℐX^(3)
U7vpTX^⊆U5vpTX^⊆U8vpTX^(4)
U7vpTX^⊆U6vpTX^⊆U8vpTX^Proof.
It is obvious from Definition7.


## 6. Proposed Decision-Making Approach

Now, we use the presented study to obtain a decision on a realistic issue.


### 6.1. Description and Process

Suppose thatΩ=u1,u2,…,un be n candidates and Γ˜=Γ˜1,Γ˜2,…,Γ˜m be a set of features. Thus, Γ˜uj denotes the experts rating value related to the candidate uj to the attribute Γ˜, and we assume that for a fixed β∈0,1, Ω,Γ˜ is an FCAS. Based on the explained covering methods, we submit an algorithm that seeks to obtain the best choice as in the following procedures:Step 1. Produce a matrix Dℳ as(26)Dℳ=e11e12⋯e1me21e22⋯e2m⋮⋮⋱⋮en1en2⋯enm,whereeij is the experts rating value of alternative ui.Step 2. Count the favorable ideal I⊕ and the unfavorable ideal I⊖ fuzzy sets with the help of the following equations:(27)I⊕=u1,∨1≤j≤me1j,u2,∨1≤j≤me2j,…,un,∨1≤j≤menj,I⊖=u1,∧1≤j≤me1j,u2,∧1≤j≤me2j,…,un,∧1≤j≤menj,where∨ and ∧ symbolize to “maximum” and “minimum,” respectively.Set the integrated ideal fuzzy setNℐ as follows:(28)Nℐ=u1,∑1≤j≤me1jwj,u2,∑1≤j≤me2jwj,…,un,∑1≤j≤menjwj,whereWV=w1,w2,…,wm is the weight vector, ∑r=1mwr=1, and wj∈0,1 for every j∈1,2,…,n.Step 3. Count the lower and upper approximations of I⊕, I⊖, and Nℐ as defined by the 7-FCFCℐTFRS model.Step 4. Count the respective distances D¯ and D¯ between the lower approximation of Nℐ and the lower approximation of I⊕ and also the distance among their parallel upper approximations as follows:(29)D¯i=νDΓ˜⊖I⊕ui,Γ˜⊖Nℐui+1−νDΓ˜⊕I⊕ui,Γ˜⊕Nℐui,D¯i=νDΓ˜⊖I⊖ui,Γ˜⊖Nℐui+1−νDΓ˜⊕I⊖ui,Γ˜⊕Nℐui,whereν is a controlling result which is determined by the experts and Dx,y=x−y.Step 5. Count the closeness coefficient grade by(30)Rlui=D¯uiD¯ui+D¯ui,and hence order the candidates.Based on these procedures, we establish an algorithm to fix the decision-making issues according to 7-FCℐTFRS paradigm. The steps corresponding to it are summarized in Algorithm 1.Algorithm 1: Algorithm for MADM.
Input: Fuzzy information systems.Output: Decision-making.(1)
GetX^, β, Γ˜, and Ω(2)
EstablishDℳ(3)
By Definition2, count ∧ℳ˜uβ(4)
By Step (3), countℳD˜uβ(5)
By Steps (3) and (4), count∧ℳD˜uβ∩ℳD˜uβ(6)
CountI⊕ and I⊖(7)
CountNℐ(8)
Count the lower and upper approximations ofI⊕,I⊖, and Nℐ, according to Definition 6(9)
CountD¯i and D¯i(10)
Countℛl(11)
Get the judgment


### 6.2. Illustrative Example

The above procedures have been clarified with a test example as follows.Table 12
Values ofDℳ.
u1u2u3u4u5u6Γ˜10.70.60.40.50.10.6Γ˜20.50.30.30.70.40.8Γ˜30.40.30.50.50.20.4Γ˜40.30.70.80.20.60.1Γ˜50.20.30.50.60.10.5Table 13
Values ofI⊕, I⊖, and Nℐ.
u1u2u3u4u5u6I⊕0.70.70.80.70.60.8I⊖0.20.30.30.20.10.1Nℐ0.4050.390.4750.5350.290.515Table 14
Values of∧ℳD˜xβ.
u1u2u3u4u5u6∧ℳD˜u10.50.50.30.30.50.10.6∧ℳD˜u20.50.30.60.40.20.10.1∧ℳD˜u30.50.20.30.50.20.10.1∧ℳD˜u40.50.20.30.30.50.10.4∧ℳD˜u50.50.30.70.80.20.60.1∧ℳD˜u60.50.20.30.30.50.10.5Table 15
Values ofℳD˜xβy.
u1u2u3u4u5u6∧ℳD˜u10.50.50.30.20.20.30.2∧ℳD˜u20.50.30.60.30.30.70.3∧ℳD˜u30.50.30.40.50.30.80.3∧ℳD˜u40.50.50.20.20.50.20.5∧ℳD˜u50.50.10.10.10.10.60.1∧ℳD˜u60.50.60.10.10.40.10.5Table 16
The result for∧ℳD˜uβ∩ℳD˜uβ.
u1u2u3u4u5u6∧ℳD˜u10.5∩ℳD˜u10.50.50.30.20.20.10.2∧ℳD˜u20.5∩ℳD˜u20.50.30.60.30.20.10.1∧ℳD˜u30.5∩ℳD˜u30.50.20.30.50.20.10.1∧ℳD˜u40.5∩ℳD˜u40.50.20.20.20.50.10.4∧ℳD˜u50.5∩ℳD˜u50.50.10.10.10.10.20.1∧ℳD˜u60.5∩ℳD˜u60.50.20.10.10.40.10.5Example 4 (see [48]).
Nominee engineers form a setΩ=u1,u2,…,u6, and the related characteristics are combined via attribute set Γ˜ = { passionate firmness Γ1, verbal connection expertise Γ2, individuality Γ3, former practice Γ4, aplomb Γ5}. Here, we proceed with the above-declared algorithm as in the following steps.Step 1. The evaluation for each nominee through the decision-makers is summarized in Table 12.Step 2. Based on the significance of these five features, we have WV=0.1,0.3,0.25,0.15,0.2, we calculate I⊕,I⊖, and Nℐ, and outcomes are listed in Table 13.Step 3. Using ℐm and Tℳ, through 7-FCℐTFRS, we obtain the the following values:(31)ℒ7ℐX^x=∧y∈Ωℐ∧ℳD˜xβy∩ℳD˜xβy,X^y,U7TX^x=∨y∈ΩT∧ℳD˜xβy∩ℳD˜xβy,X^y.So, we evaluate∧ℳD˜xβ and ℳD˜xβ, and the counted outcomes are shown in Tables 14 and 15, and hence we compute ∧ℳD˜uβ∩ℳD˜uβ whose result is listed in Table 16.Based on these information, we get(32)Γ˜⊖I⊕=0.7u1+0.7u2+0.7u3+0.7u4+0.8u5+0.7u6,Γ˜⊕I⊕=0.5u1+0.6u2+0.5u3+0.5u4+0.2u5+0.5u6,Γ˜⊖I⊖=0.5u1+0.4u2+0.5u3+0.5u4+0.8u5+0.5u6,Γ˜⊕I⊖=0.3u1+0.3u2+0.3u3+0.2u4+0.1u5+0.2u6,Γ˜⊖Nℐ=0.5u1+0.4u2+0.5u3+0.535u4+0.8u5+0.515u6,Γ˜⊕Nℐ=0.405u1+0.39u2+0.475u3+0.5u4+0.2u5+0.5u6.Step 4. Count D¯i and D¯i indicated as follows:(33)D¯i=0.1475u1+0.255u2+0.1125u3+0.0825u4+0u5+0.0925u6,D¯i=0.0525u1+0.045u2+0.0875u3+0.1675u4+0.05u5+0.1575u6.Step 5. According to the results on Step 4, we count Rl as follows:(34)Rl=0.2625u1+0.15u2+0.4375u3+0.67u4+1u5+0.63u6,and hence get the ranking order as(35)u5≥u4≥u6≥u3≥u1≥u2.
By the above procedures and counting, you can say that the 5th engineer is the best nominee among the others.


### 6.3. Comparative Analysis

This section aims to explain the merit of the proposed method by comparing it to the last study in the area which was given by Zhang et al. [48]. The purpose here is it is able to raise the lower approximation and down the upper approximation of Zhang et al.’s models as you see in Example 4. We established Tables 17 and 18 to explain the differences between Zhang et al.’s model and our models.Table 17
Values by FCITFRS.
Different processesObtain a valueu1u2u3u4u5u6Zhang’s model [48]0.26250.150.43750.670.27140.63Our model0.26250.150.43750.6710.63Table 18
Values by FCITFRS.
Different processesObtain a decisionZhang’s process [48]u4≥u6≥u3≥u5≥u1≥u2Our processu5≥u4≥u6≥u1≥u2To view another way to illustrate this comparison between Zhang et al.’s [48] and our’s, we present Figures 1 and 2.Figure 1
The lower approximation using our model and Zhang et al.’s [48] model.Figure 2
The upper approximation using our model and Zhang et al.’s [48] model.Figure1 displays the relations among Zhang-lower positive ideal, Zhang-lower negative ideal, and Zhang-lower integrated ideal and our-lower positive ideal, our-lower negative ideal, and our-lower integrated ideal. From this point of view, you can see that our-lower (positive, negative, and integrated ideal) is superior to Zhang-lower (positive, negative, and integrated ideal). This means that our-lower is better than Zhang-lower.Figure2 discusses the links among Zhang-upper positive ideal, Zhang-upper negative ideal, and Zhang-upper integrated ideal and our-upper positive ideal, our-upper negative ideal, and our-upper integrated ideal. Thus, you can see that our-upper (positive, negative, and integrated ideal) is down than Zhang-upper (positive, negative, and integrated ideal). This means that our-upper is lower than Zhang-upper.Through Tables17 and 18 and Figures 1 and 2, it is easy to say that our work is considered as an improvement of Zhang et al.’s [48] work via raising the lower approximation and down the upper approximation (for instance, see Example 4). This means that our method is effective and reliable.


## 6.1. Description and Process

Suppose thatΩ=u1,u2,…,un be n candidates and Γ˜=Γ˜1,Γ˜2,…,Γ˜m be a set of features. Thus, Γ˜uj denotes the experts rating value related to the candidate uj to the attribute Γ˜, and we assume that for a fixed β∈0,1, Ω,Γ˜ is an FCAS. Based on the explained covering methods, we submit an algorithm that seeks to obtain the best choice as in the following procedures:Step 1. Produce a matrix Dℳ as(26)Dℳ=e11e12⋯e1me21e22⋯e2m⋮⋮⋱⋮en1en2⋯enm,whereeij is the experts rating value of alternative ui.Step 2. Count the favorable ideal I⊕ and the unfavorable ideal I⊖ fuzzy sets with the help of the following equations:(27)I⊕=u1,∨1≤j≤me1j,u2,∨1≤j≤me2j,…,un,∨1≤j≤menj,I⊖=u1,∧1≤j≤me1j,u2,∧1≤j≤me2j,…,un,∧1≤j≤menj,where∨ and ∧ symbolize to “maximum” and “minimum,” respectively.Set the integrated ideal fuzzy setNℐ as follows:(28)Nℐ=u1,∑1≤j≤me1jwj,u2,∑1≤j≤me2jwj,…,un,∑1≤j≤menjwj,whereWV=w1,w2,…,wm is the weight vector, ∑r=1mwr=1, and wj∈0,1 for every j∈1,2,…,n.Step 3. Count the lower and upper approximations of I⊕, I⊖, and Nℐ as defined by the 7-FCFCℐTFRS model.Step 4. Count the respective distances D¯ and D¯ between the lower approximation of Nℐ and the lower approximation of I⊕ and also the distance among their parallel upper approximations as follows:(29)D¯i=νDΓ˜⊖I⊕ui,Γ˜⊖Nℐui+1−νDΓ˜⊕I⊕ui,Γ˜⊕Nℐui,D¯i=νDΓ˜⊖I⊖ui,Γ˜⊖Nℐui+1−νDΓ˜⊕I⊖ui,Γ˜⊕Nℐui,whereν is a controlling result which is determined by the experts and Dx,y=x−y.Step 5. Count the closeness coefficient grade by(30)Rlui=D¯uiD¯ui+D¯ui,and hence order the candidates.Based on these procedures, we establish an algorithm to fix the decision-making issues according to 7-FCℐTFRS paradigm. The steps corresponding to it are summarized in Algorithm 1.Algorithm 1: Algorithm for MADM.
Input: Fuzzy information systems.Output: Decision-making.(1)
GetX^, β, Γ˜, and Ω(2)
EstablishDℳ(3)
By Definition2, count ∧ℳ˜uβ(4)
By Step (3), countℳD˜uβ(5)
By Steps (3) and (4), count∧ℳD˜uβ∩ℳD˜uβ(6)
CountI⊕ and I⊖(7)
CountNℐ(8)
Count the lower and upper approximations ofI⊕,I⊖, and Nℐ, according to Definition 6(9)
CountD¯i and D¯i(10)
Countℛl(11)
Get the judgment


## 6.2. Illustrative Example

The above procedures have been clarified with a test example as follows.Table 12
Values ofDℳ.
u1u2u3u4u5u6Γ˜10.70.60.40.50.10.6Γ˜20.50.30.30.70.40.8Γ˜30.40.30.50.50.20.4Γ˜40.30.70.80.20.60.1Γ˜50.20.30.50.60.10.5Table 13
Values ofI⊕, I⊖, and Nℐ.
u1u2u3u4u5u6I⊕0.70.70.80.70.60.8I⊖0.20.30.30.20.10.1Nℐ0.4050.390.4750.5350.290.515Table 14
Values of∧ℳD˜xβ.
u1u2u3u4u5u6∧ℳD˜u10.50.50.30.30.50.10.6∧ℳD˜u20.50.30.60.40.20.10.1∧ℳD˜u30.50.20.30.50.20.10.1∧ℳD˜u40.50.20.30.30.50.10.4∧ℳD˜u50.50.30.70.80.20.60.1∧ℳD˜u60.50.20.30.30.50.10.5Table 15
Values ofℳD˜xβy.
u1u2u3u4u5u6∧ℳD˜u10.50.50.30.20.20.30.2∧ℳD˜u20.50.30.60.30.30.70.3∧ℳD˜u30.50.30.40.50.30.80.3∧ℳD˜u40.50.50.20.20.50.20.5∧ℳD˜u50.50.10.10.10.10.60.1∧ℳD˜u60.50.60.10.10.40.10.5Table 16
The result for∧ℳD˜uβ∩ℳD˜uβ.
u1u2u3u4u5u6∧ℳD˜u10.5∩ℳD˜u10.50.50.30.20.20.10.2∧ℳD˜u20.5∩ℳD˜u20.50.30.60.30.20.10.1∧ℳD˜u30.5∩ℳD˜u30.50.20.30.50.20.10.1∧ℳD˜u40.5∩ℳD˜u40.50.20.20.20.50.10.4∧ℳD˜u50.5∩ℳD˜u50.50.10.10.10.10.20.1∧ℳD˜u60.5∩ℳD˜u60.50.20.10.10.40.10.5Example 4 (see [48]).
Nominee engineers form a setΩ=u1,u2,…,u6, and the related characteristics are combined via attribute set Γ˜ = { passionate firmness Γ1, verbal connection expertise Γ2, individuality Γ3, former practice Γ4, aplomb Γ5}. Here, we proceed with the above-declared algorithm as in the following steps.Step 1. The evaluation for each nominee through the decision-makers is summarized in Table 12.Step 2. Based on the significance of these five features, we have WV=0.1,0.3,0.25,0.15,0.2, we calculate I⊕,I⊖, and Nℐ, and outcomes are listed in Table 13.Step 3. Using ℐm and Tℳ, through 7-FCℐTFRS, we obtain the the following values:(31)ℒ7ℐX^x=∧y∈Ωℐ∧ℳD˜xβy∩ℳD˜xβy,X^y,U7TX^x=∨y∈ΩT∧ℳD˜xβy∩ℳD˜xβy,X^y.So, we evaluate∧ℳD˜xβ and ℳD˜xβ, and the counted outcomes are shown in Tables 14 and 15, and hence we compute ∧ℳD˜uβ∩ℳD˜uβ whose result is listed in Table 16.Based on these information, we get(32)Γ˜⊖I⊕=0.7u1+0.7u2+0.7u3+0.7u4+0.8u5+0.7u6,Γ˜⊕I⊕=0.5u1+0.6u2+0.5u3+0.5u4+0.2u5+0.5u6,Γ˜⊖I⊖=0.5u1+0.4u2+0.5u3+0.5u4+0.8u5+0.5u6,Γ˜⊕I⊖=0.3u1+0.3u2+0.3u3+0.2u4+0.1u5+0.2u6,Γ˜⊖Nℐ=0.5u1+0.4u2+0.5u3+0.535u4+0.8u5+0.515u6,Γ˜⊕Nℐ=0.405u1+0.39u2+0.475u3+0.5u4+0.2u5+0.5u6.Step 4. Count D¯i and D¯i indicated as follows:(33)D¯i=0.1475u1+0.255u2+0.1125u3+0.0825u4+0u5+0.0925u6,D¯i=0.0525u1+0.045u2+0.0875u3+0.1675u4+0.05u5+0.1575u6.Step 5. According to the results on Step 4, we count Rl as follows:(34)Rl=0.2625u1+0.15u2+0.4375u3+0.67u4+1u5+0.63u6,and hence get the ranking order as(35)u5≥u4≥u6≥u3≥u1≥u2.
By the above procedures and counting, you can say that the 5th engineer is the best nominee among the others.


## 6.3. Comparative Analysis

This section aims to explain the merit of the proposed method by comparing it to the last study in the area which was given by Zhang et al. [48]. The purpose here is it is able to raise the lower approximation and down the upper approximation of Zhang et al.’s models as you see in Example 4. We established Tables 17 and 18 to explain the differences between Zhang et al.’s model and our models.Table 17
Values by FCITFRS.
Different processesObtain a valueu1u2u3u4u5u6Zhang’s model [48]0.26250.150.43750.670.27140.63Our model0.26250.150.43750.6710.63Table 18
Values by FCITFRS.
Different processesObtain a decisionZhang’s process [48]u4≥u6≥u3≥u5≥u1≥u2Our processu5≥u4≥u6≥u1≥u2To view another way to illustrate this comparison between Zhang et al.’s [48] and our’s, we present Figures 1 and 2.Figure 1
The lower approximation using our model and Zhang et al.’s [48] model.Figure 2
The upper approximation using our model and Zhang et al.’s [48] model.Figure1 displays the relations among Zhang-lower positive ideal, Zhang-lower negative ideal, and Zhang-lower integrated ideal and our-lower positive ideal, our-lower negative ideal, and our-lower integrated ideal. From this point of view, you can see that our-lower (positive, negative, and integrated ideal) is superior to Zhang-lower (positive, negative, and integrated ideal). This means that our-lower is better than Zhang-lower.Figure2 discusses the links among Zhang-upper positive ideal, Zhang-upper negative ideal, and Zhang-upper integrated ideal and our-upper positive ideal, our-upper negative ideal, and our-upper integrated ideal. Thus, you can see that our-upper (positive, negative, and integrated ideal) is down than Zhang-upper (positive, negative, and integrated ideal). This means that our-upper is lower than Zhang-upper.Through Tables17 and 18 and Figures 1 and 2, it is easy to say that our work is considered as an improvement of Zhang et al.’s [48] work via raising the lower approximation and down the upper approximation (for instance, see Example 4). This means that our method is effective and reliable.


## 7. Conclusion

The major purpose of the presented work is to improve Zhang et al.’s model [48] and Jiang et al.’s model [49]. In this paper, we considered the issue of the neighborhood-related β covering the classification of approximation spaces. Eight new paradigms of FCℐTFRSs models can be viewed as an improvement of Zhang et al. [48] model through concepts of fuzzy β-minimal description and fuzzy β-maximal description. Also, we introduce eight kinds CVPℐTFRSs as generalizations of Jiang et al. [49] model. Further, we study the relations between our models and the previous models in [48, 49]. After that, we place the present study on a real issue to show its effectiveness and reliability.



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