# Energy Saving via a Minimal Structure

Authors: B. Almarri; A. A. Azzam
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Not only mathematical statistics, differential equations, and mathematical models were used to analyze and reduce data, but a rough set model is also employed in medical, engineering, and other fields to analyze and reduce them. The goal of this paper is to introduce a minimal structure concept to produce new rough set models and show that it is suitable for analyzing most real-life problems, reduction of attributes, and decision making. We examine the effectiveness of the following method in the problem of electric power generators and decision making. We also offer a comparison of our method and Pawlak’s method. Finally, the variable precision model improves the accuracy of decision making.


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## Body

## 1. Introduction

The current era is distinguished by its ability to collect and store large amount of data on any topic we want, to make, and take a decision regarding it. Uncertainty is a characteristic of almost all real life problems and mathematics plays a vital role in processing these data to obtain the best decisions.The analysis process has many mathematical methods that we can use. Since the current statement of a rough set theory, topology and decision making have had a long history together. The theory of rough set is a useful tool for analyzing unclear data. Pawlak first introduced the rough set model in 1982 [1] and constructed a topological space using the equivalent class of equivalence relations on the items as its foundation [2–4]. These equivalence classes is a base for a special type of topological structure in which every open set is closed. This condition is a constrain which limits the range of application. Some extensions under other relations, such as 1998 [5, 6], have been proposed in an attempt to solve such unreasonableness. New types of neighborhoods were added for various goals, including improving the set’s accuracy values, such as intersection neighborhoods, which contained some topology expansions such as the minimal structure [2, 7]. The manuscript contributes to this trend by employing a simple structure to generalize rough set approximation. The data are formatted as an information system Σ and are processed in several stages. The first of which is the selection of effective features (reduction) and the second is an approximation. This method was used for many medical applications such as those in [7, 8] and engineering applications such as those in [9, 10], and others such as those in [2, 11–15].Many researchers all over the world attempted to overcome the shortcomings that the original model of rough set theory may lose in the classification or have a discrete classification in which every class has a single element. This paper aims to construct lower and upper approximations based on a similarity degree between objects ofΣ and demonstrate that a minimal structure framework can be used to solve a variety of real-world challenges. Consequently, we apply these concepts to obtain reducts and core of Σ using the accuracy measure of decision sets.We use a variable precision to improve the decision accuracy in our approach. Also, we provide a comparison between our strategy and Pawlak’s strategy.The following is the structure of this paper:We go over the fundamental ideas in the second segment. In the third section, we describe our proposed method for reducing the characteristics. In the fourth section, there are some examples that explain the method used.


## 2. Preliminaries

Definition 1.
[16] Information system
An information systemΣ can be written as a ℵ,ℏ,V,f where ℵ is the universe of finite objects, ℵ=ℵ1,ℵ2,…,ℵn,V attribute values, and ℏ is the set of attributes ℏ1,ℏ2,…,ℏm,f:ℵ×ℏ⟶V.(1)
These objects, for example: patients, electric generators, students, ….(2)
The attributes can be features or characteristics of objects.Example 1.
An information system for an exam of five students in Table1, where ℵ=ℵ1,ℵ2,ℵ3,ℵ4,ℵ5,ℏ=M,C,P,E are four courses, mathematics M, chemistry C, physics P, and English language E. The entries of the table represent the set V of the scores for the students.
[?]
Adding the columnD values B, C, C, B, and D is an example of the decision table [Table 2].Table 1
Information system table.
MCPEℵ19085ABℵ28576BCℵ37092CAℵ46284DBℵ54075CDTable 2
Decision information system table.
MCPEDℵ19085ABBℵ28576BCCℵ37092CACℵ46284DBBℵ54075CDDDefinition 2.
[17, 18]. A minimal structre (short for ℳ-structure on a set ℵ is defined as ℳ⊆2ℵ (the collection of all ℵ subsets) containing ∅,ℵ). A minimal structure space is defined by the pair ℵ,ℳ (simplify, ℳ-space). Members of ℳ′s are known as open sets, and the complement of ℳ′s is known as closed sets. .Definition 3.
[17, 18]. Let ℳ be an ℳ-structure on ℵ and ℵ be a nonempty set. Let ℳℵ be an ℳℵ-structure on N and ℵ be a non empty set.(i)
ℳ¯N=⊓ℱ:N⊑ℱ,ℵ/ℱ∈ℳℵ,(ii)
ℳ¯N=⊔O:O⊑ℬ,O∈ℳℵ.Definition 4.
[3, 19]. If ℵ is a set of objects that is not empty, and an equivalence relation on ℵ is named ℜ , the approximation space is named after the pair ℵ,ℜ. Let N⊑ℵ, the rough set approximation is as follows:(i)
The set of all objects that can be categorized asN concerning ℜ is called upper approximation of N concerning ℜ and is denoted by ℜ¯N. That is, ℜ¯N=n∈N:ℜn⊓N≠φ, where n determines the equivalence class and ℜn is the equivalence class.(ii)
The lower approximation ofN concerning ℜ is ℜ¯N, which denotes the set of all objects that may absolutely be classed as N concerning ℜ. To put it another way, ℜ¯N=n∈N:ℜn⊑D, where ℜn denotes the equivalence class determined by n.(iii)
NrℜN boundary region of N for ℜ. That is,(1)NrℜN=ℜ¯N−ℜ¯N,which is the collection of all elements that cannot be categorized as either N or ℵ/N. The set N is said to be rough concerning ℜ If ℜ¯N≠ℜ¯N. That is, NrℜN≠∅. Then, the set N is called rough to ℜ.Definition 5.
[3, 19]. Let ℵ,ℜ,D be a decision approximation space, with ℵ,ℜ being an approximation space and D being a decision equivalence relation on ℵ. Hence, ℵ/D=Z1,Z2,…,Zn is a partition on ℵ. In terms of, The lower and upper approximations of ℵ/D with respect to ℜ denoted as follows:(2)ℜ¯ℵD=ℜ¯Z1,ℜ¯Z2,…,ℜ¯Zn,and(3)ℜ¯ℵD=ℜ¯Z1,ℜ¯Z2,…,ℜ¯Zn.


### 2.1. Variable Precision Model

(1)
For allC,D⊑ℵ such that ℵ denotes a non-null finite universe set and C⊑D that all elements of C are contained in D.(2)
C⊑D with an error ratio β based on the value of measure VC,D is discussed in [4, 7]. i.e., C⊑βD if and only if VC,D≤β, where VC,D=1−cardC⊓D/cardC if cardC>0, VC,D=0 if cardC=0, where the allowable error must be 0<β≤0.5.


## 2.1. Variable Precision Model

(1)
For allC,D⊑ℵ such that ℵ denotes a non-null finite universe set and C⊑D that all elements of C are contained in D.(2)
C⊑D with an error ratio β based on the value of measure VC,D is discussed in [4, 7]. i.e., C⊑βD if and only if VC,D≤β, where VC,D=1−cardC⊓D/cardC if cardC>0, VC,D=0 if cardC=0, where the allowable error must be 0<β≤0.5.


## 3. Similarity Approach for Reduction

We demonstrate the usefulness of a minimal structural space in engineering science for use in decision-making situations in this section. Therefore, we have decided to use it to solve a technical problem. This data collection contains the results of four attributes for eight different generators. The research study was carried out at the Princess Nourah bint Abdulrahman University’s Mathematical Department in Riyadh, Saudi Arabia. The study included eight different generators presenting to this university with different presenting features, where the first two are two different transmission lines and the second two for self-generation and self-load. 1 denotes low, 2 denotes medium, and 3 denotes high. Finally, the heart failure diagnosis is normal or rest.The equivalence relation is a condition that sometimes generates a discrete equivalence classes, that is, each class contains one object, and this makes all concepts exactly as in [20].The following example for anΣ when applying Pawlak methods gives equivalence classes each of which contains singleton.Example 3.
Consider the table ofΣ (Table 3).
Using the original model, the classes area,b,c,d,e, but the similarity degrees are as follows:(4)μa,a=1,μa,b=15,μa,c=15,μa,d=25,μa,e=25,which produce no discrete classes that give nontrivial approximation.Table 3
Information system table.
lmnopa10110b01101c11001d00101e11011Definition 6.
Attribute similarity.
In anΣ=ℵ,ℏ=D∪C,V,f, for an attribute ℏi, the similarity value for two objects x,y is denoted by δix,y,(5)δix,y=1,ℏix=ℏjy0,ℏix≠ℏjy.
The similarity degree among two objectsx,y is as follows:(6)μx,y=∑i=1nδix,yn,∀ai∈ℏ,where n is ℵ’s cardinality.


### 3.1. Discussion of the Results of the Experiment

We explain the experimental results in this paragraph by introducing a preparatory study that was conducted on four features for eight generators. The study was conducted at the Mathematical department, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. The remaining 8 records were offered to this university with comparable presenting qualities, such as comprehensive history, date of manufacture, volume, power, and wren done. Due to similar features, the data in the information system for just eight features, as shown in Table4, discuss energy conservation.Table 4
Table of the attributes.
EXℏ1ℏ2ℏ3ℏ4Dℵ12222Nℵ22112Rℵ32112Nℵ41113Rℵ51222Rℵ62213Nℵ72211Rℵ82213NThe following example is taken from an engineering problem.Example 3.
In Table4, our universe contains eight different generators, ℵ=ℵ1,ℵ2,ℵ3,ℵ4,ℵ5,ℵ6,ℵ7,ℵ8. Moreover, there are four features and a decision. The feature set is ℏ=ℏ1,ℏ2,ℏ3,ℏ4 where the first two are two different transmission lines and the second two are for self-generation and self-load. 1 denotes low, 2 denotes medium, and 3 denotes high.
Now, using the connection that is relevant to the nature of the examined problem, we design a minimal structure space. It is worth noting that we define the relationship in each case based on the expert’s specifications. In this instance,aℜb↔μa,b>4/5, where μa,b is the sum of the similar features between a and b divided on the number of features.
From Table5, we have ℜℵ1=ℵ1, ℜℵ2=ℜℵ3=ℵ2,ℵ3, ℜℵ4=ℵ4, ℜℵ5=ℵ5, ℜℵ6=ℵ6,ℵ8=ℜp8, ℜℵ7=ℵ7. Then, the minimal structure space on ℵ is as follows:(i)
ℳℵ=ℵ,∅,ℵ1,ℵ2,ℵ3,ℵ4,ℵ5,ℵ6,ℵ8,ℵ7 and(ii)
ℳℵc=ℵ,∅,ℵ/ℵ1,ℵ1,ℵ4−ℵ8,ℵ/ℵ4,ℵ/ℵ5,ℵ/ℵ6,ℵ8,ℵ/ℵ7.
We can calculate the accuracy of decision making for two groups of features using the rough set theory:(1)
DN=ℵ1,ℵ3,ℵ6,ℵ8, the set of normal generators.(2)
DR=ℵ2,ℵ4,ℵ5,ℵ7, the set of rest generators.
Therefore, the lower, the upper, and the accuracy ofDN are calculated as follows:(7)ℜ−DN=ℵ1,ℵ6,ℵ8,ℜ+DN=ℵ1,ℵ2,ℵ3,ℵ6,ℵ8,(8)ΓDN=35,respectively.
Also, the lower, the upper, and the accuracy ofDR are calculated as follows:(9)ℜ−DR=ℵ2,ℵ5,ℵ7,ℜ+DR=ℵ2,ℵ3,ℵ4,ℵ5,ℵ7,(10)ΓDR=35,respectively.
Another technique to determine accuracy is to create a dissimilarity table instead of a similarity table. In dissimilarity, a variable precision model can be used to improve it. When we compare our method to Pawlak’s method [8] for determining decision-making accuracy, we get the same result.
We use the degree of similarity between futures using Definition6 to find Table 5.
Moreover, this gives the accuracy as shown in Table6.Table 5
Similarities between features of eight generators.
ℵ1ℵ2ℵ3ℵ4ℵ5ℵ6ℵ7ℵ8ℵ111/21/203/41/21/21/2ℵ21/2111/21/41/21/21/2ℵ31/2111/21/41/21/21/2ℵ401/21/211/41/21/41/2ℵ53/41/41/41/411/41/41/4ℵ61/21/21/21/21/413/41ℵ71/21/21/21/41/43/413/4ℵ81/21/21/21/21/413/41(11)Ωℏ=ℵi|ℵi∈ℵ=ℵ1,ℵ2,ℵ3,ℵ4,ℵ5,ℵ6,ℵ8,ℵ7.Table 6
Accuracy table of the similarity matrix.
Decision setDN=ℵ1,ℵ3,ℵ6,ℵ8DR=ℵ2,ℵ4,ℵ5,ℵ7ℜ−ℵ1,ℵ6,ℵ8ℵ4,ℵ5,ℵ7ℜ+ℵ1,ℵ2,ℵ3,ℵ6,ℵ8ℵ2,ℵ3,ℵ4,ℵ5,ℵ7Accuracy3/53/5


### 3.2. Reduction of Attributes

Here begins the reduction for the attributes by constructing similarity matrices by removing one of the features and noticing how that will affect the accuracy of the decision set.The results in Table7 denote the values by removing ℏ4, while Table 8 shows the accuracy.Table 7
Table ofℏ=ℏ1,ℏ2,ℏ3.
ℏ1,ℏ2,ℏ3ℵ1ℵ2ℵ3ℵ4ℵ5ℵ6ℵ7ℵ8ℵ111/31/302/32/32/32/3ℵ21/3112/302/32/32/3ℵ31/3111/21/42/32/32/3ℵ402/32/311/31/31/31/3ℵ52/3001/311/31/31/3ℵ62/32/32/31/31/3111ℵ72/32/32/31/31/3111ℵ82/32/32/31/31/3111(12)ℳℵ=ℵ,∅,ℵ1,ℵ2,ℵ3,ℵ4,ℵ5,ℵ6,ℵ7,ℵ8and,ℳℵc=ℵ,∅,Xℵ1,Xℵ2,ℵ3,Xℵ4,Xℵ5,ℵ1−ℵ5.Table 8
Accuracy table ofℏ=ℏ1,ℏ2,ℏ3.
Decision setDN=ℵ1,ℵ3,ℵ6,ℵ8DR=ℵ2,ℵ4,ℵ5,ℵ7ℜ−ℵ1ℵ4,ℵ5ℜ+ℵ1,ℵ2,ℵ3,ℵ6,ℵ7,ℵ8ℵ2,ℵ3,ℵ4,ℵ5,ℵ6,ℵ7,ℵ8Accuracy1/62/7


#### 3.2.1. Forℏ=ℏ1,ℏ2,ℏ3

Theℏ4 variable is not superfluous and ℏ1,ℏ2,ℏ3 is not a reduction. We repeat the same steps for all possible subsets of ℏ.


#### 3.2.2. Forℏ=ℏ1,ℏ2,ℏ4

Theℏ3 variable is superfluous and ℏ1,ℏ2,ℏ4 is a reduction.


#### 3.2.3. Forℏ=ℏ1,ℏ3,ℏ4

Theℏ2 variable is superfluous and ℏ1,ℏ3,ℏ4 is a reduction.


#### 3.2.4. Forℏ=ℏ2,ℏ3,ℏ4

Theℏ1 variable is not superfluous and ℏ2,ℏ3,ℏ4 is not a reduction.We conclude thatℏ1,ℏ2,ℏ4,ℏ1,ℏ3,ℏ4 is the class of 2 effective feature subsets and the core of effective features is ℏ1,ℏ4.


### 3.3. Using a Variable Precision Rough Set Model to Improve Accuracy

Because decision making is so important in engineering situations, we can use a variable precisionβ as suggested in [21, 22] to try to improve accuracy. We may see the inconsistency when we compare the definitions of ℳ-interior and ℳ-closure with a majority β to traditional definitions. As a result, we present a novel ℳ-interior and ℳ-closure generalization concept as follows:Definition 7.
Letℵ be a nonempty set and ℳ a ℳ-structure on ℵ. For a subset N of ℵ and variable precision 0<β≤0.5,(1)
ℳβ¯-N=⊓ℱ⊑ℵ:Cℱ,N<1−β.(2)
ℳβ¯-N=⊔O⊓D,CO,N≤β,O∈ℳ.Example 4.
Comparison between our improved method and Pawlak’s approach.
From Table9, at the variable β=0.5 in the original information system, then(1)
ℜ¯0.5Dℵ=ℵ1,ℵ3,ℵ6,ℵ8, ℜ¯0.5Dℵ=ℵ1,ℵ2,ℵ3,ℵ6,x8, and Γ0.5Dℵ=4/5.(2)
ℜ¯0.5Dℜ=ℵ2,ℵ4,ℵ5,ℵ7, ℜ¯0.5Dℜ=ℵ2,ℵ3,ℵ4,ℵ5,x7, and Γ0.5Dℜ=4/5.
Based on a variable precision rough set model, we can see that the accuracy is improving in Example4. Furthermore, when it comes to removing ℏ3 and ℏ2, the accuracy does not change. Table 9 illustrates that our method is improving in comparison to Pawlak’s method [1].Table 9
Comparison between our improved method and Pawlak’s approach.
NThe current approachPawlak’s approach [?]ℜ¯ℜ¯Γℬ0.5ℜ¯ℜ¯ΓDN=ℵ1,ℵ3,ℵ6,ℵ8ℵ1,ℵ3,ℵ6,ℵ8ℵ1,ℵ2,ℵ3,ℵ6,ℵ84/5ℵ1,ℵ6,ℵ8ℵ1,ℵ2,ℵ3,ℵ6,ℵ83/5DR=ℵ2,ℵ4,ℵ5,ℵ7ℵ2,ℵ4,ℵ5,ℵ7ℵ2,ℵ3,ℵ4,ℵ5,ℵ74/5ℵ4,ℵ5,ℵ7ℵ2,ℵ3,ℵ4,ℵ5,ℵ73/5


## 3.1. Discussion of the Results of the Experiment

We explain the experimental results in this paragraph by introducing a preparatory study that was conducted on four features for eight generators. The study was conducted at the Mathematical department, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. The remaining 8 records were offered to this university with comparable presenting qualities, such as comprehensive history, date of manufacture, volume, power, and wren done. Due to similar features, the data in the information system for just eight features, as shown in Table4, discuss energy conservation.Table 4
Table of the attributes.
EXℏ1ℏ2ℏ3ℏ4Dℵ12222Nℵ22112Rℵ32112Nℵ41113Rℵ51222Rℵ62213Nℵ72211Rℵ82213NThe following example is taken from an engineering problem.Example 3.
In Table4, our universe contains eight different generators, ℵ=ℵ1,ℵ2,ℵ3,ℵ4,ℵ5,ℵ6,ℵ7,ℵ8. Moreover, there are four features and a decision. The feature set is ℏ=ℏ1,ℏ2,ℏ3,ℏ4 where the first two are two different transmission lines and the second two are for self-generation and self-load. 1 denotes low, 2 denotes medium, and 3 denotes high.
Now, using the connection that is relevant to the nature of the examined problem, we design a minimal structure space. It is worth noting that we define the relationship in each case based on the expert’s specifications. In this instance,aℜb↔μa,b>4/5, where μa,b is the sum of the similar features between a and b divided on the number of features.
From Table5, we have ℜℵ1=ℵ1, ℜℵ2=ℜℵ3=ℵ2,ℵ3, ℜℵ4=ℵ4, ℜℵ5=ℵ5, ℜℵ6=ℵ6,ℵ8=ℜp8, ℜℵ7=ℵ7. Then, the minimal structure space on ℵ is as follows:(i)
ℳℵ=ℵ,∅,ℵ1,ℵ2,ℵ3,ℵ4,ℵ5,ℵ6,ℵ8,ℵ7 and(ii)
ℳℵc=ℵ,∅,ℵ/ℵ1,ℵ1,ℵ4−ℵ8,ℵ/ℵ4,ℵ/ℵ5,ℵ/ℵ6,ℵ8,ℵ/ℵ7.
We can calculate the accuracy of decision making for two groups of features using the rough set theory:(1)
DN=ℵ1,ℵ3,ℵ6,ℵ8, the set of normal generators.(2)
DR=ℵ2,ℵ4,ℵ5,ℵ7, the set of rest generators.
Therefore, the lower, the upper, and the accuracy ofDN are calculated as follows:(7)ℜ−DN=ℵ1,ℵ6,ℵ8,ℜ+DN=ℵ1,ℵ2,ℵ3,ℵ6,ℵ8,(8)ΓDN=35,respectively.
Also, the lower, the upper, and the accuracy ofDR are calculated as follows:(9)ℜ−DR=ℵ2,ℵ5,ℵ7,ℜ+DR=ℵ2,ℵ3,ℵ4,ℵ5,ℵ7,(10)ΓDR=35,respectively.
Another technique to determine accuracy is to create a dissimilarity table instead of a similarity table. In dissimilarity, a variable precision model can be used to improve it. When we compare our method to Pawlak’s method [8] for determining decision-making accuracy, we get the same result.
We use the degree of similarity between futures using Definition6 to find Table 5.
Moreover, this gives the accuracy as shown in Table6.Table 5
Similarities between features of eight generators.
ℵ1ℵ2ℵ3ℵ4ℵ5ℵ6ℵ7ℵ8ℵ111/21/203/41/21/21/2ℵ21/2111/21/41/21/21/2ℵ31/2111/21/41/21/21/2ℵ401/21/211/41/21/41/2ℵ53/41/41/41/411/41/41/4ℵ61/21/21/21/21/413/41ℵ71/21/21/21/41/43/413/4ℵ81/21/21/21/21/413/41(11)Ωℏ=ℵi|ℵi∈ℵ=ℵ1,ℵ2,ℵ3,ℵ4,ℵ5,ℵ6,ℵ8,ℵ7.Table 6
Accuracy table of the similarity matrix.
Decision setDN=ℵ1,ℵ3,ℵ6,ℵ8DR=ℵ2,ℵ4,ℵ5,ℵ7ℜ−ℵ1,ℵ6,ℵ8ℵ4,ℵ5,ℵ7ℜ+ℵ1,ℵ2,ℵ3,ℵ6,ℵ8ℵ2,ℵ3,ℵ4,ℵ5,ℵ7Accuracy3/53/5


## 3.2. Reduction of Attributes

Here begins the reduction for the attributes by constructing similarity matrices by removing one of the features and noticing how that will affect the accuracy of the decision set.The results in Table7 denote the values by removing ℏ4, while Table 8 shows the accuracy.Table 7
Table ofℏ=ℏ1,ℏ2,ℏ3.
ℏ1,ℏ2,ℏ3ℵ1ℵ2ℵ3ℵ4ℵ5ℵ6ℵ7ℵ8ℵ111/31/302/32/32/32/3ℵ21/3112/302/32/32/3ℵ31/3111/21/42/32/32/3ℵ402/32/311/31/31/31/3ℵ52/3001/311/31/31/3ℵ62/32/32/31/31/3111ℵ72/32/32/31/31/3111ℵ82/32/32/31/31/3111(12)ℳℵ=ℵ,∅,ℵ1,ℵ2,ℵ3,ℵ4,ℵ5,ℵ6,ℵ7,ℵ8and,ℳℵc=ℵ,∅,Xℵ1,Xℵ2,ℵ3,Xℵ4,Xℵ5,ℵ1−ℵ5.Table 8
Accuracy table ofℏ=ℏ1,ℏ2,ℏ3.
Decision setDN=ℵ1,ℵ3,ℵ6,ℵ8DR=ℵ2,ℵ4,ℵ5,ℵ7ℜ−ℵ1ℵ4,ℵ5ℜ+ℵ1,ℵ2,ℵ3,ℵ6,ℵ7,ℵ8ℵ2,ℵ3,ℵ4,ℵ5,ℵ6,ℵ7,ℵ8Accuracy1/62/7


### 3.2.1. Forℏ=ℏ1,ℏ2,ℏ3

Theℏ4 variable is not superfluous and ℏ1,ℏ2,ℏ3 is not a reduction. We repeat the same steps for all possible subsets of ℏ.


### 3.2.2. Forℏ=ℏ1,ℏ2,ℏ4

Theℏ3 variable is superfluous and ℏ1,ℏ2,ℏ4 is a reduction.


### 3.2.3. Forℏ=ℏ1,ℏ3,ℏ4

Theℏ2 variable is superfluous and ℏ1,ℏ3,ℏ4 is a reduction.


### 3.2.4. Forℏ=ℏ2,ℏ3,ℏ4

Theℏ1 variable is not superfluous and ℏ2,ℏ3,ℏ4 is not a reduction.We conclude thatℏ1,ℏ2,ℏ4,ℏ1,ℏ3,ℏ4 is the class of 2 effective feature subsets and the core of effective features is ℏ1,ℏ4.


## 3.2.1. Forℏ=ℏ1,ℏ2,ℏ3

Theℏ4 variable is not superfluous and ℏ1,ℏ2,ℏ3 is not a reduction. We repeat the same steps for all possible subsets of ℏ.


## 3.2.2. Forℏ=ℏ1,ℏ2,ℏ4

Theℏ3 variable is superfluous and ℏ1,ℏ2,ℏ4 is a reduction.


## 3.2.3. Forℏ=ℏ1,ℏ3,ℏ4

Theℏ2 variable is superfluous and ℏ1,ℏ3,ℏ4 is a reduction.


## 3.2.4. Forℏ=ℏ2,ℏ3,ℏ4

Theℏ1 variable is not superfluous and ℏ2,ℏ3,ℏ4 is not a reduction.We conclude thatℏ1,ℏ2,ℏ4,ℏ1,ℏ3,ℏ4 is the class of 2 effective feature subsets and the core of effective features is ℏ1,ℏ4.


## 3.3. Using a Variable Precision Rough Set Model to Improve Accuracy

Because decision making is so important in engineering situations, we can use a variable precisionβ as suggested in [21, 22] to try to improve accuracy. We may see the inconsistency when we compare the definitions of ℳ-interior and ℳ-closure with a majority β to traditional definitions. As a result, we present a novel ℳ-interior and ℳ-closure generalization concept as follows:Definition 7.
Letℵ be a nonempty set and ℳ a ℳ-structure on ℵ. For a subset N of ℵ and variable precision 0<β≤0.5,(1)
ℳβ¯-N=⊓ℱ⊑ℵ:Cℱ,N<1−β.(2)
ℳβ¯-N=⊔O⊓D,CO,N≤β,O∈ℳ.Example 4.
Comparison between our improved method and Pawlak’s approach.
From Table9, at the variable β=0.5 in the original information system, then(1)
ℜ¯0.5Dℵ=ℵ1,ℵ3,ℵ6,ℵ8, ℜ¯0.5Dℵ=ℵ1,ℵ2,ℵ3,ℵ6,x8, and Γ0.5Dℵ=4/5.(2)
ℜ¯0.5Dℜ=ℵ2,ℵ4,ℵ5,ℵ7, ℜ¯0.5Dℜ=ℵ2,ℵ3,ℵ4,ℵ5,x7, and Γ0.5Dℜ=4/5.
Based on a variable precision rough set model, we can see that the accuracy is improving in Example4. Furthermore, when it comes to removing ℏ3 and ℏ2, the accuracy does not change. Table 9 illustrates that our method is improving in comparison to Pawlak’s method [1].Table 9
Comparison between our improved method and Pawlak’s approach.
NThe current approachPawlak’s approach [?]ℜ¯ℜ¯Γℬ0.5ℜ¯ℜ¯ΓDN=ℵ1,ℵ3,ℵ6,ℵ8ℵ1,ℵ3,ℵ6,ℵ8ℵ1,ℵ2,ℵ3,ℵ6,ℵ84/5ℵ1,ℵ6,ℵ8ℵ1,ℵ2,ℵ3,ℵ6,ℵ83/5DR=ℵ2,ℵ4,ℵ5,ℵ7ℵ2,ℵ4,ℵ5,ℵ7ℵ2,ℵ3,ℵ4,ℵ5,ℵ74/5ℵ4,ℵ5,ℵ7ℵ2,ℵ3,ℵ4,ℵ5,ℵ73/5


## 4. Conclusion

The suggested approach helps in selecting effective features in some type of information system for which we cannot get reducts using the original rough set model.The approach is suitable for researchers from application fields since it does not need lot of mathematics. The subsequent work will consider control strategies (related to medicine) and mathematical models (related to fuzzy with a minimal structure) in the incipient stage of cancer [23–25].



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*Source: 5450344-2022-06-15.xml*