Communications on Applied Nonlinear Analysis 

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Rough Set Analysis for Categorizing Motorcycles based on 100 Cubic 

Centimeters (CC) Engine Displacements 
 

1R. Venmani, 2,*K. Rajendran 

1Research Scholar, Vels Institute of Science Technology and Advanced Studies, Chennai, Tamil Nadu, India. 

2Associate Professor, Vels Institute of Science Technology and Advanced Studies, Chennai, Tamil Nadu, India 

1 Email id: rvenmani@yahoo.co.in 

2 Corresponding author: gkrajendra59@gmail.com 

 

Article History: 

Received: 01-03-2024 

Revised: 19-04-2024 

Accepted: 15-05-2024 

Abstract:  

Rough Set Theory (RST) is a mathematical approach used for dealing with uncertainty and 

vagueness in decision-making and data analysis. It provides a framework for classifying 

objects into different equivalence classes based on their attributes or characteristics. In RST, 

the concept of different 𝑐𝑐 bikes can be analyzed based on their attributes or characteristics. 

Each bike can be represented as an object with a set of attributes such as engine displacement, 

weight, top speed, fuel efficiency and price. Another class may consist of bikes with lower 

engine displacement, lighter weight, and better fuel efficiency, which could be more suitable 

for daily commuting. By applying RST, we can analyze the relationship between these 

attributes and determine the essential and non-essential features of 100 𝐶𝐶 bikes. This 

analysis can help in decision making processes, such as choosing the right bike based on 

specific requirements or preferences. It's important to note that the application of RST to100 

𝑐𝑐 bikes is just one example of how this mathematical approach can be used in decision-

making and data analysis. The specific attributes and classes may vary depending on the 

context and purpure of the analysis. 

.Keywords: Rough set, motorbike, energy, decision-making, reduct 

1. Introduction 

In decision-making and data analysis, rough set theory provides a mathematical framework for 

handling ambiguity and uncertainty. Pawlak first presented it as a method of handling imprecise and 

imperfect data in the early 1980s [1]. The concept of approximations is the foundation of rough set 

theory. Separating the data into lower and upper approximations offers a formal technique for its 

analysis and classification. The lower approximation represents the set of things that unquestionably 

belong to one idea, while the higher approximation represents the set of potential concepts. 

The lower and higher approximation operators in Pawlak rough set theory [2] are based on equivalence 

relations. The need for an equivalence relation in Pawlak rough set models, however, appears to be a 

highly rigorous requirement that can constrain the applicability of the rough set models. Based on the 

comparable classes produced by the attribute values, rough set theory divides into three regions: 

boundary, lower approximation, and upper approximation. According to the data gathered, the upper 

approximation comprises all the things that can be classified presumably, while the lower 

approximation contains all the objects that are classified with certainty. The boundary is the difference 

between the upper and lower approximations [3]. 



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From the perspective of decision-making, an attained rule results in an optimistic decision. In multi-

granulation rough set theory, this model is known as the optimistic rough set model. Additionally, 

Qian [4,5] defines another model known as the pessimistic multi-granulation rough set model. 

Equivalence relations are expanded to comparable relations or generic binary relations in the extension 

of the classical rough set model to the generalized rough set model [6]. Data distribution-based rough 

set models, including the decision-theoretic, the probabilistic, the cloud rough set model, etc [7-9]. 

It is essential to have a technique for making decisions that will help in identifying and choosing the 

best motorized vehicles. There are a number of approaches for creating a decision support system 

(DSS), and one method that may be utilized for decision making systems is WP (Weighted Product) 

[10]. Although bikes with small engines spend less fuel (per person-kilometer) than cars, the situation 

may be the opposite for motorcycles with larger engines that are more potent. Additionally, if short 

non-motorized journeys are replaced with longer motorbike excursions, the overall energy efficiency 

of the transportation system declines. Pfaffenbichler and Circella [11] analyzed the conditions under 

which motorbikes can significantly contribute to the development of an energy-efficient and 

sustainable transportation system.  

Butalia et al. [12] revealed that two innovative methods are implemented in Java 1.5 to determine, in 

light of the relative attribute dependency, the best reductions of condition attributes. The first algorithm 

provides a simple reduct, while the second one provides a reduct with minimum attributes. In order to 

create the core of the attribute set or the effective reduct set, unnecessary attributes are removed using 

Vashist's [13] suggested algorithm. The space complexity and computed time of the suggested 

approach is compared to that of other known algorithms. It is proven that the suggested method reduces 

computation time and space without sacrificing the efficacy and quality of the output. Wu and Mi [14] 

investigated rough sets' mathematical structure in infinite universes of discourse. 

Nowadays, there are many various types of motorcycles, including mopeds with a 50 cubic centimeter 

engine, scooters with engines between 50 and 250 cubic centimeters, and motorbikes with engines up 

to 1,000 centimeters and even more. Based on these characteristics, we may divide bikes into various 

equivalence classes in this study using RST. Also, we discover the important and optional aspects of 

various CC motorbikes by using RST and examining the relationship between these attributes. The 

results of this research can be useful for making decisions, such as selecting the best bike for a 

particular set of requirements or desires. 

Let's consider that we have a list of bikes with the following attributes: 

1. Brand (B): The brand of the bike (e.g., Honda, Yamaha, Suzuki, etc.) as (𝑥1, 𝑥2, 𝑥3, 𝑥4…) 

2. Engine Displacement (E): The engine displacement in cubic centimeters (e.g., 100 cc, 150 cc, 

etc.). 

3. Fuel Efficiency (F): The fuel efficiency of the bike in kilometers per litre (e.g., 40 km/l, 50 

km/l, etc.). 

4. Price (P): The price of the bike in a certain currency (e.g., USD, INR, etc.). 

5. Style (S): The style of the bike (e.g., commuter, sport, cruiser, etc.). 

 



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2.  Rough Set Analysis 

Now, let's apply a simplified rough set analysis with seven types of analysis: 

2.1 Lower Approximation (Certain Knowledge) 

Identify the bikes that definitely belong to the 100 cc category based on their engine displacement of 

a set 𝑋 with respect to 𝑆. The lower approximation is represented by 𝑆∗(𝑥). 

 𝑆∗(𝑋) = {𝑥: 𝑆(𝑥) ⊆ 𝑋}. 

2.2 Upper Approximation (Potential Knowledge) 

Identify the bikes that potentially belong to the 100 cc category based on their engine displacement 𝑋 

with respect to 𝑆. The upper approximation is referred by 𝑆∗(𝑥). 

𝑆∗(𝑋) = {𝑥: 𝑆(𝑥) ⊆ 𝑋}. 

2.3 Negative Region (Contradiction) 

Identify attributes of the set of objects that are contradictory to the 100 cc category of the set 𝑋. 

𝑈 − 𝑆∗(𝑋) 

2.4 Boundary Region (Uncertainty) 

Identify attributes that are ambiguous and contribute to the uncertainty of bike categorisation of a set 

𝑋 with respect to 𝑆 and is denoted by 𝑆𝑁𝑆(𝑋). 

𝑆𝑁𝑆(𝑋) = 𝑆∗(𝑋) − 𝑆∗(𝑋). 

2.5 Indiscernibility Relation (Equivalence)  

Identify bikes that are indiscernible (indistinguishable) based on the selected attributes.  

Let 𝑅 = (𝐴, 𝐵) be an information system, and 𝑋 ⊆ 𝐴. A binary relation 𝐼𝑁𝐷𝑅(𝑋) defined in the 

following way 

𝐼𝑁𝐷𝑅⁡(𝑋) = {(𝑥1, 𝑥2) ∈ 𝐴2 ∣ ∀𝑎𝑋, 𝑎(𝑥1) = 𝑎(𝑥2)} 

is called a 𝑋-indiscernibility relation [𝑥]𝑅. If (𝑥1, 𝑥2) ∈ 𝐼𝑁𝐷𝑅(𝑋), then 𝑥1 and 𝑥2 are indiscernible 

(or indistinguishable) by attributes from 𝑋.  

2.6 Reduct (Minimal Set of Attributes) 

Identify the smallest set of attributes that are sufficient to determine the category of a bike. 

Using these analyses, you would analyze the attributes of the bikes in your list and determine their 

relationships to the 100 cc category in Table 1. This analysis would provide insights into the 

characteristics of bikes that belong to the 100 cc category and help in classifying them based on these 

attributes. 



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Please note that this example is a simplified illustration and may not reflect the full complexity of 

applying rough set theory to real-world data. In practice, rough set analysis involves more rigorous 

mathematical concepts and techniques. 

Table 1. Analysis of engine, power, torque and mileage in 100cc bikes 

BIKE  

(100 CC) ENGINE (cc) POWER TORQUE MILEAGE 

Decision 

Attributes 

𝑥1 97.2 8.02 8.05 70 Good 

𝑥2 97 8.02 8.05 70 Good 

𝑥3 97.2 8.02 8.05 70 Good 

𝑥4 97.2 8.02 8.05 70 Good 

𝑥5 102 7.9 8.3 70 Good 

𝑥6 109.7 8.29 8.7 70 Good 

𝑥7 109.51 8.79 9.3 65 Good 

𝑥8 109.51 8.79 9.3 60 Bad 

𝑥9 109.7 8.19 8.7 83.09 Good 

𝑥10 109.7 8.19 8.7 73.68 Good 

𝑥11 113.2 9.15 9.79 0 Bad 

𝑥12 113.2 9.15 9.89 0 Bad 

 

𝐴 = {𝑥1, 𝑥2, 𝑥3, 𝑥4, … , 𝑥12}  

𝐵 = {ENGINE, POWER, TORQUE,MILEAGE}⁡  

𝑋ENGINE = {97.2, 97, 102, 109.7, 109.51,113.2}  

𝑋POWER = {8.02, 7.9, 8.29, 8.79, 8.19, 9.15}  

𝑋TORQUE = {8.05, 8.3, 8.7, 9.3, 9.79, 9.89}  

𝑋MILEAGE = {60, 65, 70, 73.68, 83.09}  

𝑋DECISION = {Good, Bad}  

 

Consider 𝑃 = {ENGINE} 

𝑃 = elementary⁡sets  

[𝑥1]𝑃 = [𝑥3]𝑃 = [𝑥4]𝑃 = {𝑥1, 𝑥3, 𝑥4}  

[𝑥2]𝑃 = {𝑥2}  

[𝑥5]𝑃 = {𝑥5}  

[𝑥6]𝑃 = [𝑥9]𝑃 = [𝑥10]𝑃 = {𝑥6, 𝑥9, 𝑥10}  

[𝑥11]𝑃 = [𝑥12]𝑃 = {𝑥11, 𝑥12}  



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The Equivalence classes are 

𝑋1 = {𝑥1, 𝑥3, 𝑥4}  

𝑋2 = {𝑥2}  

𝑋3 = {𝑥5}  

𝑋4 = {𝑥6, 𝑥9, 𝑥10}  

𝑋5 = {𝑥11, 𝑥12}  

IND(ENGINE) = {𝑋1, 𝑋2, 𝑋3, 𝑋4, 𝑋5} = {{𝑥1, 𝑥3, 𝑥4}, {𝑥2}, {𝑥5}, {𝑥6, 𝑥9, 𝑥10}, {𝑥11, 𝑥12}}    

𝑆∗(𝑋) = {𝑥2} 

𝑆∗(𝑋) = {𝑥7, 𝑥8} 

𝑆𝑁𝑆(𝑋) = 𝑆∗(𝑋) − 𝑆∗(𝑋) = {𝑥7, 𝑥8} 

𝑈 − 𝑆∗(𝑋) = {𝑥1, 𝑥3, 𝑥4, 𝑥5, 𝑥6, 𝑥9, 𝑥10, 𝑥11, 𝑥12} 

This scatter plot visually represents the bikes' data points in the "ENGINE (cc)" vs. "POWER" 

space. The placement of the data points can give you an idea of how the bikes are distributed in this 

attribute space.  This concept can be extended to additional attributes or even use advanced 

visualization techniques to represent decision boundaries, discernibility, core, and reducts. However, 

visualizing rough set theory concepts comprehensively requires advanced data visualization 

techniques and tools. 

Rough set theory involves mathematical concepts and notations that can be challenging to 

represent in a single diagram. However, we can provide you with a simplified algorithm 1 that 

illustrates the basic idea of rough set theory and its components: universe of discourse, attributes, and 

approximations. 

Algorithm 1 (Python) 

Step1: Find the decision attribute. 

Step 2: Create the Concept class based on the decision attribute, identify all the tuples associated 

with that attribute. 

Step 3: Select the Condition attributes. Determine the lower approximation i.e. indiscernible with 

respect to the selected attributes. 

Step 4: Estimate the upper approximation i.e. the objects in the boundary region are indiscernible. 

Step 5: Find the rough set 

Step 6: Identify the reduct (minimal subset of relevant attributes) 

 



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3.  Implementation of Proposed Algorithm 

Consider implementing the aforementioned technique using a table-based information system. The 

condition and decision attributes make up the information system's column. The values that the 

condition and decision attribute can take from the discourse universe are represented by the rows of 

the information system. Table 2 shows that the universe consists of twelve elements in total. The 

condition attributes' value sets consist of: 

𝑎 = 𝑋1 = {𝑥1, 𝑥3, 𝑥4}  

𝑏 = 𝑋2 = {𝑥2}  

𝑐 = 𝑋3 = {𝑥5}  

𝑑 = 𝑋4 = {𝑥6, 𝑥9, 𝑥10}  

Decision Attribute can take values: 

𝑒0 = 𝑋5 = {𝑥8, 𝑥11, 𝑥12}  

Table 2. Information System 

𝑈 𝑎 𝑏 𝑐 𝑑 𝑒 

𝑥1 1 0 0 0 1 

𝑥2 0 1 0 0 1 

𝑥3 1 0 0 0 1 

𝑥4 1 0 0 0 1 

𝑥5 0 0 1 0 1 

𝑥6 0 0 0 1 1 

𝑥7 0 0 0 0 1 

𝑥8 0 0 0 0 0 

𝑥9 0 0 0 1 1 

𝑥10 0 0 0 1 1 

𝑥11 0 0 0 0 0 

𝑥12 0 0 0 0 0 

 

Using our approach, the first condition attribute, "𝑎," is removed from table 2 and the consistency of 

the remaining table, table 3, is checked. When we have identical options for two or more rows or cases 

with the same values of condition characteristics, a table is considered to be consistent; if not, it is 

inconsistent. 

Table 3. Attribute ' 𝑎 ' is eliminated 

𝑈 𝑏 𝑐 𝑑 𝑒 

𝑥1 0 0 0 1 

𝑥2 1 0 0 1 

𝑥3 0 0 0 1 

𝑥4 0 0 0 1 

𝑥5 0 1 0 1 

𝑥6 0 0 1 1 



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𝑥7 0 0 0 1 

𝑥8 0 0 0 0 

𝑥9 0 0 1 1 

𝑥10 0 0 1 1 

𝑥11 0 0 0 0 

𝑥12 0 0 0 0 

 

Table 3 is created by removing the condition attribute ' 𝑎 ' from Table 2.  

 

Upon examining table 3, it becomes evident that the removal of attribute ' 𝑎 ' from table 2 results in 

inconsistency since, for each of the following values of the condition attributes: ' 𝑏', ' 𝑐 ', ' 𝑑 ' the value 

of the decision attribute, ' 𝑒 ', differs for 𝑥1, 𝑥3, 𝑥4 and 𝑥8, 𝑥11, 𝑥12. 

 

𝑥1, 𝑥3, 𝑥4: 𝑏0𝑐0𝑑0 → 𝑒1
𝑥8, 𝑥11, 𝑥12: 𝑏0𝑐0𝑑0 → 𝑒0

 

Thus, CORE is used with condition attribute' 𝑎 '.  

Table 4 remains after removing characteristic ' 𝑏 ' from Table 2.  

Table 4. Attribute ' 𝑏 ' is eliminated 

𝑈 𝑎 𝑐 𝑑 𝑒 

𝑥1 1 0 0 1 

𝑥2 0 0 0 1 

𝑥3 1 0 0 1 

𝑥4 1 0 0 1 

𝑥5 0 1 0 1 

𝑥6 0 0 1 1 

𝑥7 0 0 0 1 

𝑥8 0 0 0 0 

𝑥9 0 0 1 1 

𝑥10 0 0 1 1 

𝑥11 0 0 0 0 

𝑥12 0 0 0 0 

 

Upon examination, Table 4 appears to be devoid of any discrepancies. Eliminating the characteristic ' 

𝑏 ' results in no inconsistency. Hence, according on the information, attribute   ' 𝑏 ' is not a CORE. 

 

Reduct 1 = {a, c, d} 

Table 5 remains after removing attribute ' 𝑐 ' from Table 2. 

 

 

 



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Table 5. Attribute ' 𝑐 ' is eliminated 

𝑈 𝑎 𝑏 𝑑 𝑒 

𝑥1 1 0 0 1 

𝑥2 0 1 0 1 

𝑥3 1 0 0 1 

𝑥4 1 0 0 1 

𝑥5 0 0 0 1 

𝑥6 0 0 1 1 

𝑥7 0 0 0 1 

𝑥8 0 0 0 0 

𝑥9 0 0 1 1 

𝑥10 0 0 1 1 

𝑥11 0 0 0 0 

𝑥12 0 0 0 0 

 

The removal of the attribute ' 𝑐 ' doesn't result in inconsistent data. As a result, condition characteristic 

' 𝑐 ' is not utilized as the core. However, Table 5's remaining condition attribute is regarded as reduct. 

 

Hence, 

Reduct 2 = {a, b, d} 

Table 6. Attribute ' 𝑑 ' is eliminated 

𝑈 𝑎 𝑏 𝑐 𝑒 

𝑥1 1 0 0 1 

𝑥2 0 1 0 1 

𝑥3 1 0 0 1 

𝑥4 1 0 0 1 

𝑥5 0 0 1 1 

𝑥6 0 0 0 1 

𝑥7 0 0 0 1 

𝑥8 0 0 0 0 

𝑥9 0 0 0 1 

𝑥10 0 0 0 1 

𝑥11 0 0 0 0 

𝑥12 0 0 0 0 

 

There is no inconsistency in the dataset when condition attribute ' 𝑑 ' is removed from table 6.  

As a result of this, attribute ' 𝑑 ' is not a CORE.  

Reduct 3 = {a, b, c} 

Characteristic ' 𝑎 ' is the only essential characteristic. ' 𝑎 '  also appears in each reduct set.  



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Additionally, we understand that  

Core =∩ Reducts 

CORE⁡(C) =∩ {{ Reduct 1}{ Reduct 2}{ Reduct 3}} 

and the Reduct sets are 

Reduct 1 = {a, c, d} 

Reduct 2 = {a, b, d} 

Reduct 3 = {a, b, c} 

CORE⁡(C) = {a} 

This further demonstrates that our algorithm is accurate.  

 

4.  Conclusion 

The application of rough set theory to the task of finding the best bike in different cubic 

centimeters (cc) has provided valuable insights in this paper. By employing rough sets, we can 

effectively analyze and categorize bikes based on various attributes and features relevant to different 

cc categories. This approach helps us to identify the essential attributes that contribute to the superiority 

of a bike within a specific cc range. Through the reduction of redundant information and the extraction 

of essential characteristics, rough set theory streamlines the decision-making process for selecting the 

optimal bike in each cc category. Overall, utilizing rough set theory enhances our ability to make 

informed choices while considering the intricate relationships between attributes and bike performance 

in different cubic centimeter ranges. 

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