ACADEMIC PERFORMANCE EVALUATION USING FUZZY C-MEANS
RAMJEET SINGH YADAV1 & P. AHMED
2
1Research Scholar, Department of Computer Science and Engineering, School Engineering and Technology, Sharda
University, Greater Noida, UP, India.
2Professor, Department of Computer Science and Engineering, School Engineering and Technology, Sharda University,
Greater Noida, UP, India
ABSTRACT
In this paper we explore the applicability of K-means and Fuzzy C-Means clustering algorithms to student
allocation problem that allocates new students to homogenous groups of specified maximum capacity, and analyze effects
of such allocations on the academic performance of students. The paper also presents a Fuzzy set and Regression analysis
based Dynamic Fuzzy Expert System model which is capable of dealing with imprecision and missing data that is
commonly inherited in the student academic performance evaluation. This model automatically converts crisp sets into
fuzzy sets by using C-Means clustering algorithm method. The comparative performance analysis indicates that the student
group formed by Fuzzy C-Means clustering algorithm performed better than groups formed by K-Means and Hard C-
Means clustering algorithms.
KEYWORDS: Fuzzy Logic, Clustering, K-Means Algorithm, Hard C-Means Algorithm, Fuzzy C-Means algorithm,
Fuzzy Expert Systems, Membership Function and Academic Performance Evaluation
INTRODUCTION
The student academic performance evaluation problem can be considered as a clustering problem where clusters (or
classes) are formed on the basis of intelligence level of students, and the class size should not exceed the predefined
capacity. The intelligence level wise grouping is essential for maintaining the homogeneity of the group otherwise it would
be difficult to provide good educational services to highly diverse student population. Moreover, homogenous grouping of
students having similar ranking (or some other measures) into classes would further make the academic performance
results fairer, realistic and comparable.
The existing practice of score aggregation based student similarity or his/her rank determination is unrealistic
because scores are assembled from different score combinations. Universities use GPA (Grade Point Average), an
example of score aggregation based measure, as a major criterion for student selection. Most universities consider 3.0 and
above GPA as an indicator of good academic performance, hence, it remains the most common factor used by the
academic planners to evaluate progression in an academic environment (S. S. Sansgiry, et al., 2006) despite its
limitations in providing a comprehensive view of the state of students’ performance evaluation and simultaneously
discovering important details from their continuous performance assessments (O.J. Oyelade, et al., 2010). Furthermore,
average score may lead to wrong conclusion. Especially, when details of data from which it is computed are not given.
It has been observed that there are factors, other than academic ones, pose barriers to students attaining and
maintaining high. Therefore, grouping or clustering students using cognitive as well as affective factors into different
categories, and then defining performance measure may be a realistic approach. For example, consider a scenario where
two students score 50, 60, 70, and 70, 60, 50 in three tests respectively. The average mark obtained by each is 60. Can we
International Journal of Computer Science Engineering
and Information Technology Research (IJCSEITR)
ISSN 2249-6831
Vol.2, Issue 4, Dec 2012 55-84 © TJPRC Pvt. Ltd.,
56 Ramjeet Singh Yadav & P. Ahmed
conclude, from the average, that intelligence level of both the students is same? Of course not! The data indicates that one
student is improving while the other is deteriorating consistentlyit may imply that one student is learning consistently
from his experience.
The example illustrates that the student ranking or modeling academic performance evaluation method should be
based on class homogeneity− a view point supported by other researchers (Z. Zukhri, et al., 2008). In addition to such
computational issues, as mentioned before, the imprecision and vagueness in data collection process also affect the
performance indicators evaluation. Unfortunately, this aspect is ignored in practice because generally hard computing
based process, procedures and techniques are used in performance evaluation. Observation shows the soft computing
techniques are more powerful and better suited in providing feasible solutions to the problems that deal with uncertainties
and vagueness. For instance, the fuzzy logic, handles, imprecision, and uncertainty in a natural manner by providing a
human oriented knowledge representation is possible, but it is weak in self learning and generalization of rules. A
combination of fuzzy logic and genetic algorithm is expected to eliminate this weakness. Now, their power is being
investigated.
In their recent work Mankad K. et al., (2011) have reported an evolving rule based model for identification of
multiple intelligence. Their genetic-fuzzy hybrid model identifies human intelligence. Zainudin Zukhri and Khairuddin
Omar (2008) have reported successful application of Genetic Algorithm for solving difficult optimization problems in
new students’ allocation problem. Vuda Sreenivasarao et al, (2012) developed a model for improving academic
performance evaluation of students based on data warehousing and data mining techniques that use soft-computing
intensively. Their analysis indicates that the group homogeneity improves students academic performance thereby
enhances education quality.
An Artificial Neural Network (ANN) model reported in Obinity Afolayan Ayodele et al., (2010) that along with
computation also derives meaning from imprecise data, extracts patterns and detects trends. This ability has added new
dimensions in comprehending the complex phenomena that is buried in students’ data otherwise might have gone
unnoticed using hard computing techniques.
In practice, whether phenomena discovery or performance indicator computation, their accuracy depends on the
data quality that in turn depends on the accuracy of data collection process and representation techniques. In order to
address the data related issues, in education domain, Biswas (1995) suggested use of fuzzy sets (Zadeh, 1965) in students’
answer-sheets evaluation. Wang H.Y. and Chen S.M. (2007) recommended use of vague sets (Gau and Buehrer, 1993)
instead of fuzzy sets to represents the vague marks of each question where the evaluator can use vague values to indicate
the degree of the evaluators’ satisfaction for each question.
In fuzzy sets the membership evaluation (characteristics function definition) is a major issue. In order to apply the
fuzzy set in education domain effectively, there have been a lot efforts in defining the effective membership. Bai S.M. and
Chen S.M. (2008) define fuzzy membership functions for fuzzy rules; Law C.K. (1996) used fuzzy numbers, and for more
information on this issue consult: Chen S.M. and Lee C.H. (1999), Wang H.Y. and Chen S.M. (2006), Stathacopoulou R.,
et al. (2004), Guh Y.Y., et al. (2008), Gokmen E., et al. (2010), Hameed I.A. (2011), Baylari A. and Montazer Gh. A.
(2009), Posey C.L. and Hawkes L.W. (1996), Stathacopoulou R., et al. (2007), Bhatt R. and Bhatt D. (2011), and Zhou
D. and Ma J. (2000). The research works cited in the preceding paragraph indicates that the fuzzy logic, neural network
and fuzzy neural network have already been employed in student modeling systems but almost nothing or very little has
been mentioned about automatic generation of fuzzy membership function. This paper describes a method for automatic
Academic Performance Evaluation Using Fuzzy C-Means 57
generation of membership function for student academic performance evaluation. For this purpose we have used fuzzy C-
means Clustering algorithm for automatic generation of membership function. In order to obtain the homogeneous clusters
(or classes) of students, we have studied the performance of Fuzzy C-Means and K-Means clustering algorithms for
student population clustering. For both the cases, we have developed students’ academic performance evaluation models.
In this research paper, the proposed dynamic Fuzzy Expert System automatically convert the crisp data into fuzzy
set and also calculate the total mark of a student sit in semsete-1 and semester-2 examination. The proposed idea is a
starting attempt to use the applicability of Fuzzy C-Means clustering algorithm to analyze and find out modeling academic
performance and to improve the quality of the students and teachers performance in educational domains. Fuzzy C-Means
Clustering algorithm is a data warehousing and data mining techniques. Due to this reason it is more effective for improve
the quality of education. The management can use some techniques to improve the course outcome according to the
improve knowledge. Such knowledge can be used to give a good understanding of student’s enrollment pattern in the
course under study, the faculty and managerial decision maker in order to utilize the necessary steps needed to provide
extra classes. On the other hand, such types of knowledge the management system can be enhance their policies, improve
their strategies and improve the quality of the system.
The paper, besides introduction, has nine sections. The next Section gives a survey on Fuzzy approaches in
academic performance evaluation. Section three describes Data Cluster Analysis for Academic Performance Evaluation.
Section four describes Expert System and their components. Section five describes the architecture of the proposed
Dynamic Fuzzy Expert System (DEFS). Section six describes experimental results of K-Means clustering technique. In
Section seven, we present the experimental results of DEFS. Section eight describes the comparison of classical, Fuzzy
Expert System, K-Means and Fuzzy C-Means Clustering methods for Modeling Academic Performance Evaluation. We
conclude paper with section nine.
SURVEY OF FUZZY APPROACHES IN ACADEMIC PERFORMANCE EVALUATION
While fuzzy logic techniques have earned their place in a variety of field ranging from engineering to financial
sector, to medicine, few efforts have been made to test the potential usefulness of these methods in the modeling academic
performance evaluation. This section discusses the literature survey about the past and current research application of fuzzy
logic. It discusses about the academic achievement of student and teacher, prediction model and academic performance
evaluation fuzzy logic approaches in academic performance evaluation.
A. Modeling Academic Performance Evaluation Using Soft Computing Techniques: A Fuzzy Logic Approach
Ramjeet Singh Yadav et al., (2011) presented a method to deal with the modeling academic performance evaluation
using fuzzy logic. Academic performance evaluation with fuzzy expert system comprised with three steps:
1. Fuzzification of inputs semester examination results and output performance value.
2. Determine of application rules and inference method.
3. Defuzzification of performance value.
Fuzzification of Semester Examination Results and Performance Value
Fuzzification of semester examinations was carried out using input variables and their membership functions of
fuzzy sets. Each student has two semester results both of which from input variables of the fuzzy logic based expert
system. Each input variable has five triangular membership functions. The fuzzy sets of the input and output variable are
given in Table-1 and Table-2 respectively.
58 Ramjeet Singh Yadav & P. Ahmed
Table 1: Fuzzy Set of Input Variable
Linguistic variable Interval
Very Low (VL) (0, 0, 25)
Low (L) (0, 25, 50)
Average (A) (25, 50, 75)
High (H) (50, 75, 100)
Very High (VH) (75, 100, 100)
Table 2: Fuzzy Set of Output Variable
Linguistic Variable Interval
Very Low (VL) (0, 0, 25)
Low (L) (0, 25, 50)
Average (A) (25, 50, 75)
High (H) (50, 75, 100)
Very High (VH) (75, 100, 100)
Experimental Results
Ramjeet Singh Yadav et al., (2011) have proposed Fuzzy Expert System was tested with 20 student’s marks
obtained in the Department of Computer Science and Applications, MG Kashi Vidyapith Varanasi, UP, India; appeared in
semester-1 and semester-2 examinations. For each student, both semester examination scores were fuzzified by means of
the triangular membership function. Active membership functions were calculated according to rule table, using Mamdani
fuzzy decision techniques. The output (performance value) was calculated and then defuzzified by calculating the centre
(centroid) of the resulting geometrical shape. This sequence was repeated using the semester examination scores for each
student. Table-1 shows the semester scores and calculated students performance value.
Table 3: Semester Score and Calculated Performance Value
S.No. Semsester-
1
Semester-
2
Performance Value
Fuzzy-1 Fuzzy-2
1. 40 65 0.530 0.627
2. 20 35 0.243 0.243
3. 50 65 0.654 0.750
4. 10 20 0.203 0.203
5. 45 65 0.576 0.676
6. 65 45 0.576 0.625
7. 34 60 0.462 0.530
8. 48 55 0.533 0.758
9. 56 90 0.759 0.759
10. 74 70 0.735 0.440
11. 45 50 0.440 0.575
12. 89 100 0.908 0.908
13. 100 100 0.920 0.920
14. 65 35 0.500 0.387
15. 48 50 0.473 0.473
16. 45 55 0.500 0.490
17. 55 25 0.310 0.310
18. 84 80 0765 0.778
19. 63 65 0.639 0.753
20. 28 30 0.310 0.241
Both inputs had same triangular membership functions. In the above Table-1, students 5 and 6 have same
performance value. We conclude that the level of intelligence of both students is same. This is a fallacious conclusion since
we find from the above Table-1 that the student 5 has improved consistently while student 6 has deteriorated consistently.
Academic Performance Evaluation Using Fuzzy C-Means 59
This is the drawback of Fuzzy Expert System proposed by Ramjeet Singh Yadav et al., (2011). Here, also pointed
out that the problem of in this method is that fuzzy membership value is fixed by the expert domain. Solve such type of
problem by the Fuzzy C-Means algorithm.
B. Evaluation of Teacher’s Performance Evaluation Using Fuzzy Logic Techniques
Sirigiri Pavani et al., (2012) presented a method to deal with the evaluation of teacher’s academic performance
evaluation using fuzzy logic techniques. The descriptions of this method are given below:
Fuzzification of Semester Examination Results and Performance Value
Fuzzification of input parameters of teacher’s performance was carried out using input variables and their
membership functions of fuzzy sets are given below in Table-4.
Table 4: Fuzzy Set of Input Variables
Input Input Name linguistic Variable Range
Input-1 Knowledge Bad 01-50
Good 25-75
Very Good 50-100
Input-2 Speed Delivery Erratic 01-50
Manageable 25-75
Optimum 50-100
Input-3 Representation Abstract 01-50
Better 25-50
Relevant 50-100
Input-4 Over All
Impression
Very Unimpression 01-50
Impression 25-75
Very Impression 50-100
The fuzzy sets of output (performance value) variable are shown in Table-5.
Table 5: Fuzzy Set of Output Variable of Teacher’s Performance
Output Performance Linguistic
Variable
Range
Output Performance Poor 01-40
Good 40-80
Excellent 90-100
Experimental Results
As per the input, output parameters fuzzified and rule base is generated by applying my own reasoning as an expert
person to observe or taking decision to evaluate the performance of teacher. For the simplicity of discussion only the
trapezoidal fuzzified are presented here for fuzzification of a real-valued variable is done with intuition, experience and
analysis of the rues and conditions associated with input data variables. Here, there are 34 numbers of rule generated using
‘AND’ and ‘OR’ operator. Some rules are below:
1. If (knowledge is bad) then (performance) is poor.
2. If (knowledge is good) and (speed of delivery is manageable) and (presentation is relevant) then (performance is
good).
3. If (knowledge is very good) and (speed of delivery is manageable) and presentation is relevant) then (performance is
good).
4. If (knowledge is very good) and (speed of delivery is optimum) and (presentation is relevant) and (overall impression
is high impressible) then (performance is excellent). The experimental results of this method are given in Table-6.
60 Ramjeet Singh Yadav & P. Ahmed
Table 6: Input Variables and Teacher’s Performance Value
S.No. Input Output
(Performance)
Knowledge Speed of
Delivery
Presentation Over All
Impression
Explanation Triangular
1. 06.0 12.9 18 15.9 20.3 20.4
2. 07.0 12.2 24.5 9.85 22.0 33.7
3. 32.6 35.6 28.0 37.0 31.1 40.4
4. 44.7 38.6 40.2 31.1 38.6 56.2
5. 40.2 47.7 52.2 41.1 55.0 67.2
6. 53.8 41.7 53.5 55.2 61.4 68.3
7. 64.4 58.3 62.8 64.4 64.4 70.4
8. 68.8 76.5 70.5 75.0 72.0 76.1
9. 78.5 81.1 70.5 70.0 84.1 83.8
10. 97.7 87.6 97.7 96.2 96.2 95.0
The above Table-6, the inference process when knowledge = 97.7, speed of delivery = 87.6, presentation = 97.7,
overall impression = 96.2 and explanation = 96.2 then performance = 95. Here, we pointed out that the membership value
of input variable and output variables are fixed by expert domain. In this method, there is no fixed set of procedure for the
fuzzification. This is another drawback. Such type of problems solved by the fuzzy C-Means Clustering Algorithm
C. Soft Computing Model for Academic Performance of Teachers Using Fuzzy Logic
O.K. Chaudhari et al., (2012) presented a method to deal with the evaluation of teacher’s academic performance
evaluation using fuzzy logic. The descriptions of this method are given below:
Fuzzy Expert System for Academic Performance Evaluation
Steps involved in the Fuzzy Expert System are as follows:
Step-1 (Crisp Value (Data)): Teachers self-appraisal forms are filled in by respective teachers with sub activity which
then recommended by the head of the department and head of the institution. The crisp data is tabulated from these forms
(Table-7).
Step-2 (Fuzzification (Fuzzy Input Value)): The input variables (elements) are then divided into linguistic variables-
excellent, very good, good, average and poor. O.K. Chaudhari, et al. (2012) has used the trapezoidal membership function
for converting the crisp set into fuzzy set.
Step-3 (Fuzzy Rule and Interference Mechanism): The rules determine input and output membership functions that will
be used in inference process. These rules are linguistics and are entitled “IF-THEN” rules. From the discussion with the
academic experts some rules are formulated from their practical and past experiences. Here, we pointed out that the
drawback of this proposed study, there is need of academic expert for the generation of fuzzy rule and membership
function.
Step-4 (Fuzzy Output (Overall Performance) and Defuzzification (Performance)): The output variable is the overall
performance of the teacher, which has five linguistic variables. The degree of membership function is given by equation
(1):
(1)
This expression determines an output membership function value for each active rule. When one rule is active an
AND operation is applied between inputs. The linguistic variables of output variable are shown in Table-7.
Academic Performance Evaluation Using Fuzzy C-Means 61
Numerical Results and Discussions
In order to test the above proposed model by using fuzzy expert system and rules defined in the this study the data
from one of the reputed engineering college have been used. From the input data the output variable overall performance of
teacher is determined by direct method and also by using the fuzzy model developed in the study. Last two columns of
Table-7 show the values of teachers’ performance by direct method and fuzzy expert system respectively.
Table 7: Teachers Overall Performance (Crisp and Fuzzy)
S.No. Input Variables Output Value
F1 F2 F3 F4 F5 F6 Direct Fuzzy
1. 86 85 70 12 13 33 86 80
2. 85 92 90 12 14 34 92 90
3. 95 98 60 09 08 26 73 80
4. 80 95 73 10 15 32 87 80
5. 89 75 60 09 08 33 77 73
6. 94 80 60 12 10 34 84 80
7. 75 80 75 12 04 28 72 71
8. 67 75 75 09 08 33 76 76
9. 70 85 75 09 13 25 74 76
10. 85 90 90 12 08 25 77 89
11. 93 100 75 10 08 28 78 80
12. 82 80 70 09 08 30 75 75
13. 83 91 70 12 00 35 76 70
14. 80 95 73 12 00 21 63 70
15. 71 89 83 12 00 21 63 72
16. 83 90 82 12 00 26 69 76
17. 97 90 95 12 01 34 81 80
18. 75 97 90 10 02 17 61 70
19. 85 96 84 12 08 34 86 84
20. 71 95 76 10 03 23 65 72
21. 73 95 94 06 04 19 60 70
22. 70 94 85 12 09 18 68 80
23. 76 89 75 12 00 24 65 71
24. 72 95 80 12 00 23 65 74
25. 79 99 84 12 00 17 61 70
26. 86 96 90 12 00 14 59 70
27. 95 95 85 12 00 28 72 80
28. 81 96 72 10 00 26 67 70
29. 83 98 85 12 01 30 75 80
30. 79 93 73 11 02 25 68 70
31. 70 100 77 09 01 28 67 71
O.K. Chaudhari et al., (2012) observed that the difference in the direct value and the values determined by using
fuzzy model. This is due to the weight age given on some important related to teaching learning process and overall
development of the institute while framing the rules. Here, we observed that the membership function values of input
variable and output variables for academic performance of teachers are fixed and decided by the domain expert. This is the
drawback of the proposed Fuzzy Expert System. In this method, we also observed that this proposed Fuzzy Expert System
cannot group or cluster the teachers’ performance. Such type of problem can solve by the fuzzy C-means clustering
algorithm.
D. Using Fuzzy Numbers in Educational Grading System
Chiu-Keung Law (1996) presented a method for using fuzzy numbers in educational grading system. They also
discussed a method to build the membership functions of several linguistic values with different weights. The description
this method is given below:
62 Ramjeet Singh Yadav & P. Ahmed
Fuzzy Numbers of Educational Grading System
Generally, Chiu-Keung Law (1996) has assigned the linguistic values A, B, C, D, and F to describe a student’s
performance. It is important that the criteria of the performance of the ideal population (students who take the same course
in the same school or district) be set before students take an examination. Thus, the criteria cannot be influenced by how
well the subjects in the samples (students in a particular class) do on examination. They try to make the linguistic values A,
B, C, D and F into corresponding reasonable normal fuzzy numbers with trapezoidal (or triangular)
membership functions.
Advantage of the Fuzzy Educational Grading System
As national Council of Teachers of Mathematics reported, only adding scores on examination will not give a full
picture of what students know. The challenge for teacher is to try different ways of grading, scoring, and reporting to
determine the best ways to describe students’ knowledge of mathematics. They list the raw scores of 10 students and their
corresponding grade in Table-8:
Table 8: The Raw Scores of 10 Students and their Corresponding Grade
S.No. S1 S2 S3 S4 S5 Total Fuzzy Performance Value Grade
1. 10 15 20 25 30 100 0.8878 A
2. 14 19 24 28 94 94 08562 A
3. 08 12 15 24 27 86 0.7978 B
4. 05 11 17 21 05 59 0.5671 B
5. 02 11 19 02 11 45 0.4386 C
6. 00 08 01 15 03 27 0.3274 C
7. 02 03 09 12 00 26 0.2945 D
8. 04 03 02 04 02 15 0.1734 D
9. 01 00 02 00 01 04 0.0980 F
10. 00 00 00 00 00 00 0.0781 F
From Table-8, although the highest and lowest degrees of membership are 0.8878 and 0.0781 known that the ideal
percentage of receiving grades A and grade B are 15% and 10%. It is important to emphasize that this approach not only
apply to an individual, but also a group of individuals. Here, we observed that the membership function values of input
variable and output variables for academic performance (grading System) of students are fixed and decided by the expert
(educational domain expert). This is the drawback of the proposed fuzzy numbers grading system for students’ academic
performance.
E. An Evaluation of Students Performance in Oral Presentation Using Fuzzy Approach
Wan Suhan Wan Daud et al., (2011) presented a method for evaluating student’s academic performance using fuzzy
logic approach. They pointed that the evaluation of students’ performance is a process of making judgment on a student
based on several elements such as examinations, assignment, test, quiz, research work and so on. They have used the
following methodology for evaluating students’ performance:
Step-1 (Normalized the Marks): The mark obtained by each student has to be converted to the normalized values.
Normalized value is referred to a range of [0, 1]. It can be obtained by dividing the mark for each criterion with the total
mark. The normalized value will be the input value of this evaluation. Table-9 shows the examples marks and the
normalized values obtained by a student for all the criteria.
Academic Performance Evaluation Using Fuzzy C-Means 63
Table 9: An Example of Mark and Normalized Value
Criteria Total
Mark
Mark
Obtained
Normalized
Mark
Introduction and Objective(C1) 15 11.67 0.78
Research(C2) 20 15.33 0.77
System Implementation(C3) 15 12.00 0.80
Results(C4) 15 12.67 0.84
Conclusion(C5) 10 08.00 0.80
Organization(C6) 05 03.67 0.73
Creativity(C7) 05 03.00 0.60
Visual Aids(C8) 05 03.12 0.62
Stage Presence(C9) 05 04.17 0.83
Report with the panels(C10) 05 03.50 0.70
The graph of membership function is developed in order to execute the fuzzification process. In this process, the
input value is mapped into the graph of membership function to obtain the fuzzy membership value of that particular input
value. Each membership value will represent the level of satisfaction. Table-10 shows 12 satisfaction levels that have been
proposed in this study.
Table 10: Standard Satisfaction Level and the Corresponding Degree of Satisfaction
Satisfaction Laves Degree of
Satisfaction
Maximum Degrees
of Satisfaction
Exceptional(E) 80-100(0.8-1.0) 1.00
Excellent(EX) 75-79(0.75-0.79) 0.79
Very Good(VG) 70-74(0.70-0.74) 0.74
Fairly Good(FG) 65-69(0.65-0.69) 0.69
Marginally Good(MG) 60-64(0.60-0.64) 0.64
Competent(C) 55-59(0.55-0.59) 0.59
Fairly Competent(FC) 50-54(0.50-0.54) 0.54
Marginally Competent(MC) 45-49(0.45-0.49) 0.49
Bad(B) 40-44(0.40-0.44) 0.44
Fairly Bad(FB) 35-39(0.35-0.39) 0.39
Marginally Bad(MB) 30-34(0.30-0.34) 0.34
Fail(B) 00-29(0.00-0.29) 0.29
Step-2: Calculate the Degree of satisfaction by formula given below:
(2)
Where yi = degree of membership value for each satisfaction level, i = 1, 2, 3,……,12.
Step-3: Compute the Final Mark.
The final mark for kth
student by the formula given below:
(3)
Where wi = the total marks of ith criteria for i = 1,2, ……..,10.
The result obtained is put into the fuzzy grade sheet (Table-11) in the appropriate columns.
64 Ramjeet Singh Yadav & P. Ahmed
Table 11: Fuzzy Grade Sheet with Contain the Overall Fuzzy Marks of Student-1
Criteria Fuzzy Membership Value Degree of
Satisfaction F MB FB MC FC CT MG FG VG EX ET
C1 0 0 0 0 0 0 0 0 0.4 0.6 0 0.770
C2 0 0 0 0 0 0 0 0 0.62 0.38 0 0.759
C3 0 0 0 0 0 0 0 0 0 0.81 0.19 0.830
C4 0 0 0 0 0 0 0 0 0.50 0.50 0 0.765
C5 0 0 0 0 0 0 0 0 0 0 1 1.000
C6 0 0 0 0 0 0 0 0.17 0.83 0 0 0.732
C7 0 0 0 0 0 0.8 0.2 0 0 0 0 0.600
C8 0 0 0 0 0 0.43 0.57 0 0 0 0 0.619
C9 0 0 0 0 0 0 0 0 0 0.2 0.8 0.958
C10 0 0 0 0 0 0 0 0.8 0.2 0 0 0.700
The Final Mark of student-1 = 0.7869
Table 12: The Results for 10 Students Obtained from Fuzzy and Non-Fuzzy Method
St. Non-Fuzzy Method Fuzzy Evaluation Method
Final Mark Linguistic Term Final Mark Linguistic Term
1. 77 Excellent 0.79 Very Good at 0.17, Excellent at 0.83
2. 89 Exceptional 0.90 Exceptional at 1.0
3. 71 Very good 0.73 Fairly Good at 0.18, Very Good at 0.82
4. 56 Competent 0.59 Competent at 1.0
5. 69 Fairly Good 0.71 Fairly Good at 0.6, Very Good at 0.4
6. 75 Excellent 0.80 Excellent at 0.81, Exceptional at 0.19
7. 73 Very Good 0.77 Very Good at 0.4, Excellent at 0.6
8. 83 Exceptional 0.87 Exceptional at 1.0
9. 51 Fairly Competent 0.54 Fairly Competent at 1.0
10. 68 Fairly Good 0.71 Fairly Good at 0.6, Very Good at 0.4
The Table-12 shows the fuzzy marks obtained are higher than the non-fuzzy marks. Here, we pointed out that the
student-1 has the performance of Very Good at 0.17 and also Excellent at 0.83. This is the drawback of the proposed
method. We also pointed out that membership function is fixed and decided by the domain expert.
F. Fuzzy Logic Based Evaluation of Performance of Students in Colleges
Mamatha S. Upadhya (2012) presented a method for evaluation of students’ performance based on fuzzy logic. The
description of this method is given below:
Details about the Set Applied
The proposed fuzzy system is dealt with, the range of possible values for the input and output variables are
determined. These (in language of fuzzy set theory) are the membership function (input variables vs. the degree of
membership function) used to map the real world measurement values to the fuzzy values. Values of the input variables are
considered in term of percentage. The membership function input and output variables are given in Table-13, 14, 15 and 16
Table 13: Fuzzy Membership Function for the Input Variable (Student Attendance)
Linguistic
variable
Interval
Medium (0, 0, 40)
Good (20, 50, 80)
Very Good (60, 100, 100)
Academic Performance Evaluation Using Fuzzy C-Means 65
Table 14: Fuzzy Membership Function for the Input Variable (Teaching Effectiveness)
Linguistic
variable
Interval
Less Effective (0, 0, 40)
Effective (20, 50, 80)
Highly Effective (60, 100, 100)
Table 15: Fuzzy Membership Function for the Input Variable (Facilities)
Linguistic
variable
Interval
Medium (0, 0, 40)
Good (20, 50, 80)
Very Good (60, 100, 100)
Table 16: Fuzzy membership Function for the Output Variable (Student Performance)
Linguistic
variable
Interval
Poor (0, 0, 30)
Medium (0, 30, 60)
Good (30, 60, 90)
Very Good (60, 100, 100)
The rules framed for this study is provided below:
1. If student attendance is medium and teaching effectiveness is Less Effective and Facilities is medium then
performance of student is Poor.
2. If student attendance is Good and teaching effectiveness is Less Effective and Facilities is medium then
performance of student is Medium.
3. If student attendance is Very Good and teaching less effective is Less Effective and Facilities is medium then
performance of student is Medium.
Defuzzification
At last, the crisp value of the ‘Performance of Students’ is obtained as an answer. This is done by defuzzifying the
fuzzy output. There are many defuzzification methods available in the literature but most commonly used are centroid and
maximum defuzzification methods. The criteria used to select suitable defuzzification method are very difficult. In this
proposed, centroid defuzzification method is used, which is given by:
(4)
Where A is the output fuzzy set and is the membership function.
RESULTS AND DISCUSSIONS
With the input values and using the above model, the inputs are fuzzified and then by using simple if-else rules and
other simple fuzzy set operations, the output fuzzy function is obtained and using the criteria, the output value for
performance of students is obtained. The fuzzy output for few different input values is provided in Table-17.
66 Ramjeet Singh Yadav & P. Ahmed
Table-17: Performance of students for Different Input Values
S.No. Student
Attendance
Teaching
Effectiveness
Facilities Performance
of Students
1. 40 60 50 60.00
2. 80 60 70 64.54
3. 80 90 70 84.70
4. 30 90 40 47.20
5. 90 90 30 72.76
6. 35 45 65 53.80
7. 65 45 35 53.80
In the above Table-17, student 6 and 7 belong to same class (cluster). We conclude that the level of intelligence of
both students is same. This is a fallacious conclusion since we find from the above Table-17 that the student 6 has
improved consistently while student 7 has deteriorated consistently. This is the drawback of proposed fuzzy model for
student academic performance. Solve such type of problem by the Fuzzy C-Means algorithm.
DATA CLUSTER ANALYSIS TECHNIQUES FOR ACADEMIC PERFORMANCE EVALUATION
The clustering problem can be stated simply as follows: Given a finite set of data, X, develop a grouping scheme for
grouping the objects into classes. In classical cluster analysis, these classes are required to form a partition of X such that
the degree of association is strong for data within blocks of the partition and weak for data in different blocks. However,
this requirement is too strong in many practical applications, and it is thus desirable to replace it with a weaker
requirement. When the requirement of a crisp partition of X is replaced with a weaker requirement of a fuzzy partition or a
fuzzy pseudo partition on X, we refer to the emerging problem area as fuzzy clustering. Fuzzy pseudo partitions are often
called fuzzy c-partitions, where c designates the number of fuzzy classes in the partition (S. Gagula-Palalic and M. Can,
2008).
Pattern recognition techniques can be classified into two broad categories: unsupervised techniques and supervise
techniques. An unsupervised technique does not use a given set of unclassified data, whereas a supervised technique uses a
dataset with known classification. These two types of techniques are complementary to each other. The Hard C-Means and
Fuzzy C-Means clustering techniques fall in unsupervised category. In this paper, we use K-Means, Hard C-Means and
Fuzzy C-Means clustering techniques for students’ academic performance evaluation.
A. K-Means Clustering
The K-means clustering technique is an iterative algorithm in which items are moved among sets of clusters until
the desired set is related. A high degree of similarity among elements in clusters is obtained, while a high degree of
dissimilarity among elements in different clusters is achieved simultaneously.
The K-Means clustering technique is used to classify data in a crisp sense. By this we mean that each data point will
be assigned to one, and only one, data cluster. In this sense these clusters as also called partitions-that is, partitions of data.
Define a family of sets as a partition of X, where the following set-theoretic forms apply to those
partitions:
(5)
(6)
(7)
Academic Performance Evaluation Using Fuzzy C-Means 67
Again, where a finite set space is comprised of the universe of data samples, and C is the
number of cases, or partitions, or clusters, into which we want to classify the data. We note the obvious,
(8)
Where C = n classes just places each data sample into its own class, and C = 1 places all data samples into the
same class; neither case requires any effort in classification, and both are intrinsically uninteresting. Equation (5) expresses
the fact that the set of all classes exhausts the universe of data samples. Equation (6) indicates that none of the classes
overlap in the sense that a data samples can belong to more than one class. Equation (7) simply express that a class cannot
be empty and it cannot contain al, the data samples. Here the objective function (or classification criteria) to be used to
classify or cluster the data. The one proposed for the hard K-Means algorithm is kwon as a within-class sum of squared
errors approach using a Euclidean norm to characterize distance. This algorithm is denoted where U is the
partition matrix, and the parameter, v, is a vector of cluster centers. This objective function is given by:
(9)
Where is a Euclidean distance measure (in m-dimensional feature space, between the kth
data sample and ith
cluster centre , is given by
(10)
Since each data sample requires m coordinates to describe its location in -space, each cluster centre also
requires m coordinates to describe its location in this same space. Therefore, the ith cluster centre is a vector of length
m, . The flow of the main optimization activities in K-Means clustering can be outlined in
the following manner:
Step-I: Start with some initial configuration of prototypes (e.g., choose them randomly).
Step-II: We compute the value for or the distance from the sample (a data set) to the centre, , of the ith
class, using
equation (4).
Step-III: construct a partition matrix by assigning numeric values to U according to the following rule:
(11)
Step-IV: Update the prototype by computing the weighted average, which involves the entries of the partition matrix:
(12)
Until convergence criteria is met.
B. Hard C-Means (HCM) Clustering Algorithms
HCM is used to classify data in a crisp sense. By this we mean that data point will be assigned to one, and only
one, data cluster. In this sense these clusters are also called partitions-that is, partitions of data. Assuming that a dataset
contains, well-separated clusters, the goals of hard C-means algorithm are twofold (J. Yen, et al., 1999).
1. To find the centre of these cluster.
2. To determine the clusters (i.e., labels) of each point in the dataset.
68 Ramjeet Singh Yadav & P. Ahmed
In fact, the second goal can easily be achieved once we accomplished the first goal, based on that clusters are
compact and well separated (J. Yen, et al., 1999). Given cluster centers, a point in the dataset belongs to the cluster whose
center is the closet, i.e.,
(13)
Where denotes the cluster of the cluster In order to achieve the first goal (i.e., finding the cluster centers),
we need to establish a criterion that can be used to search for these cluster centers. One such criterion is the sum of the
distance between points in each cluster and their center.
(14)
Where P is a vector of cluster centers to be identified. This criterion is useful because a set of true cluster centers
will give a minimal J value for s given data. Based on these observations, the hard C-means algorithm tries to find the
cluster centers V that minimizes J. However,, J is also a function of partition, P, which is determined by the cluster centers
V according equation (10). Therefore, the Hard C-means (HCM) searches for the true cluster center by iterating the
following two steps:
1. Calculating the current partition based on the current cluster.
2. Modifying the current cluster centers using a gradient descent method to minimize the J function.
The cycle terminate when the difference between clusters in two cycles is smaller than a threshold. This means
that the algorithm has converged to a local minimum of J.
C. Fuzzy C-Means (FCM) Clustering Algorithm
The fuzzy C-Means algorithm (FCM) generalizes the hard C-Means algorithm to allow a point to partially belong to
multiple clusters. Therefore, it produces a soft partition for a given dataset. In fact, it produces a constrained soft partition
(J. Yen, et al., 1999). To this, the objective function J1 of hard C-Means has been extended in two ways:
1. The fuzzy membership degrees in clusters were incorporated into the formula.
2. An additional parameter m was introduced as a weight exponent in the fuzzy membership.
The extended objective function, denoted Jm, is
(15)
Where P is a fuzzy partition of the dataset X formed by . The parameter m is a weight that
determines the degree to which partial members of a cluster affect the clustering result. Like hard c-means, fuzzy c-means
also tries to find a good partition by searching for prototypes vi that minimize the objective function Jm. Unlike hard C-
means, however, the fuzzy C-means algorithms also need to search for membership functions that minimize Jm.
The fuzzy C-means (FCM) algorithm is given below:
FCM(X, c, m, )
X : An unlabeled data set
C : the number of clusters to form
Academic Performance Evaluation Using Fuzzy C-Means 69
m : the parameter in the objective function
: A threshold for the convergence criteria
Initialize prototype
Repeat
Compute membership function using equation (9).
Update the prototype, vi in V using equation (10).
Until
Until convergence criteria is met.
Fuzzy C-Means Theorem
A constrained fuzzy partition can be a local minimum of the objective function Jm only
if the following conditions are satisfied:
(16)
(17)
Bases on this theorem, FCM updates the prototypes and the membership function iteratively using equation (16)
and (17) until a convergence criterion is reached.
D. Regression Model
Regression is one of the most common problems in statistics. It consists in exploring the association between
dependent and independent variables and in identifying their impact on the dependent variable. Ordinarily, we do not have
knowledge of the exact functional relationship between the two random variables x and y, where to each vector x sampled
according to a distribution P(x) there corresponds a scalar in accordance to a conditional distribution P(y/x). Typically we
proceed by assuming that the target variables y is given by some deterministic function of x with added Gaussian noise
that represents a measurement error or, more generally, our “ignorance” about the dependence of y on x (H. White, 1989):
(18)
The function is called the regression function and the statistical model described by the above equation is
called regression model. The error is a random variable having a normal distribution with zero mean, and a standard
deviation which does not depend on x or y, that is:
(19)
70 Ramjeet Singh Yadav & P. Ahmed
This common assumption can be partly justified by results from experimental measurements and by the central limit
theorem, which states that the sample mean of any reasonable distribution can be approximated by a normal distribution. It
follows from this assumption and from (17) that the conditional distribution of y given x will be a normal distribution with
mean and variance . Hence we obtain:
(20)
That is is the conditional mean of the output y given the input x. In other words, the regression of y on x is
that (deterministic) function of x that gives the mean value of y conditional on x. It can be demonstrated that the regression
function is an excellent solution to the problem of fitting the data, i.e. among all functions of x, the regression is the best
predictor of y given x, in the squared-error sense. Precisely, it can be shown that the minimum of the risk functional:
(21)
Is attained by the regression function . Thus the problem of regression estimation can be addressed in the
statistical learning framework, once the learning machine is assessed by a quadratic loss function:
(22)
In the case of a quadratic loss function, the empirical risk functional becomes:
(23)
Which is usually referred to as the Mean Squared Error (MSE)?
EXPERT SYSTEM
An expert system is a class of computer programs first developed by researchers in artificial intelligence (AI) during
the 1970s (J.C. Giarratano and G. Riley, 2005) and has been applied commercially throughout the 1980s. Prof. Edward
Feigenbaum of Stanford University, an early pioneer of expert systems technology, has defined an expert system as “an
intelligent computer program that uses knowledge and inference procedures to solve problem that are difficult enough to
require significant human expertise for their solution”. In other words, an expert system is a computer system that can
perform the decision-making ability as a human expert. Expert system have been combined with database for human-like
pattern recognition and automated decision systems to yield knowledge discovery through data mining and thus produce an
intelligent database. The knowledge in expert systems may be either expertise, or knowledge that is generally available
from books, magazines, and knowledgeable persons. For example, when we consult an expert (e.g., doctor, lawyer, or
teacher) about a problem, the expert asks for the current information about our condition, searches his or her knowledge
base (memory) for existing knowledge that relates to elements of the current situation, processes the information, arrives at
a decision, and presents his or her solution.
Academic Performance Evaluation Using Fuzzy C-Means 71
Figure-1 shows the basic concept of a knowledge-based expert system. The user supplies facts or other information
to the expert system and receives expert advice or expertise in response. Internally, the expert system consists of two main
components: the knowledge base and an inference engine. The former contains the knowledge which is used by to draw by
the latter to draw conclusions. These conclusions are the expert system’s responses to the user’s queries for expertise. The
expert’s knowledge about solving specific problems is called the knowledge domain of the expert. An expert’s knowledge
is commonly specific to one problem domain as opposed to general problem solving area. Inference or reasoning is
particularly important in the expert system because it is the technique by which expert system solve problems.
Numerical techniques for reasoning under uncertainty have been applied to expert system, such as Bayesian
network, the Dempster-Shafer theory of evidence and fuzzy logic. Inference engine may be called reasoning strategies. The
inference engine directs the search through the knowledge base; a process that may involve the application of inference
rules in what is called pattern matching. The control program decides which rule to investigate, which alternative to
eliminate, and which attribute to match. The most common knowledge representation in the computational format is the
IF…….THEN control structure.
PROPOSED DYNAMIC FUZZY EXPERT SYSTEM (DEFS) FOR ACADEMIC PERFORMANCE
EVALUATION
In this paper, we have proposed Dynamic Fuzzy Expert System (DEFS) for student academic performance
evaluation. This proposed Dynamic Fuzzy Expert System (DEFS) consists of Fuzzy Logic, Fuzzy C-means clustering
algorithm and Regression analysis model. The Fuzzy C-Means clustering algorithm is used for classify input space into
different classes or clusters and regression analysis model used for output estimation of the input data.
A. Dynamic Fuzzy Expert System (DFES)
The world of information is surrounded by uncertainty and imprecision. The human reasoning process can handle
inexact, uncertain, and vague concepts in an appropriate manner. Usually, the human thinking, reasoning, and perception
process cannot be expressed precisely. These types of experiences can rarely express or measured using statistical or
probability theory. Fuzzy logic provides a framework to model uncertainty, the human way of thinking, reasoning, and the
perception process. Fuzzy system was introduced by Zadeh (1965). A fuzzy expert system is simply an expert system that
uses a collection of fuzzy membership functions and rules, instead of Boolean logic, to reason about data (Schneider et al.
1996). The rules in a fuzzy expert system are usually of a form similar to the following:
If A is Low and B is High then (X = Medium).
Where A and B are input variables, X is an output variable.
Here low, high and medium are fuzzy sets defined on A, B and X respectively. The antecedent (the rule’s premise)
describes to what degree the rule applies, while the rule’s consequent assigns a membership function to each of one or
more output variables.
Let X is a space of objects and x be a generic element of X. A classical set , is defined as a collection of
elements objects, such that x can either belong or not belong to the set. A Fuzzy set A in X is defined as a set of ordered
pairs: , where is called the membership function (MF) for the fuzzy set A. The MF maps
each element of X to a membership grade (or membership value) between zero and one. Figure-2 shows the basic
architecture of proposed fuzzy expert system for modeling academic performance evaluation.
72 Ramjeet Singh Yadav & P. Ahmed
The main components of proposed dynamic fuzzy expert system are: a fuzzification interface, a fuzzy rule-base
(knowledge base), an inference engine (decision making logic), and a defuzzification interface.
1. Fuzzification Interface: The input variables are fuzzified by the Fuzzy C-Means clustering algorithm.
2. Fuzzy Rule Base (Knowledge Base): Fuzzy if-then rules and fuzzy reasoning are the backbone of fuzzy expert
systems, which are the most important modeling tools based on fuzzy set theory. The rule base is characterized in the
form of if-then rules in which the antecedents and consequents involve linguistic variables. In this paper, we use very
high, high, average, low and very low as linguistic variable. The collection of these rules forms the rule base for the
fuzzy logic system. In this proposed dynamic fuzzy expert system, we have used the following rules for finding the
knowledge base:
1. If student belong to very high then
2. If student belong to high then
3. If student belong to average then
4. If student belong to low then
5. If student belong to very low then
Where X is the students’ mark obtained in semester-1 examination. are
constant determine by the method of regression analysis model.
3. Inference Engine (Decision Making Logic): Using suitable inference procedure, the truth value for the antecedent of
each rule is computed and applied to the consequent part of each rule. Here, we have used the regression analysis
model for decision making. This results in one fuzzy subset to be assigned to each output variable for each rule. Again,
by using suitable composition procedure, all the fuzzy subsets to be assigned to each output variable are combined
together to form a single fuzzy subset for each output variable.
4. Defuzzification Interface: Defuzzification means convert fuzzy output into crisp output. Here, we have used the
height defuzzification technique for converting fuzzy output into crisp output (performance value of students). The
defuzzification formula are given below:
(24)
With the help of equation (24), we can convert the fuzzy output into crisp output (performance value of a student).
EXPERIMENTAL RESULTS OF K-MEANS TECHNIQUE
Let us consider, 20 students’ marks obtained by Semester-1 and Semester-2 examination. Table-18 shows the scores
achieved by 20 B.Tech. 2nd
year students in the Department of Computer Science and Engineering, Ashoka Institute of
Academic Performance Evaluation Using Fuzzy C-Means 73
Technology and Management, Aktha, Saranath, Varanasi-221007, Uttar Pradesh, India, appeared in semester-I and
semester-II examination.
Table 18: Data Set of Students’ Score in Semester-I and Semester-II
S.No. Sem-1 Sem-2 S.No. Sem-1 Sem-2
1. 40 65 11. 65 45
2. 20 35 12. 89 100
3. 50 65 13. 100 100
4. 10 20 14. 65 35
5. 45 65 15. 48 50
6. 34 60 16. 45 55
7. 48 55 17. 55 25
8. 56 90 18. 84 80
9. 74 70 19. 63 65
10. 45 50 20. 28 30
The above data points (Table-18) are first divided into different clusters using K-Means clustering techniques For
this purpose, we use MATLAB software for grouping (Clustering) the students’ data score in three groups (Clusters),
namely cluster (very high), cluster (high), cluster (average), cluster (low) and Cluster (very low), shown in Table-19.
Table 19: The membership functions for crisp clustering of Students Academic Performance Evaluation by K-
Means Algorithms
S.No. Sem-1 Sem-2 Classical Clustering (K-Means Clustering)
Very high
(VH)
High (V) Average
(A)
Low
(L)
Very Low
(VL)
1. 40 65 0 0 1 0 0
2. 20 35 0 0 0 1 0
3. 50 65 0 0 1 0 0
4. 10 20 0 0 0 0 1
5. 45 65 0 0 1 0 0
6. 34 60 0 0 1 0 0
7. 48 55 0 0 1 0 0
8. 56 90 1 0 0 0 0
9. 74 70 1 0 0 0 0
10. 45 50 0 0 1 0 0
11. 65 45 0 1 0 0 0
12. 89 100 1 0 0 0 0
13. 100 100 1 0 0 0 0
14. 65 35 0 1 0 0 0
15. 48 50 0 0 1 0 0
16. 45 55 0 0 1 0 0
17. 55 25 0 0 0 1 0
18. 84 80 1 0 0 0 0
19. 63 65 0 1 0 0 0
20. 28 30 0 0 0 1 0
In the above Table-19 shows that there 05 students belong to cluster (very high), 03 students belongs to cluster
(high), 08 students belongs to cluster (average), 03 students belongs to cluster (low) and 01 students belongs to cluster
(very low). Table-19 also shows that the 5th
student belongs to cluster (Average) and 11th
student belongs to cluster (high).
We conclude that the level of intelligence of both students is that 11th
student more intelligent than the 5th
student. This is a
fallacious conclusion, since we find from the above Table-19 that the 5th
student has improved consistently while 11th
student has deteriorated consistently. This is the drawback of K-means clustering algorithm. Other drawback of K-Means
clustering algorithm is that cannot calculate the total mark of a student. We have solved such types of problem by the
proposed Dynamic Fuzzy Expert System based on Fuzzy C-Means clustering algorithm and Regression model.
74 Ramjeet Singh Yadav & P. Ahmed
EXPERIMENTAL RESULT OF DYNAMIC FUZZY EXPERT SYSTEM (DFES) FOR MODELING
ACADEMIC PERFORMANCE EVALUATION
The main goal of this paper is to propose a new methodology to carry out evaluate the academic performance of the
students. In order to analyze and organize the Dynamic Fuzzy Expert System (DFES) with the help of Fuzzy set and
Fuzzy C-Means clustering technique. Figure 2 illustrates the components of Dynamic Fuzzy Expert System. The proposed
Dynamic Fuzzy Expert System is implemented using the Takagi-Sugeno-Kang (TSK) model and to defuzzify the resulting
fuzzy set, the center of gravity (COG) defuzzification method is selected. The first step in using Fuzzy C-Means clustering
within this model is to identify the parameters that will be fuzzified dynamically and to determine their respective range of
values.
The final result of this interaction is the value for each performance parameter. The proposed system has been
simulated using the Fuzzy Logic (MATLAB) toolbox. Here, we use Fuzzy C-Means clustering Algorithms for classifying
students’ scores data set (conversion of crisp score into fuzzy set), given in Table-18. For this purpose, we use Fuzzy Logic
ToolboxTM
2.2.7 by MathWorks for classifying (Clustering) the students’ data score in five classes or clusters, namely
Very High, High, Average, Low, and Very Low for modeling students’ academic performance evaluation, shown in Table-
20. Figue-3 shows the students dataset partitioned into three classes or cluster. Figue-4 shows the performance of objective
function for students’ academic performance evaluation.
Table 20: The Membership Functions for Fuzzy Clustering of Students Academic Performance Evaluation by
Fuzzy C-Means Algorithms
S.No. Sem-1 Sem-2 Classical Clustering (Fuzzy C-Means Clustering Method)
Very High
(VH)
High
(H)
Average
(A)
Low
(L)
Very Low
(VL)
1. 40 65 0.0138 0.0554 0.8574 0.0412 0.0322
2. 20 35 0.0036 0.0085 0.0290 0.0194 0.9395
3. 50 65 0.0180 0.1135 0.7891 0.0547 0.0247
4. 10 20 0.0115 0.0236 0.0563 0.0517 0.8569
5. 45 65 0.0106 0.0518 0.8862 0.0323 0.0191
6. 34 60 0.0181 0.0610 0.7755 0.0669 0.0784
7. 48 55 0.0054 0.0260 0.9163 0.0379 0.0145
8. 56 90 0.1674 0.4805 0.2206 0.0826 0.0489
9. 74 70 0.0150 0.9490 0.0184 0.0137 0.0039
10. 45 50 0.0120 0.0485 0.7708 0.1161 0.0525
11. 65 45 0.0192 0.0893 0.1196 0.7410 0.0309
12. 89 100 0.9713 0.0176 0.0052 0.0039 0.0019
13. 100 100 0.9518 0.0272 0.0092 0.0079 0.0038
14. 65 35 0.0021 0.0071 0.0107 0.9751 0.0050
15. 48 50 0.0137 0.0595 0.7240 0.1538 0.0491
16. 45 55 0.0029 0.0126 0.9566 0.0186 0.0093
17. 55 25 0.0173 0.0478 0.0975 0.7416 0.0957
18. 84 80 0.2989 0.5613 0.0661 0.0540 0.0197
19. 63 65 0.0364 0.6519 0.2004 0.0875 0.0237
20. 28 30 0.0066 0.0505 0.0543 0.0505 0.8722
Academic Performance Evaluation Using Fuzzy C-Means 75
Figure 3: Partition of the Students Score Dataset for Academic Performance Evaluation
Table 21: The cluster centers of Very High, High, Average, Low and Very Low
Cluster Center Sem.-1 Sem.-2
Cluster Centre of Very High 93.2948 98.8680
Cluster Centre of High 70.5267 72.6503
Cluster Centre of Average 44.7493 58.5596
Cluster Centre of Low 61.8312 35.7363
Cluster Centre of Very Low 19.8020 28.9976
Figure 4: Performance of Objective Function
The component value of vectors P and V are obtained by soling the fuzzy clustering problem (Academic
Performance Evaluation problem), which is basically constrained optimization problems in equation (15). A description of
each item of notation as follows:
The variable k represents the number of students sit in Semester-1 and Semester-2, who will be allocated into C
classes or clusters. The variable C represents the number of classes or clusters, the value of this variable can be determined
by the institution policy. The matrix consists of n rows and c columns, of which the element represents
the degree of membership (or the suitability level) of the kth
student. The matrix , consists of m rows and c
columns, of which the element represents the (weighted) average of students’ grade achieved by students, belong to the
cluster (or class).
In extreme condition, the value of the fundamental equation (10) is 0, which indicates the obtained clusters
are ideal, since they consist of students with the same level of mastery. Principally, the minimum the value of is,
then the better the clustering process. The application of fuzzy C-Means Algorithm (FCM) illustrated by a case described
as dataset of students score marks shown in Table-20. Table-22 gives the value of elements of vector Ui (i=1, 2, 3). As an
illustration, the values in the 11th
row of Table-20 can be interpreted as:
76 Ramjeet Singh Yadav & P. Ahmed
From those five values, 11th
student is the most suitable to be in class or cluster (Low), since he/she has the
highest degree of membership to this class or cluster compared to the other four. 5th
student is the most suitable to be in
class or cluster (average), since he/she has the highest degree of membership to this class or cluster compared to the other
four. Thus, we conclude that 5th
student has improved consistently while 11th
student has deteriorated consistently. By the
same observations, the following class or cluster was obtained for students partitioning in Semester-1 and Semester-2
examinations:
1. The first class or cluster (Very High) consists of students’ numbers 12, and 13.
2. The second class or cluster (High) consists of students’ numbers 8, 9, 18 and 19.
3. The third class or cluster (Average) consists of students’ numbers 1, 3, 5, 6, 7, 10, 15, and 16.
4. The fourth class or cluster (Low) consists of students’ numbers’ 11, 14 and 17.
5. The fifth class or cluster (Very Low) consists of students’ numbers 2, 4 and 20.
Thus, two students belong to class or cluster (Very High), four students belong to class or cluster (High), eight
students belong to class or cluster (Average), three students belong to class cluster (Low) and three students belong to class
or cluster (Very Low).
Output Estimation: Regression problems deal with estimation of an output value based on input values. When used for
classification, the input values are values from the database and the output values represents the classes. Regression can be
used to solve classification problems. In actually, regression takes a set of data and fits the data to formal. The linear
regression formula in two dimensional spaces is given bellow:
(25)
Where a and b are constant. They are determining by the normal equations for best fit of linear relationship of
input and output. This model is estimate the actual relationship between input and output. We can use the generated linear
regression model to predict an output value given an input value. Here, we use the regression analysis of output estimation
of Dynamic Fuzzy Expert System (DFES) for modeling academic performance evaluation. In this proposed research work,
we use linear regression model for estimation of output of Dynamic Fuzzy Expert System (DFES). Here we use the
MATAB software for estimating the output of DFES. The output of cluster (Very High), cluster (High), Cluster (Average),
cluster (Low) and Cluster (Very Low) are given bellow:
Average
Low
Where X is students mark of semester-1.
Academic Performance Evaluation Using Fuzzy C-Means 77
Rule Generation
1. If Student belongs to cluster (very high) then student performance is very high .
2. If student is belongs to cluster (high) then student performance is high ).
3. If student is belongs to cluster (average) then student performance is average(
4. If student belongs to cluster (low) then student performance low .
5. If student belongs to cluster very low then student performance is very low ( .
If we take the first student of Table-20, then the output of Y is given by
Defuzzification (Calculation of Student Academic Performance)
The final calculation of student academic performance is determined by the following formula:
Similarly, we can calculate the academic performance of other students’ given in Table-22.
Table 22: The Membership Functions and Students Academic Performance Calculated by the Dynamic Fuzzy
Expert System S.No Sem-1 Sem-2 Dynamic Fuzzy Expert System method
(Fuzzy C-Means Clustering Method)
Very High
(VH)
High
(H)
Average
(A)
Low (L) Very Low
(VL)
Student
Performance (SP)
1. 40 65 0.0138 0.0554 0.8574 0.0412 0.0322 58.257320
2. 20 35 0.0036 0.0085 0.0290 0.0194 0.9395 29.438568
3. 50 65 0.0180 0.1135 0.7891 0.0547 0.0247 57.545231
4. 10 20 0.0115 0.0236 0.0563 0.0517 0.8569 24.382494
5. 45 65 0.0106 0.0518 0.8862 0.0323 0.0191 57.753239
6. 34 60 0.0181 0.0610 0.7755 0.0669 0.0784 56.775181
7. 48 55 0.0054 0.0260 0.9163 0.0379 0.0145 56.118908
8. 56 90 0.1674 0.4805 0.2206 0.0826 0.0489 71.297348
9. 74 70 0.0150 0.9490 0.0184 0.0137 0.0039 74.884071
10. 45 50 0.0120 0.0485 0.7708 0.1161 0.0525 53.238884
11. 65 45 0.0192 0.0893 0.1196 0.7410 0.0309 46.385464
12. 89 100 0.9713 0.0176 0.0052 0.0039 0.0019 99.079208
13. 100 100 0.9518 0.0272 0.0092 0.0079 0.0038 98.510788
14. 65 35 0.0021 0.0071 0.0107 0.9751 0.0050 40.595856
15. 48 50 0.0137 0.0595 0.7240 0.1538 0.0491 51.915192
16. 45 55 0.0029 0.0126 0.9566 0.0186 0.0093 57.329090
17. 55 25 0.0173 0.0478 0.0975 0.7416 0.0957 34.151695
18. 84 80 0.2989 0.5613 0.0661 0.0540 0.0197 79.207535
19. 63 65 0.0364 0.6519 0.2004 0.0875 0.0237 69.206512
20. 28 30 0.0066 0.0505 0.0543 0.0505 0.8722 35.532959
78 Ramjeet Singh Yadav & P. Ahmed
From above Table-22 shows that the 11th
student is the most suitable to be in class or cluster (Low), since he/she
has the highest degree of membership to this class or cluster compared to the other four. 5th student is the most suitable to
be in class or cluster (average), since he/she has the highest degree of membership to this class or cluster compared to the
other four. Thus, we conclude that 5th
student has improved consistently while 11th
student has deteriorated consistently.
Therefore, we observed that the fuzzy C-Means clustering algorithm method is more suitable than the classical K-Means
clustering algorithms method for evaluating academic performance.
COMPARISON OF CLASSICAL, FUZZY EXPERT SYSTEM, K-MEANS, FUZZY C-MEANS
CLUSTERING ALGORITHM METHOD FOR MODELING ACADEMIC PERFORMANCE
EVALUATION
The comparison of Classical, Classical Fuzzy Expert, K-Means and Fuzzy C-Means Clustering algorithm method
for students’ academic performance are given in Table-2
Table 23: Comparison of Classical, Fuzzy Expert System, K-Means, Fuzzy C-Means Clustering Algorithm Method
S.N
o.
Sem
-1
Sem
-2
Cla
ssic
al
Met
hod
Fu
zzy E
xp
ert
Sy
stem
Met
hod
K-Means Clustering
Method
Dynamic Fuzzy Expert System method
(Fuzzy C-Means Clustering Method)
Ver
y
Hig
h
Hig
h (
H)
Aver
ag
e
(A)
Lo
w (
L)
Ver
y L
ow
(VL
)
Ver
y
Hig
h
Hig
h (
H)
Aver
ag
e
(A)
Lo
w (
L)
Ver
y L
ow
(VL
)
Stu
den
t
Per
form
a
nce
(S
P)
1.
40
65
52.5
0
62.7
0
0
0
1
0
0
0.0
138
0.0
554
0.8
574
0.0
412
0.0
322
58
.25
73
20
2.
20
35
27
.50
24
.30
0
0
0
1
0
0.0
03
6
0.0
08
5
0.0
29
0
0.0
19
4
0.9
39
5
29
.43
85
68
3.
50
65
57.5
0
75.0
0
0
0
1
0
0
0.0
180
0.1
135
0.7
891
0.0
547
0.0
247
57.5
45
23
1
4.
10
20
15
.00
20
.30
0
0
0
0
1
0.0
11
5
0.0
23
6
0.0
56
3
0.0
51
7
0.8
56
9
24.3
82
49
4
5.
45
65
55
.00
67
.60
0
0
1
0
0
0.0
10
6
0.0
51
8
0.8
86
2
0.0
32
3
0.0
19
1
57
.75
32
39
6.
34
60
47
.00
62
.50
0
0
1
0
0
0.0
181
0.0
610
0.7
755
0.0
669
0.0
784
56
.77
51
81
7.
48
55
51.5
0
53.3
0
0
0
1
0
0
0.0
054
0.0
260
0.9
163
0.0
379
0.0
145
56
.11
89
08
Academic Performance Evaluation Using Fuzzy C-Means 79
8.
56
90
73.0
0
75.8
0
1
0
0
0
0
0.1
674
0.4
805
0.2
206
0.0
826
0.0
489
71.2
97
34
8
9.
74
70
72.0
0
75.9
0
1
0
0
0
0
0.0
15
0
0.9
49
0
0.0
18
4
0.0
13
7
0.0
03
9
74.8
84
07
1
10.
45
50
47
.50
44
.00
0
0
1
0
0
0.0
12
0
0.0
48
5
0.7
70
8
0.1
16
1
0.0
52
5
53.2
38
88
4
11
.
65
45
55
.00
57
.50
0
1
0
0
0
0.0
19
2
0.0
89
3
0.1
19
6
0.7
41
0
0.0
30
9
46
.385
464
12
.
89
10
0
94
.50
90
.80
1
0
0
0
0
0.9
71
3
0.0
17
6
0.0
05
2
0.0
03
9
0.0
01
9
99
.07
92
08
13
.
10
0
10
0
10
0.0
92
.00
1
0
0
0
0
0.9
51
8
0.0
27
2
0.0
09
2
0.0
07
9
0.0
03
8
98
.51
07
88
14
.
65
35
50.0
0
38.7
0
0
1
0
0
0
0.0
021
0.0
071
0.0
107
0.9
751
0.0
050
40
.59
58
56
15.
48
50
49.0
0
47.3
0
0
0
1
0
0
0.0
137
0.0
595
0.7
240
0.1
538
0.0
491
51.9
15
19
2
16.
45
55
50.0
0
49.0
0
0
0
1
0
0
0.0
029
0.0
126
0.9
566
0.0
186
0.0
093
57.3
29
09
0
17.
55
25
40.0
0
31.0
0
0
0
0
1
0
0.0
17
3
0.0
47
8
0.0
97
5
0.7
41
6
0.0
95
7
34.1
51
69
5
18.
84
80
82
.00
77
.80
1
0
0
0
0
0.2
989
0.5
61
3
0.0
66
1
0.0
54
0
0.0
19
7
79.2
07
53
5
19
.
63
65
64
.00
75
.30
0
1
0
0
0
0.0
36
4
0.6
51
9
0.2
00
4
0.0
87
5
0.0
23
7
69
.206
512
80 Ramjeet Singh Yadav & P. Ahmed
20
.
28
30
29
.00
24
.10
0
0
0
1
0
0.0
06
6
0.0
50
5
0.0
54
3
0.0
50
5
0.8
72
2
35
.532
959
Table-23 shows that the average marks of both 11th
student and 5th
student are same in classical method. Table-23
also shows that the 5th
student belongs to cluster (average) and 11th
student belongs to the cluster (high) in K-Means
method and 5th
student belongs to cluster (average), 11th
student belongs to cluster (low) in Fuzzy C-Means method. We
conclude that the level of intelligence of both students is same in classical (Mean) method. 5th
Student is more intelligent
than 11th student in fuzzy C-Means Clustering method. Thus, we can say that the Fuzzy C-Means clustering algorithm is
more powerful clustering algorithm than the K-means clustering algorithm for academic performance evaluation. The
fuzzy C-Means Clustering algorithm automatically generates the membership value of semester-1 and semester-2
examination scores of students marks for further treatment of student academic performance such as rule generation of
fuzzy expert system. Figure-5 and Table-24 shows the comparison of K-Means and Fuzzy C-Means clustering algorithm
for academic performance evaluation.
The proposed Dynamic Fuzzy Expert System also calculates the total mark of a student sit in semester-1 and
semester-2 examination. The proposed dynamic fuzzy Expert System is based on Fuzzy C-Means Clustering algorithm
method, Regression analysis model and Fuzzy logic. Therefore, we can say that the proposed Dynamic Fuzzy Expert
System method for modeling student academic performance evaluation is more powerful method in comparison to classical
(mean) method, fuzzy logic method (Sirigiri Pavani et al., 2012, Chiu-Keung Law, 1996, Wan Suhan Wan Daud et al.,
2011, Mamatha S. Upadhya, 2012) and Fuzzy Expert System method (Ramjeet et al. 2011, O.K. Chaudhari et al., 2012).
The proposed Dynamic Fuzzy Expert System automatically converts the crisp set into fuzzy set. There is no need of
the domain expert. Thus, the proposed Dynamic Fuzzy Expert System is more powerful method for evaluating the student
academic performance. This method also evaluates the teacher academic performance for the different attributes.
Table 24: Comparison of K-Means and Fuzzy C-Means Clustering Algorithm
Clusters or
Classes
K-Means
Clustering
Fuzzy C-Means
Clustering
Very High 05 02
High 03 04
Average 08 08
Low 03 03
Very Low 01 03
CONCLUSIONS AND FUTURE WORK
In this paper, we have proposed Dynamic Fuzzy Expert system for modeling students’ academic performance
evaluation based Fuzzy C-Means Clustering Algorithm, Fuzzy Logic and Regression analysis model. The proposed
Dynamic Fuzzy Expert System automatically convert the crisp data into fuzzy set and also calculate the total marks of a
student sit in semsetr-1 and semester-2 examination.
The K-Means clustering algorithm is based on crisp set or classical logic and fuzzy C-Means clustering algorithm
based on fuzzy logic techniques. In this paper, we have provided a simple and qualitative methodology to compare the
predictive power of clustering algorithm and the Euclidean distance.
We demonstrated our techniques using K-Means and Fuzzy C-Means clustering algorithm for modeling academic
performance evaluation and combined with the deterministic model on a dataset of B.Tech. (Computer Science and
Engineering), Saranath, Varanasi, UP, India, students’, sit in semester-1 and semester-2 examination. Here, there are 20
Academic Performance Evaluation Using Fuzzy C-Means 81
students sit in semester-1 and semester-2 examination provides the numerical interpretation of the results for modeling
students’ academic performance evaluation. These both models, K-Means and Fuzzy C-Means algorithm clustering models
improved on some limitation of the existing traditional methods, such as average method and statistical method.
The Fuzzy C-Means Algorithm model based on fuzzy logic best model for modeling academic performance
evaluation in comparison in comparison to the K-Means clustering algorithm model because this algorithm based on crisp
set or classical logic. I
n this paper, we have observed that the Fuzzy C-Means algorithm is best model for modeling academic
performance in educational domain. Therefore, the fuzzy C-Means clustering algorithm serves as a good benchmark to
monitor the progression of students modeling in educational domain. It also enhances the decision making by academic
planners semester by semester by improving on the future academic results in the subsequence academic session. It worth
of future research to use combine technique of fuzzy C-Means artificial neural networks called Neuro-Dynamic Fuzzy
Expert system to evaluate student and teacher academic performance and also develop adaptive learning system and
Intelligent Tutoring System for Internet based education like Distance Education. The system is implemented by using the
Fuzzy Logic ToolboxTM
2.2.7 by MathWorks.
Figure 5: Comparison of K-Means and Fuzzy C-Means Clustering Algorithm for Modeling Academic Performance
Evaluation
ACKNOWLEDGEMENTS
I would like to express my deep sense of gratitude and respect to my supervisor Prof. Pervez Ahmed, for their
excellent guidance and suggestions. They have been to source of inspiration for me. I would like to render heartiest thanks
to various friends for their priceless help and support. Last but not the least we thank our parents and wife and the almighty
whose blessings are always there with us.
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Ramjeet Singh Yadav is working as an Associate Professor and Head in the Department of Computer Science and
Engineering, Ashoka Institute of Technology and Management, Paharia, Sarnath, Varanasi (Uttar Pradesh), India. In
addition, he is a Research Scholar in the Department of Computer Science and Engineering, Sharda University, Greater
Noida, Uttar Pradesh, India. His research interest areas are in Fuzzy Logic, Neural Networks, Genetics Algorithms, and
Neuro Fuzzy Systems and Dynamic Fuzzy Expert Systems. He has published over four journal papers (one International
and three National Journals), and fifteen papers in National and International Conference proceedings.
84 Ramjeet Singh Yadav & P. Ahmed
Professor Pervez Ahmed is working as a Professor in the Department of Computer Science and Engineering in Sharda
University, Greater Noida, Uttar Pradesh, India. Professor Ahmed has more than three decades of teaching experience of
Computer Science courses, at undergraduate and graduate levels, in universities in Iraq (1975-78), Canada (1979-88), India
(1989-89) and Saudi Arabia (1990-2010). In 1999, he was appointed as Visiting Professor of Computer Science by the
Commonwealth Secretariat, UK. He is the founder chairman of the Computer Science department of Aligarh Muslim
University, UP, India, and has served as Chairman, Computer Science and Engineering department, International Science
College, Al-Baha, Saudi Arabia. He has been a Senior Software Designer at PHILIPS/MICOM, Montreal, Canada;
Research Fellow (MRI imaging) at Montreal Neurological Institute, McGill University, Canada, and visiting Scientist,
Centre for Pattern Recognition and Machine Intelligence (CENPARMI), Montreal, Canada. His primary area of research is
Pattern Recognition and Machine Intelligence. During his Ph.D. he developed, implemented and tested a novel technique
for postal mail sorting by automatically recognizing the zip-codes that were extracted from the totally unconstrained
handwritten mail addresses. The technique was tested on real-life data collected by the US postal service department.
