Comparison of Fuzzy Clustering Algorithms on UI GreenMetric
University Rankings
OZER OZDEMIR1, ASLI KAYA KARAKUTUK2
1Department of Statistics
Eskisehir Technical University
İki Eylül Campus, 26555
TÜRKİYE
2Rectorate
Eskisehir Technical University
İki Eylül Campus, 26555
TÜRKİYE
Abstract: - That universities will supply remarkable assistance on the pathway to sustainability by providing
intellectual leadership in attaining a sustainable society is an acceptable and emphasized issue. Herewith, in this
work, we aimed to divide the campuses into homogeneous groups using fuzzy clustering techniques by
employing the UI GreenMetric World University Rankings data having ranked the world's prestigious
universities every year since 2010. In the results obtained, these universities are separated into four groups.
Generalized possibilistic fuzzy clustering algorithm formed the most homogeneous clusters. Also, this
algorithm clustered universities in a shortest time. While the "education and research" and "waste" categories
are the strength aspect of the top green campuses, "water" is determined as the improvement aspect of the less
sustainable campuses.
Key-Words: - Fuzzy C-Means, Possibilistic Fuzzy C-Means, Generalized possibilistic Fuzzy C-Means,
Sustainability, GreenMetric, Clustering.
Received: June 9, 2024. Revised: September 11, 2024. Accepted: November 12, 2024. Published: December 19, 2024.
1 Introduction
The terms “Sustainability” and its close counterpart
“Sustainable Development” have been in the public
space since the mid-80s when their original
definitions were made. Yet the unique concept is
“sustainable development”, it can be said that it is
replaced by 'sustainability' at an increasing rate
within the process. It can be stated that these
concepts, which are clarified to have emerged from
the environmental movement, reflect the acceptance
that the main environmental problems are related to
the economic and social justice issues [1].
The overscale definition of sustainable development
is made in the form of 'development that preserves
and improves the natural environment and social
equality and includes economic and social
development'. In this definition, all three dimensions
of sustainable development are clearly included.
Provided that the environment and social equality
are protected and developed, it has been stated that
all kinds of social and economic development are
sustainable. Emphasizing the importance of
ecological sustainability since the economy and
society ultimately depend on the integrity of the
biosphere and the ecological processes that develop
within it, it is presented as a constraint on the
varieties of economic and social development [2].
With the advancements in the communication and
transportation technology, in today’s world
becoming almost a “global village”, where
globalization continues at full speed, the importance
given to sustainability issue is increasing day by
day. The notion of sustainability forms one of the
most significant ideas for the 21st century on a
social, global and humanistic level.
It is seen that in the second decade of the 21st
century, an increasing number of universities and
higher education institutions have adopted more
responsible behaviors towards society and various
stakeholders and have begun to follow the
sustainable development agenda more closely due to
its wide sphere of influence. Universities are no
longer judged solely on the basis of their potential to
provide quality education; including the
community's commitment to progress, other criteria
and factors play a role in reflecting the true image of
the university [3]. In this manner, many educational
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.19
Ozer Ozdemir, Asli Kaya Karakutuk
E-ISSN: 2766-9823
230
Volume 6, 2024
institutions commit to sustainable development as
signatories of international declaration and
declaration [4]. It is seen that the universities
increasingly institutionalize sustainability practices
within the scope of curriculum, research, activity,
progress, evaluation and reporting [3]. Besides, it
can be said that the concept has become one of the
significant parameters of university ranking
systems.
In this study, it is aimed to divide the campuses into
homogeneous groups with the help of fuzzy
clustering methods through the UI GreenMetric
World University Rankings data, which evaluates
higher education institutions according to six criteria
and 39 indicators: infrastructure, recycling, energy
and climate change, water resources, transportation
and education [5].
The rest of the article is structured as follows.
Section 2 explains fuzzy clustering techniques and
GreenMetric Ranking indicators. Section 3 details
results. Section 4 summarizes the main conclusions
of the article.
2 Problem Formulation
Unsupervised learning is an important subsection of
Machine Learning that deals with the problem of
analyzing unlabeled data [6]. Among the various
techniques in this domain, clustering methods are
widely used for grouping unlabeled data. These
methods are categorized based on the structure of
their cluster assignments [7]. Studies have shown
that fuzzy clustering methods are more flexible than
hard clustering methods. In this work, we will apply
fuzzy clustering techniques.
2.1 Fuzzy Clustering Techniques
2.1.1 Fuzzy C-Means and Possibilistic C-Means
(FCM-PCM)
Fuzzy c-means (FCM) algorithm is the best known
and widely used method among fuzzy division
clustering techniques [8]. The fuzzy c-means
method allows objects to belong to two or more
clusters [9]. Fuzzy logic says that each data or
object is connected to clusters with a membership
value ranging from [0,1], and the sum of the
membership values of a data or object to all classes
must be “1”. Whichever cluster center the object is
close to; the membership of that cluster will be
larger than the membership of other clusters. The
membership matrix, which is the most important
feature of the fuzzy c-means algorithm, has positive
effects on clustering. This matrix facilitates the
identification of uncertain situations [9]. In addition,
due to low membership degrees, the impact of
extraordinary data is small. The ability of the FCM
algorithm, which has a flexible structure, to find
overlapping sets is higher than other divisional
algorithms.
In addition to the advantages mentioned above, the
fuzzy c-means algorithm also has some
disadvantages. Since membership function increases
computational complexity, it is a time-consuming
divisional clustering algorithm. FCM is also an
objective function-based method. The algorithm
tries to minimize by shifting following objective
function, which it implements as a generalization of
the least squares method [10].
U membership matrix with fuzzy values reflects the
clustering result. If necessary, these values obtained
after clustering can be rounded to 0 and 1 by
defuzzification. Not very good performance for
noisy data is a disadvantage of the FCM algorithm.
Possibilistic C-Means (PCM) have been introduced
to cope with this disadvantage [11] but suffer from
overlapping clusters [11].
The first step in Fuzzy C-Means is to determine the
cluster center, which will mark the average position
for each cluster. Under initial conditions, this cluster
center is not yet accurate. Each data point has a
membership degree for each cluster. By repeatedly
fixing the center of the cluster and the membership
degree of each data point, it will be seen that the
center of the cluster will move to the correct
position. This iteration is based on minimizing the
objective function that defines the distance from a
given data point to the center of the cluster,
weighted by the membership degree of the data
point [12]. The function of the FCM algorithm to
find clusters with maximum purity is given by Eq.
(1) [13].
where is number of data vectors, is
number of clusters, is center of the cluster,
is data vector ( column of the data matrix
), is degree of fuzziness, is
membership grade of the data vector in the
cluster,
is distance.
Partition matrix and cluster centers matrix
minimize Eq. (2) [13].
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.19
Ozer Ozdemir, Asli Kaya Karakutuk
E-ISSN: 2766-9823
231
Volume 6, 2024
PCM uses the objective function with c_FCM=0 in
which cluster centers and typicalities t_ijs are
calculated without considering u_ijs [14].
2.1.2 Fuzzy Possibilistic C-Means (FPCM)
The objective of FPCM model is to find by
minimizing the Eq 3.
where .
The constraints that Fuzzy Possibilistic C-Means
must satisfy as follows;
2.1.3 Possibilistic Fuzzy C-Means (PFCM)
The PFCM algorithm that minimizes the objective
function by crossing PCM and FCM is given by Eq.
(4) [15].
where and refer membership
grade and typicality and is a number as
[15].
Euclidean distance norm compliance is the major
issue with FCM, PCM, FPCM and PFCM.
Ellipsoidal sets consist of the covariance norm that
better fits with the patterns and structures of the
data. Euclidean distance measure is not suitable
when data rows have different types of units, but
covariance gives non- dimension distance and could
be used.
2.1.4 Generalized Possibilistic Fuzzy C-Means
(GPFCM)
The function of the GPFCM with constraints is
given by Eq. (5) and optimized [16].
Eq. 5 is optimized through Lagrange Multipliers.
Centroids are calculated as Eq 7.
′
′
′
′
′
′
The algorithm converges when
where z refers iteration number and is a
threshold [16]. is calculated by Eq. (8):
And for the typicality is calculated by Eq. (9);
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.19
Ozer Ozdemir, Asli Kaya Karakutuk
E-ISSN: 2766-9823
232
Volume 6, 2024
s, and are calculated using PFCM to initialize
GPFCM [16].
2.2 UI GreenMetric
The sustainability ranking designed by the
University of Indonesia and put into practice in
2010 is the Universitas Indonesia GreenMetric
World University Rankings. This ranking is
accepted as the first and only global assessment tool
for sustainability. With the UI GreenMetric WUR, it
is aimed to evaluate the policies and activities of
green campuses to promote a culture of
sustainability in universities. In this direction,
criteria have been created in some headings to
measure universities in terms of sustainability. The
fact that the ranking is suitable for universities in
developed and developing countries is the reason
why it is considered as a global ranking system [17].
In addition to measuring universities' efforts to
improve the sustainability of their different
campuses, one of the main purposes of the UI
GreenMetric World University Rankings is to
provide a comparative tool for assessing campus
sustainability worldwide [18]. GreenMetric has
identified 6 categories for the evaluation of
universities based on certain criteria considered
important in sustainability issues [19]. Each
category and their specific weights are shown in
Fig.1
Figure 1. UI GreenMetric Category
The number of universities participating in the
rankings has seen a steep increase over the years,
from 95 universities in 35 countries in 2010 to 956
universities in 80 countries in 2021 [20].
3 Problem Solution
The Sustainable Development Goals aim to generate
global action for the implementation of the 2030
Agenda for Sustainable Development [21], and the
international rankings for Sustainable Development
provide a baseline, list areas of strength and identify
needs and areas for improvement [22]. In this
context, this study seeks to address the hypothesis of
whether it is possible to group universities
according to their environmental sustainability
performance by utilizing the GreenMetric ranking,
an international ranking.
At this stage, a different fuzzy clustering algorithm
was applied to group the different universities into
homogeneous groups. We used the 2021 UI
GreenMetric World Rankings for Universities as a
data set. The application carried out within the
scope of the study was developed in the MATLAB
R2021b version. Codes were created for FCM,
FPCM, PFCM and GPFCM algorithms. All
algorithms were applied within the framework of
the same parameters (initial value, stopping
criterion, etc.). In terms of ease of operation, the
Fuzzifier parameter (m) is preferred as 2. The
initialization of the typicality matrix T is randomly
generated. First, the FCM algorithm was run. The
reason for this is to avoid random assignment of the
initial U-matrix in the other algorithm. According to
the result of the membership matrix of FCM,
FPCM, PFCM and GPFCM algorithms were run
respectively. In addition, Separation and Partition
validity indices were used to select the optimal
number of clusters for the four algorithms.
Separation and Partition indices are objective
methods that comprehensively evaluate clustering
quality, provide comparability between algorithms
and are suitable for fuzzy clustering. These indices,
which are widely used in literature, aim to increase
the reliability of the study.
Separation Index (S): Separation index directly
measures the magnitude of gap between pair of
clusters is easy to compute and interpret and hast the
scale equivariance property [23].
Partition Index (SC): By this measure, good
clustering results in homogeneity within each cluster
and heterogeneity between clusters. Variability in
clusters is measured by distance calculation. Internal
variability is determined by taking the mean
distance between each pair of data in the cluster and
then the average across all clusters. The distance
between clusters is obtained by averaging all
pairwise distances between clusters. [23].
Setting
and
Infrastruct
ure (SI)
;
15%
Energy
and
Climate
Change
(EC)
;
21%
Waste
(WS) ;
18%
Water
(WR) ;
10%
Transport
ation (TR)
; 18%
Educatio
n and
Research
(ED)
;
18%
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.19
Ozer Ozdemir, Asli Kaya Karakutuk
E-ISSN: 2766-9823
233
Volume 6, 2024
Figure 2.Optimal number of clusters by using validity
indices
Obtained results are shown on Fig.2. The optimal
number of clusters was taken as 4. These 4 clusters
can be classified as follows:
Cluster 1: “Low Sustainable Universities”
Cluster 2: “Middle Sustainable universities”
Cluster 3: “High Sustainable universities”
Cluster 4: “Top Sustainable universities”
Table 1.Cluster size for all algorithms
Cluster1
Cluster2
Cluster3
Cluster4
FCM
177
272
243
264
PCM
956
0
0
0
FPCM
232
237
229
258
PFCM
220
91
348
297
GPFCM
177
272
243
264
Applying the FCM and the improved algorithm
GPFCM procedure, we obtained four clusters of
very similar size. Since the PCM algorithm collects all
the data in a single cluster, it could not make the
correct classification.
Table 2.Comparison of algorithms according to calculation
time and number of iterations
Algorithms
Comp.
Time
İteration
FCM
6,78
93
PCM
34,25
205
FPCM
98,61
100
PFCM
8,75
94
GPFCM
0,33
100
Although the GPFCM algorithm obtained clusters
like the FCM algorithm, much better results have
been obtained from the FCM algorithm in terms of
iteration and computation time.
Figure 3. Distribution of universities by Region
Although FCM and GPFCM algorithm obtained
almost the same clusters, the GPFCM algorithm was
completed in a much shorter time than all other
algorithms. The mean value calculated for each
category was found to provide within-group
homogeneity in GPFCM and FCM algorithms. In
other words, the correct application of GPFCM and
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.19
Ozer Ozdemir, Asli Kaya Karakutuk
E-ISSN: 2766-9823
234
Volume 6, 2024
FCM in the groups obtained because of clustering
resulted in low variability.
Table 3. Descriptive statistics of clusters obtained by all
algorithms
FCM
FPCM
PFCM
GPFCM
Cluster 1
SI
Mean
542,66
574,03
879,66
542,66
Std. Dev.
223,45
227,59
201,03
227,28
EC
Mean
546,89
571,77
1027,73
546,89
Std. Dev.
249,30
267,13
283,96
262,59
WS
Mean
416,95
484,91
1111,70
416,95
Std. Dev.
264,16
296,23
268,18
255,57
WR
Mean
260,17
287,72
610,68
260,17
Std. Dev.
160,52
163,93
167,68
163,36
TR
Mean
549,58
607,11
1153,18
549,58
Std. Dev.
273,92
281,39
229,82
254,86
ED
Mean
548,59
624,68
1202,50
548,59
Std. Dev.
304,90
318,76
227,77
273,34
Cluster 2
SI
Mean
786,21
814,35
832,97
786,21
Std. Dev.
183,05
172,33
153,70
183,56
EC
Mean
752,21
788,61
878,85
752,21
Std. Dev.
247,79
247,62
214,70
247,55
WS
Mean
788,60
827,22
875,27
788,60
Std. Dev.
260,12
254,94
228,32
259,65
WR
Mean
465,44
492,19
537,36
465,44
Std. Dev.
156,04
160,88
134,49
156,71
TR
Mean
931,43
974,89
1008,52
931,43
Std. Dev.
233,02
208,63
174,61
232,68
ED
Mean
965,35
1001,37
1046,70
965,35
Std. Dev.
248,23
226,08
195,19
248,27
Cluster 3
SI
Mean
896,30
907,21
648,35
896,30
Std. Dev.
197,06
193,49
234,76
196,99
EC
Mean
1078,1
1093,45
621,62
1078,09
Std. Dev.
256,17
243,02
260,74
255,66
WS
Mean
1107,1
1122,05
567,03
1107,10
Std. Dev.
250,71
245,99
294,37
250,34
WR
Mean
629,84
640,39
342,39
629,84
Std. Dev.
157,89
150,86
177,74
158,91
TR
Mean
1191,8
1203,82
718,89
1191,77
Std. Dev.
213,81
207,24
303,53
214,29
ED
Mean
1226,4
1239,74
722,13
1226,44
Std. Dev.
221,34
212,90
310,95
221,35
Cluster 4
SI
Mean
1083,5
1084,59
1073,40
1083,52
Std. Dev.
174,64
175,87
175,17
174,64
EC
Mean
1394,1
1401,07
1377,19
1394,13
Std. Dev.
235,97
230,73
230,72
235,97
FCM
FPCM
PFCM
GPFCM
WS
Mean
1457,4
1459,59
1415,91
1457,39
Std. Dev.
233,30
230,44
248,66
233,30
WR
Mean
813,07
815,12
801,18
813,07
Std. Dev.
132,36
131,45
134,97
132,36
TR
Mean
1384,4
1388,08
1380,64
1384,38
Std. Dev.
171,19
169,32
165,15
171,19
ED
Mean
1523,8
1526,45
1511,36
1523,77
Std. Dev.
159,97
159,05
160,48
159,97
Table 3 shows the basic statistics of different groups
and clusters by all algorithms. According to the
results of all algorithms, universities in the 4th
cluster (i.e., high sustainability level) with high
values in the Waste, Education and Research
categories, while the universities in the 1st cluster
have lowest values in the water treatment category.
In this context, these results can be a useful
indicator for improving the ranking of universities
and investing in these categories.
4 Conclusion
In this study, both the comparison of fuzzy
clustering algorithms and the current sustainability
levels of universities are revealed. In addition, it
aims to reveal the aspects that are open to
development in terms of sustainability and to
contribute to universities. This article compares
fuzzy clustering methods to divide the campuses
into homogeneous groups based on the UI
GreenMetric World University Ranking data. In this
context, we used the UI GreenMetric 2021 as a
dataset. This analysis allowed us to identify four
levels of sustainability campuses: top, high,
medium, and low green.
It can be said that PCM clustering analysis occurred
an artificial cluster as all data is associated in a
cluster. It can also be said that the classification
based on fuzzy c-means and generalized fuzzy
possibilistic c means algorithm reflects the natural
classification resulting from the combination of
variables in the data matrix.
Generalized fuzzy possibilistic c- means algorithm
worked in harmony with the validity indices and
clustered the data in an optimal computation time
than other basic algorithms. Thus, we have saved
time.
It revealed that in universities with the least
sensitivity to sustainability, the category of the
treatment of water and waste was seen as open to
improvement. On the other hand, higher and
medium-high universities (the most stable) managed
to score maximum points in the treatment of water
and waste, as well as in the research and education
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.19
Ozer Ozdemir, Asli Kaya Karakutuk
E-ISSN: 2766-9823
235
Volume 6, 2024
aspects. We offer some suggestions to support the
improvement of activities of universities that are
less successful in water and waste areas.
“Sustainability Advisory Center” can be
established where guidance on water
treatment and waste management can be
provided through best practices.
Sensor-based systems that can monitor
water consumption on campus in real time
can be installed and interventions can be
made to prevent water waste.
Rainwater collection systems can be
installed on university campuses and
recycled water can be used for needs such
as irrigation and cleaning.
Separation bins can be placed throughout
the campus to include paper, plastic, glass
and organic waste in recycling processes.
Reusable products can be preferred over
single-use products, for example by
distributing refillable water bottles to
students and employees.
Collaborations could be established with
successful universities in the UI
GreenMetric rankings to encourage the
sharing of good practices in these areas.
It is thought that this study will bring transparency
in terms of creating a more social awareness of
sustainability and a model for the Universities that
are left behind.
References:
[1] Soyka, P.A., Creating a Sustainable
Organization: Approaches for Enhancing
Corporate Value Through Sustainability,
Prentice Hall, 2012.
[2] Dunphy, D., Benveniste, J., Griffiths, A.,
Sutton, P., Sustainability: The Corporate
Challenge of the 21st Century, Allen & Unwin,
2000.
[3] Nejati, M., Nejati, M., "Assessment of
Sustainable University Factors from the
Perspective of University Students," Journal of
Cleaner Production, Vol. 48, 2013, pp. 101–
107.
[4] Weenen, H.V., "Towards a Vision of a
Sustainable University," International Journal
of Sustainability in Higher Education, Vol. 1,
No. 1, 2000, pp. 20–34.
[5] UI GreenMetric, Guideline 2021,
https://greenmetric.ui.ac.id/publications/guideli
nes/2021/english.
[6] Achmad Efendi, Yusi Tyroni Mursityo, Ninik
Wahju Hidajati, Nur Andajani, Zuraidah
Zuraidah, Samingun Handoyo, "Multiple Time
Series Modeling of Autoregressive Distributed
Lags with Forward Variable Selection for
Prediction," WSEAS Transactions on Business
and Economics, vol. 21, pp. 1012-1026, 2024,
[7] Hartigan, J.A., Clustering Algorithms, John
Wiley & Sons, 1975.
[8] Dave, R.N., Sen, S., "Robust Fuzzy Clustering
of Relational Data," IEEE Transactions on
Fuzzy Systems, Vol. 10, 2002, pp. 713–727.
[9] El Fahssi Khalid, Ounasser Saida, Mohamed
Taj Bennani, Abenaou Abdenbi, "Development
of a Medical Image Segmentation Algorithm
based on Fuzzy C-Means Clustering," WSEAS
Transactions on Signal Processing, vol. 19, pp.
215-220, 2023,
[10] Bezdek, J.C., Ehrlich, R., Full, W., "FCM: The
Fuzzy c-means Clustering Algorithm,"
Computers & Geosciences, Vol. 10, 1984, pp.
191–203.
[11] Krishnapuram, R., Keller, J., "A Possibilistic
Approach to Clustering," IEEE Transactions on
Fuzzy Systems, Vol. 1, 1993, pp. 98–110.
[12] Aisyah Aryandani, Solimun, Nurjannah, Adji
Achmad Rinaldo Fernandes, Achmad Efendi,
"Implementation of Fuzzy C-Means in Investor
Group in the Stock Market Post-Covid-19
Pandemic," WSEAS Transactions on
Mathematics, vol. 21, pp. 415-423, 2022,
DOI:10.37394/23206.2022.21.49
[13] Graves, D., Pedrycz, W., "Kernel-based Fuzzy
Clustering and Fuzzy Clustering: A
Comparative Experimental Study," Fuzzy Sets
and Systems, Vol. 161, 2010, pp. 522–543.
[14] Krishnapuram, R., Keller, J., "The Possibilistic
c-Means Algorithm: Insights and
Recommendations," IEEE Transactions on
Fuzzy Systems, Vol. 4, 1996, pp. 385–393.
[15] Pal, N.R., Pal, K., Keller, J.M., Bezdek, J.C.,
"A Possibilistic Fuzzy c-Means Clustering
Algorithm," IEEE Transactions on Fuzzy
Systems, Vol. 13, 2005, pp. 517–530.
[16] Askaria, S., Montazerina, N., Zarandib,
M.H.F., "Generalized Possibilistic Fuzzy c-
Means with Novel Cluster Validity Indices for
Clustering Noisy Data," Applied Soft
Computing, Vol. 53, 2017, pp. 262–283.
[17] Ragazzi, M., Ghidini, F., "Environmental
Sustainability of Universities: Critical Analysis
of a Green Ranking," Energy Procedia, Vol.
119, 2017, pp. 111–120.
[18] Puertas, R., Martí, L., “Sustainability in
Universities: DEA-GreenMetric. Sustainability,
Vol.11, 2019.
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.19
Ozer Ozdemir, Asli Kaya Karakutuk
E-ISSN: 2766-9823
236
Volume 6, 2024
[19] Perchinunno, P., Cazzolle, M., "A Clustering
Approach for Classifying Universities in a
World Sustainability Ranking," Environmental
Impact Assessment Review, Vol. 85, 2020,
Article 106471.
[20] UI GreenMetric, "Greenmetric Overall
Ranking,"https://greenmetric.ui.ac.id/rankings/
overall-rankings-2021.
[21] Raihan, A., Sarker, T., Zimon, G., “An
Investigation on the Prospects, Challenges and
Policy Consequences of Renewable Energy
Technology Development for India’s
Environmental Sustainability”, WSEAS
TRANSACTIONS ON ENVIRONMENT
AND DEVELOPMENT, Vol. 20, 2024, pp.
365-390.
[22] Eppinga, M., Mijts, E., Santos, M., “Ranking
the sustainable development goals: perceived
sustainability priorities in small island states”,
Sustainability Science, Vol. 17, 2022.
[23] Rezae, B., "A Cluster Validity Index for Fuzzy
Clustering," Fuzzy Sets and Systems, Vol. 161,
No. 23, 2010, pp. 3014–3025.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.19
Ozer Ozdemir, Asli Kaya Karakutuk
E-ISSN: 2766-9823
237
Volume 6, 2024
