  Fuzzy clustering methods identify naturally occurring clusters in a dataset,
where the extent to which different clusters are overlapped can differ. Most
methods have a parameter to fix the level of fuzziness. However, the
appropriate level of fuzziness depends on the application at hand. This paper
presents Entropy $c$-Means (ECM), a method of fuzzy clustering that
simultaneously optimizes two contradictory objective functions, resulting in
the creation of fuzzy clusters with different levels of fuzziness. This allows
ECM to identify clusters with different degrees of overlap. ECM optimizes the
two objective functions using two multi-objective optimization methods,
Non-dominated Sorting Genetic Algorithm II (NSGA-II), and Multiobjective
Evolutionary Algorithm based on Decomposition (MOEA/D). We also propose a
method to select a suitable trade-off clustering from the Pareto front.
Experiments on challenging synthetic datasets as well as real-world datasets
show that ECM leads to better cluster detection compared to the conventional
fuzzy clustering methods as well as previously used multi-objective methods for
fuzzy clustering.
