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

Fuzzy Clustering to Identify Clusters at Different Levels of Fuzziness: An Evolutionary Multi-Objective Optimization Approach