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In the fuzzy c-means (FCM) clustering algorithm, almost none of the data points have a membership value of 1. Moreover, noise and outliers may cause difficulties in obtaining appropriate clustering results from the FCM algorithm. The embedding of FCM into switching regressions, called the fuzzy c-regressions (FCRs), still has the same drawbacks as FCM. In this paper, we propose the -cut implemented fuzzy clustering algorithms, referred to as , which allow the data points being able to completely belong to one cluster. The proposed algorithms can form a cluster core for each cluster, where data points inside a cluster core will have a membership value of 1 so that it can resolve the drawbacks of FCM. On the other hand, the fuzziness index plays different roles for FCM and . We find that the clustering results obtained by are more robust to noise and outliers than FCM when a larger is used. Moreover, the cluster cores generated by are workable for various data shape clusters, so that is very suitable for embedding into switching regressions. The embedding of into switching regressions is called . The proposed provides better results than FCR for environments with noise or outliers. Numerical examples show the robustness and the superiority of our proposed methods.