###### Abstract  
Clustering is an unsupervised machine learning method with many practical applications that has gathered extensive research interest. It is a technique of dividing data elements into clusters such that elements in the same cluster are similar. Clustering belongs to the group of unsupervised machine learning techniques, meaning that there is no information about the labels of the elements. However, when knowledge of data points is known in advance, it will be beneficial to use a semi-supervised algorithm. Within many clustering techniques available, fuzzy C-means clustering (FCM) is a common one. To make the FCM algorithm a semi-supervised method, it was proposed in the literature to use an auxiliary matrix to adjust the membership grade of the elements to force them into certain clusters during the computation. In this study, instead of using the auxiliary matrix, we proposed to use multiple fuzzification coefficients to implement the semi-supervision component. After deriving the proposed semi-supervised fuzzy C-means clustering algorithm with multiple fuzzification coefficients (sSMC-FCM), we demonstrated the convergence of the algorithm and validated the efficiency of the method through a numerical example. clustering technique; fuzzy C-means clustering; semi-supervised clustering; fuzzification coefficient; objective function +
Footnote †: : 2021  
Footnote †: 2021: 14.258. [[https://doi.org/10.3390/a14090258](https://doi.org/10.3390/a14090258)]([https://doi.org/10.3390/a14090258](https://doi.org/10.3390/a14090258))  
Footnote †: 2021: 14.258. [[https://doi.org/10.33390/a14090258](https://doi.org/10.33390/a14090258)]([https://doi.org/10.33390/a14090258](https://doi.org/10.33390/a14090258))  
Footnote †: 2021: 14.258. [[https://doi.org/10.3390/a14090258](https://doi.org/10.3390/a14090258)]([https://doi.org/10.3390/a14090258](https://doi.org/10.3390/a14090258))knowledge needs to be introduced, and the clustering results will be expected to be better. The disadvantage of this method is that certain measures of \"similarity\" may not be better, but this is acceptable when there are data points that are \"shaped\" by the knowledge; hence, the ability to update the cluster center is also improved [12]. There is knowledge about data that can be considered to be relatively accurate. It will be known to a degree in advance if a data element should or should not belong to a certain cluster. Such factors are dependent on the related expertise of the person conducting the clustering [13]. [PERSON] et al. [14] proposed to improve the FCM algorithm by adding an auxiliary supervisory matrix, representing the supervised membership grade, into the objective function. Although the authors did not outline a specific method to determine the value of the supervisory matrix, the study made it clear that semi-supervised fuzzy clustering is completely feasible. The authors called this algorithm semi-supervised standard Fuzzy C-means clustering (sSFCM). This algorithm is mainly based on the approach that the supervised data point must belong to a certain cluster. The membership grade on that cluster then must be larger than the value in other unsupervised clusters. The FCM method uses an exponential parameter \(m\), also known as the fuzzification coefficient, in the objective function, which adjusts the membership grade \(U_{lk}\) of element \(i\) belonging to cluster \(k\). In the standard FCM algorithm, the value of parameter \(m\) is selected from the beginning, for example, \(m=2\). In recent years, there have been studies that extended the selection of parameter \(m\) as an interval \([m_{1},m_{2}]\) or a fuzzy value that, when reducing the type, would essentially be the selection of different \(m\in[m_{1},m_{2}]\) values, applied to each iteration [15]. The extension to use multiple fuzzification coefficients \(m\) instead of only one value in FCM was presented in [16].