Comparison and Weighted Summation Type of Fuzzy Cluster Validity Indices

Finding the optimal cluster number and validating the partition resultsof a data set are difficult tasks since clustering is an unsupervised learning process.Cluster validity index (CVI) is a kind of criterion function for evaluating the clusteringresults and determining the optimal number of clusters. In this paper, we present anextensive comparison of ten well-known CVIs for fuzzy clustering. Then we extendtraditional single CVIs by introducing the weighted method and propose a weightedsummation type of CVI (WSCVI). Experiments on nine synthetic data sets and fourreal-world UCI data sets demonstrate that no one CVI performs better on all datasets than others. Nevertheless, the proposed WSCVI is more effective by properlysetting the weights

Frank Aggregation Operators for Triangular Interval Type-2 Fuzzy Set and Its Application in Multiple Attribute Group Decision Making

This paper investigates an approach to multiple attribute group decision-making (MAGDM) problems, in which the individual assessments are in the form of triangle interval type-2 fuzzy numbers (TIT2FNs). Firstly, some Frank operation laws of triangle interval type-2 fuzzy set (TIT2FS) are defined. Secondly, some Frank aggregation operators such as the triangle interval type-2 fuzzy Frank weighted averaging (TIT2FFWA) operator and the triangle interval type-2 fuzzy Frank weighted geometric (TIT2FFWG) operator are developed for aggregation TIT2FNs. Furthermore, some desirable properties of the two aggregation operators are analyzed in detail. Finally, an approach based on TIT2FFWA (or TIT2FFWG) operator to solve MAGDM is developed. An illustrative example about supplier selection is provided to illustrate the developed procedures. The results demonstrate the practicality and effectiveness of our new method