The distance measures and cross-entropy based on complex fuzzy sets and their application in decision making

The information measures (IMs) of complex fuzzy information are very useful tools in the areas of machine learning and decision making. In some multi-attribute group decision making (MAGDM) problems, the decision makers can make a decision mostly according to IMs such as similarity measures (SMs), distance measures (DIMs), entropy measures (EMs) and cross-entropy measures (C-EMs) in order to choose the best one. However, the relation between C-EMs and DIMs in the environment of complex fuzzy sets (CFSs) has not been developed and verified. In this manuscript, the notions of DIMs and C-EMs in the environment of CFSs are investigated and the relation between DIMs and EMs in the environment of CFSs is also discussed. The complex fuzzy discrimination measures (CFDMs), the complex fuzzy cross-entropy measures (CFC-EMs), and the symmetry complex fuzzy cross-entropy measures (SCFC-EMs) are proposed. We also examined that the C-EMs satisfied all the conditions of DIMs, and finally proved that C-EMs including CFC-EMs were also a DIMs. In last, we used some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods.