The Consistency between Cross-Entropy and Distance Measures in Fuzzy Sets

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.

An Improved Butterfly Optimization Algorithm for Engineering Design Problems Using the Cross-Entropy Method

Engineering design optimization in real life is a challenging global optimization problem, and many meta-heuristic algorithms have been proposed to obtain the global best solutions. An excellent meta-heuristic algorithm has two symmetric search capabilities: local search and global search. In this paper, an improved Butterfly Optimization Algorithm (BOA) is developed by embedding the cross-entropy (CE) method into the original BOA. Based on a co-evolution technique, this new method achieves a proper balance between exploration and exploitation to enhance its global search capability, and effectively avoid it falling into a local optimum. The performance of the proposed approach was evaluated on 19 well-known benchmark test functions and three classical engineering design problems. The results of the test functions show that the proposed algorithm can provide very competitive results in terms of improved exploration, local optima avoidance, exploitation, and convergence rate. The results of the engineering problems prove that the new approach is applicable to challenging problems with constrained and unknown search spaces.