Some Picture Fuzzy Aggregation Operators based on Frank t-norm and t-conorm: Application to MADM Process

In this paper, we develop some new operational laws and their corresponding aggregation operators for picture fuzzy sets (PFSs). PFS is a powerful tool to deal with vagueness, which is a generalization of a fuzzy set and intuitionistic fuzzy set (IFS). PFSs can model uncertainty in such situations, which consist of more than two answers like yes, refusal, neutral, and no. The operations of t-norm and t-conorms, developed by Frank, are usually a better application with its flexibility. From that point of view, the concepts of Frank t-norm and t-conorms are introduced to aggregate picture fuzzy information. We propose some new operational laws of picture fuzzy numbers (PFNs) based on Frank t-norm and t-conorm. Further, with the assistance of these operational laws, we introduce picture fuzzy Frank weighted averaging (PFFWA) operator, picture fuzzy Frank order weighted averaging (PFFOWA) operator, picture fuzzy Frank hybrid averaging (PFFHA) operator, picture fuzzy Frank weighted geometric (PFFWG) operator, picture fuzzy Frank order weighted geometric (PFFOWG) operator, picture fuzzy Frank hybrid geometric (PFFHG) operator and discussed with their suitable properties. Then, with the help of PFFWA and PFFWG Operators, we have presented an algorithm to solve multiple-attribute decision making (MADM) problems under the picture fuzzy environment. Finally, we have used a numerical example to illustrate the flexibility and validity of the proposed method, and have compared the results with other existing methods.