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

Abstract

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.