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.
Highlights
In real-life situations, the fuzzy set theory [48] plays a vital role in handling the vagueness of human choices
In Definition 3.2, we find that the weights associated with the picture fuzzy Frank weighted averaging (PFFWA) operator are the simplest form of picture fuzzy (PF) value and in Definition 3.3 the weights associated with the picture fuzzy Frank order weighted averaging (PFFOWA) operator is the original form of the ordered positions of the PF values
In Definition 3.5, we find that the weights associated with the picture fuzzy Frank weighted geometric (PFFWG) operator are in the simplest form of PF value and in Definition 3.6 the weights associated with the picture fuzzy Frank order weighted geometric (PFFOWG) operator are in the actual form of the ordered positions of the PF values
Summary
In real-life situations, the fuzzy set theory [48] plays a vital role in handling the vagueness of human choices. Xing et al [44] introduced aggregation operators for Pythagorean fuzzy set based on Frank t-norm and t-conorm and applied them to solve MADM problems. Qin and Liu [24] introduced Frank aggregation operators for a triangular interval type fuzzy set and applied it to solve multiple attribute group decision making (MAGDM) problems. The investigation on the applications of Frank operators is rare, in the area of information aggregation and decision making Keeping this in mind, it is worthy to prolong Frank t-norm and t-conorm to handle the PF environment.
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