RETRACTED: Extension principles for picture fuzzy sets

Abstract

Picture fuzzy set (PFS), is a newly developed apparatus to treaty with uncertainties in problems where the opinions are yes, no, neutral, and refusal types. Extension principle is one of the key tools for describing uncertainties. It provides a general method for existing classical mathematical concepts to address fuzzy quantities. It has numerous applications in various arena of our real life. However, there are less works on extension principle for picture fuzzy sets. In this article, new extension principles namely minimal extension principle and average extension principle are proposed for picture fuzzy sets. Various properties of the minimal extension principle and the average extension principle for PFSs are also established. We also prove some properties of Zadeh’s extension principle for PFSs. Finally, arithmetic operations for PFSs based on the average extension principle are developed with numerical illustrations.

Enhancing Urban Development with Picture Fuzzy Sets: A Strategic Decision Support Framework

“…In general, models based on PFS [13] may be suitable for dealing with human opinions that involve a greater number of responses of the following type: yes, abstain, no, and refuse. Therefore, in order to deal with these types of situations, in this paper we introduce the concept of similarity measures for PFS, which is a new extension of the similarity measure of Atanassov's IFS.…”