Abstract
Picture fuzzy sets (PFSs) can be used to handle real-life problems with uncertainty and vagueness more effectively than intuitionistic fuzzy sets (IFSs). In the process of information aggregation, many aggregation operators under PFSs are used by different authors in different fields. In this article, a multi-attribute decision-making (MADM) problem is introduced utilizing harmonic mean aggregation operators with trapezoidal fuzzy number (TrFN) under picture fuzzy information. Three harmonic mean operators are developed namely trapezoidal picture fuzzy weighted harmonic mean (TrPFWHM) operator, trapezoidal picture fuzzy order weighted harmonic mean (TrPFOWHM) operator and trapezoidal picture fuzzy hybrid harmonic mean (TrPFHHM) operator. The related properties about these operators are also studied. At last, an MADM problem is considered to interrelate among these operators. Furthermore, a numerical instance is considered to explain the productivity of the proposed operators.
Highlights
multi-attribute decision-making (MADM) plays a vital role in decision-making science
We have introduced the score and accuracy function to rank the trapezoidal picture fuzzy number (TrPFN) and developed trapezoidal picture fuzzy weighted harmonic mean (TrPFWHM), TrPFOWHM, and trapezoidal picture fuzzy hybrid harmonic mean (TrPFHHM) operators which are discussed in the upcoming section
The paper is arranged as follows: After the introduction, Section 2 contains some preliminary concepts of Picture fuzzy sets (PFSs), TrPFNs, harmonic mean (HM), and weighted harmonic mean (WHM), and some useful operations related to this paper
Summary
MADM plays a vital role in decision-making science. It is important to various field such as economics, engineering and management. FSs theory fixed this issue by considering the membership grades of the elements. In an MADM problem information aggregation is a common process to ranking the alternatives. Research Background In 1965, Zadeh [1] proposed FSs theory which is an augmentation of crisp set and can deal with uncertainty and vagueness. The group “yes” means that the voters select the candidate; the group “no” means that the voters do not select the candidate; the group “refusal” means that the voters neither select nor reject the candidate For this issue, Cuong [4,5] introduced PFSs in which positive membership grade (μ), negative membership grade (ν) as well as neutral membership grade (η) is present, such that their sum does not exceed 1—that is, μ + η + ν ≤ 1
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