Generalized -rung orthopair fuzzy interactive Hamacher power average and Heronian means for MADM.

In this paper, we establish a novel -rung orthopair fuzzy (-ROF) multi-attribute decision making (MADM) model on the basis of the proposed -ROF interactive Hamacher weighted adjustable power average (-ROFIHWAPA) and -ROF interactive Hamacher weighted coordinated Heronian means (HMs), which (1) can reflect the correlations among multiple attributes; (2) weakens the impacts of the extreme evaluation values more reasonably; (3) considers the interactions between the membership degree (MD) and non-membership degree (N-MD) of different -ROF numbers (-ROFNs); (4) has the characteristic of generality (It can generate different methods by different operations). Firstly, the -ROF interactive Hamacher operations, improved score function and new -ROF entropy (-ROFE) formula, which are the necessary raw materials for the implementation of MADM, are presented. Secondly, we introduce the adjustable power average (APA) and its weight form (WAPA) to remedy the deficiencies of the classical power averages (PAs). Afterwards we extend the WAPA to -ROF circumstance and propose the -ROF interactive Hamacher WAPA (-ROFIHWAPA), and its basic properties are analyzed. Further, the entropy weight fitting method is presented to determine the parameter carried by the -ROFIHWAPA. Thirdly, inspired by the evolutionary process of Bonferroni means (BMs), we define the weighted coordinated HM (WCHM) and weighted geometric coordinated HM (WGCHM) based on the traditional HMs, respectively, which eliminate the redundancy of the dual generalized weighted BM (DGWBM) and dual generalized weighted Bonferroni geometric mean (DGWBGM), i.e., the case of Then we develop the -ROF interactive Hamacher WCHM (-ROFIHWCHM) and -ROF interactive Hamacher WGCHM (-ROFIHWGCHM) by combining them with the -ROF interactive Hamacher operations, and the common properties and special cases are also investigated. Finally, we create a MADM algorithm relied on the -ROFIHWAPA and -ROFIHWCHM (resp. -ROFIHWGCHM), and a practical example is introduced to illustrate the effectiveness and superiority of the proposed method.