A New Standardization-Based Ranking Method for Generalized Trapezoidal Picture Fuzzy Numbers

Abstract

A generalized fuzzy number (GFN), whose height is not necessarily 1, is used in situations when expert opinions are not completely reliable. This subnormality complicates operations based on the extension principle. Moreover, complexity is inherited in non-standard fuzzy numbers (FNs). This paper aims to present a unified approach for comparing generalized and trapezoidal types of FNs, intuitionistic FNs (IFNs), and picture FNs (PFNs). If some of the hesitation, neutrality, and refusal are assumed to be resolved, then the uncertainty is reduced while making a non-standard FN standardized. The method uses the weighted average membership function (WAMF) to standardize generalized IFNs (GIFNs) and generalized PFNs (GPFNs). WAMF employs parameters describing the behavioral patterns when decision-makers encounter situations involving risk. Then, the ranking process can be continued with the calculation of the centroid point of the resulting GFN. One of the main advantages of this approach is that the computations are straightforward due to the presence of piecewise linearity, enabling us to employ numerical integration. Furthermore, we adapt operations for generalized trapezoidal PFNs (GTPFNs) to mitigate the counter-intuitive consequences resulting from utilizing the minimum operator. The effectiveness of the method is discussed through benchmarks and its implementation in multi-attribute decision-making (MADM).