Asymmetric hesitant fuzzy sigmoid preference relations in the analytic hierarchy process

This study considers the practical phenomena in the process of preference elicitation and proposes an asymmetric sigmoid numerical scale (ASNS) based on a generalized sigmoid function. It also offers proof of the scale's asymmetry, variability, consistency, and diminishing utility properties. Further, this study introduces the hesitant fuzzy preference format and defines the hesitant fuzzy continuous preference term. Based on this approach, the asymmetric hesitant fuzzy sigmoid preference relation (AHSPR) is developed and used in the analytic hierarchy process (AHP). The results show that the AHSPR is a general and optimal preference relation. Additionally, this study constructs a discrete fitting technology and an approximate translation method as the applied bases of the new scale and the preference relation. Following this, a model framework of the AHSPR in the AHP is provided. Finally, this study re-examines a well-known numerical example in order to demonstrate the application and advantages of the proposed numerical scale, the preference format, and the modeling framework.