Multiple attribute decision making using an enhanced complex proportional assessment method based on probabilistic generalized orthopair fuzzy soft sets

As a common generalization of intuitionistic fuzzy sets (IFSs), Pythagorean fuzzy sets (PFSs) and Fermatean fuzzy sets (FFSs), generalized orthopair fuzzy sets (GOFSs) have received worldwide attention in past few years. These extended fuzzy sets are all built on the basis that the information described by the membership grade (MG) and non-membership grade (NMG) is of equal importance. In some real-life decision-making applications, however, membership and non-membership grades may have different degrees of importance. To address this issue, probability information is used to enhance Yager's GOFSs, which gives rise to the notion of probabilistic generalized orthopair fuzzy sets (PGOFSs). Following this line of exploration, we put forth probabilistic generalized orthopair fuzzy soft sets (PGOFSSs) in this study, and consider their applications to multiple attribute decision making (MADM) under uncertainty. Firstly, we integrate PGOFSs with Molodtsov's soft sets to define the notion of PGOFSSs. Some basic operations of PGOFSSs are presented and related algebraic properties are investigated. We also define two different types of probabilistic generalized orthopair fuzzy soft subsets and consider the probabilistic generalized orthopair fuzzy soft equal relation derived from them. Secondly, we extend the traditional complex proportional assessment (COPRAS) method by virtue of PGOFSSs, and propose the probabilistic generalized orthopair fuzzy soft complex proportional assessment (PGOFSoft-COPRAS) approach to solve MADM problems based on PGOFSSs. We also extend the maximizing deviation method (MDM) within the framework of PGOFSSs for attribute weight determination. Finally, the PGOFSoft-COPRAS approach is utilized to address a problem regarding cloud service provider (CSP) selection. The effectiveness and superiority of the PGOFSoft-COPRAS approach are verified through a brief comparison with several existing MADM methods. Moreover, we propose a Levenshtein distance based similarity measure to quantify the consistency between the ranking results obtained by disparate MADM methods.