Maximal consistent blocks-based optimistic and pessimistic probabilistic rough fuzzy sets and their applications in three-way multiple attribute decision-making

The integration of three-way decision (3WD) into multiple attribute decision-making (MADM) problems has emerged as a pivotal research area. 3WD can effectively manage the inherent uncertainty within the decision-making process. Additionally, it offers a semantic interpretation of the outcomes. In this paper, we introduce two innovative 3WD-MADM approaches, with a focus on granule selection and the handling of multi-type information in the framework of three-way decisions. Firstly, we construct maximal consistent blocks (MCBs)-based pessimistic and optimistic probabilistic rough fuzzy set (RFS) models and investigate their properties to ascertain their efficacy and reliability in decision-making contexts. Then, we define relative loss functions associated with “good state” and “bad state” scenarios. Building on this, we introduce four types of 3WDs based on our newly proposed optimistic and pessimistic probabilistic RFSs. Furthermore, we integrate the 3WDs information from both scenarios to formulate optimistic and pessimistic 3WD-MADM approaches, handling both single-valued fuzzy and intuitionistic fuzzy information. Finally, we contrast our proposed methodologies with the current MADM methods, and demonstrate their validity, significance and generalization ability.