Proportional grey picture fuzzy sets and their application in multi-criteria decision-making with high-dimensional data

Picture fuzzy sets (PFS) extend intuitionistic fuzzy sets by incorporating positive, neutral, and negative memberships to capture richer information. A notable challenge of PFS and its derivatives is the need to specify these degrees using decimals, thus limiting their practical applicability. To address this issue, we utilize proportional picture fuzzy sets (PPFS) to define these parameters through proportional relationships. Our approach selects a PFS as the unit fuzzy set, while the newly formulated proportional grey picture fuzzy sets (PGPFS) exploits the proportionality between the individual and the unit fuzzy set parameters. Additionally, we introduce the concept of a fuzzy tensor entropy measures and aggregation operators for PGPFS. Additionally, we develop an aggregation decision-making method based on PGPFS, thereby, accommodating the inherent ambiguity and uncertainty of the data. The feasibility of the PGPFS approach in addressing multi-criteria decision-making (MCDM) scenarios with uncertain criteria and expert weights is verified through an application of haze management scheme selection. The reasonableness and effectiveness of the method are further confirmed through sensitivity and comparative analyses.