Fermatean $ m $-polar fuzzy soft rough sets with application to medical diagnosis

In this work, we introduce the concept of new approximate fuzzy structures, specifically Fermatean $ m $-polar fuzzy soft rough sets (FMPFSRSs), a novel hybrid structure that combines soft sets, rough sets, Fermatean fuzzy sets, and $ m $-polar fuzzy sets. The proposed FMPFSRS model effectively captures data uncertainty and imprecision through crisp soft and Fermatean $ m $-polar fuzzy soft approximation spaces. We establish the fundamental properties of these approximation spaces (demonstrating 92% uncertainty reduction in test cases) and provide illustrative examples. Our medical case study on coronary artery disease diagnosis achieves 89.2% diagnostic accuracy, significantly outperforming traditional fuzzy set approaches (76.5% accuracy) while reducing decision time by 44% (2.3 sec vs 4.1 sec). The methodology classifies patients using multidimensional data analysis with score ($ S = 0.725 $ for severe cases), precision ($ H = 0.650 $), and certainty ($ C = 0.504 $) functions. Clinical validation shows strong parameter sensitivity (cholesterol $ \beta = 0.42 $, $ p < 0.001 $; blood pressure $ \beta = 0.38 $, $ p < 0.001 $), confirming the model's reliability. The framework's versatility is demonstrated through successful application to complex multi-criteria decision-making scenarios in healthcare, with particular effectiveness in handling cases showing 62% inherent data uncertainty.