This study explores the tribological performance of bump foil journal bearing (FJB) equipped with micro-pocket. The influence of micro-pocket on the load capacity (LC) and friction factor (FF) of FJBs is evaluated for various foil materials, including Inconel X750 (IN), Spring steel EN42J (SP), SS 316, Beryllium copper (BC), and Phosphor bronze (PB). The compressible Reynolds equation is solved numerically to assess the tribological behavior of bump FJB under various operating conditions. Initially, the impact of micro pocket on the tribological characteristics such as load capacity and friction factor of bump FJB is examined by comparing their performance with plain FJB. Subsequently, the effect variable parameters such as pocket depth (PD), bearing number (Λ), and eccentricity ratio (ε), on the tribological behavior is analysed by using response surface methodology (RSM). The multi-objective grey relational analysis (GRA) technique is employed to determine the optimal values of these variable parameters for each foils material. The findings reveal that micro-pocket enhance the dimensionless LC of bump FJB beyond a bearing number of 2.061. Moreover, micro-pocket reduce the friction factor of bump FJBs. Within the presence of micro pocket, SS 316 exhibits the most significant foil material for the better tribological characteristics of bump FJB, while phosphor bronze is the least signifcant. For SS 316, the optimal values of pocket depth (PD), bearing number (Λ), and eccentricity ratio (ε) are 3 μm, 2.061, and 0.5 respectively. Under these optimal conditions, the load capacity (LC) and friction factor (FF) of bump FJBs are 0.7740 and 13.645, respectively. Micro-pocket reduces the friction factor (FF) by 2 to 15%. The impact of pocket depth on the LC and FF of bump FJB is significant for the phosphor bronze. Additionally, increasing the pocket depth enhances the dimensionless LC of bump FJBs while decreasing the friction factor.