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Fused deposition modeling (FDM) is an additive manufacturing (AM) technology capable of producing functional parts with complex geometries. However, optimizing both dimensional and mechanical quality characteristics is challenging due to the influence of multiple process parameters. This study aims to determine how key FDM parameters affect the multivariate dimensional and mechanical quality characteristics and to establish an integrated framework for optimizing these responses simultaneously. An experimental design based on response surface methodology (RSM) was implemented to optimize four process parameters: layer thickness, extruder temperature, plate temperature, and printing speed. Cylindrical PLA samples adhering to compression standards were fabricated, and both dimensional characteristics (length and diameter) and mechanical characteristics (compressive strength and modulus) were evaluated. Multivariate process capability indices (MPCIs) were then estimated to assess overall process capability. The results revealed that layer thickness and extruder temperature are the most influential parameters affecting MPCIs. The optimal dimensional MPCI was achieved with a low layer thickness, high extruder temperature, low plate temperature, and low printing speed. Also, the proposed model explained 88% of the total variability in mechanical MPCI. This research introduces an integrated RSM–multivariate process capability analysis approach for the simultaneous optimization of multiple correlated quality characteristics in FDM, which improves both dimensional precision and mechanical performance in AM processes.