This study addresses the magnetohydrodynamic flow of a squeezed ternary nanofluid between two horizontal parallel Riga plates. The importance of this problem lies in understanding the complex interactions between magnetic fields, nanofluid dynamics, and heat transfer, which are crucial for optimizing thermal management systems. This study utilizes a numerical approach, specifically a collocation method implemented in MATLAB, to solve the governing equations with high precision. Key results acquired indicate that the magnetic field and Riga plate actuator significantly enhance fluid velocity, whereas the variation in thermal conductivity, radiation, and viscous dissipation increases the temperature distribution. Quantitative analysis illustrates the impact of all these factors on skin friction and Nusselt number. Sensitivity analysis using the response surface methodology exhibits the conditions for optimized heat transfer. The novelty of this work lies in its comprehensive analysis of the magnetohydrodynamic flow in the presence of a microcantilever sensor, which provides deep understanding of optimization of heat transfer rates. This research offers a detailed examination of the combined effects of various physical phenomena and also validates them through graphical comparisons with existing studies.