Irreversible losses and heat transport in a magnetohydrodynamic flow of a viscous, steady, incompressible, and fully developed couple stress Al 2 O 3-water nanofluid through a sloping permeable wall channel with porous medium and under the effect of radiation heat flux and slip were analyzed. The fundamental equations were solved numerically by using Runge-Kutta together with the shooting technique and the results were in qualitative agreement with an exact solution obtained for a limit case. The impacts of couple stress, Darcy number, solid nanoparticle concentrations, conduction-radiation parameter, Hartmann number and hydrodynamic slip on flow, temperature, heat transport, and entropy production were examined. It was possible to achieve values of minimum entropy production not yet reported in previous studies. In this way, optimal values of couple stress and slip were obtained. The heat transport was also explored and optimal values of slip flow and conduction-radiation parameter with maximum heat transfer were found. Finally, in addition to the alumina, the distributions of velocity, temperature, and entropy generation in TiO 2-water and Cu-water were presented for different solid nanoparticle concentrations. It was obtained that the local entropy of TiO 2-water was lower than Cu-water and Al 2 O 3-water in the channel bottom region while it was greater in the upper region.