A steady, incompressible, two-dimensional Sisko-nanofluid flow towards the horizontal direction with no movement in the vertical direction is considered on a stretching/shrinking surface. The power law component (Sisko model) is incorporated under the regime of the porous medium. A magnetic impact is included coming from the MHD in the surface normal direction. In addition, thermal radiation, Brownian diffusion, and thermophoresis are involved in the governing system of equations obtained from the Navier–Stokes model in two-dimensional flow systems. The PDEs are converted into the one-dimensional system using suitable transformations and solved by Galerkin weighted residual method validated with the spectral collocation method. The optimization analysis is performed on heat transfer and skin-friction factors using response surface methodology. The impact of the parameters involved in the model has been testified and is provided in graphical forms. The outcomes indicate that for the values of the porosity factor fluctuating between [0, 2.5], the velocity profile and corresponding boundary layer thickness are lesser towards the maximum value of the parameter, and the results are opposite as the parameter approaches zero. The optimization and sensitivity analysis shows that the transport of heat sensitivity towards thermal radiation, Brownian diffusion, and thermophoresis declined whenever the Nt and Nb increased from low to high and at the medium level of thermal radiation. An increment in the Forchheimer parameter increases the sensitivity of the rate of friction factor, whereas increasing the Sisk-fluid parameter has the reverse effect. Elongation processes like those of pseudopods and bubbles make use of such models. The idea is also widely used in other sectors, such as the textile industry, glass fiber production, cooling baths, paper manufacture, and many more.