The main investigation of the paper is to develop the restricted fixed charge solid transportation problem under a hybrid uncertain environment where fuzziness and roughness coexist. A fuzzy rough STP model is developed by integrating the classical STP, fuzzy set theory, and rough set theory, which apparently provide a way to accommodate the uncertainty. For solving the problem, we apply the fuzzy rough expected value operator and propose the possibility based STP model with fuzzy rough parameters on a rough space. A numerical example is presented to describe the fuzzy rough approach using Lingo 13.0 optimization software. Finally, a graphical presentation has also shown to describe the comparison between two proposed approaches. Some important managerial decisions are also drawn by observing the optimal result. © Springer Nature Switzerland AG 2020.