Research Project Planning (RPP) is a central task routinely tackled by research institutions, which aims at planning the dedication of a set of researchers involved in a set of research projects, with the goal of meeting the preferences of each researcher as much as possible while efficiently utilizing the available budget. Despite its importance, in real-world institutions, RPP is solved manually due to the lack of automated solutions. To overcome this limitation, we put forward a flexible and scalable approach to provide robust project plans to users (e.g., the administrative staff of a research institution) by modeling the RPP as a constrained norm approximation problem, hence enabling the use of modern off-the-shelf optimization solvers. Results on real-world data provided by our research institution show that our approach can compute optimal plans that reflect the preferences of users (i.e., meeting the preferred dedication of researchers and efficiently spending the available budget) in a matter of seconds. Furthermore, we show that our approach can compute plans for thousands of projects and researchers within minutes, hence being able to solve RPP problem instances much larger than the ones typically encountered in an average-sized institution.