Planning the defossilization of energy systems while maintaining access to abundant primary energy resources is a nontrivial multi-objective problem encompassing economic, technical, environmental, and social aspects. However, most longterm policies consider the cost of the system as the leading indicator in the energy system models to decrease the carbon footprint. This paper is the first to develop a novel approach by adding a surrogate indicator for the social and economic aspects, the energy return on investment (EROI), in a whole-energy system optimization model. In addition, we conducted a global sensitivity analysis to identify the main parameters driving the EROI uncertainty. This method is illustrated in the 2035 Belgian energy system for several greenhouse gas (GHG) emissions targets. Nevertheless, it can be applied to any worldwide or country energy system. The main results are threefold when the GHG emissions are reduced by 80%: (i) the EROI decreases from 8.9 to 3.9; (ii) the imported renewable gas (methane) represents 60 % of the system primary energy mix; (iii) the sensitivity analysis reveals this fuel drives 67% of the variation of the EROI. These results raise questions about meeting the climate targets without adverse socio-economic impact, demonstrating the importance of considering the EROI in energy system models.