The P-graph framework introduced by Friedler et al. (Chem Eng Sci. 1992;47:1973-1988) is a general mathematical methodology based on Graph Theory which is applicable to many process design problems. We propose an extension of the P-graph framework and the associated MILP model to account for operating units and systems where the inputs and outputs are variable. This is important because the P-graph framework in its current form would otherwise apply only to systems where the ratios of inputs to outputs are fixed. Consequently, it is difficult to apply the method to many emerging systems without the new mathematical model presented here. We discuss and develop the model in detail, and we then illustrate its application with the case study of an energy system. We establish a structure with optimal cost for a baseline heat demand, and we further explore how the energy system structure and the feedstocks change as the heat demand increases. (c) 2015 American Institute of Chemical Engineers AIChE J, 62: 1143-1153, 2016