In this paper we expand the work of Brandt and Dale (2011) on ERRs (energy return ratios) such as EROI (energy return on investment). This paper describes a "bottom-up" mathematical formulation which uses matrix-based computations adapted from the LCA (life cycle assessment) literature. The framework allows multiple energy pathways and flexible inclusion of non-energy sectors. This framework is then used to define a variety of ERRs that measure the amount of energy supplied by an energy extraction and processing pathway compared to the amount of energy consumed in producing the energy. ERRs that were previously defined in the literature are cast in our framework for calculation and comparison. For illustration, our framework is applied to include oil production and processing and generation of electricity from PV (photovoltaic) systems. Results show that ERR values will decline as system boundaries expand to include more processes. NERs (net energy return ratios) tend to be lower than GERs (gross energy return ratios). External energy return ratios (such as net external energy return, or NEER (net external energy ratio)) tend to be higher than their equivalent total energy return ratios. (C) 2013 Elsevier Ltd. All rights reserved.