In this paper we consider different models that assess eco-efficiency with production frontier estimation when both desirable outputs and undesirable outputs (or residuals) are considered. These models are confronted to livestock farm data (sheep meat farms) and greenhouse gas (GHG) emissions, to discuss their suitability in eco-efficiency measurement. The application is to French sheep meat farms. Our results show that under certain conditions the existing models, except for the by-production, yield the same results as when residuals are treated as inputs. The results also reveal that the by-production model augmented with dependence constraints offer some promising opportunities. Besides, environmental inefficiency appears to be the main contributor of eco-inefficiency in the sheep meat production.