Abstract The thermodynamic analysis of open systems is fundamental in engineering. For the open systems at their steady state, two apparently opposed principles for the rate of entropy production have been used: the minimum entropy production rate derived by Prigogine, used in the description of various processes in physics, chemistry and biology, and the maximum entropy production, used in many other cases and now considered more general. Both principles involve an extreme value of the rate of entropy production in an open system at the steady state under non-equilibrium conditions. In this paper, a link between these two approaches is developed and their synthesis with the constructal law is proposed. An application to ATP synthesis in anaerobic fermentation for biogas production is presented.