In this work we consider multi-objective optimization problems arising from the domain of biochemical systems, namely metabolic pathways, with the ethanol production by Saccharomyces cerevisiae as a case study. The goals are to maximize the rate of production of ethanol and simultaneously minimize several internal metabolite concentrations, imposing additional constraints in order to ensure cell viability. As a result, the Pareto-optimal set is obtained for different formulations of the optimization problem. Starting from a detailed, nonlinear, kinetic model of the pathway, several recent solution strategies and other well-known techniques are compared, highlighting the advantages and drawbacks of each one.