In practical decision making, one often is interested in solutions that balance multiple objectives. In this study we focus on generating efficient solutions for optimization problems with two objectives and a large but finite number of feasible solutions. Two classical approaches exist, being the constraint method and the weighting method, for which a specific implementation is required for this problem class. This paper elaborates specific straightforward implementations and applies them to a practical allocation problem, in which transportation cost and risk of shortage in supplied livestock quality are balanced. The variability in delivered quality is modelled using a scenario-based model that exploits historical farmer quality delivery data. The behaviour of both implementations is illustrated on this specific case, providing insight in (i) the obtained solutions, (ii) their computational efficiency. Our results indicate how efficient trade-offs in bi-criterion problems can be found in practical problems.