Pareto optimality and robustness in bi-blending problems

The mixture design problem for two products concerns finding simultaneously two recipes of a blending problem with linear, quadratic and semi-continuity constraints. A solution of the blending problem minimizes a linear cost objective and an integer valued objective that keeps track of the number of raw materials that are used by the two recipes, i.e. this is a bi-objective problem. Additionally, the solution must be robust. We focus on possible solution approaches that provide a guarantee to solve bi-blending problems with a certain accuracy, where two products are using (partly) the same scarce raw materials. The bi-blending problem is described, and a search strategy based on Branch-and-Bound is analysed. Specific tests are developed for the bi-blending aspect of the problem. The whole is illustrated numerically.