Optimal Product Portfolio Design by Means of Semi-infinite Programming

A new type of product portfolio design task where the products are identified with geometrical objects representing the efficiency of a product, is introduced. The sizes and shapes of these objects are determined by multiple constraints whose activity cannot be easily predicted. Hence, a discretization of the parameter spaces could obfuscate some advantageous portfolio configurations. Therefore, the classical optimal product portfolio problem is not suitable for this task. As a new mathematical formulation, the continuous set covering problem is presented which transfers into a semi-infinite optimization problem (SIP). A solution approach combining adaptive discretization of the infinite index set with regularization of the non-smooth constraint function is suggested. Numerical examples based on questions from pump industry show that the approach is capable to work with real-world applications.