Uncertainty analysis of trade‐offs between multiple responses using hypervolume

Abstract

When multiple responses are considered in process optimization, the degree to which they can be simultaneously optimized depends on the optimization objectives and the amount of trade‐offs between the responses. The normalized hypervolume of the Pareto front is a useful summary to quantify the amount of trade‐offs required to balance performance across the multiple responses. To quantify the impact of uncertainty of the estimated response surfaces and add realism to what future data to expect, 2 versions of the scaled normalized hypervolume of the Pareto front are presented. To demonstrate the variation of the hypervolume distributions, we explore a case study for a chemical process involving 3 responses, each with a different type of optimization goal. Results show that the global normalized hypervolume characterizes the proximity to the ideal results possible, while the instance‐specific summary considers the richness of the front and the severity of trade‐offs between alternatives. The 2 scaling schemes complement each other and highlight different features of the Pareto front and hence are useful to quantify what solutions are possible for simultaneous optimization of multiple responses.