Multi-objective Optimization under Uncertain Objectives: Application to Engineering Design Problem

In the process of multi-objective optimization of real-world systems, uncertainties have to be taken into account. We focus on a particular type of uncertainties, related to uncertain objective functions. In the literature, such uncertainties are considered as noise that should be eliminated to ensure convergence of the optimization process to the most accurate solutions. In this paper, we adopt a different point of view and propose a new framework to handle uncertain objective functions in a Pareto-based multi-objective optimization process: we consider that uncertain objective functions are not only biasing errors due to the optimization, but also contain useful information on the impact of uncertainties on the system to optimize. From the Probability Density Function (PDF) of random variables modeling uncertainties of objective functions, we determine the "Uncertain Pareto Front", defined as a "tradeoff probability function" in objective space and a "solution probability function" in decision space. Then, from the "Uncertain Pareto Front", we show how the reliable solutions, i.e. the most probable solutions, can be identified. We propose a Monte Carlo process to approximate the "Uncertain Pareto Front". The proposed process is illustrated through a case study of a famous engineering problem: the welded beam design problem aimed at identifying solutions featuring at the same time low cost and low deflection with respect to an uncertain Young's modulus.