Uncertainty quantifications of Pareto optima in multiobjective problems

Design is a multi-objective decision-making process considering manufacturing, cost, aesthetics, usability among many other product attributes. The set of optimal solutions, the Pareto set, indicates the tradeoffs between objectives. Decision-makers generally select their own optima from the Pareto set based on personal preferences or other judgements. However, uncertainties from manufacturing processes and from operating conditions will change the performances of the Pareto optima. Evaluating the impacts of uncertainties on Pareto optima requires a large amount of data and resources. Comparing multiple Pareto solutions under uncertainty are also very costly. In this work, local Pareto set approximation is integrated with uncertainty propagation technique to quantify design variations in the objective space. An optimality influence range is proposed using linear combinations of objective functions that creates a more accurate polygon object variation subspace. A set of ‘virtual samples’ is then generated to form two quantifications of the objective variation subspace, namely an influence noise indicates how the design maintain Pareto, and an influence range that quantifies the overall variations of a design. In most engineering practices, a Pareto optimum with a smaller influence noise and a smaller influence area is preferred. We also extend the influence noise/range concept to nonlinear Pareto set with the second-order approximations. The quadratic local Pareto approximation method in the literature is also extended in this work to solve multi-objective engineering problems with black-box functions. The usefulness of the proposed quantification method is demonstrated using numerical examples as well as using engineering problems in structural design.