The primary goal of this work is to provide easy-to-use and cutting-edge optimization software designed to handle uncertainties, intended for use in research and education. Robustimizer offers efficient uncertainty quantification through exact analytic formulas using specific surrogate models, such as Gaussian Processes. Moreover, it supports integration with other software packages, and automatic updating of initial design space through exploration–exploitation techniques, among other features. This software has proven its value in sustainable manufacturing, where optimizing processes to reduce environmental impact while managing uncertainties is critical. In this article, the Robustimizer graphical user interface is introduced as a domain-independent optimization tool for surrogate-model-based robust and reliability-based design or process optimization.

In this work, metamodel-based robust optimization is performed using measured scatter of noise variables. Principal component analysis is used to describe the input noise using linearly uncorrelated principal components. Some of these principal components follow a normal probability distribution, others however deviate from a normal probability distribution. In that case, for more accurate description of material scatter, a multimodal distribution is used. An analytical method is implemented to propagate the noise distribution via metamodel and to calculate the statistics of the response accurately and efficiently. The robust optimization criterion as well as the constraints evaluation are adjusted to properly deal with multimodal response. Two problems are presented to show the effectiveness of the proposed approach and to validate the method. A basketball free throw in windy weather condition and forming of B-pillar component are presented. The significance of accounting for non-normal distribution of input variables using multimodal distributions is investigated. Moreover, analytical calculation of response statistics, and adjustment of the robust optimization problem are presented and discussed.

Production efficiency in metal forming processes can be improved by implementing robust optimization. In a robust optimization method, the material and process scatter are taken into account to predict and to minimize the product variability around the target mean. For this purpose, the scatter of input parameters are propagated to predict the product variability. Consequently, a design setting is selected at which product variation due to input scatter is minimized. If the minimum product variation is still higher than the specific tolerance, then the input noise must be adjusted accordingly. For example this means that materials with a tighter specification must be ordered, which often results in additional costs. In this article, an inverse robust optimization approach is presented to tailor the variation of material and process noise parameters based on the specified product tolerance. Both robust optimization and tailoring of material and process scatter are performed on the metamodel of an automotive part. Although the robust optimization method facilitates finding a design setting at which the product to product variation is minimized, the tighter product tolerance is only achievable by requiring less scatter of noise parameters. It is shown that the presented inverse approach is able to predict the required adjustment for each noise parameter to obtain the specified product tolerance. Additionally, the developed method can equally be used to relax material specifications and thus obtain the same product tolerance, ultimately resulting in a cheaper process. A strategy for updating the metamodel on a wider (noise) base is presented and implemented to obtain a larger noise scatter while maintaining the same product tolerance.

A robustness criterion that employs skewness of output is presented for a metamodel-based robust optimization. The propagation of a normally distributed noise variable via nonlinear functions leads to a non-normal output distribution. To consider the non-normality of the output, a skew-normal distribution is used. Mean, standard deviation, and skewness of the output are calculated by applying an analytical approach. To show the applicability of the proposed method, a metal forming process is optimized. The optimization is defined by an objective and a constraint, which are both nonlinear. A Kriging metamodel is used as nonlinear model of that forming process. It is shown that the new robustness criterion is effective at reducing the output variability. Additionally, the results demonstrate that taking into account the skewness of the output helps to satisfy the constraints at the desired level accurately.

Optimization under uncertainty requires proper handling of those input parameters that contain scatter. Scatter in input parameters propagates through the process and causes scatter in the output. Stochastic methods (e.g. Monte Carlo) are very popular for assessing uncertainty propagation using black-box function metamodels. However, they are expensive. Therefore, in this article a direct method of calculating uncertainty propagation has been employed based on the analytical integration of a metamodel of a process. Analytical handling of noise variables not only improves the accuracy of the results but also provides the gradients of the output with respect to input variables. This is advantageous in the case of gradient-based optimization. Additionally, it is shown that the analytical approach can be applied during sequential improvement of the metamodel to obtain a more accurate representative model of the black-box function and to enhance the search for the robust optimum.

Robust optimization is a powerful method to find the parameters for a process at which its output is least sensitive to the variation of the input parameters. In this method, measured or estimated noise parameters are used to estimate the scatter of the output. At the optimum design, the variation in noise parameters leads to a minimum scatter of the output. If this minimum scatter of the output does not meet the specified tolerance, then the input noise must be adjusted accordingly. This means for example that materials with a tighter specification must be ordered, which usually incurs additional costs. In this article, an inverse method is presented to tailor the variation of noise parameters based on the allowable tolerance in the output. This method is successfully applied to a non-linear process, lab-type B-pillar part. The results show how to adjust the input noise parameters at a minimum cost to meet the required output tolerance.