Bayesian optimization (BO) is a low-cost global optimization tool for expensive black-box objective functions, where we learn from prior evaluated designs, update a posterior surrogate Gaussian process model, and select new designs for future evaluation using an acquisition function. This research focuses upon developing a BO model with multiple black-box objective functions. In the standard multi-objective optimization problem, the weighted Tchebycheff method is efficiently used to find both convex and non-convex Pareto frontier. This approach requires knowledge of utopia values before we start optimization. However, in the BO framework, since the functions are expensive to evaluate, it is very expensive to obtain the utopia values as a priori knowledge. Therefore, in this paper, we develop a Multi-Objective Bayesian Optimization (MO-BO) framework where we calibrate with Multiple Linear Regression (MLR) models to estimate the utopia value for each objective as a function of design input variables; the models are updated iteratively with sampled training data from the proposed multi-objective BO. The iteratively estimated mean utopia values are used to formulate the weighted Tchebycheff multi-objective acquisition function. The proposed approach is implemented in optimizing a thin tube design under constant loading of temperature and pressure, with multiple objectives such as minimizing the risk of creep-fatigue failure and design cost along with risk-based and manufacturing constraints. Finally, the model accuracy with and without MLR-based calibration is compared to the true Pareto solutions. The results show potential broader impacts, future research directions for further improving the proposed MO-BO model, and potential extensions to the application of large-scale design problems.

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