A Framework for the Determination of Weak Pareto Frontier Solutions under Probabilistic Constraints

Abstract

A framework is proposed that combines separately developed multidisciplinary optimization, multi-objective optimization, and joint probability assessment methods together but in a decoupled way, to solve joint probabilistic constraint, multi-objective, multidisciplinary optimization problems that are representative of realistic conceptual design problems of design alternative generation and selection. The intent here is to find the Weak Pareto Frontier (WPF) solutions that include additional compromised solutions besides the ones identified by a conventional Pareto frontier. This framework starts with constructing fast and accurate surrogate models of different disciplinary analyses. A new hybrid method is formed that consists of the second order Response Surface Methodology (RSM) and the Support Vector Regression (SVR) method. The three parameters needed by SVR to be pre-specified are automatically selected using a modified information criterion based on model fitting error, predicting error, and model complexity information. The model predicting error is estimated inexpensively with a new method called Random Cross Validation. This modified information criterion is also used to select the best surrogate model for a given problem out of the RSM, SVR, and the hybrid methods. A new neighborhood search method based on Monte Carlo simulation is proposed to find valid designs that satisfy the deterministic constraints and are consistent for the coupling variables featured in a multidisciplinary design problem, and at the same time decouple the three loops required by the multidisciplinary, multi-objective, and probabilistic features. Two schemes have been developed. One scheme finds the WPF by finding a large enough number of valid design solutions such that some WPF solutions are included in those valid solutions. Another scheme finds the WPF by directly finding the WPF of each consistent design zone. Then the probabilities of the PCs are estimated, and the WPF and corresponding design solutions are found. Various examples demonstrate the feasibility of this framework.