A framework for the determination of weak Pareto frontier solutions under probabilistic constraints.

摘要

The purpose of this research is to provide such a framework. The proposed framework 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 in order to reduce the computational time and expense to a manageable level so that the design space can be explored quickly, obtain trustworthy probabilities of the probabilistic constraints (PC) and WPF, and so as to enable conceptual design decision making in shorter time period.; A new hybrid method is formed that consists of the second order Response Surface Methodology (RSM) and the Support Vector Regression (SVR) method capturing the global tendency and the local nonlinear behavior respectively. The purpose of forming this hybrid method is to provide a method that can achieve high accuracy for many kinds of problems with a small training sample. The three parameters needed by SVR to be pre-specified are selected using practical methods and a modified information criterion that makes use of model fitting error, predicting error, and model complexity information. The model predicting error is estimated inexpensively with a new method called Random Cross Validation. In order to select a surrogate model without unnecessary complexity from RSM, SVR, and the hybrid method, this modified information criterion is also used as a surrogate model advisor to select the best surrogate model for a given problem.; 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 that is made up of consistent design solutions. Then the probabilities of the PC's are estimated, and the WPF and corresponding design solutions are found.; Three pure mathematical model fitting problems are used to demonstrate that the hybrid method of RSM and SVR really can obtain more accurate surrogate models with better results where sometimes the (second order) RSM, SVR, and Neural Network methods can not fit a given problem well with a small training sample. This illustrates the need for the hybrid method.; Three two-objective and one three-objective deterministic optimization problems are used to demonstrate that this framework can find the true weak Pareto frontier. The results show this framework can be used for many types of problems, such as cases of multiple-to-one mapping from design solutions in the design space to objective points in the objective space, problems of which the WPF is made up of spatially disjointed segments, and problems with constraints and more than two objectives.; A typical aircraft design problem and a reusable launch vehicle design problem under probabilistic constraints are solved to demonstrate the feasibility of this framework for engineering-based problems. The results of these two design problems show that both neighborhood search schemes can find the WPF. These results also show the methods to select the pre-specified paramete