A Clustering‐based Methodology for Discontinuous Pareto Frontier Representation

Abstract

ABSTRACTThis paper presents a methodology for capturing complex trade‐off relationships among objective functions and the underlying design principles through Pareto frontier representations, for Multi‐Objective Optimization problems in complex engineering design processes. The methodology provides engineers a way to quickly assess achievability of target values on objective functions (referred to as f‐feasibility), and propose design solutions that meet or exceed these targets. Obtaining Pareto frontier representations can be challenging for cases when there are discontinuities in the Pareto frontiers and when the trade‐off relationships vary across these discontinuities. The proposed methodology addresses this issue by first identifying discontinuities in the Pareto frontier through a clustering procedure, and then obtaining functional approximations of the trade‐off relationships for each of the discontinuous portion of the Pareto frontier. A two‐stage approach for clustering is employed. In the first stage, a DBSCAN clustering technique is used to classify disjoint Pareto sets in the objective space (f‐space) into different groups. In the second stage, a frequent set mining algorithm named ECLAT and a subsequent filtering procedure are used to extract design constraint combinations that are active (binding) within each of these groups to form subgroups. Approximations are obtained for each of the subgroups using a constrained least squares technique, which are then combined to obtain an overall equation for assessing f‐feasibility of any arbitrary point in objective space. In the two implementation studies, it is observed that the two‐stage approach for clustering is highly effective in yielding good quality Pareto frontier representations and f‐feasibility assessments. Copyright © 2013 John Wiley & Sons, Ltd.