Parametric Optimization: Applications in Systems Design

Abstract

The aim of this research is to introduce the notion of parametric optimization (PO) as a useful approach for solving systems design challenges. In this research, we define PO as the process of finding the optimal solution as a function of one or more parameters. Parameters are variables that affect the optimal solution but, unlike the decision variables, are not directly controlled by the designer. The principal contributions of this research are (1) a novel formulation of the PO problem relevant to systems design, (2) a strategy for empirically assessing the performance of parametric search algorithms, (3) the development and evaluation of novel algorithms for PO, and (4) a demonstration of the use of PO for two real-world systems design challenges. The real-world demonstrations, include the design of (i) a multi-ratio vehicle transmission, and (ii) a Liquid Metal Magnetohydrodynamic Pump. A practical challenge of applying the notion PO to systems design is that existing methods are limited to problems where the models are accessible algebraic equations and single objectives. However, many challenges in systems design involve inaccessible models or are too complicated to be manipulated algebraically and have multiple objectives. If PO is to be used widely in systems design, there is a need for search methods that can approximate the solution to a general PO problem. As a step toward this goal, a strategy for performance assessment is developed. The use of the mean Hausdorff distance is proposed as a measure of solution quality for the PO problem. The mean Hausdorff distance has desirable properties from a mathematical and decision theoretic basis. Using the proposed performance assessment strategy, two algorithms for parametric optimization are evaluated, (a) p-NSGAII which is a straightforward extension of existing methods to the case with parameters, and (b) P3GA an algorithm intended to exploit the parametric structure of the problem. The results of the study indicate that a considered approach, P3GA, to the PO problem results in considerable computational advantage.