Designing experiments in Robust Design problems

In the last decades the method of Robust Design that was originally suggested by Taguchi has been widely applied to various engineering areas. Usually, when a designer aims for a robust design of a system with unknown analytical form, he or she follows a two-step procedure. First, he or she fits a response function for the unknown system by using experimental arrays that are based on known design of experiments (DOE) criteria. For example, using D-optimal designs to minimize the variance-covariance matrix of the response's coefficients. Second, once the response function has been established, he or she formulates a Robust-Design criterion and solves it to obtain an optimal robust configuration. In this work, we aim to combine both steps in a unified, yet sequential, DOE protocol. In particular, we suggest a methodology for designing experiments that minimize the variance of the optimal robust configuration. In other words, the variance of the optimal solution for a robust system is minimized already at the DOE stage. This new DOE optimal criteria prioritizes the various response's coefficients and enables the designer to indicate which coefficients should be estimated more accurately with respect to others in order to obtain a reliable robust solution. As it is based on the Response Surface Methodology (RSM), the suggested methodology is iterative and gradually refines the optimal robust solution. Every iteration includes the following steps: (i) fitting a response model with various types of factors, including noise factors that are represented by random variables; (ii) obtaining the robust-design optimal solution, which is based on the mean and on the variance of the fitted response, either analytically for a first-order response model or numerically for higher-order models; and (iii) using the new DOE optimal criteria to minimize the variance of the robust solution rather than, for example, minimizing the variance of the model's coefficients. The suggested methodology is based on known techniques such as Fractional-Factorial Experiments, Response Surface Methodology, Monte-Carlo simulation, Bootstrap and on mathematical optimization. When comparing the suggested methodology to conventional methods, its main advantages are twofold. First, in a case for which a fixed experimental budget is given, it enables the designer to obtain a Robust Design solution more accurately. Second, it provides more information on the optimal robust solution by generating a (multidimensional) distribution of it. The suggested method can be implemented under different criteria than Robust Design. Numerical examples will be presented for illustration purposes.