An algorithm for the optimization of response surface designs

Abstract

An algorithm is developed for the optimization of second order response surface designs. The technique of response surface analysis consists of two stages: (a) the design of the experimental plan consisting of N individual trials and (b) using the data from (a) to estimate the coefficients in a mathematical formula representing the relationship between inputs and responses. This research is concerned with stage (a) adopting the approach to construct an optimal design for an experiment rather than tailoring the experiment so as to fit a known design. The response surfaces considered are quadratic in nature; that is, each of the n 'input' variables appears in the model as zero, first and second order variables. Each of these variables is limited to some region of interest for any given experiment. The criterion adopted here is the minimization of the 'generalized variance' of the estimated coefficients. Thus the statistical problem in summary is to determine the levels of the n 'input' variables for each of the N trials so as to make the generalized variance as small as possible. This translates to the mathematical programming problem of maximizing an objective function which is the reciprocal of the generalized variance subject to restraining the variables in the experimental region of Interest. This mathematical programming problem is solved using the technique of Spherical Programming which is capable of finding the global maximum of a concave function subject to linear constraints. This is essentially a steepest ascent algorithm which creates and solves a sequence of subproblems leading to the solution of the main problem. The objective function for this problem, however, is not concave; none the less, this technique may be used to solve the problem at the risk of the algorithm terminating at a relative optimum rather than a global optimum. A stochastic control procedure is introduced which allows comparison of the algorithm result with an estimated global optimum. A package computer program is assembled that allows one to enter the statistical parameters involved and receive as an answer an optimized design. The program does not compare this result with a predicted, global optimum although this could be incorporated.