The optimization of response surface designs

Abstract

"Optimal" response surface design of order three are obtained for the full cubic polynomial response function. The designs are optimal in the sense that the generalized variance of the estimates of the coefficients is minimized in a convex region, R. In the optimization, R is the region defined by the inequalities -1<=xi<=1, i=1, 2, & k where k is equal to the number of factors in the response function. "Optimal" designs are also obtained for a cubic polynomial response function composed of the full quadratic plus the cubic terms of the form x3i, i=1, 2, & k. Although the main emphasis in the research is on the cubic function, a section concerned with the replication of saturated quadratic designs is also included. Specifically, the "optimal" six-point design for the full quadratic polynomial in two factors is replicated to form a twelve-point design, then this design is compared with the "optimal" twelve-point design. It is seen that, as indicated in similar studies by other authors, that the replicated designs compare favorably with the "optimal" twelve-point design. Although it is anticipated the designs optimized with respect to the generalized variance will usually b used non-sequentially, a section is included that illustrates a method of obtaining two-stage sequential designs. In order to obtain the "optimal" designs a computer program was written employing the basic algorithm of the gradient projection method for non-linear programming developed by J.B. Rosen. The convergence of the algorithm to either a global or a constrained global optimum is based on the assumption that the objective function is concave. Since the objective function for minimized the generalized variance is not concave, nor convex, global optimality cannot be claimed for the "optimal" designs. However, as Rosen points out, if the same optimum is obtained for a general non-linear objective function from many widely separated starting points it can be concluded that either a constrained or an interior local optimum has been found. Furthermore, using theory developed by Hartley and Pfaffenberger, confidence statements concerning the global maximum in mathematical programming can be obtained. Various authors have suggested other criteria for choosing response surface designs. Box and Draper suggested the minimization of the mean square error averaged over the region of interest. However, their implementation of the criterion is not consistent with the strategy which they advocate. Basically, their strategy is to add more experimental points and fit a higher order polynomial if the lack of fit is significant. The cause of this inconsistency is illustrated and discussed by means of example.