Variance exchange process: overcoming the problem of singular information matrices in quadratic three−variable response designs

Many a time, in a variance exchange process looking for D-optimal designs, the initial designs of all quadratic components of three-variable response polynomials produce non-invertible information matrices. For such matrices, the variances of predicted responses at variance points cannot be evaluated, and the variance exchange process, not possible. D-optimality is a design criterion that seeks to maximize the determinant of the information matrix, or equivalently minimize the determinant of the inverse information matrix of the design. This work seeks to address the challenges posed by initial quadratic designs with zero-determinant information matrices for three-variable response polynomials to allow for the possibility of the variance exchange. The singular value decomposition (SUV) method was adopted and an algorithm was constructed for a variance exchange process involving quadratic designs of threevariable response functions. The study considered generated data for quadratic threevariable designs of sizes 12 and 13 for the analysis. MATLAB 7.5.0 (R2007b) was used to obtain the Penrose inverses. The results show that a variance exchange process was possible, evaluating the variances of the predicted responses at the design points, thereby overcoming the problem of singular information matrices on the initial quadratic designs. The D-optimal designs, computer-generated optimal designs, provide ready alternatives for finding optimum conditions for factors in engineering optimization problems with response surface functions that require structured data collection using experimental design when the experimental design space is constrained owing to zero determinant of the information matrices of the initial designs.