Central composite designs for estimating the optimum conditions for a second-order model

Central composite designs which maximize both the precision and the accuracy of estimates of the extremal point of a second-order response surface for fixed values of the model parameters are constructed. Two optimality criteria are developed, the one relating to precision and based on the sum of the first-order approximations to the asymptotic variances and the other to accuracy and based on the sum of squares of the second-order approximations to the asymptotic biases of the estimates of the coordinates of the extremal point. Exact and continuous central composite designs are introduced and in particular designs which place no restriction on the pattern of the weights, termed benchmark designs, and designs which comprise equally weighted factorial and equally weighted axial points, termed axial-factorial designs, are explored. Algebraic results proved somewhat elusive and the requisite designs are obtained by a mix of algebra and numeric calculation or simply numerically. An illustrative example is presented and some interesting features which emerge from that example are discussed.