Practical efficient alpha: Encompassing all alphas of central composite design

Abstract

The positioning of two levels, namely the factorial level and the axial level in small or central composite designs is an important step for the second order response surface modeling and optimization of a process. Generally, the factorial levels are set for the region of interest in estimating factorial effects, and the axial levels, often called ’alpha’, are set reasonably. The reason could be adequate operability and/or model prediction properties. In this paper we propose the practical efficient (PE) alpha which addresses both the aspects. It is based on two parameters unlike all the three literature cited alphas, the practical alpha, the rotatable alpha and the spherical alpha which are given by single parameter. The PE alpha is given by the eighth root of the product of the number of factors involved in the process and the number of factorial runs considered in the selected composite design. It generalizes all the three alphas closely. Use of the PE alpha gives better performance than the practical alpha in terms of the D and G efficiencies. Even the use of an approximate value of PE alpha still gives better performance.