Construction of modified optimal second order rotatable designs

Abstract

Optimal rotatable designs are experimental designs that are applicable in agricultural, industrial and pharmaceutical processes to provide optimum responses subject to various input variables. Most researchers in these fields seek alternative experimental measures to improve productivity by optimizing the scarce resources available in order to cut on the cost of experimentation. The purpose of the study was to construct optimal modified Second Order Rotatable Designs (SORD) with reduced number of designs points from the existing SORD. The objectives were: Construction of SORD with reduced number of designs points from existing SORD; construction of Modified Second Order Slope Rotatable Designs (MSOSRD) with reduced designs points from existing modified slope rotatable designs, evaluation of Average (A-), Determinant (D-), Eigenvalue (E-) and Trace (T-) optimality criteria and illustrate the application of the constructed reduced designs points with hypothetical example. Construction of reduced SORD was done by taking a fraction of a suitable set of points for existing design points while keeping the other set of points constant and subjecting them to rotatability conditions. The MSOSRD with reduced designs points were obtained by taking fractions of suitable factorial combination obtained from 2t(v) fractional factorial designs. The parameter system of interest considered linear, pure and mixed quadratic factors to determine the moment matrix used for the evaluation of the alphabetic optimality criteria (A-, D-, E- and T-). A practical hypothetical example of sixteen design points was used in the analysis of response surface design using Mintab version 17.All the SORD and MSOSRD considered in this study were reducible and rotatable. From the evaluation of optimality criteria in three dimensions; the design with fourteen (14) points was E- optimal design with an eigenvalue of 0.004492. In four dimensions, the design with twenty four ( ) design points was A- optimal with an optimal trace value of 0.001437 while in five dimensions, the forty two (42) design points was also A- optimal with an optimal trace value of 0.000357. Designs in three, four and five dimensions with reduced number of design points were constructed and their optimality criteria evaluated. The reduced number of designs points imply fewer experimental runs therefore minimizes the cost of experimentation. The study recommends utilization of optimal reduced designs for cost effectiveness in designing of experiments for production processes in various sectors of the economy. Further study can be done in higher order rotatable designs if the second order is established to be inadequate.