Construction of optimal second order rotatable designs through simplex and factorial designs with application in petroleum biodegradation

Abstract

Rotatable designs are mainly used for the exploration of response surfaces and estimation of parameters. These designs provide constant prediction variance at all points that are equidistant from the design center, thus improving the quality of the prediction. Rotatable designs were constructed through geometrical configurations and several second order designs obtained. The designs can be used in Biodegradation which is the process by which microorganisms degrade organic compounds to non – toxic substances. The rate and extent are affected by the nature of the environment and the interactions between different factors. Full Factorial Design of Experiment provides the most response information about factor main effects and interactions, the process model’s coefficients for all factors and interactions, and when validated, allows process to be optimized. On the other hand, mixture designs are a special case of response surface designs where prediction and optimization are the main goals. Despite the fact that mixture designs usually are intended to predict all possible formulations of the ingredients, no research has been done concerning rotatability with mixture designs. This study examined the construction of optimal rotatable designs through Simplex Designs (SDs) and Full Factorial Designs (FFDs). The main objective of the study was to Construct Optimal Rotatable Designs through Simplex and Full Factorial Designs with a practical illustration of their utility in petroleum biodegradation. Specifically it constructed Rotatable Designs (RDs) through Simplex - Lattice Designs (SLDs) and Full Factorial Designs (FFDs); constructed Weighted Rotatable Designs (WRDs); obtained D- optimal, A- optimal and I- optimal values for the constructed designs and considered the efficiencies of the Rotatable Designs (RDs) and the Weighted Rotatable Designs (WRDs). A Response Surface Analysis (RSA) of percentage crude oil removed by three factors using one of the constructed Rotatable Designs (RD) was carried out. The method involved augmenting Simplex Lattice Designs for two and three factors with their Full Factorial portions. For the response surface of the second degree, rotatability was obtained by selecting design points to satisfy some rotatability relations. Using moment matrices based on the Simplex and Factorial Designs, and the General Equivalence Theorem (GET) for D-, A- and I- optimality, weighted rotatable designs (WRDs) were obtained. The generalized D-optimal and A-optimal values from the family of matrix means for the RDs were obtained. The D-, A- and I- optimality critera were used to establish the efficiency of the designs. To illustrate the use of RD, analysis for a practical example was carried out on a Second Order model by locating the stationary point and characterizing the response surface. The RDs were constructed through SLD and FFD for two and three factors each of level 2. A generalized form of the designs for k factors of level 2 was obtained. A detailed comparison of the original RD and the new WRD in terms of overall efficiency was then provided in tabular form. The results showed that WRD was D- and A- more efficient than the ordinary RD, whereas RD was more I- efficient than WRD. This research recommended the use of RDs for greater precision results with reduced computational cost.