Construction of some third order optimum sequential rotatable designs in three and four dimensions

Abstract

Response Surface Methodology (RSM) is a collection of statistical and mathematical techniques useful for developing, improving and optimizing processes. Response Surface Methodology has gained recognition as a useful tool in a number of fields, including industry, agriculture, and medicine. Given the limited resources currently accessible on the planet, researchers are looking for strategies to maximize resource utilization in order to meet the continually increasing demands at both the individual and society levels. This can only be achieved through an appropriate design of experiments such as in this study. The problem of this study was to construct some third order optimum sequential rotatable designs in three and four dimensions. The specific objectives of the study were; to construct third order sequential rotatable designs in three dimensions by combining pairs of second order rotatable point sets; to construct third order sequential rotatable designs in four dimensions by appending an extra factor in each of the second order rotatable point sets and; to obtain the A-, D-, T-, E- optimality criteria of the sequential third order rotatable designs in three and four dimensions. The third order rotatable arrangement in three and four dimensions were established after all variables were proved to be real and positive and their excess functions were found to be zero. These arrangements formed TORDs after they satisfied the non-singularity conditions required for rotatability, yielding 44, 58, and 46 points TORDs for the three-dimensional designs and 80a and 80b points TORDs for the four-dimensional designs. Most of researches in RSM are theoretical especially in third order rotatability. So, there is need to give hypothetical examples to these designs and existing designs for presentation in applicable formats for the three and four-dimensional rotatable designs to give maximum produce. The study also identified and presented A-, D-, T-, E- optimality criteria in order to obtain the effectiveness of the constructed third order rotatable designs. The design with the smallest (least) optimality criterion among these is considered to be optimal. Design B12 was determined to be D-optimal for TORDs in three dimensions, while design was identified as the D-optimal design for TORDs in four dimensions. Design B12 was found to be T-optimal for TORDs in three dimensions, while design was considered T-optimal for TORDs in four dimensions. Regarding the A-criterion, design B13 was deemed optimal in three dimensions, whereas design was identified as the A-optimal design for TORDs in four dimensions. Both designs B12 and were found to be E-optimal for the designs in three dimensions and four dimensions, respectively. In order to obtain optimality criteria and confirm the existence of optimal solutions in these and other design settings, the study recommends employing various methodologies. These methods include balanced incomplete block design and pairwise block design.