Construction of some economical optimum second order rotatable designs in four and five dimensions

Abstract

The technique of fitting a response surface is one widely used especially in the chemical industry to aid in the statistical analysis of experimental work in which the yield of a product depends in some unknown fashion on one or more controllable variables. The world is facing depletion of resources and search for alternative measures is inevitable in all human endeavor. Since the resources are scarce, we need to produce maximally by utilizing design of experiments like in this study. In the current study, optimal economical second order rotatable designs SORDs in four and five dimensions were constructed. The objectives of the study were; to construct economical SORDs in four, and five dimensions, evaluate the alphabetic optimality criteria for the designs, and determine the A- and E- efficiencies for the designs. The sets of points for a sequential rotatable arrangement in four and five dimensions formed rotatable arrangements if their excess functions were zero. All the variables determined were real and positive and this confirmed the existence of a rotatable arrangement. The designs were considered to be SORDs after satisfying both the moment and non-singularity conditions. The moments which formed the elements of the moment matrix were determined by taking the parameter system of interest to be that of a second order model. The moment matrices formed the basis for determination of the optimality criteria for every design considered. The determinant criterion (D-), Average variance criterion (A-), Eigen value criterion (E-) and the trace criterion (T-). Were considered. For each criterion the design with the least value will be optimal to the specific criteria under consideration. The study yielded; 32, 40, 48a, and 48b points SORDs in four dimensions and; 52, 74,100a, and 100b points SORDs in five dimensions respectively. From the Table 2 the design G 2 (40 points SORD in four dimensions) is A-, D-, T- and E-optimal. The design G 6 (74 points SORD in five dimensions) is A-, D- and-optimal and designs G 7 was found to be E-optimal. The analysis of efficiencies facilitated the choice of the most desirable design from the other designs under consideration. In conclusion, the more homogenous the design is, the more optimal it became, and thus the designs obtained provides very essential tools for use in various fields such as in medicine, agriculture and industry. The study recommends evaluation of robustness of missing data for these designs and all other designs to enable researchers make informed decisions whenever missing information is anticipated.