Optimal and efficient production of Rose Coco Beans through the twenty four points second order rotatable design

Abstract

Response Surface Methodology (RSM) is a collection of mathematical and statistical techniques useful for the modeling and analysis of problems in which a response of interest is influenced by several variables and the objective is to optimize the response. The yield results of the twenty four points RSM design permitted a response surface to be fitted easily and provided spherical information contours besides the realizations of an optimum combination of the fertilizers in rose coco beans, which resulted in economic use of scarce resources for optimal production of rose coco beans. In this study an existing A-optimum and D-efficient second order rotatable design in three dimensions was used in the production of rose coco beans optimally and efficiently. The general objective of the study was to produce rose coco beans (Phaseolus vulgaris) optimally and efficiently using an existing A-optimum and D-efficient twenty four points second order rotatable design in three dimensions in a greenhouse setting using three inorganic fertilizers, namely, nitrogen, phosphorus and potassium. Thus the study was accomplished using the calculus optimum value of the free/letter parameter f=1.1072569. The specific objectives were to estimate the linear parameters, thereby making available the yield response of rose coco beans at calculus optimum value design for the first time. The generalized variance of the estimated linear parameters was also obtained, fitted and tested the three models adequacies via lack of fit test, and then found the settings of the experimental factors that produces optimal response using contour plots to assist visualizes the response surfaces. This study demonstrated the importance of statistical methods in the optimal and efficient production of rose coco beans. The results showed that the three factors: nitrogen, phosphorus, and potassium contributed significantly on the yield of rose coco beans (p<0.05). The regression coefficients were determined by employing least squares techniques to predict quadratic polynomial models for group 1 greenhouse (GP1G), group 2 greenhouse (GP2G) and group 3 greenhouse (GP3G) for the three fertilizer combinations. In GP1G second order model was inadequate with a p value of 0.3178, in GP2G and GP3G, the second order model was adequate at 1% level of significance with p values of 0.0065 and 0.0034 respectively. The analysis of variance (ANOVA) of response surface for rose coco yield showed that this design was adequate due to satisfactory levels of coefficient of determination, R 2 , 0.6810 (GP1G), 0.6704 (GP2G), and 0.8066 (GP3G) and coefficient variation, CV was 13.48, 14.47 and 10.30 for GP1G, GP2G, GP3G respectively. The canonical analysis showed that there were saddle points for the three groups, meaning there was no unique optimum; therefore ridge analysis was used to overcome the saddle problem. The results from ridge analysis provided the maximum yield of 58.78grams, 48.36grams and 70.25grams in GP1G, GP2G and GP3G respectively for the various fertilizer combinations at radii of one. We therefore recommend the use of GP3G design since it gave above board the required coefficient of determination (R 2 =80.66) and the maximum yield (70.25grams) was achieved.