Some sequential third order response surface designs

Sequential third order designs are very important due to their nice sequential property. These designs become even more important in case of second order model lack of fit. There is possibility that an experimenter uses a second order design to explore the second order response surfaces, however, lack of fit problem may arise. If one is using the sequential third order design, then there is an opportunity to use the next portion of that design to estimate the third order terms and remove the second order model lack of fit. Das and Narasimham (1962) and Arshad, Akhtar, and Gilmour (2012) have proposed sequential third order designs that can be used to estimate a complete third order model, in case we face second order model lack of fit. However, these designs have very large sizes. Sometime, it becomes difficult for an experimenter, in real world situation, to use a design with such a large size due to different issues like high cost of experimental material. Arshad, Akhtar, and Gilmour (2012) augmented the very popular designs of Box and Behnken (1958and 1960) and made them third order. Those designs, called augmented Box-Behnken designs (ABBDs), are sequential in nature and can be used for estimation of a complete third order model in case of second order model lack of fit. These designs have been developed by augmenting the Box-Behnken designs using different combinations of factorial points, axial points and complementary design points. In this research paper, we have also selected the Box-Behnken designs due to their popularity and augmented them using the concept of doubly balanced incomplete block design provided by Calvin (1954). We have developed 9 designs one for each from 4 to 12 number of factors (design for 3 factors was not achievable). These third order designs are also sequential in nature, and can be used for estimation of complete third order model in case the second order model lack of fit is exhibited. The newly proposed designs have smaller sizes as compared to Das-Narasimham designs (DNDs) in all cases, and smaller than ABBDs in many cases. In addition to the designs sizes, some other design optimality criteria have also been explored for these designs, and compared with the DNDs and ABBDs. Mostly the new designs have been found better than DNDs and ABBDs.