Industrial Engineering Letters www.iiste.org ISSN 2224-6096 (Paper) ISSN 2225-0581 (online) Vol.4, No.12, 2014 28 Prediction Variance Assessment of Variations of Two Second-Order Response Surface Designs Eugene C. Ukaegbu (Corresponding author) Department of Statistics,University of Nigeria, Nsukka, Nigeria E-mail: eugndu@yahoo.com Polycarp E. Chigbu Department of Statistics,University of Nigeria, Nsukka, Nigeria E-mail: polycarp.chigbu@unn.edu.ng Abstract Two second-order response surface designs have been evaluated. The designs are the small composite designs and the minimum-run resolution V designs. The cube and star portions of these second-order designs are replicated with different amounts and the variations of the designs generated by replication are compared independently to assess the performance of the prediction variances for each of the second-order design under consideration. Two optimality criteria, G- and I-optimality, that are prediction variance-oriented are used to evaluate the maximum and average prediction variance of the designs while fraction of designs space plots are constructed to track the prediction variance performance of these designs throughout the design space. For the two second-order designs, the results indicate that it is advantageous to replicate the star than replicating the cube. Keywords: Optimality criteria, fraction of design space plot, small composite design, minimum-run resolution V design, design replication, cube, star. 1. Introduction Response surface methodology (RSM) is a collection of mathematical and statistical techniques that are very useful when modeling and analyzing experimental situations. The objective is, by careful design of experiments, a response variable (output variable) that is being influenced by several independent variables (input variables) is optimized (see Montgomery, 2005). Designs used to describe these experimental situations are called response surface designs. A second-order response surface design is often chosen based on many considerations such as those identified by Box and Draper (1959), Montgomery (2005), Myers et al (2009) and Anderson-Cook et al (2009a). As the number of factors in a second-order model increases, the number of terms also increases. Therefore, economic second-order designs with reasonable prediction variance are highly desirable. Two second- order response surface designs with similar components (cube, star and centre point) and used as smaller alternatives to the central composite designs are considered. They are the small composite designs (SCD) and minimum-run resolution V (MinResV) designs. Hartley (1959) and Oehlert and Whitcomb (2002) are useful references for detailed discussion on the two designs. Several other second-order response surface designs have been evaluated and compared using various criteria: see, for example, Zahran et al (2003) ad Ozol-Godfrey (2004). Replication of experimental observations is considered indispensible for efficient and optimal performance of the second-order designs. Traditionally, the centre point of the design is replicated to ensure proper estimation of the experimental error with 0 1 n − degrees of freedom as it is assumed that the optimum response is at the centre of the design. However, recent researches have shown that replicating at the centre alone may lead to estimating error that may be too small for correct evaluation of the model. Since there is no assurance that variability will remain constant throughout the design region, Dykstra (1960) posits that it is sound experimental strategy to replicate at other locations in the design region. See also, Giovannitti-Jensen and Myers (1989) for further contributions on replication at other design locations apart from the centre point. Several works on replicating at other design locations have been focused on the central composite designs (CCD). Such works include Dykstra (1960), Draper (1982), Borkowski (1995), Borkowski and Valeroso (2001) and recently, Chigbu and Ohaegbulem (2011). In this study, we extend this idea to the SCD and MinResV designs since these designs share similar components (cube, star and centre point) with the CCD. We adopt the replication procedure introduced by Draper (1982). The distance of the star points (axial distance), α , from the centre of the design plays significant role in the distribution of the prediction variance in the design region of interest. Several axial distances have been proposed in the literature and each axial distance affects the structure and performance of the design. Some of the available values of α can be found in Box and Hunter (1957), Montgomery (2005), Myers et al (2009) and