On the construction of asymmetric third-order rotatable designs

Response surface methodology (RSM) has broader applicability, where numerous input variables may impact a given performance metric or quality attribute of the final product or process. It involves designing experiments, collecting data, and developing models to optimize the response. Rotatable designs have the property to generate information about the response surface equally in all locations, even when no or little prior knowledge is available about the nature of the response surface. These designs are constructed by imposing certain restrictions on the moment matrix of the design to achieve constancy in the variance of predicted response at all points equidistant from the design center and is invariant to rotation of axis with respect to any angle. Most rotatable response surface designs are symmetric in nature, although factors with mixed-level have more practical utility as it can explore more regions in the design space. In this article, we have proposed a procedure for creating asymmetric third-order rotatable designs (ATORDs) as well as a strategy for creating them with fewer design points when time and resources are the main limitations. Two classes of orthogonal transformation-based ATORDs viz., ATORD-I and ATORD-II have been obtained. ATORD-II does not completely satisfy the moment matrix constraints, although both ATORD-I and ATORD-II have constant prediction variance. ATORD-II is more cost-efficient for experimentation when resource and financial constraints are the primary factors to be taken into account. A comparison of the designs developed is also made using efficiency criterion and dispersion graphs.