Construction and optimality of a special class of balanced designs

The use of balanced designs is generally advisable in experimental practice. In technological experiments, balanced designs optimize the exploitation of experimental resources, whereas in marketing research experiments they avoid erroneous conclusions caused by the misinterpretation of interviewed customers. In general, the balancing property assures the minimum variance of first-order effect estimates. In this work the authors consider situations in which all factors are categorical and minimum run size is required. In a symmetrical case, it is often possible to find an economical balanced design by means of algebraic methods. Conversely, in an asymmetrical case algebraic methods lead to expensive designs, and therefore it is necessary to adopt heuristic methods. The existing methods implemented in widespread statistical packages do not guarantee the balancing property as they are designed to pursue other optimality criteria. To deal with this problem, the authors recently proposed a new method to generate balanced asymmetrical designs aimed at estimating first- and second-order effects. To reduce the run size as much as possible, the orthogonality cannot be guaranteed. However, the method enables designs that approach the orthogonality as much as possible (near orthogonality). A collection of designs with two- and three-level factors and run size lower than 100 was prepared. In this work an empirical study was conducted to understand how much is lost in terms of other optimality criteria when pursuing balancing. In order to show the potential applications of these designs, an illustrative example is provided.