The construction of robust mixture‐process experimental designs via genetic algorithm

Mixture experiments with the presence of process variables are commonly encountered in the manufacturing industry. The experimenter who plans to conduct mixture experiments in which a process involves the combination of machines, methods, and other resources will try to find condition of design factors which make the product/process insensitive or robust to the variability transmitted into the response variable. We propose the genetic algorithm (GA) for generating robust mixture‐process experimental designs involving control and noise variables. When the noise variables, which are extremely difficult to control or not routinely controlled during the manufacturing process and may change without warning, are considered in a mixture experiment, we propose the robust design setting. When considering a robust design, the design that has a lower and flatter faction of design space curves for all levels of the controllable process variables at varying noise interaction is preferable. We evaluate the designs with respect to these criteria for both the mean model and the slope model. The evaluation demonstrates that the proposed GA designs are robust to the contribution of the interactions involving the noise variables.     

Using a Genetic Algorithm to Generate D‐optimal Designs for Mixture Experiments

We propose and develop a genetic algorithm (GA) for generating D‐optimal designs where the experimental region is an irregularly shaped polyhedral region. Our approach does not require selection of points from a user‐defined candidate set of mixtures and allows movement through a continuous region that includes highly constrained mixture regions. This approach is useful in situations where extreme vertices (EV) designs or conventional exchange algorithms fail to find a near‐optimal design. For illustration, examples with three and four components are presented with comparisons of our GA designs with those obtained using EV designs and exchange‐point algorithms over an irregularly shaped polyhedral region. The results show that the designs produced by the GA perform better than, if not as well as, the designs produced by the exchange‐point algorithms; however, the designs produced by the GA perform better than the designs produced by the EV. This suggests that GA is an alternative approach for constructing the D‐optimal designs in problems of mixture experiments when EV designs or exchange‐point algorithms are insufficient. Copyright © 2012 John Wiley & Sons, Ltd.     

Using a genetic algorithm to generate D‐optimal designs for mixture‐process variable experiments

This article presents and develops a genetic algorithm (GA) to generate D‐efficient designs for mixture‐process variable experiments. It is assumed the levels of a process variable are controlled during the process. The GA approach searches design points from a set of possible points over a continuous region and works without having a finite user‐defined candidate set. We compare the performance of designs generated by the GA with designs generated by two exchange algorithms (DETMAX and k‐exchange) in terms of D‐efficiencies and fraction of design space (FDS) plots which are used to evaluate a design's prediction variance properties. To illustrate the methodology, examples involving three and four mixture components and one process variable are proposed for creating the optimal designs. The results show that GA designs have superior prediction variance properties in comparison with the DETMAX and k‐exchange algorithm designs when the design space is the simplex or is a highly‐constrained subspace of the simplex.     

Using geometric mean to compute robust mixture designs

Mixture experiments involve developing a dedicated formulation for specific applications. We propose the weighted optimality criterion using the geometric mean as the objective function for the genetic algorithms. We generate a robust mixture design using genetic algorithms (GAs) of which the region of interest is an irregularly shaped polyhedral region formed by constraints on proportions of the mixture component. When specific terms in the initial model display unimportant effects, it is assumed that they are removed. The design generation objective requires model robustness across the set of the reduced models of the design. Proposing an alternative way to tackle the problem, we find that the proposed GA designs based on G- or/and IV-efficiency are robust to model misspecification.