Genetic Algorithms for the Construction of D-Optimal Designs - ASQ

Genetic Algorithms for the Construction of D-Optimal Designs

Computer-generated designs are useful for situations where standard factorial, fractional factorial or response surface designs cannot be easily employed. Alphabetically-optimal designs are the most widely used type of computer-generated designs, and of these, the D-optimal (or D-efficient) class of designs is extremely popular. D-optimal designs are usually constructed by algorithms that sequentially add and delete points from a potential design by using a candidate set of points spaced over the region of interest. We present a technique to generate D-efficient designs using genetic algorithms (GA). This approach eliminates the need to explicitly consider a candidate set of experimental points and it can be used in highly constrained regions while maintaining a level of performance comparable to more traditional design construction techniques.

Key Words: Contrasts, Factorial Designs, Screening.

By ALEJANDRO HEREDIA-LANGNER, Pacific Northwest National Laboratory, Richland, WA 99352
W. M. CARLYLE, Naval Postgraduate School, Monterey, CA 93943-5219
D. C. MONTGOMERY, C. M. BORROR, and G. C. RUNGER, Arizona State University, Tempe, AZ 85287-5906

Introduction

Statistically designed experiments find extensive application in scientific and industrial settings. Textbooks such as Box, Hunter, and Hunter (1978) and Montgomery (2001) describe the strategy and tactics of using statistically designed experiments. These texts concentrate on the familiar factorial, fractional factorial and response surface designs. In the 1950s, research began on a new class of designs. These designs would satisfy certain optimality criteria related to the precision with which model parameters of the response variable were estimated. For example, in using the D-optimality criterion one selects design points to minimize the volume of the joint confidence ellipsoid for the regressor coeficients. With A-optimality one selects design points so that the sum of the variances of the estimated regression coeficients is minimized and the G-optimality criterion requires that the maximum variance of the predicted response in the design region is minimized. Collectively, these criteria are often referred to as alphabetic-optimality criteria.

Alphabetically-optimal designs are often generated by an algorithm implemented using a computer. The most commonly used algorithms for their construction, described by Dykstra (1971), Wynn (1970), St. John and Draper (1975), Johnson and Nachtsheim (1983), and Meyer and Nachtsheim (1995), are based on the exchange of points from an initial design and a suitable candidate set. This sequential exchange can be performed one coordinate or experimental run at a time or by simultaneously exchanging multiple design points. Other approaches, like simulated annealing (Haines (1987)), also perform searches of the type just described but have not been as successful as the more structured methodologies.

We apply a technique developed in the field of computer science that employs a very effective and efficient approach to optimize the D-efficiency of experimentaldesigns. This solution procedure offers a degree of flexibility in its way of constructing designs that allows it to overcome restrictions that may limit the applicability of other more commonly used algorithms. Constrained regions are easily handled, and there is no need to explicitly generate a candidate set of points.

In the following sections, we present the general idea behind D-optimal or D-efficient designs and illustrate how they can be constructed using genetic algorithms. We illustrate the procedure on a variety of problems, including some challenging mixture problems. The designs obtained using genetic algorithms (GA) are compared to those obtained using exchange-point or coordinate-exchange techniques.

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