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# D-optimality of non-regular design spaces by using a Bayesian modification and a hybrid method

Research Paper

## Abstract

In this work a hybrid method of a genetic algorithm and sequential linear programming is suggested to obtain a D-optimal design of experiments. Regular as well as non-regular design spaces are considered. A D-optimal design of experiments maximizes the determinant of the information matrix, which appears in the normal equation. It is known that D-optimal design of experiments sometimes include duplicate design points. This is, of course, not preferable since duplicates do not add any new information to the response surface approximation and the computational effort is therefore wasted. In this work a Bayesian modification, where higher order terms are added to the response surface approximation, is used in case of duplicates in the design of experiments. In such manner, the draw-back with duplicates might be eliminated. The D-optimal problem, which is obtained by using the Bayesian modification, is then solved by a hybrid method. A hybrid method of a genetic algorithm that generates a starting point for sequential linear programming is developed. The genetic algorithm performs genetic operators such as cross-over and mutation on a binary version of the design of experiments, while the real valued version is used to evaluate the fitness. Next, by taking the gradient of the objective, a LP-problem is formulated which is solved by an interior point method that is available in Matlab. This is repeated in a sequence until convergence is reached. The hybrid method is tested for five numerical examples. Results from the numerical examples show a very robust convergence to a global optimum. Furthermore, the results show that the problem with duplicates is eliminated by using the Bayesian modification.

## Keywords

D-optimality Design of experiments (DoE) Sequential linear programming (SLP)  Genetic algorithms (GA) Response surface methodology (RSM) Bayesian modification (BM)

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## Copyright information

© Springer-Verlag 2009

## Authors and Affiliations

1. 1.Department of Mechanical EngineeringJönköping UniversityJönköpingSweden
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