Chapter 4 - Chapter 4 Design of Experiments 4.1...

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48 Chapter 4 Design of Experiments 4.1 Introduction In Chapter 3 we have considered the location of the data points fixed and studied how to pass a good response surface through the given data. However, the choice of points where experiments (whether numerical or physical) are performed has very large effect on the accuracy of the response surface, and in this chapter we will explore methods for selecting a good set of points for carrying out experiments. The selection of these points is known as design of experiments. Design of experiments is inherently a multi-objective optimization problem. We would like to select points so that we maximize the accuracy of the information that we get from the experiments. We usually also would like to minimize the number of experiments, because these are expensive. In some cases the objective of the experiments is to estimate some physical characteristics, and in these cases, we would like to maximize the accuracy of these characteristics. However, in the design applications, which are of primary interest to us, we would like to construct a response surface that could be used to predict the performance of other designs. In this case, our primary goal is to choose the points for the experiments so as to maximize the predictive capability of the model. A lot of work has been done on experimental designs in regular design domains. Such domains occur when each design variable is bounded by simple lower and upper limits, so that the design domain is box like. Occasionally, spherical domains are also considered. Sometimes each design variable can take only two or three values, often called levels. These levels are termed low, nominal and high. In other cases, the design space is approximately box like, but it is possible to carry experiments with the design variables taking values outside the box for the purpose of improving the properties of the response surface. In the next section we will summarize briefly some of the properties of experimental design in box-like domains, and present some of the more popular experimental designs in such domains. For design optimization, however, it is common for us to try and create response surfaces in irregularly shaped domains. In that case, we have to create our own experimental design. Section 4.3 will discuss several techniques available for finding good designs in an irregular shaped domain.
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49 4.2 Design of Experiments in Boxlike Domains In this case the region of interest is defined by simple lower and upper limits on each of the design variables , ,.... , 1 , ' n i x x x iu i il (4.2.1) where il x , and iu x are the lower and upper limits, respectively, on the design variable ' i x . The prime indicates that the design variable has not been normalized. For convenience we normalize the design variable as , 2 ' il iu in il i i x x x x x x ( 4 . 2 . 2 ) The normalized variables are then all bound in the cube , 1 1 i x ( 4 . 2 . 3 ) 4.2.1 Interpolation, extrapolation and prediction variance
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Chapter 4 - Chapter 4 Design of Experiments 4.1...

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