CHAPTER 3
 RESPONSE SURFACE APPROXIMATIONS FOR RELIABILITY-BASED DESIGN
 OPTIMIZATION

 Response surface approximation (RSA) methods are used to construct an

approximate relationship between a dependent variable f(the response) and a vector x of

n independent variables (the predictor variables). The response is generally evaluated

experimentally (these experiments may be numerical in nature), in which case denotes

the mean or expected response value. It is assumed that the true model of the response

may be written as a linear combination of basis functions Z(x) with some unknown

coefficients (3 in the form of Z(xy (3 Response surface model can be expressed as

 Y(x) = Z(x)T b (3-1)

where Z(x) is the assumed basis function vector that usually consists of monomial

functions, and b is the least square estimate of p For example, if the a linear response

surface model is employed to approximate the response in terms of two independent

variables, xl anS X2, the response surface approximation is

 Y(x)= b, +bhx, +bh2x: (3 -2)

The three major steps of response surface approximation as summarized by Khuri and

Cornell (1996) are

 *Selecting design points where responses must be evaluated. The points are chosen
 by statistical design of experiment (DOE), which is performed in such a way that
 the input parameters are varied in a structured pattern so as to maximize the
 information that can be extracted from the resulting simulations. Typical DOE for
 quadratic RSA is central composite design (CCD, Khuri and Cornell 1996).