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MANE 6313

Week 13, Module A

Student Learning Outcome

  • Select an appropriate experimental design with one or more factors,
  • Select an appropriate model with one or more factors,
  • Evaluate statistical analyses of experimental designs,
  • Assess the model adequacy of any experimental design, and
  • Interpret model results.

Module Learning Outcome

Describe response surface methodology.

Introduction to RSM

  • Response Surface Methodology is a collection of mathematical and statistical techniques that are useful for the modeling and analysis of problems in which a response of interest is influenced by several variables and the objective is to optimize this response

  • Consider a two-variable function where

\[ y=f(x_1,x_2)+\epsilon \]
  • The expected response function is
\[ E(y)=f(x_1,x_2)=\eta \]

  • Thus the function \(\eta\) is often called the response surface.

  • The response surface is often shown graphically

Textbook Figure 11.2

  • In general the function \(\eta\) is unknown.

  • We will approximate \(\eta\) with low-order polynomial functions.

  • A first-order model is

\[ y=\beta_0+\beta_1x_1+\beta_2x_2+\cdots+\beta_kx_k+\epsilon \]
  • A second-order model is
\[ y=\beta_0+\sum_{i=1}^k\beta_ix_i+\sum_{i=1}^k\beta_{ii}x^2_i+\sum_{i<j}\sum\beta_{ij}x_ix_j+\epsilon \]
  • The method of least squares will be used to estimate the parameters, \(\beta\)

Sequential Approach

  • The use of RSM often requires sequential analysis

  • Most of the time, you will not be operating at (or possibly near) an optimal region

  • Perform an initial experiment, often first-order design

  • Determine direction towards optimum point

  • Conduct another experiment nearer to the optimum point

  • Repeat until in the neighborhood of the optimum

  • Conduct an experiment using a second-order design

Textbook Figure 11.3

Analysis of the 2nd-order Response Surface

  • Suppose that we wish to find the levels of \(x_1,x_2,\ldots,x_k\) that optimize the predicted response

  • The point, if it exists, will be the set of \(x_1,x_2,\ldots,x_k\) for which \(\partial\hat{y}/\partial x_1=\partial\hat{y}/\partial x_2=\cdots=\partial\hat{y}/\partial x_k=0\)

  • This point is called the stationary point.

  • Based upon our knowledge of calculus what are the possible types of stationary points?

  • How do we determine if a stationary point is an optimal point?