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Design of Engineering Experiments Part 8 – Overview of

Response Surface Methods

• Text reference, Chapter 11, Sections 11-1 through 11-4

• Primary focus of previous chapters is factor screening– Two-level factorials, fractional factorials are widely used

• Objective of RSM is optimization• RSM dates from the 1950s; early applications in

chemical industry

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RSM is a Sequential Procedure

• Factor screening

• Finding the region of the optimum

• Modeling & Optimization of the response

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Response Surface Models

• Screening

• Steepest ascent

• Optimization

0 1 1 2 2 12 1 2y x x x x

0 1 1 2 2y x x

2 20 1 1 2 2 12 1 2 11 1 22 2y x x x x x x

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The Method of Steepest Ascent

Text, page 407

A procedure for moving sequentially from an initial “guess” towards to region of the optimum

Based on the fitted first-order model

Steepest ascent is a gradient procedure

0 1 1 2 2ˆ ˆ ˆy x x

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An Example of Steepest AscentExample 11-1, pg. 409

1 2ˆ 40.44 0.775 0.325y x x

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An Example of Steepest AscentExample 11-1, pg. 409

DESIGN-EXPERT Plot

yieldX = A: Reaction TimeY = B: Temp

Design Points

yield

A: Reaction Time

B: T

em

p

-1.000 -0.250 0.500 1.250 2.000

-1.000

-0.250

0.500

1.250

2.000

39.5

40

40.5 41 41.5

5

An approximate step size and path can be determined graphically

Formal methods can also be used (pp. 407-412)

Types of experiments along the path:

•Single runs

•Replicated runs

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Results from the Example (pg. 434)

The step size is 5 minutes of reaction time and 2 degrees F

What happens at the conclusion of steepest ascent?

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Second-Order Models in RSM

• These models are used widely in practice

• The Taylor series analogy

• Fitting the model is easy, some nice designs are available

• Optimization is easy

• There is a lot of empirical evidence that they work very well

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Analysis of the Second-Order Response Surface Model (pg. 413)

This is a central composite design

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Example 11-2

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Example 11-2

2 21 2 1 2 1 2ˆ 79.94 0.99 0.52 0.25 1.38 1.00y x x x x x x

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Contour Plots for Example 11-2The contour plot is given in the natural variables

The optimum is at about 87 minutes and 176.5 degrees

Formal optimization methods can also be used (particularly when k > 2)

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Multiple Responses

• Example 11-2 illustrated three response variables (yield, viscosity and molecular weight)

• Multiple responses are common in practice• Typically, we want to simultaneously optimize all

responses, or find a set of conditions where certain product properties are achieved

• A simple approach is to model all responses and overlay the contour plots

• See Section 11-3.4, pp. 423 -427.

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Designs for Fitting Response Surface Models

• Section 11-4, page 427• For the first-order model, two-level factorials (and

fractional factorials) augmented with center points are appropriate choices

• The central composite design is the most widely used design for fitting the second-order model

• Selection of a second-order design is an interesting problem

• There are numerous excellent second-order designs available

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Other Aspects of RSM

• Robust parameter design and process robustness studies (Chapter 12)– Find levels of controllable variables that optimize mean response and

minimize variability in the response transmitted from “noise” variables

– Original approaches due to Taguchi (1980s)– Modern approach based on RSM

• Experiments with mixtures– Special type of RSM problem– Design factors are components (ingredients) of a mixture– Response depends only on the proportions– Many applications in product formulation