+

Miss Hanna Ilyani Zulhaimi

DESIGN OF EXPERIMENT

ERT 427 Response Surface

Methodology (RSM)

+Outline

n  Definition of Response Surface Methodology

n  Method of Steepest Ascent

n  Second-Order Response Surface

n  Response Surface Characterization

n  Location of Stationary Point

n  Experimental Design for Fitting First Order and Second Order Model (CCD and Box-Behnken Design)

n  Blocking in Response Surface Design

+Response Surface Methods and Designs n  Primary focus of previous chapters is factor screening

Ø  Two-level factorials, fractional factorials are widely used

n  Response Surface Methodology (RSM) is useful for the modeling and analysis of programs in which a response of interest is influenced by several variables.

n  Objective of RSM is optimization

n  For example: Find the levels of temperature (x1) and pressure (x2) to maximize the yield (y) of a process.

y = f (x1, x2 )+ε

+Example of Response Surface Graph

+(continued)

n  RSM a sequential procedure

n  The objective is to lead the experimenter rapidly and efficiently along a path of improvement toward the general vicinity of the optimum.

n  First-order model goes to Second-order model

n  ‘’Climb a hill’’

+Response Surface Model

+Method of Steepest Ascent

n  Assume that the first-order model is an adequate approximation to the true surface in a small ragion of the x’s.

n  A p r o c e d u r e f o r m o v i n g sequentially from an initial “guess” towards to region of the optimum.

n  Based on the fitted first-order model

n  Steepest ascent is a gradient procedure

n  The path of steepest ascent is the regression coefficients

Figure 11.4: First-order response surface and path of steepest ascent

+Example 11.1:

n  A chemical engineer is interested in determining the operating conditions that maximize the yield of a process. Two controllable variables: reaction time (ξ1) and reaction temperature (ξ2). The engineer is currently operating the process with reaction time of 35 minutes and temperature of 155 °F, which result in yield around 40 percent. Because it is unlikely that this region contains the optimum, she fits a first-order model and applies the method of steepest ascent.

+(continue…)

The  design  employed  was  a  22  factorial  design  with  6ive  centerpoints  

+(Continued)

n  A first-order model may be fit to these data by least squares.

+

n  To move away from the design center, the point (x1=0,x2=0) along the path of steepest ascent, we would move 0.775 units in the x1 direction for every 0.325 units in the x2 direction. If we decide to use 5 minutes of reaction time as the basic step size, then.

Δx1 =1.0000

Δx2= [(0.325/0.775)Δx1] =0.4194

n  The engineer computes points along this path and observes yields at these points until a decrease in response is noted.The results are shown in Table1- 3.The Steps are shown in both coded and natural variables.

12

The step size is 5 minutes of reaction time and 2 degrees F We can see from the table1-3,increases in response are observed through the tenth step. However the eleventh step produces a decrease in yield. Therefore, another first-order model must be fit in the general vicinity of the point( ξ1 = 85, ξ2 =175) 2

+Data for second First-Order Model

The  first-­‐order  model  fit  to  the  coded  data  in  Table  1-­‐4  is      y=78.97+1.00X1  +0.50X2    

+Analysis of data for second first-model

+Summary of Steepest Ascent

q  Points on the path of steepest ascent are proportional to the magnitudes of the model regression coefficients.

q  The direction depends on the sign of the regression coefficient. q  Step-by-step procedure:

+The Second-Order Model

n When  the  experimenter  is  relative  closed  to  the  optimum,  the  second-­‐order  model  is  used  to  approximate  the  response.  

n  It  is  used  to  6ind  the  stationary  point.    

n  Determine  whether  the  stationary  point  is  a  point  of  maximum  or  minimum  response  or  a  saddle  point.    

n  There  is  a  lot  of  empirical  evidence  that  they  work  very  well.  These  models  are  used  widely  in  practice.  

n  Optimization  is  easy.  

+Characterization of the Response Surface

n  Find the stationary point

n  Find what type of surface we have

Ø  Graphical Analysis

Ø  Canonical Analysis

n  Determine the sensitivity of the response variable to the optimum value

Ø  Canonical Analysis

+Second Order Model: Stationary Point

n  Writing the second-order model in matrix notation, we have:

bxy

bBx

Bbx

Bxxbx

s

1s

'0

222

11211

2

1

2

1

0

21ˆˆ

21

ˆ

2/ˆˆ2/ˆ2/ˆˆ

and

ˆ

ˆˆ

,

,''ˆˆ

s

kk

k

k

kkx

xx

y

+=

−=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

++=

−

β

β

ββ

βββ

β

β

β

β

ç  Stationary point

+Second Order Model: Stationary Point

n  In derivatives form:

n  Stationary point represents:

Ø  Maximum Point

Ø  Minimum Point

Ø  Saddle Point (minimax)

+Response Surface with Maximum Point

+Response Surface with minimum point

+Response Surface with Saddle point

+Canonical Analysis

The canonical form:

t h e {λ i } a re t h e e i g e n v a l u e s o r characteristic roots of the matrix B

+Canonical Analysis

n  The nature of the response surface can be determined from the stationary point & the signs and magnitudes of the {λi }. Ø  all positive: a minimum is found

Ø  all negative: a maximum is found

Ø  mixed: a saddle point is found

n  The response surface is steepest in the direction (canonical) corresponding to the largest absolute eigenvalue

+In Example 11.2

n  We continue in Example 11.1. However, we try to transfer first-order model with second-order.

n  Central composite design (CCD) was chosen (Table 11.6 & Figure 11.10).

n  The coefficients are as in Table 11.7

Ø  Find the stationary point, xs

Ø  Change coded variable to natural variables (ξ1= 87 minutes; ξ2= 176.5 F)

Ø  Characterize response surface by using canonical analysis

Ø  Rewrite the fitted model in canonical form:

+Fitting Design for First-Order Model

n  Suppose we have first-order model;

n  The class of orthogonal includes 2k full and fractional design in which main effects are not aliased with each other.

Addition of center points is usually a good idea

+Fitting Design for Second-Order Model

n  Suppose we have second-order model;

n  Design for second-order model includes:

Ø  Central Composite Design Ø  Box-Behnken design Ø  Face-centered design

Ø  Equiradial design

Ø  small composite design & hybrid design

+Central Composite Design

n  The CCD consists of a 2k factorial with nF factorial runs, 2k axial or star runs, and nc center runs.

+Rotatable CCD,α= nF 1/4

u A second-order response surface design is rotatable if the variance of the predicted response V[yˆ(x)] is the same at all the points of x that are at the same distance form the design center: α = (nF )1/4

u Gives the same prediction in all directions.

+Spherical CCD, α= (k)1/2

n  All factorial runs and axial runs have the same distance to the center: α = (k)1/2

n  Rotatability is a spherical property. However, it is not important to have exact rotatability to have spherical.

n  Spherical CCD: puts all the factorial and axial points on the surface of a sphere of radius √k.

+Box –Behnken Design

n  Three-level designs for small number of runs.

n  All points are on the distance 21/2 = 0.707 from the center.

n  No points at the vertices.

n  Either rotatable or nearly rotatable.

+Blocking in Response Surface Methodology n  It is often necessary to consider blocking to eliminate nuisance.

n  For second-order design to block orthogonally, two conditions must be satisfied:

1.  Each block must be a first-order orthogonal design; that is:

2.  The fraction of the total sum of square for each variable contributed by every block must be equal to the fraction of of the total observation that occur in the block; that is:

+Blocking in Response Surface Methodology

n  In general, when two blocks are required blocking, there should be an axial block and a factorial block.

n  For three blocks, the factorial block is divided into two blocks and the axial block is not split.

+CCD of Two Factors, 2 Blocks

+CCD of Three Factors, 3 Blocks

+

Thank you J