cahpter xi cyril

7/25/2019 cahpter XI cyril

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1

Chapter 11 Response Surface

Methods and Other Approaches to

Process Optimization



2

11.1 Introduction to Response

Surface Methodology Response Surface Methodology !RSM" is useful

for the modeling and analysis of programs in

#hich a response of interest is influenced $yse%eral %aria$les and the o$&ecti%e is to optimize

this response.

'or e(ample) 'ind the le%els of temperature !(1"

and pressure !(2" to ma(imize the yield !y" of a

process.

ε += "*! 21 x x f y



+

Response surface) !see 'igure 11.1 , 11.2"

-he function f is unno#n Appro(imate the true relationship $et#een y and

the independent %aria$les $y the lo#er/order

polynomial model.

Response surface design

"*!"! 21 x x f y E ==η

ε β β β β

ε β β β

++++=

++++=

∑∑∑<== ji

jiij

k

i

iii

k

i

ii

k k

x x x x y

x x y

1

2

1

0

110



A seuential procedure

-he o$&ecti%e is to lead

the e(perimenter rapidly

and efficiently along a

path of impro%ement

to#ard the general%icinity of the optimum.

'irst/order model 34

Second/order model

Clim$ a hill



5

11.2 -he Method of Steepest Ascent

Assume that the first/order model is an adeuate

appro(imation to the true surface in a small ragion

of the (6s. -he method of steepest ascent) A procedure for

mo%ing seuentially along the path of steepest

ascent.



7

8ased on the first/order

model*

-he path of steepest ascent 99

the regression coefficients

-he actual step size is

determined $y the

e(perimenter $ased on process no#ledge or other

practical considerations

∑=

+=k

i

ii x y1

0::: β β



;

<(ample 11.1

= -#o factors* reaction time , reaction

temperature = >se a full factorial design and center points

!see -a$le 11.1")

1. O$tain an estimate of error 2. Chec for interactions in the model

+. Chec for uadratic effect

A?O@A ta$le !see -a$le 11.2"

-a$le 11.+ , 'igure 11.5

-a$le 11. , 11.5



Factor 1 Factor 2 ResponseStd Run Block A:Time B:Temp yield

minutes degC percent

1 7 { 1 } 1 1 !"#!

2 $ { 1 } 1 1 %&#"

! ' { 1 } 1 1 %&% 2 { 1 } 1 1 %1#'

' " { 1 } & & %&#!

$ % { 1 } & & %&#'

7 1 { 1 } & & %&#7

( ! { 1 } & & %&#2" ( { 1 } & & %&#$

1 2: 0. 0.;;5 0.+25 y x x= + +



B

-he step size is 5 minutes of reaction time and 2 degrees '

hat happens at the conclusion of steepest ascentD



10

Assume the first/order model

1. Choose a step size in one process %aria$le* ( &.

2. -he step size in the other %aria$le*

+. Con%ert the ( & from coded %aria$les to the

natural %aria$le

∑=

+=k

iii

x y1

0

::: β β

j j

ii

x x

∆=∆

9:

:

β β



11

11.+ Analysis of a Second/order

Response Surface hen the e(perimenter is relati%e closed to the

optimum* the second/order model is used to

appro(imate the response.

'ind the stationary point. Ma(imum response*

Minimum response or saddle point.

Eetermine #hether the stationary point is a point

of ma(imum or minimum response or a saddle

point.

2 2

0 1 1 2 2 12 1 2 11 1 22 2 y x x x x x xβ β β β β β ε = + + + + + +



12

-he second/order model)

bxy

bBx

Bbx

Bxxbx

s

1

s

F0

222

11211

2

1

2

1

0

21::

2

1

:

29::

29:29::

and

:

:

:

*

*FF::

s

kk

k

k

k k x

x

x

y

+=

−=

=

=

=

++=

−

β

β

β β

β β β

β

β

β

β



1+

Characterizing the response surface)

= Contour plot or Canonical analysis

= Canonical form !see 'igure 11.B"

= Minimum response) i are all positi%e

= Ma(imum response) i are all negati%e

= Saddle point) i ha%e different signs

22

11::

k k s ww y y λ λ +++=



1

<(ample 11.2

= Continue <(ample 11.1

= Central composite design !CCE" !-a$le 11.7 ,'igure 11.10"

= -a$le 11.; ANOVA for Response Surface Quadratic Model

Analysis of variance table [Partial sum of squares]Sum of Mean

Source Squares ! Square Value Prob "

)odel 2(#2' ' '#$' 7"#(' * &#&&&1

A 7.92 1 7.92 111.93 < 0.0001

B 2.12 1 2.12 30.01 0.0009

A2 13.18 1 13.18 186.22 < 0.0001

B2 6.97 1 6.97 98.56 < 0.0001

AB 0.25 1 0.25 3.53 0.1022

Residual &#'& 7 &#&71

Lack of Fit 0.28 3 0.094 1.78 0.2897

Pure Error 0.21 4 0.053

Cor Total 2(#7% 12

1 2 1 2

2 21 2

: ;B.B 0.BB 0.52 0.25

1.+ 1.00

y x x x x

x x

= + + +

− + −



15

-he contour plot is gi%enin the natural %aria$les

!see 'igure 11.11"

-he optimum is at a$out

; minutes and 1;7.5

degrees

yield

A : ti m e

B

: t

e

m

p

(&#&& (2#'& ('#&& (7#'& "&#&&

17&#&&

172#'&

17'#&&

177#'&

1(&#&&

7 $ # " ' %

7 7 # $ & ' $

7 ( # 2 ' 7 !

7 ( # 2 ' 7 !

7 ( # " & ( "

7 " # ' $ & $

'



17

-he relationship $et#een ( and #)

= M is an orthogonal matri( and the columns of

M are the normalized eigen%ectors of 8.

Multiple response)

= -ypically* #e #ant to simultaneously optimizeall responses* or find a set of conditions #here

certain product properties are achie%ed

= O%erlay the contour plots !'igure 11.17"

= Constrained optimization pro$lem

"!F sxxMw −=



1;

11. <(perimental Eesigns for

'itting Response Surfaces Eesigns for fitting the first/order model

= -he orthogonal first/order designs

= G6G is a diagonal matri(

= 2 factorial and fractions of the 2 series in

#hich main effects are not aliased #ith each

others = 8esides factorial designs* include se%eral

o$ser%ations at the center.

= Simple( design



1

Eesigns for fitting the second/order model

= Central composite design !CCE"

= n' runs on 2 a(ial or star points* and nC centerruns

= Seuential e(perimentation

= -#o parameters) nC and α

= -he %ariance of the predicted response at ()

= Rotata$le design) -he %ariance of predicted

response is constant on spheres

= -he purpose of RSM is optimization and the

location of the optimum is unno#n prior to

running the e(periment.

xX)(X'x'x 1−= 2""!:! σ yVar



1B

3 !n'"19 yields a rotata$le central composite

design

= -he spherical CCE) Set 3 !"192

= Center runs in the CCE* nC) + to 5 center runs

= -he 8o(/8ehnen design) three/le%el designs

!see -a$le 11."

= Cu$oidal region)

face/centered central composite design !or

face/centered cu$e" 3 1

nC32 or +

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