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Box and Draper: Empirical Model-Building andResponse Surfaces

Khuri and Cornell: Response Surfaces

Myers and Montgomery: Response SurfaceMethodology

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An example from Box, Hunter and Hunter

Objective: Find settings of time ( ) and temperture>( ) that produced maximum yield.X

Current setting: 75 minutes, 130 C> œ X œ ‰

Past experience suggested that 1.5.5 œ

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First-order model:

C œ B " " %! = ==œ"

5

Second-order model:

C œ B B B B " " " " %! = = == => = >=œ" =œ"

5 5

=#

"Ÿ=>Ÿ5

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First first-order design

Runs time (min) Temperature ( C) (grams)‰" #B B C

" (! "#(Þ& " " &%Þ$# )! "#(Þ& " " '!Þ$$ (! "$#Þ& " " '%Þ'% )! "$#Þ& " " ')Þ!& (& "$!Þ! ! ! '!Þ$' (& "$!Þ! ! ! '%Þ$( (& "$!Þ! ! ! '#Þ$

The random order in which runs were actuallyperformed was 5,4,2,6,1,7,3

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C œ '#Þ!" #Þ$&B %Þ&Bs " #=Þ.Þ Ð „ !Þ&(Ñ Ð „ !Þ(&Ñ Ð „ !Þ(&Ñ

Interaction test:

, œ !Þ'& =Þ.Þ œ"# , 0.75

Curvature test:

, , œ Ð&%Þ$ '!Þ$ '%Þ' ')Þ!Ñ Ð'!Þ$"" ##" "% $

'%Þ$ '#Þ$Ñ œ !Þ&! œ "Þ"&, s.d.

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Estimation of error (from the three observations atthe center point): = œ %Þ!#

= œ #.0 agrees well with the preliminary value of5 œ 1.5

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Contour lines:

'#Þ!" C #Þ$&B %Þ&B œ !s " #

Direction of steepest ascent is perpendicular to thecontour lines. Starting at the center of theexperimental region, simultaneously move%Þ&Î#Þ$& œ "Þ*" B units in for every unit moved in#

B".

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Points on the path of steepest ascent

B B > X C

! ! (& "$!Þ! &ß 'ß ( '#Þ$" "Þ*" )! "$%Þ) ) ($Þ$# $Þ)$ )& "$*Þ'$ &Þ(% *! "%%Þ% "! )'Þ)% (Þ'' *& "%*Þ"& *Þ&( "!! "&$Þ* * &)Þ#

" # run

(mean)

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Second first-order design

Runs time (min) Temperature ( C) (grams)‰" #B B C

"" )! "%! " " ()Þ)"# "!! "%! " " )%Þ&"$ )! "&! " " *"Þ#"% "!! "&! " " ((Þ%"& *! "%& ! ! )*Þ("' *! "%& ! ! )'Þ)

The random order in which runs were actuallyperformed was 11,15,13,12,16,14

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C œ )%Þ($ #Þ!#&B "Þ$#&Bs " #=Þ.Þ Ð „ !Þ&(Ñ Ð „ !Þ(&Ñ Ð „ !Þ(&Ñ

Interaction test:

, œ =Þ.Þ œ"# 4.88, 0.75

Curvature test:

, , œ &Þ#) œ "Þ$!"" ## , s.d.

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A first-order model is not adequate to represent thelocal response function.

Second-order design (Star (axial) points are added toform a central composite design)

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Runs time (min) Temperature ( C) (grams)‰" #B B C

"" )! "%! " " ()Þ)"# "!! "%! " " )%Þ&"$ )! "&! " " *"Þ#"% "!! "&! " " ((Þ%"& *! "%& ! ! )*Þ("' *! "%& ! ! )'Þ)

"( (' "%& # ! )$Þ$

") "!% "%& # ! )"Þ#

"* *! "$) ! # )"Þ#

#! *! "&# ! # (*Þ&#" *! "%& ! ! )(Þ!## *! "%& ! ! )'Þ!

ÈÈ ÈÈ

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The random order in which the last 6 runs wereactually performed was 1 , , , , ,* #! "( #" ## ")

C œ )(Þ$' "Þ$*B !Þ$(B #Þ"&Bs " # "#

$Þ"#B %Þ))B B##

" #

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