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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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