Optimum Design
169
use the form Eq. (3.70)
~,6i = 1
fll
Ul -__
~2 ~f12X2X3
XlX2X3
U 3 ~ fl3XlX3
U4~fl4XIX2
So
fll
-- + f12X2X3
+ fl3XIX3
+ fl4XIX2
>~
XIX2X3
. fl~l .~ 61 (flaX2X3~ 62 (fi3XlX3~ 63 (fi4XlX2" ~ 64
61XIX2X3,]
~k 62 ) ~k 63 J ~k 64
61 +62+63
+64 = 1
Look at the right hand side of Eq. (3.73) and rearrange
A minimum is obtained if
X’{61q-630VC~4 X~6’ q-62q-64 X; 61"}-~)2q-~3 = 1
or the exponents are zero
--61
+0 +63 +64 = 0
--61
+62 +0+64
=0
61 -I-62 +63 +64 ~- 1
and in Eq. (3.77)
6~ = 2/5 6 2 = 1/5 63 -~- 1/5 64 = 1/5
Examine the degrees of difficulty
DD = T- (N + 1)
T = Number of terms
N = Number of variable
(3.72)
(3.73)
(3.74)
(3.75)
(3.76)
(3.77)
(3.78)
169
use the form Eq. (3.70)
~,6i = 1
fll
Ul -__
~2 ~f12X2X3
XlX2X3
U 3 ~ fl3XlX3
U4~fl4XIX2
So
fll
-- + f12X2X3
+ fl3XIX3
+ fl4XIX2
>~
XIX2X3
. fl~l .~ 61 (flaX2X3~ 62 (fi3XlX3~ 63 (fi4XlX2" ~ 64
61XIX2X3,]
~k 62 ) ~k 63 J ~k 64
61 +62+63
+64 = 1
Look at the right hand side of Eq. (3.73) and rearrange
A minimum is obtained if
X’{61q-630VC~4 X~6’ q-62q-64 X; 61"}-~)2q-~3 = 1
or the exponents are zero
--61
+0 +63 +64 = 0
--61
+62 +0+64
=0
61 -I-62 +63 +64 ~- 1
and in Eq. (3.77)
6~ = 2/5 6 2 = 1/5 63 -~- 1/5 64 = 1/5
Examine the degrees of difficulty
DD = T- (N + 1)
T = Number of terms
N = Number of variable
(3.72)
(3.73)
(3.74)
(3.75)
(3.76)
(3.77)
(3.78)