CHAPTER
18
Multi-objective Optimum Design
Concepts and Methods
18. 1 ______________________________________________________________________________
0
1
2
3
4
0
1
2
3
f
1
=3.0
f
1
=1.0
f
1
=0.25
f
2
=3.0
f
2
=1.0
f
2
=0.25
x
1
x
2
18. 2 _____________________________________________________________________________
10
20
30
40
50
60
70
80
10
20
30
40
50
60
70
80
Pareto
Optimal Set
f1
f2
18. 3 ______________________________________________________________________________
0
2
4
6
8
10
12
0
2
4
6
8
10
12
x
1
x
2
g
1
g
2
S
f
1
=10.5
f
1
=6.0
f
1
=2.15
f
1
=9.0
f
1
=3.5
f
1
=1.8
g
3
Pareto
Optimal
Set
Chapter 18 Multi–objective Optimum Design Concepts and Methods
Arora, Introduction to Optimum Design, 4e
18-4
18. 4 ______________________________________________________________________________
Chapter 18 Multi–objective Optimum Design Concepts and Methods
18. 5 ______________________________________________________________________________
0.5
1.0
1.5
2.0
0.5
1.0
1.5
2.0
p=20
p=5
p=2
p=1
f
1
f
2
Chapter 18 Multi–objective Optimum Design Concepts and Methods
18. 6 ____________________________________________________________________________
0.5
1.0
1.5
2.0
0.5
1.0
1.5
2.0
f
1
f
2
18. 7 _____________________________________________________________________________
( ) ( ) ( ) ( ) ( )
22
22
11 2 21 2
x 37 98f wx x wx x
 
= −+− + −+−
 
( ) ( )
11 21
12 32 90
fwx wx
x
∂= −+ −=
∂
() ( )
12 21
2
2 72 80
fwx wx
x
∂= −+ −=
∂
21
21
1
93
ww
ww
x+
+
=
12
212
78
ww
xww
+
=+
18. 8 _____________________________________________________________________________
( ) (
) ( )
( )
22 2 2
1 1 2 2 1 2 1 21
20 0.75 2 2 5 1.6 2Lw x x w x x u x s
 
= − + − + − + + −+
 
()( )
075
.0
406
.1
10
11
1
2
1
=
−+
−=
∂
∂x
w
xw
x
L
( )
0
2242 2112
2=−+−=
∂
∂xwuw
x
L
0
2
12
=+
−sx
11 0
su =
1 2 1 2 11
0.1, 0.9 1.33846, 0.0, 1.0, 0.0ww x x us= = ⇒= = = =
1 2 1 2 11
0.9, 0.1 0.772973, 0.972222, 0.0, 0.986013ww x x us= = ⇒= = = =±
18. 9 _____________________________________________________________________________
Find: h, r; to:
2
2
:22
Surface Area SA rh r
=π +π
subject to:
0
1.
0
1≤
−⇒ h
g
2
20
gh⇒−≤
0
2
3≤−⇒ rg
40.5 0g hr⇒ −≤
w = (1.0, 0.0)
Þ
r * = 2.0, h* =2.0, V * = 25.132742, SA = 50.265483
Þ
Chapter 18 Multi–objective Optimum Design Concepts and Methods
Arora, Introduction to Optimum Design, 4e
18-9
18. 10 ____________________________________________________________________________
18. 11 ____________________________________________________________________________
0
2
4
6
8
10
12
14
16
18
0
1
2
3
4
5
6
x
f (x)
f
1
f
2
f
3
Optimum
Pareto
Optimal
Set