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```									      University of Illinois-Chicago

Chapter 6
Optimization Techniques

Principles of
Computer-Aided
Design and
Manufacturing
Second Edition 2004
ISBN 0-13-064631-8

Author: Prof. Farid. Amirouche
University of Illinois-Chicago
CHAPTER 6                                                                        6.2 System Modeling

6.1 INTRODUCTION
6.2 SYSTEM MODELING
Analysis and design

Figure 6.1 Simple beam
My
                          (6.2)
I

PL3
                            (6.3)
3EI
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                        6.2 System Modeling

Example 6.1
Consider a tree trunk, which can be
modeled by a beam as shown in figure
6.2. On a windy day, one can determine
the wind speed or force that would cause
the tree to break.

r 4
I                                                                          F

4

h
4 Fh
 max              (6.4)
r 3                                                     2r

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                      6.2 System Modeling

Example 6.2
A second example involves the modeling of an automobile
wheel and tire.

..
J   K  M r             (6.5)

Figure 6.3 (a) Automobile wheel and tire
Figure 6.3 (b) Model representing the tire and shaft

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                      6.2 System Modeling

GI p
K                                           (6.6)
l
d 4
Ip                                         (6.7)
32
G  80  109 N / m 2

K  2.455N.m / rad

I p  0.006136108 m4

K
n                                            (6.8)
J

M = MO sin (ω t)
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                     6.3 Design Optimization

6.3 DESIGN OPTIMIZATION
Formulation of an optimum design

There are four steps to the formulation of an optimum design:
1. Identifying the design parameters.
2. Defining the design constraints.
3. Defining the objective functions
4. Evaluating alternatives.

Design Parameters
Constraints

x1 , x 2 ,......... , x n   0            (i  1, m)               (6.9)

 ( x1 , x2 )  x12 x2  V  0                             (6.10)

 j  x1 , x2 ,......... , xn   L j                      (6.11)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                    6.3 Design Optimization

Evaluation
Example 6.3

Consider a rectangular box used for storing important
documents. Define the objective function if C denotes the cost
per unit area of the metal used for fabrication of the box. Define
the constraint equations and the limits on their design variables.

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                     6.3 Design Optimization

Solution:
A = 2LH + 2HW + 2WL

T = C (2LH + 2HW + 2WL)
V =LWH

 = V – LWH = 0

L  L1           W  W1               H H 1
As shown in the example, the steps that generally follow in formulation of
a design problem are
Identification of the design variables.
Selection of a cost function and developing an expression for it in terms
of the design variables.
Identification of constraints and developing expressions for them in terms
of design variables.
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                    6.4 Optimal Design Concept

6.4 OPTIMAL DESIGN CONCEPT
Questions:
How is each design described?
What is the criterion for best “design”?
What are the available means?
Design optimization
Select a set of variables.
Select an objective function.
Determine a set of constraints.
Solutions will be based on finding the values for the
variables that would minimize or maximize the objective
function and satisfy the constraints at the same time.
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                    6.4 Optimal Design Concept

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                    6.4 Optimal Design Concept

Example 6.4

Figure 6.5 A hollow cylinder with thickness t

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                      6.4 Optimal Design Concept

d o  d i  2t
d o  d i  2t  0

h(d o , d i , t )  d 0  d i  2t       (6.12)
 m ax  s         (6.13)
M t (d o / 2)
 max   max                            (6.14)
J
 m ax  s  0         (6.15)

 m ax  M t (d o / 2 J )  0            (6.16)



J    d o4  d i4  0                (6.17)
 32 
16 M t d o
s0                    (6.18)
 (d 0  d i )
4      4

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                 6.5 Unconstrained Optimization

6.5 UNCONSTRAINED OPTIMIZATION
Single Variable minimization

f ( x * )  0    (6.19)

f ( x * )  0
f ( x * )  0

Figure 6.6 Function f showing local, global and strong minimum

             
f  (q1 , q2 ,...., qn , q1 , q2 ,...., qn , t )                           (6.20)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                              6.5 Unconstrained Optimization

Example 6.5

Find whether the function f  x has a minimum or maximum for x  [1,1]
3

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                              6.5 Unconstrained Optimization

Solution:

Figure 6.7 Tank

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                              6.5 Unconstrained Optimization

xL
  x 2                                  2x 2       
f  2 * 
               * 20  10xL   20
                       4      xL 
                       (6.21)
  4                                                 
f  10 x 2  10 xL  10 x 2  20 xL
f  20 x 2  30 xL                                                                (6.22)

x 2
* L  120                                                                       (6.23)
4
480                                                                          (6.24)
L
x 2
 480                                                          (6.25)
f  20 x 2  30 x    2 
 x 
14400
f  20x       2                                                                  (6.26)
x
Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                              6.5 Unconstrained Optimization

f x   40x 
14400
2
0
x
40x 3  14400                           (6.27)
Example 6.7                                         x 3  114.59
x  4.86m

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                              6.5 Unconstrained Optimization

(6.28)

(6.29)

(6.30)

(6.31)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                              6.5 Unconstrained Optimization

(6.32)

(6.33)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                              6.5 Unconstrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                              6.5 Unconstrained Optimization

(6.34)

(6.35)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                           6.6 Constrained Optimization

6.6 CONSTRAINED OPTIMIZATION
m
T  To    j  j                  (6.36)
j 1

(6.37)

Example 6.9

Solve for the optimal design problem of example 6.6 by the
method of Lagrange multipliers.

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                    6.6 Constrained Optimization

(6.38)

(6.39)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                     6.6 Constrained Optimization

(6.40)

(6.41)

(6.42)
(6.41)

(6.42)                                         (6.43)

(6.43)     (6.44)                        (6.44)          (6.40)

(6.43)              (6.44)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                    6.6 Constrained Optimization

(6.45)

(6.46)

(6.47)

(6.48)

(6.49)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                    6.6 Constrained Optimization

(6.50)

(6.51)

(6.52)

(6.53)

(6.54)

(6.55)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                    6.6 Constrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                    6.6 Constrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                    6.6 Constrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                     6.7 Fibonacci Method

6.7 FIBONACCI METHOD
6.7.1 Fibonacci algorithm

Period 1

Period 2

Period 3

Period 4

Period 5

Figure 6.10 Rabbit’s multiplication at each mature period

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                     6.7 Fibonacci Method

Figure 6.11 Initial setup of the uncertainty interval

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                     6.7 Fibonacci Method

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                     6.7 Fibonacci Method

v0sinΘt

gx2

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                     6.7 Fibonacci Method

80

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                     6.7 Fibonacci Method

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                             6.8 Newton’s Method

6.8 NEWTON’S METHOD
fk
x k 1     xk                                    (6.72)
f k

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                              6.9 Linear Programming

f

Linear approximation
to f at x
fk fk
'

xk                  x
' -1
=
x k+1 x k- (f k) f k

Figure 6.13 Newton’s method

6.9 LINEAR PROGRAMMING
T   k i xi                                   (6.74)
i

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                    6.9 Linear Programming

(6.75)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                    6.9 Linear Programming

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                         6.9 Linear Programming

X2

C
5

D
4

3

2

1

X1   Figure 6.14 Graphical solution
1      2       3       4       5    6

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                                    6.10 Geometric Programming

6.10 GEOMETRIC PROGRAMMING

3
T  x1  2 x1 x2 
1
x1 x2  x3
3
(6.76)
5

T  u1  u 2  ......  u p                                              (6.77)

sp
 u2   u p 
s1           s2
 u1 
G  
s                   ... 
s  s                                          (6.78)
 1                 2  p

where the        sp 's     are chosen properly to minimize T.

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8
Author: Prof. Farid. Amirouche, University of Illinois-Chicago
CHAPTER 6                                                       6.11 Other Optimization Techniques

6.11 OTHER OPTIMIZATION TECHNIQUES
Search methods
Exhaustive Search.
Grid search.
Random search.
Simplex search.