Engineering Optimization: Theory and Practice, Fourth Edition

22 Introduction to Optimization

Finally, the lower bounds on the design variables can be taken as

w≥ 0 (E 8 )

di≥ 0 , i= 1 , 2 , 3 , 4 (E 9 )

As the objective function, (E 1 ), and most of the constraints, (E 2 ) to (E 9 ), are nonlinear

functionsof the design variablesd 1 , d 2 , d 3 , d 4 , and w,this problem is a nonlinear

programming problem.

Geometric Programming Problem.

Definition A functionh(X)is called aposynomialifhcan be expressed as the sum

of power terms each of the form

cixai 11 x 2 ai^2 · · ·xnain

whereciandaijare constants withci> 0 andxj> 0. Thus a posynomial withNterms

can be expressed as

h(X)=c 1 x^11 a^1 x 2 2 a^1 · · ·xna^1 n+ · · · +cNx 1 aN^1 xaN 2 2 · · ·xnaN n (1.7)

Ageometric programming(GMP)problemis one in which the objective function

and constraints are expressed as posynomials inX. Thus GMP problem can be posed

as follows [1.59]:

FindXwhich minimizes

f (X)=

∑N^0

i= 1

ci





∏n

j= 1

x

pij

j



, ci> 0 , xj> 0 (1.8)

subject to

gk(X)=

∑Nk

i= 1

aik





∏n

j= 1

x

qij k

j



> 0 , aik> 0 , xj> 0 ,k= 1 , 2 ,... , m

whereN 0 andNkdenote the number of posynomial terms in the objective andkth

constraint function, respectively.

Example 1.4 Four identical helical springs are used to support a milling machine

weighing 5000 lb. Formulate the problem of finding the wire diameter(d), coil diameter

(D), and the number of turns (N) of each spring (Fig. 1.11) for minimum weight by

limiting the deflection to 0.1 in. and the shear stress to 10,000 psi in the spring. In

addition, the natural frequency of vibration of the spring is to be greater than 100 Hz.

The stiffness of the spring (k), the shear stress in the spring (τ), and the natural

frequency of vibration of the spring (fn) are given by

k=

d^4 G

8 D^3 N

τ=Ks

8 FD

π d^3

fn=

1

2

√

kg

w

=

1

2

√

d^4 G

8 D^3 N

g

ρ(π d^2 / 4 )π DN

=

√

Gg d

2

√

2 ρπ D^2 N