Engineering Optimization: Theory and Practice, Fourth Edition

24 Introduction to Optimization

that is,

g 2 (X)=

1250 π d^3

KsFD

> 1 (E 3 )

natural frequency=

√

Gg

2

√

2 ρπ

d

D^2 N

≥ 100

that is,

g 3 (X)=

√

Ggd

200

√

2 ρπ D^2 N

> 1 (E 4 )

Since the equality sign is not included (along with the inequality symbol,>) in the

constraints of Eqs. (E 2 ) to (E 4 ), the design variables are to be restricted to positive

values as

d > 0 , D > 0 , N > 0 (E 5 )

By substituting the known data,F=weight of the milling machine/4=1250 lb, ρ=

0 .3 lb/in^3 , G = 12 × 106 psi, andKs= 1. 0 5, Eqs. (E 1 ) to (E 4 ) become

f(X)=^14 π^2 ( 0. 3 )d^2 DN = 0. 7402 x 12 x 2 x 3 (E 6 )

g 1 (X)=

d^4 ( 21 × 106 )

80 ( 1250 )D^3 N

= 120 x 14 x 2 −^3 x− 31 > 1 (E 7 )

g 2 (X)=

1250 π d^3

1. 05 ( 1250 )D

= 2. 992 x 13 x− 21 > 1 (E 8 )

g 3 (X)=

√

Gg d

200

√

2 ρπ D^2 N

= 139. 8388 x 1 x− 22 x 3 −^1 > 1 (E 9 )

It can be seen that the objective function,f (X), and the constraint functions,g 1 ( X)to

g 3 ( X),are posynomials and hence the problem is a GMP problem.

Quadratic Programming Problem. A quadratic programming problem is a nonlinear

programming problem with a quadratic objective function and linear constraints. It is

usually formulated as follows:

F (X)=c+

∑n

i= 1

qixi+

∑n

i= 1

∑n

j= 1

Qijxixj (1.9)

subjectto

∑n

i= 1

aijxi=bj, j= 1 , 2 ,... , m

xi≥ 0 , i= 1 , 2 ,... , n

wherec, qi, Qij, aij, andbjare constants.