Engineering Optimization: Theory and Practice, Fourth Edition

508 Geometric Programming

The optimum values of the design variables can be found from

U 1 ∗=∗ 1 f∗= ( 0. 385 )( 242 )= 92. 2

U 2 ∗=∗ 2 f∗= ( 0. 175 )( 242 )= 42. 4

U 3 ∗=∗ 3 f∗= ( 0. 294 )( 242 )= 71. 1

U 4 ∗=∗ 4 f∗= ( 0. 147 )( 242 )= 35. 6

(E 4 )

From Eqs. (E 1 ) nd (Ea 4 ) we have,

U 1 ∗= 001 D∗= 29. 2

U 2 ∗= 05 D∗^2 = 24. 4

U 3 ∗=

20

Q∗

= 17. 1

U 4 ∗=

300 Q∗^2

D∗^5

= 53. 6

These equations can be solved to find the desired solutionD∗= 0. 9 22 cm,Q∗=

0. 2 81 m^3 /s.

8.7 Constrained Minimization

Most engineering optimization problems are subject to constraints. If the objective

function and all the constraints are expressible in the form of posynomials, geometric

programming can be used most conveniently to solve the optimization problem. Let

the constrained minimization problem be stated as

FindX=











x 1

x 2

..

.

xn











which minimizes the objective function

f (X)=

∑N^0

j= 1

c 0 j

∏n

i= 1

x

a 0 ji

i (8.52)

and satisfies the constraints

gk(X)=

∑Nk

j= 1

ckj

∏n

i= 1

x

akij

i ⋚^1 , k=^1 ,^2 ,... , m (8.53)

where the coefficientsc 0 j (j= 1 , 2 ,... , N 0 ) nda ckj (k = 1 , 2 ,... , m; j= 1 , 2 ,

... , Nk) re positive numbers, the exponentsa a 0 ji (i = 1 , 2 ,... , n;j= 1 , 2 ,... , N 0 )