Engineering Optimization: Theory and Practice, Fourth Edition

496 Geometric Programming

Degree of Difficulty. The quantityN−n−1 is termed adegree of difficulty in

geometric programming. In the case of a constrained geometric programming problem,

Ndenotes the total number of terms in all the posynomials andnrepresents the number

of design variables. IfN−n− 1 =0, the problem is said to have a zero degree of

difficulty. In this case, the unknowns∗j( j= 1 , 2 ,... , N) can be determined uniquely

from the orthogonality and normality conditions. IfNis greater thann+1, we have

more number of variables(∗js han the equations, and the method of solution for)t

this case will be discussed in subsequent sections. It is to be noted that geometric

programming is not applicable to problems with negative degree of difficulty.

Sufficiency Condition. We can see that∗jare found by solving Eqs. (8.7) and (8.9),

which in turn are obtained by using the necessary conditions only. We can show that

these conditions are also sufficient.

Finding the Optimal Values of Design Variables. Since f∗ and ∗j (j=

1 , 2 ,... , N) are known, we can determine the optimal values of the design variables

from the relations

Uj∗=∗jf∗=cj

∏n

i= 1

(x∗i)aij, j= 1 , 2 ,... , N (8.14)

The simultaneous solution of these equations will yield the desired quantitiesxi∗(i=

1 , 2 ,... , n). It can be seen that Eqs. (8.14) are nonlinear in terms of the variables

x∗ 1 , x∗ 2 ,... , x∗n, and hence their simultaneous solution is not easy if we want to solve

them directly. To simplify the simultaneous solution of Eqs. (8.14), we rewrite them as

∗jf∗

cj

= (x 1 ∗)a^1 j(x∗ 2 )a^2 j·· ·(x∗n)anj, j= 1 , 2 ,... , N (8.15)

By taking logarithms on both the sides of Eqs. (8.15), we obtain

ln

∗jf∗

cj

=a 1 jlnx 1 ∗+a 2 jlnx 2 ∗+ · · · +anjlnx∗n,

j= 1 , 2 ,... , N (8.16)

By letting

wi= nl xi∗, i= 1 , 2 ,... , n (8.17)

Eqs. (8.16) can be written as

a 11 w 1 +a 21 w 2 + · · · +an 1 wn= nl

f∗∗ 1

c 1

a 12 w 1 +a 22 w 2 + · · · +an 2 wn= nl

f∗∗ 2

c 2

..

.

a 1 Nw 1 +a 2 Nw 2 + · · · +anNwn= nl

f∗∗N

cN

(8.18)