Engineering Optimization: Theory and Practice, Fourth Edition

622 Integer Programming

where

∇Pk=











∂Pk/∂x 1

∂Pk/∂x 2

..

.

∂Pk/∂xn











(10.82)

The initial value ofs 1 , according to the requirement of Eq. (10.78), is given by

s 1 =c 1

P 1 ′(X 1 , r 1 )

Q′ 1 (X(d) 1 , β 1 )

(10.83)

whereX 1 is the initial starting point for the minimization ofφ 1 ,X(d) 1 the set of starting

values of integer-restricted variables, andc 1 a constant whose value is generally taken

inthe range 0.001 and 0.1.

To choose the weighting factor r 1 , the same consideration as discussed in

Section7.13 are to be taken into account. Accordingly, the value ofr 1 is chosen as

r 1 =c 2

f (X 1 )

+

∑m

j= 11 g/ j(X^1 )

(10.84)

with the value ofc 2 ranging between 0.1 and 1.0. Finally, the parameterβkmust be

takengreater than 1 to maintain the continuity of the first derivative of the functionφk

over the discretization points. Although no systematic study has been conducted to find

the effect of choosing different values forβk, the value ofβ 1 ≃ 2. 2 has been found to

give satisfactory convergence in some of the design problems.

Once the initial values ofrk,sk, andβk(fork=1)are chosen, the subsequent

values also have to be chosen carefully based on the numerical results obtained on

similar formulations. The sequence of valuesrkare usually determined by using the

relation

rk+ 1 =c 3 rk, k= 1 , 2 ,... (10.85)

wherec 3 <. Generally, the value of 1 c 3 is taken in the range 0.05 to 0.5. To select the

values ofsk, we first notice that the effect of the termQk(Xd) s somewhat similar toi

that of an equality constraint. Hence the method used in finding the weighting factors

for equality constraints can be used to find the factorsk+ 1. For equality constraints,

we use

sk+ 1

sk

=

rk^1 /^2

r^1 k+/^21

(10.86)

From Eqs. (10.85) and (10.86), we can take

sk+ 1 =c 4 sk (10.87)

withc 4 approximately lying in the range

√

1 / 0 .5 and

√

1 / 0 .05 (i.e., 1.4 and 4.5). The

values ofβkcan be selected according to the relation

βk+ 1 =c 5 βk (10.88)

withc 5 lying in the range 0.7 to 0.9.