Page 351 - Applied Numerical Methods Using MATLAB

340 OPTIMIZATION

Np = 30; %population size
Nb = [12 12]; %the numbers of bits for representing each variable
Pc = 0.5; Pm = 0.01; %Probability of crossover/mutation
eta = 1; kmax = 100; %learning rate and the maximum # of iterations
[xo_gen,fo_gen] = genetic(f,x0,l,u,Np,Nb,Pc,Pm,eta,kmax)

HYBRID GENETIC ALGORITHM
Step 0. Pick the initial guess x 0 = [x 01 ...x 0N ](N: the dimension of the vari-
able), the lower bound l = [l 1 .. .l N ], the upper bound u = [u 1 ... u N ],
the population size N p , the vector N b = [N b1 ... N bN ] consisting of the
numbers of bits assigned for the representation of each variable x i ,the
probability of crossover P c , the probability of mutation P m , the learn-
ing rate η(0 <η ≤ 1, to be made small/large for slow/fast learning),
and the maximum number of iterations k max > 0. Note that the dimen-
sions of x 0 , u,and l are all the same as N, which is the dimension
of the variable x to be found and the population size N p can not be
greater than 2 N b in order to avoid duplicated chromosomes and should
be an even integer for constituting the mating pool in the crossover
stage.
Step 1. Random Generation of Initial Population
o
o
o
Set x = x 0 , f = f(x ) and construct in a random way the initial pop-
ulation array X 1 that consists of N p states (in the admissible region
bounded by u and l) including the initial state x 0 , by setting
X 1 (1) = x 0 and X 1 (k) = l + rand. (u − l) for k = 2: Np (7.1.26)
∗
where rand is a random vector of the same dimension N as x 0 , u,
and l. Then, encode each number of this population array into a binary
string by

m−1 m
P 1 (n, 1 + N bi : N bi )
i=1 i=1
= binary representation of X 1 (n, m) with N bm bits
X 1 (n, m) − l(m)
= (2 N bm − 1)
u(m) − l(m)
for n = 1: N p and m = 1: N (7.1.27)

so that the whole population array becomes a pool array, each row of
which is a chromosome represented by a binary string of
N N bi bits.
i=1