Modified Harmony Methods for Slope Stability Problems - Music-Inspired Harmony Search Algorithm: Theory and Applications - page 148

The second new harmony search method NHS2 differs from the original method in
two aspects. The first difference is the probability of each harmony. Instead of the use
of uniform probability in the original harmony search method, the better the objective
function value of one harmony, the more probable will it be chosen for the generation
of a new harmony. A parameter δ (0<δ≤1) is introduced and all the harmonies in HM

are sorted by ascending order, and a probability is assigned to each of them. For in-
stance, par(i) means the probability to choose the

th

i harmony as follows:

() ( )

i − 1

par i =× − δ 1 δ (11)

where i =1,2,…, M . From Eq. 11, it can be seen that the larger the value of δ, the more
probable is the first harmony being chosen. An array ST(i) ( i =0, 1,…, M ) should be
used to implement the above procedure for choosing the harmony.

i

()

∑ (12)

( )

ST i = par j

j = 1

where ST(i) represents the accumulating probability for t i harmony. ST(0) is de-
fined as 0.0 for the sake of implementation. A random number

r is given form the

th

range [0, ST(M) ], and the k harmony in HM is chosen if the following criterion is

satisfied.

( ) ( )

ST k −<≤ 1 r ST k , k = 1, 2, ..., M (13)

c

The second modification in NHS2 is that rather than one new harmony, a certain
number of new harmonies ( Nhm ) are generated during each iteration step in the modi-
fied algorithm. The utilization of HM is intuitively more exhaustive by generating
several new harmonies than by generating only one harmony. In order to retain the
structure of HM unchanged, the M harmonies with the lower objective functions (for
the minimization optimization problem) from M+Nhm harmonies are included in the
HM again and Nhm harmonies of higher objective functions are rejected.
Based on extensive tests on different types of problems, the author has found that
the two improved harmony search methods usually perform better for more difficult
problems (presence of multi local minima, large number of control variables). The
number of evaluations and the results are better than the original harmony search
method. On the other hand, for simple problems with smaller number of control vari-
ables, the original harmony search method is usually more efficient. The new methods
require more solution parameters than the original harmony search method, but based
on extensive internal tests by the author, it is found that the new methods are stable
over wide range of problems and are more effective than the original method in most
cases which will be illustrated by the following examples.

4 Weighted Random Number for Soft Band Problem

In natural slopes, a soft thin band of soil is sometimes found. The design parameters
of this thin band of soil are much lower than those soils above and below the soft