Modified Harmony Methods for Slope Stability Problems - Music-Inspired Harmony Search Algorithm: Theory and Applications - page 145

Step 3: Update the HM. If the new harmony is better than the worst harmony in the
HM in terms of safety factor, the worst harmony is replaced with the new harmony.
Step 4: Repeat Steps 2 and 3 until the termination criterion is achieved.

Here, two random numbers rd and r in the range [0, 1] are used within different
stages of the modified harmony search algorithm. In addition, the number of evalua-
tions of objective function during the search, denoted as NEOF, can represent the
computation time required by the optimization algorithm.
Based on many trials by the author, it is found that the original simple harmony
(OHS) search algorithm works well for simple optimization problem with less than 25
control variables in slope stability problems where the objective function is not con-
tinuous over the solution domain. For more complicated problems with a large num-
ber of control variables, it is found that the results from the original harmony search
algorithm may be unsatisfactory for some difficult problems as shown in a later sec-
tion. The author has developed two modified harmony search algorithms for such dif-
ficult cases which differ from the original method in two aspects. The first difference
is the probability of each harmony. The better the objective function value of one
harmony, the more probable will the harmony be chosen for the generation of a new
harmony. The second modification is that instead of one new harmony generated in
the original method, several new harmonies ( Nhm ) are generated during each iteration
step in the modified algorithm. In general, the modified harmony search method
(NHS) is much more effective than the original harmony search method for large
scale optimization but will be slightly less efficient for small scale problem. NHS is
also more stable for small to large scale problems and is recommended for use.
In the original harmony search method (OHS), only one harmony is obtained from
M harmonies in the current HM, and each harmony is used with the same probability.
This is the main reason why the original harmony is highly efficient for small scale
problems but can be trapped by the local minimum easily for large scale optimization
problems. Actually, the M harmonies in HM can be classified into groups based on
their objective function values, and the probability of the better harmony should be
higher than the worse ones. In the new harmony search algorithm (NHS) proposed in
this chapter, the harmonies are rearranged into M /2 pairs as illustrated in Figure 5.
In Figure 5, m i ( i = 1,…, M /2) is randomly chosen from M /2+1 to M . The pairs lo-
cated at the front of HM have greater probability to generate new harmonies. We can
introduce a parameter

1)) to decide which pair is used to generate new har-
monies. An array BB () is used to represent the probabilities of all the pairs, where
()

λ (0< λ≤

( )

M

()

−
=−

i 1

. The accumulated probability array is cal- BB i λλ 1 , i = 1, ..., AC

2

i

∑ . A random number r is obtained within the range of

() ( )

culated as AC i = BB j

j = 1

( ) ( )

−<≤ , then the t i pair is used to generate one new
harmony. The procedures for generating the new harmony by one given pair are out-
lined in Figue 6. Suppose t i pair of harmonies, namely, hm i and hm k , are used to ob-
tain a new harmony. A random number

0 to AC ( M/ 2). If AC i 1 r AC i

a

δ in the range of 0.2 to 0.8 (the lower and

upper bounds to δ are based on the facts that the larger the value of δ , the more