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m
1
1
2
Pair 1
1
Pair 2
2
m
2
M
/2
M
/2+1
Pair
M
/2
m
M
/2
M
M
/2
Fig. 5.
Rearrangement for harmonies in HM
probable the better individuals in the current pair is chosen, and vice versa) is first
randomly determined and the probabilities of
hm
i
and
hm
k
are
ρ
k
respectively
using Eq. 10. The accumulated probabilities of
hm
i
and
hm
k
are
ac
i
and
ac
k
.
(
ρ
i
and
)
ρ
=
δ
;
ρ
=
δ
1
−
δ
⎧
⎨
⎩
i
k
(10)
ac
=
ρ
;
ac
=
ρ
+
ρ
i
i
k
i
k
If one random number
r
b
within the range of 0 to
ac
k
is smaller than
ac
i
, then
hm
i
is
used, and if
r
b
is higher than
ac
i
and smaller than
ac
k
, then
hm
k
is used. This procedure
is called the choosing procedure.
Instead of only one new harmony is obtained in the original Harmony search algo-
rithm, several new harmonies are generated in NHS. Two different versions of new
harmony search methods are proposed here. In the first method NHS1, the iterative
steps for NHS1 are as follows:
,
M
and randomly
generate
M
harmonies (slip surfaces). When the counter
js
= 0, the optimal harmony
is called
hm
0
, its objective function value is
f
0
, and its initial value is set to an arbitrary
large value. For example, a value of 100 (much greater than normal factors of safety)
is taken for slope stability analysis.
Step 2:
As shown in Figure 5, the harmonies in HM are grouped in to
M
/2 pairs.
Step 3:
Generate
M/2
random numbers
rn
i
,
i
=1,…,
M
/2 within the range of 0 to 1.
If
rn
i
is lower than
cp
, a new harmony is obtained as illustrated in Figure 6, and alto-
gether
D
new harmonies are obtained. One iteration step is finished and
js
=
js
+1.
Step 4:
Evaluate
D
new harmonies and choose
M
harmonies into HM from
M
old
harmonies and
D
new harmonies. The best harmony in HM is called
hm
g
and its cor-
responding value is
f
g
.
Step 1:
Initialize the algorithm parameters:
HMCR
,
PAR
,
cp
,
λ
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