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A
C
D
B
B'
A
C
D
1
D
2
D
3
D
4
Fig. 7.
Trick to generate a random number with greater probability for a special region
band. There is a much greater chance that the slip surface will fall within this soft
band of soil. Since the location of this soft band is irregular in general, it is difficult to
define a random number which will give greater probability to this narrow region at
different x-ordinate. Thus, a special trick is adopted by the author so as to simulate a
random number with special preference to a special region.
As shown in Figure 7, a classical random number is to be used within domain AB.
In the figure, the actual domain for a control variable x
i
(N+1> i >2) is represented by
the segment AB with a soft band CD in between AB. For control variables x
j
where
i≠j, the location of the soft band CD and the solution bound AB for control variable x
i
will be different from that for control variable x
j
. For segment AB, several virtual
domains with a width of CD for each domain are added adjacent to CD shown in
Figure 7. The transformed domain AB' is used as the control domain of variable x
i
.
Every point generated within the virtual domain D1-D2, D2-D3, D3-D4 is mapped to
the corresponding point in segment CD1. This technique is effectively equivalent to
giving more chances to those control variables within the soft band. The weighting to
the random number within the soft band zone can be controlled easily by the number
of extra domains added as suggested in Figure 7.
5 Examples with New Harmony Search Methods
Based on the authors' internal study, the original harmony search method is found to
be more efficient for small scale optimization problems. When the number of control
variable is large and there is a small region in the solution domain where there is a
major change in the soil parameters, it is found that the original harmony search
method can be trapped by the presence of local minimum easily. For the present pro-
posals, it is also found to work well for normal problems over wide range of soil pa-
rameters where there are no special geotechnical features. For problems with special
geotechnical features, the robustness of the present optimization algorithm will be
demonstrated by five relatively difficult examples where the precise location of the
critical failure surface has a strong influence on the factor of safety and many global
optimization methods may be trapped by the local minima easily.
5.1 Example 1
To illustrate the applicability of the proposed modified harmony search methods, four
examples will be considered. Example 1 is a slope with four layers of soils shown in
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