Improved ant colony optimization algorithms for continuous ... - CoDE

50 Ant Colony Optimization for Mixed Variable Problems
et al [52] and Abramson et al [3], which are able to handle such categorical
variables properly. we show that ACOMV can performs comparably to
MVP [52] and FMGPS [3]. Table 4.11 present the new best-so-far solution
of ACOMV after 10000 function evalutations.
4.6.5 Group IV: Welded Beam Design Problem Case B
Welded beam design problem case B is taken from Deb and Goyal [21] and
Dimopoulos [23]. It is a variation of case A and is extended to include ordered
discrete and categorical variables together. Table 4.12 shows ACOMV
obtained a new best-so-far solution with a 100% success rate. The maximum,
mean and minimum number of evaluations of is 4883, 1436 and 692,
respectively. Table 4.14 verifies the best results obtained by ACOMV not to
violate the constraints.
4.6.6 Related Work on Engineering Optimization Problems
For a detail level analysis on engineering optimization problems, we investigate
ACOMV RLDs on fighting against stagnation by restart mechanism.
An experiment is also conducted to compare the performance of the generic
restart mechanism of ACOMV with a problem tailored restart mechanism,
called cut-off restart. The later is based on an approximation of exponential
distribution. It is possible to estimate, from an empirically estimated
RLD, the number of function evaluations needed to find the required solution
with a probability greater than or equal to z if an optimal restart policy
is supposed to be used. This estimation is sometimes called computational
effort [69, 73] and it is defined as
effort = min(l)
l ·
ln(1 − z)
ln(1 − RLq(l))
(4.5)
The solution l of the computation effort is the cut-off evaluations to periodically
restart in a simulation of estimate model. RLq(l) is the algorithm’s
RLD, defined as RLq(l) = P (Lq ≤ l), where Lq is the random variable
representing the number of function evaluations needed to find a solution
of quality q, and P (Lq ≤ l) is the probability that Lq takes a value less
than or equal to l function evaluations. The cut-off restart improves the
performance of algorithms as seen from Figure 4.7. However we also see
that the tuned restart mechanism of ACOMV needs less functions evaluations
to have 100% success rate than the cut-off restart, even if latter one
is problem-tailored. Taking the pressure vessel design problem case D of
Figure 4.7 for example, the cut-off restart starts at the point (2243 function
evaluations with a 23% success rate), and obtain a 99% success rate with
44400 function evaluations, while the tuned restart mechanism of ACOMV