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

4.6 Application in Engineering Optimization Problems 47
Table 4.6: Statistical results for welded bean design problem case A. The
infeasible solutions are highlighted in italics
Methods fBest fMean fworst Sd F Es
GA1 [17] 1.748309 1.771973 1.785835 1.12E-02 N/A
GA2 [19] 1.728226 1.792654 1.993408 7.47E-02 80000
EP [18] 1.724852 1.971809 3.179709 4.43E-01 N/A
(µ + λ)ES [66] 1.724852 1.777692 NA 8.80E-02 30000
CPSO [45] 1.728024 1.748831 1.782143 1.29E-02 200000
HPSO [44] 1.724852 1.749040 1.814295 4.01E-02 81000
NM-PSO [98] 1.724717 1.72637 3 1.733393 3.50E-03 80000
PSOLVER [50] 1.724717 1.724717 1.724717 1.62E-11 297
SS [59] 1.724852 1.747429 1.928811 4.67E-02 83703
ABC [5] 1.724852 1.741913 NA 3.10E-02 30000
ACOMV 1.724852 1.724852 1.724852 1.74E-12 2303
4.6.2 Group II: Pressure Vessel Design Problem Case A, B,
C and D
There are four distinctive cases (A, B, C and D) of pressure vessel design
problem defined in the literature. These cases differ by the constraints posed
on the thickness of the steel used for the heads and the main cylinder. In
case A, B, C (see Table 4.7), ACOMV obtained the best results in a 100%
success rate. The number of evaluations are also the smallest. Case D is
more difficult to solve because of the larger range of side constraints for
decision variables. It should be noted that the solution of NM-PSO is not
feasible for this problem because the values of x1 and x2 given for NM-PSO
are not integer multiples of 0.0625. Table 4.8 illustrates ACOMV obtained
the best-so-far solution except the infeasible solution reported by NM-PSO .
Table 4.9 illustrates ACOMV has 100% success rate to obtain the best-so-far
results with smallest standard deviation, which is competitive to PSOLVER.
ACOMV require 30717 function evaluations. The mean and minimum number
of evaluations is 9448 and 1726. PSOLVER is more efficient in the aspect
of the number of functions evaluations. However, it should be noted that
In [50] PSOLVER is only designed for continuous optimization rather than
mixed-variable optimization, therefore, PSOLVER is difficult to solve categorical
variables. Moreover, PSOLVER ever reported an infeasible solution
in the previous welded beam design problem case A.
4.6.3 Group II: Coil Spring Design Problem
In coil spring design problem, most of the research reported in the literature
focused on finding the best solution. Only the recent work by [54] and [20]