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

6.1 Mathematical formulation of engineering problems 63
Table 6.5: The mathematical formulation for the coil spring design problem.
min fc(N, D, d) = π2 Dd2 (N+2)
4
No Constraint
g1
g2
g3
g4
g5
g6
g7
g8
8 Cf FmaxD
π d
lf − lmax ≤ 0
− S ≤ 0
dmin − d ≤ 0
D − Dmax ≤ 0
3.0 − D ≤ 0
d
σp − σpm ≤ 0
σp + Fmax−Fp
K + 1.05(N + 2)d − lf ≤ 0
σw − Fmax−Fp
≤ 0
where Cf =
lf = Fmax
K
K
4 D
d −1
4 D
d
0.615 d +
−4 D
K = Gd4
8 ND 3
σp = Fp
K
+ 1.05(N + 2)d
6.1.5 Thermal Insulation Systems Design Problem
The schema is shown in Figure 6.4. The basic mathematical formulation
of the classic model of thermal insulation systems is defined in Table 6.6.
The effective thermal conductivity k of all these insulators varies with the
temperature and does so differently for different materials. Considering that
the number of intercepts n is defined in advance, and based on the presented
model, we may define the following problem variables:
• Ii ∈ M, i = 1, ..., n+1 — the material used for the insulation between
the (i − 1)-st and the i-th intercepts (from a set M of materials).
• ∆xi ∈ R+, i = 1, ..., n + 1 — the thickness of the insulation between
the (i − 1)-st and the i-th intercepts.
• ∆Ti ∈ R+, i = 1, ..., n + 1 — the temperature difference of the insulation
between the (i − 1)-st and the i-th intercepts.
This way, there are n + 1 categorical variables chosen form a set M of
available materials. The remaining 2n + 2 variables are continuous.