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Constrained case: the welded beam design problem. HKA was compared to other
metaheuristics specifically designed for solving constrained problems. In these experiments,
HKA handles constraints via a new objective function which includes penalty functions (see
relation (2)).
A welded beam is designed for minimum cost subject to constraints on shear stress τ (q),
bending stress in the beam σ (q), buckling load on the bar Pc, end deflection of the beam δ (q), and
side constraints (Rao, 1996). There are four design variables as shown in Figure 10: h (denoted
q1), l (denoted q2), t (denoted q3) and b (denoted q4), q = [q1, q2, q3, q4] T .
Figure 10 - Welded beam design problem.
The problem can be mathematically formulated as follows:
Minimize 2
J ( q)
= ( 1+
c1)
q1
q2
+ c2q3q4
( L + q2)
Subject to: g ( q)
= τ ( q)
−τ
≤ 0
Where:
τ ( q)
=
6PL
σ ( q)
= , 2
q q
1
g ( q)
= σ ( q)
−σ
2
g ( q)
= q − q
3
4
5
6
7
1
1
1
4
≤ 0
2
1
3
≤ 0
g ( q)
= c q + c q q ( L + q ) − 5 ≤ 0
g ( q)
= h
min
max
4
− q ≤ 0
g ( q)
= δ ( q)
−δ
max
max
≤ 0
g ( q)
= P − Pc(
q)
≤ 0
2 q2
2
τ1
+ 2τ
1τ
2 + τ 2 ,
2R
⎛ q2
⎞
M = P⎜
L + ⎟,
⎝ 2 ⎠
4
3
R =
3
4PL
δ ( q)
= , 2
Eq q
τ =
P
2
q2
⎛ q1
+ q3
⎞
+ ⎜ ⎟ ,
4 ⎝ 2 ⎠
4
3
h
1
t
2
2q
q
1
2
l
2
,
MR
τ 2 =
I
⎪⎧
I = 2⎨
⎪⎩
2
4.
013E
q3
q
Pc(
q)
=
2
L
⎡ 2
2
q
⎤⎪⎫
2 ⎛ q1
+ q3
⎞
2q1q2
⎢ + ⎜ ⎟ ⎥⎬
⎢⎣
12 ⎝ 2 ⎠ ⎥⎦
⎪⎭
6
4
L
/ 36
P
b
⎛ q
⎜ 3
⎜
1−
⎝ 2L
E
4G
⎞
⎟
⎠
(29)
(30)