Harmony Search Methods for Multi-modal and Constrained Optimization - Music-Inspired Harmony Search Algorithm: Theory and Applications - page 49

x G Minimize f( ) = 1 . 10471 x

2
1

x + 0 04811 x x ( x + 14 ) ,

2 3 4 2

subject to

G G G G

g ( x ) = τ ( x ) − τ ≤ 0 , g ( x ) = σ ( x ) − σ ≤ 0 ,

1 max 2 max

x G

G G G G

g ( ) = x − x ≤ 0 , g ( x ) = δ ( x ) − δ ≤ 0 , g 5 ( x ) = P − P ( x ) ≤ 0 ,

3 1 4 4 max C

where

x G

x P MR

x

τ ( ) = ( τ ′ )

2

+ 2 τ ′ τ ′ + ( τ ′ )

2

, τ ′ = , τ ′ = , M = P ( L + ) ,

2

2 R J 2

2 x x

1 2

⎩ ⎨ ⎧ ⎭ ⎬ ⎫

G

x

2
2

x + x ⎡ x

2
2

x + x ⎤ 6 PL

R = + ( )

2

, J = 2 2 x x + ( )

2

, σ ( x ) = ,

1 3 1 3

⎢ ⎣ ⎥ ⎦

1 2

4 2 12 2 x

2
3

x

4

x x

2
3

6
4

4 013 E

G 4 PL

3

G x E

36

δ ( x ) = , P ( x ) = ( − ) ,

3

C

Ex

3
3

x L

2

2 L 4 G

4

P = 6000 lb, L

= 14

in, δ

= 0 25 in,

E = 30 × 10

6

psi,

max

G = 12 × 10

6

psi,

τ

= 13 , 600 psi,

σ

= 30 , 000 psi,

max max

0 . 125 ≤

x

≤ 5 , 0 . 1 ≤

x

,

x

≤ 10 , 0 . 1 ≤

x

≤ 5 .

1 2 3 4

Again, our modified HS method found good solutions without violating any con-
straint for this practical design problem.

Fig. 7. Schematic of welded beam