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[0
,
1]
,
and bounds :
x
(
lo
)
,
x
(
hi
)
.
Input :
D
,
G
max
,
NP
≥
4
,
F
∈
(0
,
1+)
,
CR
∈
⎨
⎩
∀
i
≤
NP
∧∀
j
≤
D
x
i
,
j
,
G
=0
=
x
(
lo
)
x
(
hi
)
j
−
x
(
lo
)
j
+
rand
j
[0
,
1]
•
j
if x
j
,
i
∈
x
0
,
i
,
x
1
,
i
,...,
x
j
−
1
,
i
Initialize :
i
=
{
1
,
2
,...,
NP
},
j
=
{
1
,
2
,...,
D
},
G
= 0
,
rand
j
[0
,
1]
∈
[0
,
1]
NP
:
f
x
i
,
G
=0
Cost :
∀
i
≤
⎨
⎩
While
G
<
G
max
⎨
⎩
Mutate and recombine :
r
1
,
r
2
,
r
3
∈{
1
,
2
,....,
NP
}
,
randomly selected
,
except :
r
1
=
r
2
=
r
3
=
i
j
rand
∈{
1
,
2
,...,
D
},
randomly selected once each
i
⎨
⎩
γ
j
,
r
3
,
G
←
x
j
,
r
3
,
G
:
γ
j
,
r
1
,
G
←
x
j
,
r
1
,
G
:
γ
j
,
r
2
,
G
←
x
j
,
r
2
,
G
Forward Transformation
γ
j
,
r
3
,
G
+
F
·
(γ
j
,
r
1
,
G
−
γ
j
,
r
2
,
G
)
if (
rand
j
[0
,
1]
<
CR
∨
j
=
j
rand
)
∀
j
≤
D
,
u
j
,
i
,
G
+1
=
γ
j
,
i
,
G
←
x
j
,
i
,
G
otherwise
ρ
j
,
i
,
G
+1
←
ϕ
j
,
i
,
G
+1
Backward Transformation
u
j
,
i
,
G
+1
mutate
⎨
⎩
∀
i
≤
NP
u
i
,
G
+1
=
ρ
j
,
i
,
G
+1
Mutate Schema
←
if
u
j
,
i
,
G
+1
∈
u
0
,
i
,
G
+1
,
u
1
,
i
,
G
+1
,..
u
j
−
1
,
i
,
G
+1
u
i
,
G
+1
Standard Mutation
u
j
,
i
,
G
+1
←
u
i
,
G
+1
Insertion
u
j
,
i
,
G
+1
←
Select :
x
i
,
G
+1
=
u
i
,
G
+1
if
f
(
u
i
,
G
+1
)
≤
f
(
x
i
,
G
)
x
i
,
G
otherwise
G
=
G
+ 1
Local Search
x
best
= Δ (
x
best
,
i
,
j
)
if stagnation
Fig. 3.9.
EDE Template
x
1
=
}
Each value in
x
1
and
x
2
are paired up and considered.
{
2
,
5
,
4
,
3
,
1
}
and
x
1
=
{
2
,
5
,
4
,
3
,
1
⎧
⎨
⎫
⎬
{
2
,
4
}
,
{
2
,
2
}
,
{
2
,
1
}
,
{
2
,
5
}
,
{
2
,
3
}
,
{
5
,
4
}
,
{
5
,
2
}
,
{
5
,
1
}
,
{
5
,
5
}
,
{
5
,
3
}
,
Δ
(
i
,
j
)=
{
3
,
4
}
,
{
3
,
2
}
,
{
3
,
1
}
,
{
3
,
5
}
,
{
3
,
3
}
,
⎩
⎭
{
1
,
4
} ,{
1
,
2
},{
1
,
1
} ,{
1
,
5
} ,{
1
,
3
}
(
x
,
i
,
j
) is evaluated. If this value is negative the objective
function value for the problem is decrement by
The cost of the move
Δ
Δ
(
x
,
i
,
j
). Hence the solution is im-
proved to a near optimal solution.
The complete template of Enhanced Differential Evolution is given in Fig 3.9.
3.5
Worked Example
This worked example outlines how EDE is used to solve the flowshop scheduling prob-
lem. The problem to be solved is the one represented in Table 3.26 (Section 3.6.1: Flow
Shop Scheduling Example).
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