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The total cost is
cost
=
m
=1
M
i
=1
N
k
=1
K
(
C
hkim
+
C
pkim
+
C
rkim
+
C
akim
−
P
kim
)
where
C
hkim
=
holdinv×h
,
C
pkim
=
shortage×s
,
C
rkim
=
resource×r
,
η
kim
=
1
,
if
IOP <r
0
,
otherwise
install
,
sellout
kim
= min(
demand
kim
,
IOP
kim
),
P
kim
=
C
akim
=
η
kim
×
sellout
kim
×
p
.
Then
cvar
=
u
+
1
u
]
+
,u
R.
In bi-objective model, the objectives
are min (
cost,cvar
) Solving this bi-objective model, we can get several combi-
nations of (cost, cvar) for the decision makers to choose rather than a single
solution as in the one-objective model.
α
E
[
C
−
∈
1
−
3.3 Optimization and Algorithm
3.4 Problem-Based NSGA-II
Analytic Property of Problem.
When using CVaR as the risk measurement,
we use scenario analysis method to simulate risk element
u
.
u
and (
r,Q
)in
combination will become the gene in GA and NSGA-II. We here use the property
of this problem to simplify algorithms through linear programming.
Suppose that limited samples of
u
,whichare
u
1
,u
2
,
,u
n
are known. We use
the law of large number to generate normal distribution. According to central
limit theorem for independent and identically distribution, we have
···
nμ
/
√
nσ
∼
nX
k
−
N
(0
,
1)
k
=1
It is proven that satisfying approximation effect can be gained when
n
=12. Since
this is standard normal distribution that we get, we should use linear trans-
formation for non-standard normal distribution
N
(
a,b
). This transformation is
x
=
x
0
×
b
+
a
.
In testing process, through changing the
μ
and
σ
namely, the mean and vari-
ance to have different demand functions. And thus we can see whether the result
would be different in relatively stable and unstable market environments.
We have
cvar
=
u
+
1
u
]
+
,u
R.
In this,
u
is also a decision variable.
One way to solve
u
is to combine it with (
r,Q
) in gene and get them through
crossover, mutation and selection. However, we find that when stochastic de-
mand and sample have been fixed, we can use linear programming to calculate
u
.
α
E
[
C
−
∈
1
−
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