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C p ðÞ¼x p C p ðÞ¼x p X
a∈L
c a fð ʴ ap ,
8p
∈P ,
ð 15
:
20 Þ
with notice to ( 15.8 ), where C p denotes the unit operational cost on path p .
The disaster relief supply chain network optimization problem can be expressed
as follows. The organization seeks to determine the optimal levels of the disaster
relief item processed on each supply chain link as well as the optimal amounts of
the time deviations on paths, subject to the minimization of the total operational
cost while satisfying the uncertain demand as closely as possible. Therefore, the
optimization problem is constructed as:
Minimize X
p∈P
C p ðÞþ X
n R
1 λ
þ X
n R
1 ʳ k ðÞ ,
k þ λ
k
k E
k E
ʔ
ʔ
ð 15
:
21 Þ
subject to: constraints ( 15.6 ), ( 15.18 ), and ( 15.19 ).
Next, we present the partial derivatives of the shortages and the surpluses solely
in terms of path flows, which will be used later in developing the variational
inequality formulation of the problem. The respective partial derivatives of the
expected values of shortage and surplus of the disaster relief item at each demand
point with respect to the path flows, derived in Dong et al. ( 2004 ), Nagurney
et al. ( 2011 ), and Nagurney et al. ( 2012a ), are given by:
!
1,
ʔ k
x p ¼ P k X
q
E
x q
8p
∈P k ;
k ¼ 1,
...
, n R ,
ð 15
:
22a Þ
∈P k
and
,
ʔ k
x p ¼ P k X
q
E
x q
8p
∈P k ;
k ¼ 1,
...
, n R :
ð 15
:
22b Þ
∈P k
Let K denote the feasible set such that:
Þ x
,
R n p
R n p
R n p
þ
K ¼
ð
x
;
z
; ˉ
þ , z
þ , and
ˉ∈
ð 15
:
23 Þ
where x is the vector of path flows of the relief item, z is the vector of time
deviations on paths, and
ˉ
is the vector of Lagrange multipliers corresponding to
the constraints in ( 15.19 ).
We now derive the variational inequality of the integrated disaster relief supply
chain network problem.
Theorem 1 The optimization problem ( 15.21 ), subject to its constraints ( 15.6 ),
( 15.18 ), and ( 15.19 ), is equivalent to the variational inequality problem: determine
the vector of optimal path flows, the vector of optimal path time deviations, and the
vector of optimal Lagrange multipliers ( x , z ,
ˉ )
K , such that:
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