Geoscience Reference
In-Depth Information
the author considers the existence of variable production costs at the facilities as
well as revenues associated with demand satisfaction. Denote by r
j
the unitary
revenue obtained from customer j
2
J. Additionally, assume that c
ij
(i
2
I,
j
2
J) includes the production costs. A new decision variable
z
i
(i
2
I)must
be considered, representing the capacity to be installed at location i
2
I.Now,it
may not be rewarding to satisfy all the demand; the trade-off between revenues and
costs will decide the best service level for each customer. The capacitated model
formulated above, can be easily adapted to the new conditions, leading to the model
proposed by Louveaux (
1986
):
Minimize
X
i2I
f
i
y
i
C
X
i2I
g
i
z
i
C
Q.y;
z
/
(8.40)
subject to
(
8.27
)
z
i
0; i
2
I;
(8.41)
with Q.y;
z
/
D
E
ŒQ.y;
z
;/,andQ.y;
z
;/ denoting the optimal value of the
following problem:
Minimize
X
i2I
X
c
ij
r
j
d
j
x
ij
(8.42)
j2J
subject to
X
i2I
x
ij
1; j
2
J
(8.43)
(
8.30
), (
8.31
)
X
d
j
x
ij
z
i
; i
2
I:
(8.44)
j2J
Louveaux and Peeters (
1992
) consider a finite set of scenarios for this problem
and propose a dual-based procedure for the extensive form of the deterministic
equivalent.
A different type of models emerge when the distribution decisions (represented
by x-variables) become first-stage decisions. In this case, penalties are paid in the
second stage for excess and shortage inventory. In addition to the notation already
presented, we denote by
j
the excess inventory of customer j
2
J and by
j
the corresponding unitary cost. Assuming deterministic distribution costs (as they
are associated with an
ex ante
decision), we can formulate the stochastic facility
location problem as follows:
Minimize
X
i2I
f
i
y
i
C
X
i2I
X
c
ij
x
ij
C
Q.x/
(8.45)
j2J
subject to
(
8.27
), (
8.30
), (
8.31
),
