Geoscience Reference
In-Depth Information
with Q.x/
D
E
ŒQ.x;/,andQ.x;/ denoting the optimal value of the following
problem:
Minimize
X
j2J
j
j
C
X
j2J
j
j
(8.46)
subject to d
j
X
i2I
x
ij
C
j
j
D
d
j
;
j
2
J
(8.47)
j
;
j
0;
j
2
J:
(8.48)
Capacities can be easily included in the above model leading to the so-called
stochastic transportation-location problem which has been investigated by several
authors (e.g., França and Luna
1982
; Holmberg and Tuy
1999
).
So far in this section, we have assumed that the allocation and distribution
decisions are made simultaneously, either after or before uncertainty is disclosed.
In some problems, these decisions can be made separately. We now consider the
situation in which the allocation of the customers to the facilities is a here-and-
now decision but the quantities to ship from the facilities to the customers are to
be decided after uncertainty is revealed. This situation is motivated, for instance,
by logistics applications, when a contract has to be previously signed, determining
a priori the distribution channels but leaving the distribution decisions dependent
on the observed values of the stochastic parameters. Such case can also occur
in companies providing some service and that need to define a priori groups of
customers that will be allocated to some server or facility. In this case, we need to
explicitly consider allocation decision variables. In particular, we denote by
w
ij
the
binary variable equal to 1 if customer j
2
J is allocated to facility i
2
I and 0
otherwise. The single-allocation version of the problem was introduced by Laporte
et al. (
1994
) and has the following formulation:
Minimize
X
i2I
f
i
y
i
C
X
i2I
X
b
ij
w
ij
C
Q.
w
/
(8.49)
j2J
subject to
w
ij
y
i
; i
2
I; j
2
J
(8.50)
X
w
ij
1; j
2
J
(8.51)
i2I
y
i
;
w
ij
2f
0;1
g
; i
2
I; j
2
J;
(8.52)
with Q.
w
/
D
E
ŒQ.
w
;/,andQ.
w
;/denoting the optimal value of the following
problem:
Minimize
X
i2I
X
c
ij
r
j
d
j
x
ij
(8.53)
j2J
subject to x
ij
w
ij
; i
2
I; j
2
J
(8.54)
