Geoscience Reference
In-Depth Information
subject to
X
i2I
x
ij
D
1; j
2
J
(8.29)
x
ij
y
i
; i
2
I; j
2
J
(8.30)
x
ij
0; i
2
I; j
2
J:
(8.31)
Model (
8.28
)-(
8.31
) is defined for every realization, ,of, i.e., for every
realization of costs and demands. Accordingly, the allocation decisions x
ij
(i
2
I,
j
2
J), which do not appear in the first-stage problem, can change with different
realizations of the random vector. For this reason, they are referred to as recourse
decisions. Regarding the variables associated with the location of the facilities, y
i
,
they correspond to
ex ante
(first-stage) decisions and thus, they must hold for all
possible realizations of the random variables. The expectation defining the recourse
function Q.y/, implicitly conveys a neutral attitude of the decision maker towards
risk. Later in this section, we discuss another possible attitude and the corresponding
consequences from a modeling point of view. It is also important to emphasize that
constraints (
8.30
)and(
8.31
) together assure that at least one facility is installed.
Finally, it should be noted that we are dealing with a problem that has relatively
complete recourse, i.e., for every first-stage feasible solution, y
i
(i
2
I)thereisat
least one second-stage feasible completion (solution), x
ij
(i
2
I, j
2
J)forevery
possible realization of the random quantities.
If we have a finite set of scenarios, say ǝ, we can go farther with the above
model. In order to do so, we consider scenario-indexed parameters and variables.
Denote by c
ij
!
the cost for supplying customer j
2
J from facility i
2
I under
scenario !
2
ǝ,andletd
j!
be the demand of customer j
2
J under scenario
!
2
ǝ.Ifx
ij
!
is the fraction of the demand of customer j
2
J satisfied from
facility i
2
I under scenario !
2
ǝ, then we can consider the following extensive
form of the deterministic equivalent:
0
@
X
i2I
1
Minimize
X
i2I
f
i
y
i
C
X
!2ǝ
X
A
!
c
ij
!
d
j!
x
ij
!
(8.32)
j2J
subject to
(
8.28
)
X
x
ij
!
D
1; j
2
J; !
2
ǝ
(8.33)
i2I
x
ij
!
y
i
; i
2
I; j
2
J; !
2
ǝ
(8.34)
x
ij
!
0; i
2
I; j
2
J; !
2
ǝ:
(8.35)
