Geoscience Reference
In-Depth Information
To formulate RPMP, the following assignment variables are defined:
x
ijl
D
1 if customer j is assigned to facility i at level l
0 otherwise
The idea behind the RPMP formulation is that each customer is assigned to
facilities depending upon their operational status. Accordingly, several assignment
levels can be associated with each customer. Level-0 assignments are those made
to primary facilities that serve the customers under normal circumstances. Level-l
assignments (l>0) are those made to alternative facilities that can serve a customer
if the l closer facilities have failed.
TheRPMPmodelisasfollows.
2
4
X
3
p
minimize
X
j2J
X
5
c
ij
l
.1
/x
ijl
C
c
mj
l
x
mjl
d
j
(24.27)
lD0
i2Inm
l1
subject to
X
i2I
X
x
ijl
C
x
mjt
D
1
8
j
2
J;l
D
0;:::;p
(24.28)
tD0
p
X
x
ijl
1
8
i
2
I;j
2
J
(24.29)
lD0
x
ijl
y
i
8
i
2
I;j
2
J;r
D
0;:::;p
(24.30)
X
y
i
D
p
C
1
(24.31)
i2I
y
m
D
1
(24.32)
y
i
2f
0;1
g8
i
2
I
(24.33)
x
ijl
2f
0;1
g8
i
2
I;j
2
J;l
D
0;:::;p:
(24.34)
The objective function (
24.27
) minimizes the demand-weighted expected trans-
portation and lost-sales costs. These are computed as a function of the assignment
variables by taking into account that each customer j is served by its level-l facility
i if the l closer facilities have failed, which occurs with probability
l
, and facility
i has not failed, which occurs with probability 1
for each i
2
I
n
m and
with probability 1 if i
D
m. Constraints (
24.28
) state that each customer j must be
assigned to some facility at each level l, unless j has been assigned to the emergency
facility at level t<l. Constraints (
24.29
) prevent the assignment of a customer to a
given facility at more than one level. Constraints (
24.30
) prohibit the assignment
to facilities which are not open, whereas constraint (
24.31
) state that exactly p
facilities must be opened in addition to the emergency facility, which is forced to be
open by constraint (
24.32
). Constraints (
24.33
)and(
24.34
) are standard integrality
constraints (note that the integrality constraints on the assignment variables x
ijl
can
be relaxed).
