Geoscience Reference
In-Depth Information
X
x
ijt
n
it
y
it
; t
2
T; i
2
I
(11.30)
j2P
it
z
it
y
it
y
i;t1
; t
2
T; i
2
I
(11.31)
z
it
2f
0;1
g
; t
2
T; i
2
I;
(11.32)
with y
i0
D
0, i
2
I. In this model, r
ijt
represents the revenue obtained when
supplying all the demand of customer j
2
J in period t
2
T from facility i
2
I.
For each facility i
2
I there is a maximum number of customers, n
it
, it can supply
in period t
2
T . Furthermore, not all the facilities can supply all customers. In
particular, P
it
represents the set of customers that can be served from facility i
2
I
in period t
2
T . As we will see below, constraints (
11.30
) had been proposed
before for another problem. Canel and Khumawala (
1997
) developed a branch-and-
bound procedure for this problem adapting the algorithm proposed by Khumawala
(
1972
), and Canel and Khumawala (
2001
) proposed a heuristic approach for the
same problem.
In all of the above problems, facilities can be opened and closed more than once
during the planning horizon. Dias et al. (
2007
) point out that these models ignore
the fact that re-opening a facility has in general a smaller cost than opening it for the
first time (for instance, land acquisition costs are incurred only once). They propose
a model taking this aspect into account. Additional decision variables are required
to distinguish whether a facility is being opened for the first time or is being re-
opened. A primal-dual heuristic is proposed for obtaining lower and upper bounds
for the problem. The gap is closed using a branch-and-bound procedure.
11.5
Modular Construction of Intrinsic Multi-Period Facility
Location Models
In many practical situations it is not acceptable to install and remove a facility,
say, in consecutive periods. This may make sense for seasonal facilities, such as
warehouses if, for instance, they can be rented for short time intervals. Nevertheless,
this cannot be assumed in general. Accordingly, the models presented in the
previous section may be short for capturing some real-world problems. Early,
researchers have noticed this fact and have considered models involving constraints
that impose a limit on the number of changes performed in each location during
the planning horizon. Often, such constraints state that once a facility is installed
(removed), it must remain opened (closed) until the end of the planning horizon.
We consider again the multi-period p-median problem, i.e., we assume that a
plan is to be made for locating exactly p facilities in a finite multi-period planning
horizon T . Let us assume that removing facilities is not allowed. One additional
feature that may be worth considering for this type of problem is the speed at which
p changes. The adequate model is the following (the notation was introduced in
