Geoscience Reference
In-Depth Information
Drezner and Wesolowsky (
1991
), investigated a different type of problem. Like
in all of the above works, a single facility is considered, which can be relocated
over time as a reaction to predictable changes in the demand. The set J of demand
nodes is the same throughout the planning horizon. The demand of each node j
2
J, is represented by a continuous function of time
w
j
.:/. A planning horizon T
divided into several time periods is assumed. The following optimization model can
be considered for each period t
2
T :
8
<
9
=
X
C
t
D
min
x
t
;y
t
W
jt
d
j
.x
t
;y
t
/
:
(11.4)
:
;
j2J
In this expression, .x
t
;y
t
/ denotes the coordinates of the facility in period t
2
T ;
W
jt
D
R
a
t
a
t1
w
j
./d; a
t1
and a
t
are the lower and upper time limits for period t,
respectively; d
j
.x
t
;y
t
/ denotes the distance between demand point j
2
J and point
.x
t
;y
t
/. The cost for the entire planning horizon is given by
P
t2T
C
t
. Drezner and
We s o l ow s ky (
1991
) made use of the above model to solve a more general problem
which consists of making a decision about the division of the planning horizon into
time periods. In this case, the number of time periods and the “break points” are
decisions to make. This work was later extended by Zanjirani Farahani et al. (
2009
)
that included a cost for relocating the facility.
Scott (
1971
) studied a multi-facility, multi-period continuous location problem,
assuming a finite planning horizon T divided into several time periods, and a set of
demand nodes, J. In each time period, a single facility is to be located and must
remain operating until the end of the planning horizon. A sequence of
j
T
j
problems
can be considered. In particular, the following mathematical model holds for period
t
2
T (the coordinates .x
;y
/,
D
1;:::;t
1, were already determined):
Minimize
X
j2J
X
t1
u
j
d
j
.x
;y
/
C
X
j2J
u
jt
d
j
.x
t
;y
t
/
(11.5)
D1
X
t
subject to
u
j
D
1; j
2
J
(11.6)
D1
u
j
2f
0;1
g
;
D
1;:::;t;j
2
J:
(11.7)
In this model, .x
t
;y
t
/ are the coordinates (to be determined) of the facility to install
at the beginning of period t
2
T ;
u
jt
is a binary variable equal to 1 if demand point
j
2
J is allocated to the facility installed in period t
2
T (such allocation can only
occur in periods t;:::;
j
T
j
), and 0 otherwise; d
j
.x
t
;y
t
/ is the Euclidean distance
between demand node j
2
J and the facility to be installed in period t
2
T .By
solving the full sequence of problems (one for each t
2
T ), a solution is obtained for
the multi-period problem. Nevertheless, using such a myopic procedure, optimality
cannot be guaranteed for the whole planning horizon.
