One-Dimensional Facility Location-Allocation Using Dynamic Programming
Author(s): Robert F. Love
Source: Management Science, Vol. 22, No. 5 (Jan., 1976), pp. 614-617
Published by: INFORMS
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MANAGEMENT SCIENCE
Vol. 22, No. 5, January, 1976
Printed in U.SA.
ONE-DIMENSIONAL FACILITY
LOCATION-ALLOCATION USING DYNAMIC
PROGRAMMINGt
ROBERT F. LOVEI
Universityof Wisconsin,Madison
A dynamic programming algorithm is presented to locate m variable facilities in relation to
n existing facilities situated on one route. Each of the n existing facilities has a requirement
flow which must be supplied by the variable facilities. The algorithm does the allocation
simultaneously with the location. Computational experience indicates that relatively large
problems may be solved optimally.
1. Introduction
We consider a one-dimensional location-allocation problem that arises when a set
of new facilities is to be added in relation to a set of existing facilities. The structure of
the problem is given as:
(1) A set of n points a>, j = 1, . . . , n, on the Euclidean line called existing facilities,
(2) Known nonnegative weights rj, j = 1,
,
...
n, representing requirements at each
existing facility point, and
(3) A set of m unlocated or variable facilities.
The problem is to determine
(1) the optimal location for each of the m variable facilities and
(2) the optimal allocation of existing facility requirements to the variable facilities
so that all requirements are satisfied.
Let the m unknown facility locations be given by xi, i= 1, ..., m. When the
criterion to be satisfied is the minimization of total weighted distances, the problem
may be stated as
m
minimize R (x, w)
iz
E wij xi-aj
=
i=l
j=1
subject to
m
EWuj
Wuj> O,
i=
ri,
j=
,...,m;j=
,.*
n,(1
,...
n,
where x
= (xl, ..
, xm), and w = (wII, w12, .., W.lm
W21,
Wmn).
W
* All Notes are refereed.
t Processed by Professor Edward Ignall, Departmental Editor for Dynamic Programming and Inventory
Theory; received July 1973, revised December 1973, June 1974, May 1975. This paper has been with the
author 4 months for revision.
$ The author would like to thank Jane Hsu, Ph.D. student in quantitative analysis at the University of
Wisconsin, for assistance with the computational aspects of this research. Use of the University of
Wisconsin Computing Center was made possible through support, in part, from the National Science
Foundation, other United States Government agencies and the Wisconsin Alumni Research Foundation
through the University of Wisconsin Research Committee.
The reviewers of the original version of the paper made many suggestions which improved the notation
and ?4 on computation results.
614
Copyright ? 1976, The Institute of Management Sciences
615
NOTES
During the course of studying an actual location-allocation problem, no efficient
solution methods could be found in the literature to solve one-dimensional problems
for n > 20. A review of the location-allocation literature may be found in [2].
2. Dynamic ProgrammingSolution Procedure
We number the fixed facilities so that aj < a>+ 1, j = 1, ..., n-1. Assume the fixed
facilities are to be allocated to the variable facilities with the fixed facilities considered
in ascending order. When the variable facility coordinates are given, the requirement
at each fixed facility is totally allocated to the nearest variable facility. Let the stages
of the dynamic programming formulation be the number of variable facilities yet to
be located.' The stage number is given by t, t = 1, . . . , m. During the course of the
computations, let S be the index of the lowest numbered unallocated fixed facility,
where m-t + 1 < S < n -t + if t < m, and S = l if t = m. Let At(S) be the set of
all possibly optimal subsets of allocations of fixed facilities to the tth variable facility,
given S. Each such subset is represented by an integer i, S < i < n-t + 1 if t
1I
and i = n otherwise. Let the set X(S, i) be defined by X(S, i) = {S, S + 1, . . .i, i,
and c[X(S, i)] be the minimum weighted-distance cost of one variable facility optimally located to serve the set of points X(S, i). Then
J"(S)
=
min { C[X(S, i)] + fJ* (i + 1)),
(2)
wheref*(S) is the minimum cost solution of supplying fixed facilities S, S + 1, . .. I n,
using m - t + 1 variable facilities.
Let yt(S, i) be the optimal location of the tth variable facility in relation to the set
X(S, i). The value of y1(S, i) is determined by minimizing the function g(z)
The final optimal locations of the m variable facilities are denoted by
sri1z -ij.
y)-
y,
t= 1, .. .,
m. Since g(z) is a convex piecewise
linear function with points of
discontinuity in the first derivatives at the aj values, new facility locations always
coincide with existing facility locations.
3. Example
The locations and requirements of 7 fixed points are given in Table 1. It is required
to locate and allocate three new facilities.
The dynamic programming solution results for the three stages are shown in Tables
2, 3, and 4 where ft(S, i) = c[X(S, i)] + It(i + 1), and i* denotes the value of i
which minimizes ft (S, i).
In the notation of the dynamic programming formulation of the problem the
optimal solution variables are given by:
J3*(S)
= 10.5,
i3
8, i*
i2=6,Y2
3,
4, y*
7,yl = 10.
TABLE 1
Fixed Facility No. (j)
1
2
3
4
5
6
7
Locationa1
1
2
3
5
7
8
10
Requirements (rj)
2
1
2.5
1.5
2.5
4
3
lThe following notation was suggested by referee 1 of the original version of the paper.
616
NOTES
TABLE 2.
t = 1, ii = i = 7.
S
fM(S, 7)
fr(S)
yl(S, 7)
7
6
5
4
3
0
6
8.5
13
25.5
0
6
8.5
13
25.5
10
8
8
8
8
TABLE 3.
t = 2.
\i2
3
2
S
6
5
4
3
2
f2(S, i2)
4
8.5
11.5
12.5
13
14
25.5
5
6
6
9
16
19
0
2.5
7
17
25
]2(S)
i2
Y2(S i2)
6
6
6
4
4
8
8
7or8
3
3
0
2.5
7
11.5
12.5
TABLE 4.
t = 3.
\3
SX
1
1
2
12.5
12.5
f3(S, i3)
3
1 1.5
4
5
10.5
18
f()
10.5
i
4
4. ComputationResults
The algorithm has been programmed in Fortran for the Univac 1108 at the
University of Wisconsin Computing Center. The present version is relatively unsophisticated. The value of y,(S, i) is determined by evaluating g(z) at successive fixed
facility locations in X(S, i) until the minimum point is found. For m = 7, computation
time is about 1 minute for n = 60 and 11 minutes for n = 150. For n = 100, plotting
computation time as a function of m revealed a unimodal curve with a maximum of 6
minutes at approximately m = 33. This maximum can be predicted fairly accurately.
The number of minimizations of the g(z) function at each stage is of order n - m and
there are m stages in the dynamic programming procedure. The time to do one
minimization will be of order n - m since i ranges from S to n - t + I and S ranges
from m - t + 1 to n - t + 1. Total computation time then should be of order
n(n - m)2. For fixed n, the maximum of m(n - m)2 occurs at m = n/3. A more
sophisticated routine which would tend to keep the average minimization search fairly
constant should have the effect of making Km(n - m) a reasonably good predictor of
computation time. More detailed computation results and discussion are found in [1].
5. Summary and Conclusions
For situations that are essentially one-dimensional in nature, the dynamic programming algorithm given here will solve relatively large problems. A final-comment can
be made in relation to fixed costs. When each variable facility has a constant
NOTES
617
operating cost per unit time associated with it, the method can be extended to
determine the optimal number of facilities to be located as well as the optimal
locations and allocations (for a warehouse location problem the operating cost would
correspond to the monthly fixed rental charges). The total optimal weighted distance
cost is computed for successively increasing numbers of variable facilities and added
to the total variable facility operating cost [1].
References
1.
2.
R. L., "One-Dimensional Facility Location-Allocation Using Dynamic Programming," Wisconsin Working Paper 5-75-28, Graduate School of Business, University of Wisconsin-Madison, May,
1975.
AND MORRIS, J. G., "A Computation Procedure for the Exact Solution of Location-Allocation
Problems with Rectangular Distances," Naval Research Logistics Quarterly,Vol. 22, No. 3 (1975), pp.
441-453.
LOVE,