An Improved Long-Run Model for Multiple Warehouse Location
Author(s): Dennis J. Sweeney and Ronad L. Tatham
Source: Management Science, Vol. 22, No. 7 (Mar., 1976), pp. 748-758
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MANAGEMENT SCIENCE
Vol. 22, No. 7, March, 1976
Printed in USA.
AN IMPROVED LONG-RUN MODEL FOR
MULTIPLE WAREHOUSE LOCATION*
DENNIS J. SWEENEY
AND
RONAD L. TATHAMt
University of Cincinnati
This paper proposes an improved model for solving the long-run multiple warehouse
location problem. The approach used provides a synthesis of a mixed integer programming
formulation for the single-period warehouse location model with a dynamic programming
procedure for finding the optimal sequence of configurations over multiple periods. We show
that only the Rt best rank order solutions in any single period need be considered as
candidates for inclusion in the optimal multi-period solution. Thus the computational
feasibility of the dynamic programming procedure is enhanced by restricting the state space
to these Rt best solutions. Computational results on the ranking procedure are presented, and
a problem involving two plants, five warehouses, 15 customer zones, and five periods is solved
to illustrate the application of the method.
1. Introduction
A problem commonly faced in the management of distribution systems is that of
determining a set of geographical warehouse locations such that demand is satisfied
and a satisfactory level of customer service is maintained with a minimum total
distribution cost over a relatively long planning period with varying levels of demand
over the period. Various criteria have been suggested for a satisfactory solution
procedure to the problem posed [9].
Apparently a good solution procedure for locating warehouses should meet these
objectives:
1. be capable of evaluating a reasonable number of possible warehouse configurations to determine how many there should be and where they should be located;
2. be capable of evaluating warehouse configurations over time and hence indicating when it is desirable to change configurations in response to shifting demand
patterns, supply costs, etc.;
3. allow for interdependence in costs among the warehouse sites during a single
period and across multiple periods; i.e., the sites cannot be selected independently of
one another nor of the sites chosen in other planning periods;
4. cope with nonlinearities due to both the fixed costs associated with alternative
configurations and the variable costs associated with the system throughput;
5. be computationally feasible and efficient.
Both static and dynamic procedures have been used to find solutions meeting some,
but not all, of the above criteria. Static warehouse location models deal with the single
period problem of finding the optimal warehouse configuration at a particular point
in time [3], [4], [8], [10], [14]. These models have experienced varying degrees of
success in meeting objectives 1, 3, 4 and 5 above, but simply ignore 2. For this reason
they may be suboptimal over a longer planning horizon. Even in the short run, these
models ignore the interdependence with the previous planning period. It is implicitly
assumed that the existing configuration can be immediately redesigned to implement
the optimal solution. However, in many practical problems, by the time the static
solution can be implemented it is no longer optimal.
* Processed by Professor Warren H. Hausman, Departmental Editor for Logistics; received September
1973; revised March 1974 and January 1975. This paper has been with the authors 7 months for revision.
t This research was supported by a grant from the George A. Ramlose Foundation, Inc.
748
Copyright C) 1976, The Institute of Management Sciences
IMPROVED
LONG-RUN
MODEL FOR MULTIPLE WAREHOUSE
LOCATION
749
One approach to finding a solution that satisfies criteria 2 and 3 is a multiperiod
model proposed by Ballou [2] which employs a heuristic solution procedure based on
dynamic programming.' In this procedure, the number of alternative locations considered in each period is equal to T, where T is the number of periods in the planning
horizon. The T alternatives considered are the optimal solutions to the static
warehouse location problem (SWLP) for each of the T periods. These T alternatives
are generated by any of the static models referenced earlier, and are evaluated in each
of the other periods providing T distinct alternative configurations for each period.
One then chooses by dynamic programming that sequence from these T configurations which minimizes the total cost of operation and relocation. However, Ballou's
approach is limited since it can only guarantee suboptimal solutions. This is because
there is no guarantee that the long-run optimal solution will be composed of a
sequence of configurations derived only from static optimal solutions. For example, a
configuration which yielded the second best static solution in each planning period
could quite possibly yield the long-run optimal location since no relocation costs
would be necessary over the planning horizon.
Lodish has pointed out [12], for large practical problems, it is not computationally
feasible to consider in a dynamic programming procedure all of the existing
warehouse configurations in each period. Ballou's approach overcomes this computational difficulty by the heuristic approach of limiting the number of configurations
considered in each period to the optimal static solutions in each of the T periods. In
the next section, we show how an optimal multiperiod solution can be obtained by
limiting the number of alternatives considered in each period to the R, best rank order
(ranked from lowest to highest) static configurations for that period. This preserves
the computational feasibility of the dynamic programming approach without resorting
to suboptimal solutions. In the following section we show how these R, best solutions
can be found as an extension of the solution procedure for the static model.
2. The Multiperiod Model
In Table 1 each column represents a stage (period) in the process and the entries in
these columns represent the values the state variable (the single-period warehouse
configuration) may take on. The values of these static solutions in each period
(column entries) are presented in rank order. Given that these entries are the only
alternative solutions considered (we show shortly that it is not necessary to consider
any others) the optimal long-run solution may be found by using dynamic programming to find the minimum cost path through the matrix, taking into consideration at
each stage the costs of moving from one warehouse configuration to any other.
The computational feasibility of this procedure is dependent on the number of
stages and the number of alternative solutions that must be considered at each stage
(the number of values the state variable takes on). The computational time required
by any dynamic programming procedure increases linearly with the number of stages
and exponentially with the number of states. Therefore the number of configurations
to be considered at each stage is the most critical factor from a computational point
of view.2 Thus, it is to our advantage to minimize the number of static solutions it is
necessary to consider at each stage. We now show how this can be done.
Let vtr denote the value of the rth best static configuration in period t. Then
vinf = T_ v is the sum of the minimum cost warehouse configurations for the entire
l Another example of a multiperiod model is the LREP simulator devised by Bowersox, et al. [6]. This
model is heuristic and the portion of it dealing with the warehouse problem (selection, location, flows)
provides only a heuristically satisfactory solution.
2 A problem with 10 stages and 100 alternative configurations can be solved in 20 seconds on an IBM
7094. A problem with 50 stages and 1000 configurations would require approximately 2.5 hours on the same
machine [13, p. 77].
750
DENNIS J. SWEENEY AND RONALD L. TATHAM
TABLE 1
R, Best Solutions to the Static WarehouseLocation Problem by
Planning Period
Rank Order
Solutions
(low to
high cost)
Period
1
2
3 ...
I
Viil
V21
V31 . ..
2
V 12
V22
V32
V13
*
*
*
V33---.
3
T
VT
V34...
v2R2
.
VTRT
VIR,
R,
V3R3
planning horizon. (v1l is the least cost static solution in period t). Since no relocation
expenses are considered, vinf is a lower bound on the value of the optimal multiperiod
solution. If the same configuration was optimal for each period, this would obviously
be the optimal multiperiod solution.
Let v* be an upper bound corresponding to any feasible solution to the multiperiod
problem. The following theorem can be used to determine how many static solutions
it is necessary to rank in each period.
Let K= v* -v inf. Also let R, be such that vR v,- l K and
K. In period t, no static solution with value vTr may become part of an
optimal multiperiodsolution if r > Rt.
THEOREM.
Vt R +I -Vt
I>
PROOF. Suppose r > RT.The value of the best multiperiod solution containing the
rth best configuration in year t is bounded below by Et 7, TvT + vTr.Now,
T
v
tt
+V
I
+ vm - VT, >
v
t=
vinf +
K= v*.
I
We see from the theorem that K is the maximum possible improvement that can be
made over the solution corresponding to v*. Thus it is only necessary to consider the
R, best static solutions in each period for possible inclusion in an optimal multiperiod
solution. This results in a state space reduction and makes dynamic programming a
feasible solution procedure for practical sized problems.
The values of the R, best configurations for each period are the entries in Table 1. It
is important to note that a particular static solution may be considered in one period,
and not another, since in general a particular warehouse configuration will not have
the same rank in any two periods and the same number of alternatives are not
considered in each period; i.e., Ri is not necessarily equal to Rj.
Since the number of static solutions it is necessary to rank in each period depends
upon K, it is desirable to have a good upper bound, v*, available. Any feasible
solution, such as maintaining the current configuration over the entire planning
period, can be used to determine an initial value for v*. Better upper bounds can be
generated as the solution procedure progresses. We recommend the following
approach. Using the initial value for v*, rank order the P1 best solutions in each
period where F, S R,. This may be some preset number of solutions, say 50, or we
IMPROVED
MODEL FOR MULTIPLE
LONG-RUN
WAREHOUSE
LOCATION
751
t - vt, 1 > constant. In the former case P, will be
may rank order solutions until v,,
fixed across periods and in the latter P, will vary. A new (and hopefully better) upper
bound can then be generated by using dynamic programming to find the multiperiod
solution considering as alternatives in each period the P, best static solutions. This
new upper bound is clearly a feasible solution and if it consists of a sequence of
optimal static solutions it is the Ballou solution [2]. Using the new value for v* we can
- vt, > K for all periods we are finished, and
recompute K = v* -vinf. If now v
v* is the value of the optimal multiperiod solution. If not, more solutions must be
rankorderedin those periodswherevt P - vt, I < K.
Ideally one would initially choose P, so that exactly the right number of static
solutions was rank ordered the first time through. Unfortunately the right number, P,
is data dependent and there is no way to predict this in advance. The most we can
hope for is that knowledge of the firm's distribution system will allow a judicious
choice. Since the maximum possible improvement from further ranking is given by
I = max (K-Vt
pt + vt 1, 0) t E ( 1, 2, . ., T}, one might choose to terminate if I
was sufficiently small. The value of I is a measure of the maximum possible
opportunity loss associated with implementing the solution associated with v*.
3. Finding the R, Best Solutions
Since our dynamic programming procedure depends critically on finding the R, best
static solutions for the short-run problem, it is important that the static warehouse
location model used be such that this information can easily be obtained. We show
here how this information can be obtained when a mixed integer programming
formulation of the static problem is employed. For notational convenience we restrict
our attention here to a static model for a single commodity class. For an extension to
multiple commodity classes we refer the reader to the excellent article by Geoffrion
and Graves [8].
The static warehouse location problem may be formulated as the following mixed
integer linear program.
I
n
mi>
z-O, 1
(SWLP)
s Et
J
=1
i=1
I
J
1=1
j=1
EY
i=
m
M
E
J
(A + Bjm+ C,)xqjm+
xijm = Qm
XJ
mj-
E
J1
Fjz
1, 25,... ., M
where:
xijm =the quantity of goods shipped from factory i (i = 1, ... , I) through
warehouse ] (j = 1, . . ., J) to customer m (m = 1, . .. , M),
Au = per unit cost of shipping from factory i to warehouse j,
B;m= per unit cost of shipping from warehouse j to customer m,
Cj =per unit variable cost of storing and handling goods at warehouse j,
Fj =fixed cost of operating warehouse j,
Qm = the demand for customer m,
Si = the capacity at factory i, and
Wj = the capacity at warehouse j.
z=0
=
if warehouse j closed
1 if warehouse j open.
752
DENNIS
J. SWEENEY AND RONALD
L. TATHAM
Here we have formulated the static warehouse location problem as one of minimizing total distribution costs subject to the usual capacity restrictions on the warehouses
and factories, and the constraint that demand must be satisfied. Note that in this
model the nonlinearities due to the fixed costs have been included but that we have
assumed that transportation and variable warehouse costs are linear. Also note as in
[8] that additional linear configuration constraints involving the binary z variables
may be included. In addition, the model may be expanded to consider alternative
plant locations by introducing another binary variable hi. The second constraint then
becomes
M
J
i =- 1, 2. ... I
xim <_ Sihi
I
j=I m=1
and an additional term must be added to the objective function to reflect the fixed
costs of the various plant locations. Customer service constraints may also be
incorporated by adding a constraint of the form
I
M
J
i=1j=1
m=1
d
Qm< D
amXu,,n/
where damis the distance from warehouse j to customer m and D is the firm's desired
bound on average delivery distance. The addition of these constraints requires a slight
modification of the static solution procedure we are about to present (see [7], [11]).
Benders partitioning procedure [5], [7], [8], [11] has been applied to solve variations
of the above mixed integer formulation. We have used the following specialized
version of Benders partitioning procedure to obtain the best solution to problem
(SWLP) and then applied a simple modification to obtain higher rank order solutions.
Step 0. (Iteration 0.) Pick an arbitrary feasible configuration zo (i.e., such that
constraint (2) below is satisfied). Set the lower bound LB equal to - cc and go to step
2.
Step 1. (Iteration N.) Solve problem (IP) below for a (possibly) new configuration.
min v
y; z=0,
(FQ- W)Vn)zjE
s.t. y>
(IP)
1I
I
J
i=l
M
QmV n =O,
Siu/-
,...,N-
1
(1)
m=1
i-l
WjZ >
(2)
Qm
m=l
j=l
the optimal solution. Set the lower bound, LB, equal to yN and
Denote by (yN, ZN)
go to step 2.
Step 2. Solve the primal linear program (LP) below obtained by fixing z at zN and
record the values of all the dual variables tjN
, Vm. These will be used to generate a
new constraint of form (1) above for problem (IP). Assuming I =1Si> Em= Qm,
(LP) must be feasible because of constraint (2) on (IP).
I
mini
1E1
s.t.
=1
I
E
l
x
(LP)
J
JE
]=1
J
(Ai + Bjm + Cj)XUm
In=
x
E
1
-Qm
m = 1, 2, .
,M
-l
E EY
..
<
xijm
Si
i = 1, 2,.
,I
m=lI
M
j=l
I
'i
i=
M
E1
EY'
1
rn=1I
X;m ?W;Vz;N
j-1,
2, ...,J.
IMPROVED LONG-RUN MODEL FOR MULTIPLE WAREHOUSE LOCATION
753
Denote by (xN} the optimal solution and r(xN) its value. Set the upper bound
UB = r(xN) + EJ 1FizN and go to step 3.
Step 3. If UB - e < LB, stop. The optimal solution is given by (zN, x N). If not, set
N= N + 1 and go to step 1.
Once the optimal static solution has been found using the above algorithm, the
second best solution can be found by adding a constraint to (IP), resetting UB = 00
and continuing with step 1. The constraint added is one developed by Balas and
Jeroslow [1] for making canonical cuts on the unit hypercube.
E z;
j (-O,
-
(3)
E N(q)-1
z; <
j (-C,
where
01 = (j
Zj= 1 in the best solution);
Cl = {j
Zj=0 in the best solution);
N(01)
= number of elements in 01.
This cut, see [1], will make infeasible the best solution but no others. The optimal
solution after constraint (3) has been added is the second best solution. Another
constraint of the form (3) is then added, and so on, until the R, best solutions have
been obtained.
Once 'the optimal solution has been obtained successive solutions can be ranked in
considerably less time. We have obtained computational results on an IBM 370/168
for sixteen years of data on problems involving two plants, five warehouse locations,
and 15 customer zones. These results are summarized in Table 2. An average of 6.9
TABLE 2
ComputationalResultsfor Sixteen Years of Data for Two Plants, Five WarehouseLocations, and
Fifteen CustomerZones
Optimal Solution
No. of Benders' CPU Seconds
per Solution
Iterations
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
15
17
6
5
4
5
7
3
8
8
3
7
5
7
3
7
34.5
39.6
12.0
10.1
16.7
18.3
11.7
5.1
16.1
12.5
4.2
12.7
10.7
15.4
7.2
16.1
Averages
6.9
15.2
Higher Ranked Solutions
Average No.
No. of
of Benders' Average CPU
Seconds per
Solutions
Iterations
Ranked
per solution
Solution
19
19
12
12
12
12
12
12
12
12
12
11
12
12
19
19
1.5
1.2
1.8
1.8
1.8
1.6
2.0
2.3
1.9
2.1
2.4
2.2
1.8
1.6
1.9
1.8
3.4
2.8
3.7
3.7
7.6
3.7
3.3
3.8
3.9
3.3
3.4
4.3
3.9
3.7
4.5
4.2
1.9
3.9
754
DENNIS
J. SWEENEY AND RONALD
L. TATHAM
Benders iterations, each involving the solution of (IP) and (LP) once, were required to
find the 16 optimal solutions. An average of 1.9 Benders iterations were required in
going from the rth to the (r + 1) best solution for 219 rankings over the sixteen years
of data. Approximately 28% of the computational work required to find an optimal
solution was required to rank solutions. The amount of computer execution time
needed is obviously a function of the efficiency of the subroutines used to solve (IP)
and (LP). We do not consider our code to be efficient, but for the 16 years of data the
average CPU time to find the optimal solution was 15.2 seconds. The average CPU
time for ranking was 3.9 seconds per solution.
Geoffrion and Graves [8] have reported computer execution times ranging up to
approximately three minutes on an IBM 360/91 for finding the optimal solution to a
quite large multicommodity problem using Benders decomposition. Based on our
results (approximately 28% computational effort), it would appear that a reasonable
number of configurations can be ranked using our approach even for very large
problems. We note also that the ability to rank order a reasonable number of
alternatives is often of interest even when it is only desired to solve the static problem.
4. An Illustrative Application
We demonstrate the application of our approach on a problem with two plants, five
possible warehouse locations, 15 customer zones and a five year planning horizon.
The demand forecast, together with other relevant data, are contained in Tables 3, 4
and 5. In this example problem the shipping costs from plant to warehouse, Aij, were
taken to be 8.006 per mile traveled and the shipping costs from warehouse to
customer, Bj,,?,were taken to be $.01 per mile traveled. The variable costs of storing
and handling goods, Cj, were taken to be constant and equal for all warehouses and
thus were not included in the numerical calculations. During the first two periods,
plants are located in Indianapolis and Akron, but because of plans currently in
progress, plant one (the Akron plant) will be moved to Toledo at the beginning of
period 3.
We note that the demand forecast indicates that the company plans to begin
marketing its product in the northern and western regions at the beginning of period
2. Throughout the planning period, demand is expected to increase slightly in the
TABLE 3
Demand Forecasts
Customer Zone
1
2
Period
3
4
5
Milwaukee, Wisc.
Chicago, Ill.
Bloomington, Ill.
Detroit, Mich.
Lansing, Mich.
Kalamazoo, Mich.
Cleveland, Ohio
Columbus, Ohio
Toledo, Ohio
Cincinnati, Ohio
Dayton, Ohio
Akron, Ohio
Benton Harbor, Mich.
Indianapolis, Ind.
Ft. Wayne, Ind.
0.0
0.0
0.0
4,459,800
353,426
283,870
2,756,360
1,142,360
1,264,290
1,445,380
881,624
859,025
0.0
1,166,790
388,384
433,235
2,145,070
35,074
4,459,800
353,426
283,870
2,756,360
1,142,360
1,264,290
1,589,920
969,786
859,025
62,668
1,283,470
388,384
866,471
4,290,140
70,148
4,459,800
353,426
283,870
2,756,360
1,142,360
1,264,290
1,734,450
1,057,950
859,025
125,337
1,400,150
388,384
1,299,710
6,435,200
105,223
4,459,800
353,426
283,870
2,756,360
1,142,360
1,264,290
1,878,990
1,146,110
859,025
188,005
1,156,830
388,384
1,732,940
8,580,270
140,297
4,459,800
353,426
283,870
2,756,360
1,142,360
1,264,290
2,023,530
1,234,270
859,025
250,673
1,633,510
388,384
IMPROVED
LONG-RUN
MODEL FOR MULTIPLE WAREHOUSE
755
LOCATION
TABLE 4
Warehouseand Plant Capacities (lbs. x 1000)
Warehouse
Location
No.
1
2
3
4
5
Toledo
Chicago
Detroit
Cleveland
Cincinnati
Plant
Location
1
2
Period
3
4
5
10,000
7,000
8,500
7,500
12,000
10,000
7,000
8,500
7,500
12,000
10,000
10,000
8,500
7,500
12,000
10,000
10,000
8,500
7,500
12,000
10,000
10,000
8,500
7,500
12,000
No.
Akron
1
12,000
12,000
Indianapolis
2
18,000
18,000
Toledo
3
-
18,000
18,000
18,000
12,000
12,000
12,000
TABLE 5
Fixed Costsfor Warehouses
Warehouse
Location
No./Year
Toledo
Chicago
Detroit
Cleveland
Cincinnati
Fixed Cost (X1000)
1
2
3
4
5
1
2
3
4
5
300
300
300
320
360
300
300
300
320
360
300
300
300
320
360
300
400
300
320
360
300
400
300
320
360
southern regions and heavily in the new market areas of Chicago and Milwaukee. It is
precisely this type of shifting demand pattern that makes a multiperiod model such as
that proposed here appropriate.
In Table 6 below we present results obtained by using the algorithm of the previous
section to rank order static solutions for each of the five periods. Initially we set
Pt = 10, t
=
1, 2,,3, 4, 5; and obtained the-ten best static solutions in each year.
Using these ten best static solutions and relocation costs of $100,000 for closing a
warehouse site and S200,000 for opening a new site as input to our dynamic
programming model yields the following solution.
Year
Configuration
1
(1,5)
2
3
4
(1, 5)
(1,2,5)
(1,2,5)
5
Total Cost
(1,2,5)
$10,569,000.
It remains to verify the optimality of the solution. Summing the minimum cost
static solutions for each period yields vinf= $10,210,000. An upper bound is given by
and it is only necessary to rank
v* = S10,569,000, thus K = v* - vinf=S359,000
further configurations in years where v1 1-Vt, l < $359,000. It was necessary to rank
756
DENNIS J. SWEENEY AND RONALD L. TATHAM
TABLE 6
Rt Best Solutions to the Static WarehouseLocation Problemfor the Five Period Illustration
Period
Rank
Order
Solutions
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
* Loc.=
5
4
3
2
1
Loc.*
Costt
Loc.
Cost
Loc.
Cost
Loc.
Cost
Loc.
Cost
(4, 5)
(3, 4)
(1, 4)
(3, 4, 5)
(3, 5)
(1, 5)
(1, 4, 5)
(1, 3, 4)
(1, 3)
(2, 3, 4)
(1,3,5)
(1, 3, 4, 5)
(2, 4, 5)
(1, 2, 4)
(2, 3, 4, 5)
1564
1575
1576
1586
1644
1645
1650
1769
1786
1820
1827
1847
1853
1865
1886
(4, 5)
(3, 4, 5)
(2,3, 4)
(2, 4, 5)
(2, 3, 4, 5)
(1, 2, 4)
(1, 4, 5)
(1, 5)
(3, 5)
(2, 3, 5)
(1,2,5)
(1, 2, 4, 5)
(1, 3, 5)
(1, 3, 4)
(1, 2, 3, 4)
(1, 3, 4, 5)
(1, 2, 3)
(1, 3)
(1, 2, 3, 5)
(1, 2, 3, 4, 5)
2015
2022
2029
2030
2063
2071
2078
2094
2095
2112
2120
2126
2245
2257
2264
2278
2282
2283
2304
2324
(1, 2, 5)
(2, 3, 5)
(1, 2,4)
(1, 2, 3)
(1, 2, 4, 5)
(1, 2, 3, 5)
(1, 5)
(2, 3, 4, 5)
(2, 3, 4)
(1, 4, 5)
1950
2108
2128
2133
2155
2160
2199
2206
2296
2323
(1, 2, 5)
(2, 3, 5)
(1, 2,4, 5)
(1, 2, 3, 5)
(1, 2, 4)
(1, 2, 3)
(2, 3, 4, 5)
(2, 3, 4)
(1, 2, 3, 4)
(2, 4, 5)
2227
2384
2431
2436
2439
2440
2483
2611
2670
2671
(1, 2, 5)
(1, 2, 4, 5)
(2,3, 5)
(1, 2, 3, 5)
(2,;3, 4, 5)
(1, 2, 3)
(1, 2, 4)
(1, 2, 3, 4, 5)
(1, 2, 3, 4)
(2, 4, 5)
2454
2623
2628
2629
2683
2744
2821
2878
2929
2930
Locations or warehouses in warehouse configuration.
t Cost = $ x 1000.
five more solutions in period 1 and ten more solutions in period 2 in order to obtain
1- v1 > S359,000 and V221 - V2, > S359,000. The dynamic programming solution considering these additional configurations yielded the previously obtained
optimal sequence of configurations, verifying optimality.
We note here another possibly more important situation in which our dynamic
model is appropriate. That is when the existing configuration is suboptimal and the
company would like to plan relocations over a relatively long horizon to improve the
efficiency of the distribution system. The dynamic programming procedure we have
employed yields the optimal sequence of configurations given any initial configuration. For example, our results indicate that if the existing configuration was (1, 4) the
following different sequence of configurations would be optimal.
Year
Configuration
1
2
3
4
5
Total Cost
(1,4)
(1,2,4)
(1,2,5)
(1,2,5)
(1,2,5)
810,778,000.
IMPROVED
LONG-RUN
MODEL FOR MULTIPLE WAREHOUSE
LOCATION
757
5. Discussion
The solution procedure we have proposed provides a means for using the output of
a static warehouse location model as input to a dynamic programming procedure for
finding a long-run optimal multiple warehouse configuration. As we have shown, this
long-run optimum will consist of some sequence of configurations selected from the R,
best solutions to each of the single period static problems. When compared to the
solution reached through Ballou's procedure, our solution is better by S 125,000.
Ballou's procedure would result in the firm remaining in warehouse configuration
(1, 2, 5) for all the five periods at a cost of S10,694,000. Our solution would cause the
firm to place warehouses at locations 1 and 5 (a configuration that is not a static
optimal solution in any period) for the first two periods and to add a warehouse at
location 2 for the last three periods at a total cost of S10,569,000. If the firm were to
employ the optimal static configuration for each period the cost would be higher than
either Ballou's or our solution, i.e., S10,710,000.
The reader will note that the solution we obtained after ranking the 10 best static
solutions in each period was optimal. However we had to rank an additional five
solutions in period one and an additional ten in period two in order to prove
optimality. When it is necessary to rank a large number of alternative solutions it may
be wise to adopt the heuristic of ranking only a predetermined number of static
solutions, P , in each period and if I = max{K - vt, p + vt, 1, 0} t E { 1, 2, ... , T} is
not too large terminate calculations. The determination of "too large" must be made
subjectively. Of course, one can always generate more rank order solutions and
continue the calculations to prove optimality if desired.
Several assumptions were made to clarify the illustration but not all are necessary
for the solution procedure. The static model formulated in this paper has been
illustrated for a single product or a set of products with similar (if not equal)
transportation and warehousing costs. In some situations (such as the case of a
dominant product or a totally private transportation system) this assumption is not far
from reality. However, our solution procedure can also be used in conjunction with
multiple product models such as that demonstrated by Geoffrion and Graves [8].
Another assumption which is implicit in our model is that a forecast of the expense
associated with moving from one warehouse configuration to another is available.
Obviously, these costs significantly influence the sequence of configurations that our
dynamic programming procedure will sele,ct as optimal for the multiperiod problem.
One practical approach is to assume a constant cost per warehouse added or deleted.
This was the approach used in our illustration. If this assumption is not palatable,
then separate forecasts of the cost of opening and closing each warehouse in each
period must be prepared.
6. Conclusions
The procedure presented in this paper provides a synthesis of the static and
dynamic approaches to the warehouse location problem into a computationally
efficient algorithm for finding an optimal solution to the multiperiod problem. The
computational feasibility of the approach depends on limiting the number of static
configurations it is necessary to consider in each period. We have shown how this can
be done by presenting a procedure for rank ordering the R, best solutions in each
period. These R, best static solutions will often be of interest to management for their
own sake when criteria other than those incorporated in the static model must be
considered in making the final decision on which solution to implement.
758
DENNIS J. SWEENEY AND RONALD L. TATHAM
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