Modification of a heuristic method

Operations Research and Applications : An International Journal (ORAJ), Vol.2, No.2, May 2015
1
MODIFICATION OF A HEURISTIC METHOD
FOR THE CAPACITATED FACILITY
LOCATION PROBLEM: A NOTE
Pritibhushan Sinha
6, A J C Bose Road – Thakurpukur
Kolkata – 700063 West Benagl INDIA
ABSTRACT
We give a modified version of a heuristic, available in the relevant literature, of the capacitated facility
location problem. A numerical experiment is performed to compare the two heuristics. The study would
help to design heuristics for different generalizations of the problem.
KEYWORDS
Capacitated Facility Location Problem, Heuristics, Performance
Modification of a Heuristic Method for the Capacitated Facility Location Problem: A Note
1. INTRODUCTION
We may describe the capacitated facility location problem (CFLP) as follows. In an m × n CFLP
we have m possible locations for setting up some facilities from where an item would be supplied
to n customers or demand points. There is a fixed cost to locate a facility at the i-th location and it
is given by fi. Variable or operational cost of supplying one unit from a facility, set up at i-th
location, to the j-th customer is cij. A facility at i-th location has a capacity of Si and j-th customer
has a demand of Dj. The problem is to find an optimal solution, specifying the number and
locations of facilities and supply quantity from each facility to each customer, minimizing the
total of fixed and variable costs. The CFLP may be written, with the preceding notation, and xij
denoting the supply from the i-th location to the j-th customer, as the following mixed integer
linear program (P1):
 
m
i iiij
n
j ij
m
i yfxcMinimize 111
Subject to,
;,...,1,1 miySx ii
n
j ij  
;...,,1=∀,=∑ 1= njDx j
m
i ij
yi = 1 or 0, ∀i = 1, …, m;
xij ≥ 0, ∀i, ∀j.

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The preceding mathematical programming formulation, or an equivalent one, has been often used
in the literature (see, e.g., [2, 4]) to describe the CFLP. If all Si are very large, larger than equal to
the total demand ∑j Dj, then we have an un-capacitated facility location problem (UFLP) or, as
sometimes called, a simple facility / plant location problem. Both of CFLP and UFLP are NP-
complete problems. There can be many variants and extensions of the CFLP. CFLP is a versatile
optimization model, with a wide scope of application.
Heuristic methods are often the only possible solution approach for the problem, particularly for
large instances. A type of heuristic methods considers improving the solution, beginning with an
initial solution, with dropping open locations, adding a closed location or interchanging a closed
location with an open one. Different variants of such add-drop-interchange (ADI) heuristics have
been discussed by Kuehn & Hamburger [3], Jacobsen [2], Domschke & Drexl [1], Sinha [5] and
others. Sinha [5] has given a heuristic method (subsequently called as Method 1), which in
numerical experiments has shown noteworthy performance. However, some kind of critical
instances where the heuristic methods may fail for the CFLP are described by Sinha [6].
In this article, we consider some modifications of Method 1, in order to increase solution
efficiency. The modified version is also an ADI heuristic, but varies in some steps. Not
possible theoretically, numerical experiments have been performed to compare the two
methods.
This article is organized in the following manner. We describe the modified heuristic
method (Method 2) in the next section. Results of numerical experiments on the method
are given in Section 3. This is followed by concluding remarks.
2. THE MODIFIED HEURISTIC
We shall use, for convenience, the same notation and the manner of description as used for
Method 1. We consider the transportation problem that is obtained if fixed costs are omitted and a
subset of the locations is considered. Let the set of all possible locations be I = {1, 2, ..., m} and
the set of customers be as J = {1, 2, ..., n}. Denote the problem as (P2).
xcMinimize ijJj ijKi  
Subject to,
KiSx i
n
j ij   ,1
JjDx j
m
i ij   ,1
xij ≥ 0, ∀i∊ K, ∀j ∊ J,
with K ⊆ I as a set of some possible locations. A feasible solution of (P2) is also feasible for (P1).
In the heuristic method, which is described next, (P2) is solved to get an initial solution which is
then improved in subsequent iterations. It is assumed that, fi ≥ 0, Si > 0, ∀ i; and, .> ∑∑ j ji i DS
Method 2
A near optimal solution is obtained by solving an instance in the following two phases.

3
Phase 1
Let Ti = minimum {Si, Σ Dj}. An initial solution is obtained with solving problem (P2) with K = I
and costs modified as, ,/
Tfcc iiijij  ∀ i, ∀ j. A CFLP solution, i.e., a solution of (P1), is
obtained by considering the locations which have nonzero supply to at least one demand point, in
the optimal solution obtained for (P2). K is updated accordingly to include only such locations.
Phase 2
The solution obtained in Phase 1 is tried to be improved through adding/ dropping/ interchanging
locations. Let I1 and I2 be the sets of used and unused locations, corresponding to a CFLP
solution, at the beginning of an iteration. The following steps are carried out in an iteration.
Step 1. (i) Try dropping the used locations in the following manner. For ∀ k ∈ I1, the steps, as
given next, are performed.
(a) Initialize as, ri = Si − ∑∈Jj ijx , ∀ i ∈ I1 k; Dk = 0, p = 0.
(b) If p < n, p = p + 1; R = xkp and go to (c). Else, stop.
(c) If R > 0, go to (d), else go to (b).
(d) Let }.0>,∈|min{= 1* rIcc iippi ki If such i*
exists then, z = },min{ *riR , Dk = Dk + z
× ( *c pi − ckp), R = R – z, = ** rr ii − z and go to (c). Else, location k cannot be dropped, stop
setting Dk at a large value.
Let C(drop) = max {fk − =}∈| *1 fID kk k − .*Dk
(ii) Try adding the unused locations in the following manner. For ∀ l ∈ I2, the subsequent steps
are carried out.
(a) Initialize as, R = Sl, Dl = 0, A = ∅.
(b) If R = 0, stop. Else go to (c).
(c) Let ),(,∈,∈,0,|-max{ 1
** jiJjIixccccc ijljijljijji  ∉ A}. If such i*
and j*
exist then,
z = },min{ ** Rx ji and Dl = Dl + z × ( **c ji − clj), A = A + (i*
,j*
), R = R – z, and go to (b). Else, stop.
Let C(add) = max{Dl − =}∈| *
2 DIf ll l − .*f l
(iii) Try to interchange an unused location with a used location, with ∀ k ∈ I1, ∀ l ∈ I2. In
interchanging, first, k is dropped, considering l is also used, as in (ii). Then, if all supply capacity
of l has not been used, it is attempted to be added, as in (i), with the remaining capacity.
Let C(interchange) = max{fk − Dk − =}∈,∈|+ *21 fIIDf kll lk − Dk* − .+ ** Df ll

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Step 2. Calculate }.,,max{= e)interchang(drop)(add)( CCCC If 0>C , add, drop or interchange the
corresponding locations k*
, l*
to update K, and solve (P2) with costs cij. Return to Step 1. Else,
i.e., 0≤C , stop with the current solution, with used locations in K.
Specifically, the modifications made are as described. In Phase 1, in the cost transformation, we
use Ti, instead of Si. This is done to avoid the situation that the effect of fixed cost is
unnecessarily reduced, when a supply location has very high supply capacity. In actual problems,
although, such data may not be found much. Another modification is done in step (ii), (c). Instead
of taking maximum of the costs (cij) of the arcs, we take the maximum of the difference of the
costs (cij – clj). This would more correctly identify the improvement through an add; although,
that does not ensure a better or equally efficient final solution.
3. NUMERICAL EXPERIMENT AND OBSERVATIONS
It is not possible to identify theoretically which method would work better when. We have
conducted a numerical experiment, with random instances, to compare the two methods. Such
instances have been generated in the following way.
i. Get m, n, CR as inputs (Capacity ratio (CR) = Total supply ( i iS ) / Total demand ( j jD ));
ii. cij = u, for i = 1, 2, …, m; j = 1, 2, …, n;
iii. ,100/CR5 umnSi
 rounded to the nearest integer, for i = 1, 2, …, m;
iv. Get total demand A = i iS /CR, rounded to the nearest integer;
v. Dj = 5 + (A − 5n) × ( uj /  
n
l lu1 ), rounded to the nearest integer, for j = 1, 2, …, n−1; Dn = A
−  

1
1
n
j jD . If Dn is less than 5, it is increased to 5; S1 is increased by the same quantity.
Fixed costs are generated in three ways, giving three types of random instances; Type 1: fi = 0.5 ×
Si × u; Type 2: fi = 5.0 + Si × u; Type 3: fi = 25. In the preceding, each u is an independent
random deviate in (0, 1).
Efficiency of a solution is calculated as,
Efficiency = (1 − (Cost of the solution – Cost of an optimal solution) / Cost of an optimal
solution) × 100%.
We have first evaluated the methods with 15 × 100 instances of Type 1, with CR = 4. For these
instances, optimal solutions have been obtained by considering exhaustively all combinations of
open facility locations. In all the 30 randomly generated instances, both the heuristic methods
have given the same solution. In 20 instances, the solution obtained is optimal. Average
Efficiency is 99.4% for both the methods, minimum being 88.9%. This indicates satisfactory
performance of both the methods.
The methods are also compared with the three types of randomly generated instances with size
100 × 250, with CR = 2, 4, 6 for each type. For a particular type and a particular CR, 15 instances
are used. The types of such random instances are summarized in Table 1.

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Table 1: Random Instances (Large) in the Numerical Experiment
As we have large sized instances here, the methods are compared only between those-selves, but
not with an optimal solution. For Type 1, the methods have yielded the same solutions for all
instances. For Type 2, differences have occurred only for two instances, the instances having CR
= 6. For Type 3 (constant fixed cost), there are more differences. For the highest deviation, the
difference reaches 10% (with respect to the better solution; Method 2 being better for the
particular instance). But there is no indication of clear superiority of any of the method. For Type
3 instances, paired t test comparisons show that there is not enough evidence to reject the null
hypothesis of equality of average solution values, at the level of significance 5%. This holds for
all CR values. In Table 2, we give the experimental observations for the 100 × 250 instances. The
numerical experiments have been done with Microsoft Excel, the heuristic methods being
implemented in Visual Basic. A Pentium IV personal computer with 1.86 GHz processor, 1 GB
RAM internal memory, with Windows XP Professional operating system has been used. The
maximum time taken to solve the 100 × 250 instances has been 1092.8 second (s) for Method 1;
whereas this has been 1054.5 s for Method 2 (which has occurred for the same instance) (it may
be possible to improve the time requirement by better implementation of the methods). Time
requirements are near and no method has less time requirement for all the instances uniformly.
4. CONCLUDING REMARKS
We have considered some modifications of an add-drop-interchange heuristic method, presented
in the relevant literature earlier. Apparently, such modifications should further improve the
performance of the method. Numerical experiments with randomly generated instances have been
done to compare the two methods. Observations suggest that, both the methods have satisfactory
performance and suitable for practical applications. There are sometimes differences, but there is
no indication of clear superiority of any of the method. Any one of the methods may be used; or,
to increase solution efficiency both the methods may be used, if possible.
We feel that, the numerical experiment involving the two heuristic methods has given some
insights about the CFLP. Such insights and the methods may be of help to obtain more efficient
methods for the many possible generalizations of the CFLP.

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Table 2: Comparison of the Two Heuristic Methods
Obs. No Instance
Type
CR Number of
Instances
Method 1
Better
Number
of
Instances
Method 2
Better
Average Ratio of
Solution Costs
(Method1/Method
2)
1 Type 1 2 0 0 100%
2 ,, 4 0 0 100%
3 ,, 6 0 0 100%
4 Type 2 2 0 0 100%
5 ,, 4 0 1 100.04%
6 ,, 6 0 1 100.08%
7 Type 3 2 4 5 100.41%
8 ,, 4 4 8 100.41%
9 ,, 6 4 1 99.94%
REFERENCES
[1] W. Domschke, A. Drexl, “ADD-heuristics’ starting procedures for capacitated plant location
problems”, European Journal of Operational Research 21 (1985) 47-53.
[2] S. K. Jacobsen, “Heuristics for the capacitated plant location problem”, European Journal of
Operational Research 2 (1983) 253-261.
[3] A. A. Kuehn, M. J. Hamburger, “A Heuristic Program for Locating Warehouses”, Management
Science 9 (9) (1963) 643-666.
[4] M. Sun,.“A tabu search heuristic procedure for the capacitated facility location problem”, Journal of
Heuristics 18(1) (2012) 91-118.
[5] P. Sinha, “Performance of an add-drop-interchange heuristic for the capacitated facility location
problem”, International Journal of Applied Management Science 1(4) (2009) 388-400.
[6] P. Sinha, “Observations on heuristic methods for the capacitated facility location problem”, Opsearch,
49(1) (2012) 86-93.

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