In this work we present a local search algorithm for the single row facility location problem. In contrast to other local search algorithms for the problem, our algorithm uses an exponential neighborhood structure. Our computations indicate that our local search algorithm generates solutions to benchmark instances of the problem whose costs are on average within 2% of costs of optimal solutions within reasonable execution time.

Ghosh, Diptesh

English

Research Papers in Economics

INDIAN INSTITUTE OF MANAGEMENT
AHMEDABAD  INDIA
Research and Publications
An Exponential Neighborhood Local Search Algorithm
for the Single Row Facility Location Problem
Diptesh Ghosh
W.P. No. 2011-08-01
August 2011
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INDIAN INSTITUTE OF MANAGEMENT
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INDIA
W.P. No. 2011-08-01 Page No. 1IIMA  INDIA
Research and Publications
An Exponential Neighborhood Local Search Algorithm
for the Single Row Facility Location Problem
Diptesh Ghosh
Abstract
In this work we present a local search algorithm for the single row facility location problem.
In contrast to other local search algorithms for the problem, our algorithm uses an exponential
neighborhood structure. Our computations indicate that our local search algorithm generates
solutions to benchmark instances of the problem whose costs are on average within 2% of costs
of optimal solutions within reasonable execution time.
1 Introduction
The single row facility location problem (SRFLP) is dened in the following manner. We are given
a set F = f1;2;:::;ng of n facilities. Each facility i 2 F has a length li. We are also given
a transmission intensity value cij for each pair fi, jg of facilities i;j 2 F. The objective of the
SRFLP is to arrange the facilities in a line, such that the sum of the transmission costs between
all pairs of facilities in F is minimized. The transmission cost between a given pair of facilities is
the product of the corresponding transmission intensity and the distance between the centroids of
the two facilities. Hence a solution to the SRFLP is a permutation of the facilities in F. The size
of a SRFLP instance is the cardinality of the set F. The SRFLP nds practical applications in
physical arrangements of facilities (Simmons, 1969), in computer systems (Picard and Queyranne,
1981) and in automated guided vehicle systems (Heragu and Kusiak, 1988). The problem is known
to be NP-hard (Beghin-Picavet and Hansen, 1982).
Love and Wong (1976) proposed the rst integer programming formulation for the SRFLP.
Subsequently Heragu and Kusiak (1991) and Amaral (2006) proposed other integer programming
formulations for the problem. Solution methods that have been used to obtain optimal solutions
to the SRFLP include branch and bound methods (Simmons, 1969, 1971), dynamic programming
(Picard and Queyranne, 1981), mixed integer linear programming (Amaral, 2006, 2008; Heragu and
Kusiak, 1991; Love and Wong, 1976), and semi-denite programming (Anjos et al., 2005; Anjos and
Vannelli, 2008; Anjos and Yen, 2009; Hungerl ander and Rendl, 2011). Most of these methods are
suitable for SRFLP instances with 40 facilities or less, but are prohibitively expensive for larger
problems. Lower bounds for the problem have also been reported in the literature (see, e.g. Anjos
et al., 2005; Amaral, 2009; Anjos and Yen, 2009; Hungerl ander and Rendl, 2011). Amaral and
Letchford (2011) provides a comprehensive polyhedral study of the SRFLP, deriving ve exponential
sized classes of valid inequalities with conditions under which they become facet-inducing. It also
provides a branch and cut algorithm for the SRFLP. Hungerl ander and Rendl (2011) presents a
thorough computational study of the SRFLP which includes improved lower and upper bounds to
the problem.
Since the SRFLP is NP-hard, the literature on the problem also includes heuristic approaches to
solve large sized SRFLP instances. Kumar et al. (1995) reports a greedy heuristic for the problem.
Interestingly, the heuristic does not take the lengths of the facilities into account during its execution.
Among improvement heuristics, the methods reported in the literature include simulated annealing
W.P. No. 2011-08-01 Page No. 2IIMA  INDIA
Research and Publications
(Heragu and Alfa, 1992), tabu search (Samarghandi and Eshghi, 2010), genetic algorithm (Datta
et al., 2011), scatter search (Kumar et al., 2008), ant colony algorithm (Solimanpur et al., 2005),
and particle swarm algorithm (Samarghandi et al., 2010). Among heuristics, the genetic algorithm
described in Datta et al. (2011) generates the best solutions for benchmark SRFLP instances.
In this work, we propose an exponential neighborhood structure for the SRFLP. We also provide
a heuristic to search this neighborhood to obtain good quality solutions. We use this neighborhood
structure in a simple local search algorithm and apply the algorithm on large sized benchmark
instances of the SRFLP with sizes ranging from 60 to 100. The remainder of the paper is organized
as follows. In Section 2 we describe the local search algorithm that we present in this work, and in
Section 3 we present the results of our computational experiments with this algorithm. We conclude
the paper in Section 4 with a summary of our contribution and directions for future work.
2 The Local Search Algorithm
A generic local search algorithm starts with an initial solution as its input, and then proceeds to
iteratively improve the solution by searching among solutions genereated using a pre-dened set of
operations (called moves) until it reaches a stage in which it cannot improve the solution further. It
then outputs the improved solution and terminates. The set of moves used by local search denes
a neighborhood structure in the space of all feasible solutions. The performance of a local search
algorithm critically depends on the neighborhood it searches, and the initial solution which it takes
as input. The neighborhood which is commonly used in the literature to solve SRFLP instances is
the 2-opt neighborhood (see, e.g., Samarghandi and Eshghi, 2010). In this neighborhood, a move
is an operation which swaps the positions of two facilities in a given solution. All n(n + 1)=2
solutions that are generated from a given solution by swapping the positions of two facilities form
the neighborhood of the given solution.
In our algorithm we propose an exponential neighborhood called the k-FIX neighborhood, where
k is pre-specied. The set of solutions which are neighbors to a solution, say , in this neighborhood
is generated using a two step process. In the rst step, we create a list of all n!=(n   k)! subsets Fs
of F with cardinality k. In the second step, for each of the subsets Fs that we created in the rst
step, we generate all solutions in which the positions of the k elements in Fs relative to each other
are the same as that in . Notice that the sets of solutions obtained in the second step from any
two dierent subets obtained in the rst step have solutions in common. All the solutions that are
generated in the second step form the neighborhood of the solution .
Clearly, the size of the k-FIX neighborhood is exponential in the size of the instance. Hence in
any reasonable sized SRFLP instance, it is impractical to evaluate all neighbors in order to identify
the best solution in the neighborhood . In this paper, for every subset Fs that we generate in the
rst step, we use a greedy algorithm, which we call GREEDY(;Fs) to nd a good quality solution
among those generated from Fs in the second step. GREEDY(;Fs) is dierent from the algorithm
proposed in Kumar et al. (1995) in that it takes the lengths of the facilities into account. We then
choose the best among the n!=(n   k)! solutions thus generated as an approximation of the best
solution in the neighborhood of . We describe GREEDY(;Fs) below. In the description we use
the term \partial solution" to denote a permutation of facilities in a subset of F.
Let us suppose without loss of generality that we want a good quality neighboring solution to
the solution  = (1;2;:::;n) using a subset Fs = f1, 2;:::;kg generated in the rst step.
The greedy algorithm starts with a partial solution in which the relative positions of 1 through
k are unchanged, i.e., it starts with the partial solution (1;2;:::;k). For each of the elements
j, j = k + 1;:::;n, the algorithm computes a weight wj =
Pn
i=1 cij and orders the facilities in
non-increasing order of these weights. It then performs n k iterations. At the beginning of the r-th
iteration, the partial solution has k + (r   1) facilities. During the iteration, the algorithm chooses
W.P. No. 2011-08-01 Page No. 3IIMA  INDIA
Research and Publications
the r-th facility in the ordering and inserts it in the best of all possible k +r positions in the partial
solution. Thus at the end of the iteration, the cardinality of the partial solution increases by 1. The
greedy algorithm ends when it has inserted all the n   k facilities in F n Fs.
The choice of initial solutions is also important for a local search algorithm. In our local search
algorithm we use the GREEDY(;;) to obtain an initial solution.
Given these details, Algorithm LOCALSEARCH formally describes the local search algorithm
that we use to solve SRFLP instances. In the description we use f() to denote the cost of the
solution .
ALGORITHM LOCALSEARCH
Input: A SRFLP instance with a set F of n; an integer k.
Output: A solution to the instance, i.e., a permutation  of facilities in F.
Steps:
Step 1: Set     GREEDY(;;), c    f().
Step 2: Choose Sr?  F with jFr?j = k, for which
fr? = GREEDY(;Fr?)  min
FrF;jFrj=k
fGREEDY(;Fr)g:
Step 3: If fr? < f(), set     GREEDY(;Fr?) and go to Step 2. Else output  and terminate.
3 Computational Experience
We coded the LOCALSEARCH algorithm presented in Section 2 in C and executed it on a Linux
machine with a 2.2GHz processor and 4GB RAM. This computer has speed characteristics similar
to the one used in Hungerl ander and Rendl (2011)1. We chose a 3-FIX neighborhood for our
experiments, since larger values of k in a k-FIX neighborhood require very large amounts of execution
time. Since most of the heuristics generated optimal solutions when the problem sizes were small,
we used problem instances with sizes ranging from 60 to 100 for our computational experiments.
The instances were Anjos instances which have regularly been used in computational experiments
for the SRFLP and the sko instances which have been used in Hungerl ander and Rendl (2011).
Tables 1 and 2 presents the results of our computational experiments on the Anjos instances and
the sko instances respectively. The results are compared with the results presented in Hungerl ander
and Rendl (2011).
Notice that the objective value of optimal solutions to the problems that we experimented with
are not known, but lower bounds to these values are available (Hungerl ander and Rendl, 2011). If
we dene the suboptimality value for solutions obtained by a heuristic as the excess of the ratio of
objective value of the solution output by the heuristic to the objective value of an optimal solution
over unity, then the solutions obtained by LOCALSEARCH have suboptimality values of at most
0.68% on average for the Anjos instances, and of at most 2.2% on average for the sko instances. The
suboptimality values of the upper bounds reported in Hungerl ander and Rendl (2011) are better.
However, a fair comparison will take the execution times into account; LOCALSEARCH requires
1.83% of the execution time required by the algorithm in Hungerl ander and Rendl (2011) on average
for the Anjos instances and 2.24% of the execution time on average for the sko instances. Given
1see http://www.cpubenchmark.net/midlow_range_cpus.html for exact speed comparison.
W.P. No. 2011-08-01 Page No. 4IIMA  INDIA
Research and Publications
Table 1: Results from LOCALSEARCH on the Anjos benchmark problem instances
Hungerl ander and Rendl (2011) LOCALSEARCH
Instance Size Lower Bound Upper Bound Time Objective Time
Anjos 60 01 60 1477295.0 1477834.0 103169 1479941.0 682
Anjos 60 02 60 841559.0 841776.0 111126 844892.0 687
Anjos 60 03 60 647283.5 648337.5 85021 650563.5 680
Anjos 60 04 60 398095.0 398406.0 95439 401202.0 678
Anjos 60 05 60 318801.0 318805.0 99106 320381.0 684
Anjos 70 01 70 1526359.0 1528560.0 96094 1536600.0 1959
Anjos 70 02 70 1439122.0 1441028.0 94287 1446222.0 1957
Anjos 70 03 70 1517803.5 1518993.5 94514 1531658.5 1976
Anjos 70 04 70 967316.0 969150.0 98928 976197.0 1979
Anjos 70 05 70 4213774.5 4218002.5 101765 4236723.5 1977
Anjos 75 01 75 2387590.5 2393600.5 136673 2406772.5 3151
Anjos 75 02 75 4309185.0 4322492.0 142118 4329962.0 3163
Anjos 75 03 75 1243136.0 1249251.0 138066 1253995.0 3189
Anjos 75 04 75 3936460.5 3941845.5 139378 3953651.5 3124
Anjos 75 05 75 1786154.0 1791469.0 148237 1805235.0 3189
Anjos 80 01 80 2063346.5 2070391.5 210289 2083649.5 4968
Anjos 80 02 80 1918945.0 1921202.0 211635 1933450.0 4894
Anjos 80 03 80 3245254.0 3251413.0 209839 3263348.0 4994
Anjos 80 04 80 3739657.0 3747829.0 211847 3763970.0 4984
Anjos 80 05 80 1585491.0 1590847.0 210630 1602981.0 4964
Table 2: Results from LOCALSEARCH on the sko benchmark problem instances
Hungerl ander and Rendl (2011) LOCALSEARCH
Instance Size Lower Bound Upper Bound Time Objective Time
sko64 01 64 96607.0 97194.0 76179 98707.0 1070
sko64 02 64 633694.5 634332.5 115080 641910.5 1063
sko64 03 64 413079.5 414384.5 126675 421714.5 1070
sko64 04 64 295423.0 298155.0 105809 301558.0 1070
sko64 05 64 501342.5 502063.5 119158 511693.5 1068
sko72 01 72 138885.0 139231.0 106399 141812.0 2383
sko72 02 72 707643.0 715611.0 106841 723070.0 2400
sko72 03 72 1048930.5 1061762.5 117527 1068710.5 2394
sko72 04 72 916229.5 924019.5 122308 933273.5 2293
sko72 05 72 426224.5 430288.5 113983 435754.5 2382
sko81 01 81 203424.0 207063.0 189850 208538.0 5434
sko81 02 81 518711.5 526157.5 215888 529954.5 5416
sko81 03 81 962886.0 979281.0 209860 981498.0 5358
sko81 04 81 2019058.0 2035569.0 206509 2066083.0 5357
sko81 05 81 1293905.0 1311166.0 212368 1321658.0 5421
sko100 01 100 375999.0 380562.0 690441 387863.0 23585
sko100 02 100 2056997.5 2084924.5 726412 2106591.5 23547
sko100 03 100 15987840.5 16216076.5 765534 16334235.0 23601
sko100 04 100 3200643.0 3263493.0 735279 3281817.0 23576
sko100 05 100 1021584.5 1040929.5 725367 1049581.5 23631
that the execution times required by the algorithm in Hungerl ander and Rendl (2011) are prohibitive
(e.g., for sko instances with 100 facilities the algorithm in Hungerl ander and Rendl (2011) requires
approximately 202.38 hours on average, while LOCALSEARCH requires 6.55 hours on average) and
also given that LOCALSEARCH is a much easier algorithm to implement, LOCALSEARCH seems
to be a more practical algorithm to obtain near-optimal solutions to large SRFLP instances.
W.P. No. 2011-08-01 Page No. 5IIMA  INDIA
Research and Publications
4 Summary and Future Research Directions
In this work we present a local search algorithm for the single row facility location problem (SRFLP).
Our algorithm uses a k-FIX neighborhood that is exponential in the size of the problem. To the
best of our knowledge, this is the rst work that uses exponential neighborhoods in an algorithm to
solve the SRFLP; all other algorithms are based on local search use a 2-opt neighborhood in which
the positions of two of the facilities are interchanged to generate a neighbor. Our computational
experiments show that the local search algorithm that we propose using a 3-FIX neighborhood can
generate solutions within 2 to 3% of the optimum in reasonable time.
There are several directions in which the work presented here can be extended. We provide
three immediate extensions here. First, note that the algorithm we present here is a simple local
search algorithm which terminates at the rst local optimum that it encounters. It is easy to extend
such an algorithm into a more sophisticated algorithm like tabu search, which is likely to generate
better quality solutions, although the execution time that it will require will be more. Secondly, the
search through the k-FIX neighborhood of a solution can be made more ecient through careful
book-keeping. Third, it will be interesting to use dierent starting solutions and/or implement
LOCALSEARCH in a multistart mode to see if its performance can be improved signicantly.
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An Exponential Neighborhood Local Search Algorithm for the Single Row Facility Location Problem

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An Exponential Neighborhood Local Search Algorithm for the Single Row Facility Location Problem

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