J Heuristics (2014) 20:125–142DOI 10.1007/s10732-013-9234-x

A scatter search algorithm for the single row facilitylayout problem

Ravi Kothari · Diptesh Ghosh

Received: 14 August 2012 / Revised: 28 September 2013 / Accepted: 5 November 2013 /Published online: 3 December 2013© Springer Science+Business Media New York 2013

Abstract The single row facility layout problem (SRFLP) is the problem of arrangingfacilities with given lengths on a line, with the objective of minimizing the weightedsum of the distances between all pairs of facilities. The problem is NP-hard andresearch has focused on heuristics to solve large instances of the problem. In thispaper we present a scatter search algorithm to solve large size SRFLP instances. Ourcomputational experiments show that the scatter search algorithm is an algorithm ofchoice when solving large size SRFLP instances within limited time.

Keywords Facilities planning and design · Single row facility layout ·Scatter search · Heuristics

1 Introduction

In the single row facility layout problem (SRFLP), we are given a set F = {1, 2, . . . , n}of n > 2 facilities, the length l j of each facility j ∈ F , and a weight ci j for each pair(i, j) of facilities, i, j ∈ F, i �= j . The objective of the problem is to find a permutationΠ of facilities in F that minimizes the total cost given by the expression

z(Π) =∑

1≤i< j≤n

ci j di j ,

R. Kothari (B)HSBC, HSBC House, HDPI, Kolkata 700091, Indiae-mail: ravikothari@iimahd.ernet.in

D. GhoshP&QM Area, IIM Ahmedabad, Ahmedabad 380015, Indiae-mail: diptesh@iimahd.ernet.in

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where di j is the distance between the centroids of the facilities i and j when arrangedaccording to the permutation Π . The cardinality of F is called the size of the problem.The objective of the SRFLP is to arrange the facilities in F on a line so as to minimizethe weighted sum of the distances between all pairs of facilities. This problem is knownto be NP-hard (Beghin-Picavet and Hansen 1982). The SRFLP was first proposed bySimmons (1969) and since then it has been used to model a number of applications.It has been used to design the arrangements of rooms in hospitals, departments inoffice buildings and supermarkets (Simmons 1969), to arrange machines in flexiblemanufacturing systems (Heragu and Kusiak 1988), to assign files to disk cylinders incomputer storage, and to design warehouse layouts (Picard and Queyranne 1981). Adetailed review of the literature on SRFLP has been provided by Kothari and Ghosh(2012b) and Anjos and Liers (2012).

Since the SRFLP is NP-Hard, researchers have used many exact and approximateapproaches to solve the problem. Exact approaches include branch and bound (Sim-mons 1969), integer programming (Love and Wong 1976; Heragu and Kusiak 1991;Amaral 2006, 2008), cutting planes (Amaral 2009), dynamic programming (Picardand Queyranne 1981), branch and cut (Amaral and Letchford 2012), and semidefi-nite programming (Anjos et al. 2005; Anjos and Vannelli 2008; Anjos and Yen 2009;Hungerländer and Rendl 2013). Currently, the best exact algorithms are those of Ama-ral (2009) and Hungerländer and Rendl (2013). The method used by Hungerländer andRendl (2013) has been able to obtain optimal solutions to SRFLP instances with upto 40 facilities and to one instance with 42 facilities. The 42 facility SRFLP instanceis the largest one for which an optimal solution is known in the literature. Polyhe-dral studies for the SRFLP are presented by Sanjeevi and Kianfar (2010) and Amaraland Letchford (2012). In addition, the heuristics presented by Amaral and Letchford(2012) and Hungerländer and Rendl (2013) output high-quality feasible solutions forthe SRFLP.

Researchers have focused on approximate approaches or heuristics to solve largesize SRFLP instances. Heuristics are of two types, construction and improvement.Construction heuristics for the SRFLP have been proposed by Djellab and Gourgand(2001), Heragu and Kusiak (1988), and Kumar et al. (1995). However, improvementheuristics have superseded these construction heuristics in the literature and haveyielded the best known solutions for large size SRFLP instances. Most improvementheuristics for the SRFLP are metaheuristics which are either single solution approachesor population approaches. Single solution based heuristics include simulated anneal-ing (Gomes de Alvarenga et al. 2000; Romero and Sánchez-Flores 1990; Heragu andAlfa 1992), tabu search (Gomes de Alvarenga et al. 2000; Samarghandi and Eshghi2010; Kothari and Ghosh 2013c), and local search with Lin–Kernighan neighbor-hood (Kothari and Ghosh 2013b). Population heuristics for SRFLP include ant colonyoptimization (Solimanpur et al. 2005), scatter search (Kumar et al. 2008), particleswarm optimization (Samarghandi et al. 2010), and genetic algorithm (Datta et al.2011; Ozcelik 2011; Kothari and Ghosh 2013a). Among these the genetic algorithmsby Ozcelik (2011) and Kothari and Ghosh (2013a) yield best results for benchmarkSRFLP instances of large sizes.

Scatter search has been shown to yield promising results for solving various non-linear and combinatorial optimization problems (see Laguna and Marti 2003; Marti

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A scatter search algorithm for the single row 127

et al. 2006; Glover 1977, 1998). Although it is a population based heuristic, it issignificantly different from genetic algorithms. Similarities and differences betweenscatter search and genetic algorithms have been previously discussed by Glover (1994,1995). To the best of our knowledge, the only study which has applied scatter searchto the SRFLP is by Kumar et al. (2008). It deals with SRFLP instances of size up to30 facilities only. We have not encountered any scatter search algorithm for larger sizeSRFLP instances.

Our paper is organized as follows. In Sect. 2 we describe a generic scatter searchalgorithm for the SRFLP. We then describe our specific scatter search algorithm andpresent results of our computational experiments with it in Sect. 3. We conclude thepaper in Sect. 4 with summary of the work and future research directions.

2 Scatter search for the SRFLP

Scatter search for a SRFLP is an evolutionary technique which orients its explorationrelative to a set of permutations, called the reference set. The reference set typicallyconsists of good permutations obtained by other methods. The criterion of “good” mayrefer not only to the cost of the permutation, but also to certain other properties that thepermutations or a set of permutations may possess. Most scatter search algorithms forcombinatorial optimization problems use the scatter search template by Glover (1998)as a reference for building the algorithm. We also draw ideas from the same templatein our candidate scatter search algorithms to solve large size SRFLP instances. Asper the template, our scatter search algorithms consist of five steps which we describebelow.

2.1 Diversification generation method

This element of scatter search determines the quality of permutations in the referenceset and ensures diversification in the search process. We use the concept of deviationdistances (see Sörensen 2007) to measure the diversification in a set of permutations.The deviation distance between two permutations is defined as the sum total of theabsolute values of the positional deviation between matching facilities in the twopermutations. For example, if we consider two permutations (1 2 3 4 5) and (4 3 5 1 2),then the positional deviation for the first facility 1 is |1 − 4| = 3 as the first facilityis in the first and the fourth location in the permutations. Summing up the positionaldeviation for all the facilities the deviation distance between the two permutations is

|1− 4| + |2− 5| + |3− 2| + |4− 1| + |5− 3| = 12.

In the SRFLP a permutation is identical to the permutation obtained by reversing thepositions of all the facilities in the permutation. So we define the distance between twopermutations Π1 and Π2 as the minimum of the deviation distance between Π1 and Π2and the deviation distance between Π1 and the permutation obtained by reversing Π2.Using this distance measure, we develop two methods, DIV-1 and DIV-2, to generatea diversified set of good permutations which serves as the initial population for our

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scatter search algorithms. Both these methods require an initial seed permutation offacilities in F as an input for generating the initial population.

In the DIV-1 method, starting from the initial seed permutation Π = (1, 2, . . . , n),we generate a large set of permutations with cardinality U_Size by randomly inter-changing facilities in Π . We then choose E_Size lowest cost permutations from theset to form a set of elite permutations. We generate the initial population of sizeP_Size by choosing permutations from the elite set in a way such that the mini-mum distance between any two permutations in the population is as high as possible.To do this, we first find out those two permutations among the elite permutationsbetween which the deviation distance is the largest. We move these two permutationsfrom the set of elite permutations to the initial population. We then perform the fol-lowing iteration until the we have P_Size permutations in the initial population. Inan iteration, for each permutation in the elite permutation set we compute the min-imum of the distances between it and members of the initial population. We thenmove that permutation which has the maximum value of minimum distance from theelite permutation set to the initial population. The parameters U_Size, E_Size, andP_Size are specified by the user. The pseudocode for the DIV-1 method is givenbelow.

ALGORITHM DIV-1

Input: An SRFLP instance with n facilities, U_Size, E_Size, and P_Size.Output: The initial population with P_Size permutations.Code

1. begin2. set Π ← {1, 2, 3, . . . , n}; U_List, E_List, P_List ← ∅;3. for i from 1 to U_Size do begin4. set Pi∗ ← Π ;5. for j from 1 to �n/2 do begin6. r ← random number chosen from [0, 1];7. if (r ≤ 0.5) then interchange Pi∗[ j] and Pi∗[n − j];8. end;9. U_List ← U_List ∪ {Pi∗};

10. end;11. sort the permutations in U_List in non-decreasing order of their costs;12. copy the first E_Size permutations in U_list to E_List ;13. for each pair of permutations Πi and Π j in E_List compute the

deviation distance Di j between them;14. set {i, j} = arg max{Di j |Πi ,Π j ∈ E_List};15. set P_List ← {Πi ,Π j }; PCount ← 2;16. while PCount < P_Size do begin17. for each Πi ∈ E_List do18. set M Di ← min{Di j |Π j ∈ P_List};19. set k ← arg max{M Di }; P_List ← P_List ∪ {Πk};

E_List ← E_List \ {Πk}; PCount ← PCount + 1;20. end;21. output P_List ;22. end.

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In the DIV-2 method, we generate the initial seed permutation Π using Theorem 1by Samarghandi and Eshghi (2010). A distance value H_Dist is taken as an input. Wefirst include Π in the initial population. We then generate permutations by randomlyinterchanging facilities in Π , and include these permutations in the initial populationonly if the distance between the newly generated permutation and each of the per-mutations in the initial population is at least H_Dist . The method stops when wehave the required number of permutations in the initial population. The pseudocodefor the DIV-2 method is given below. Note that steps 2 through 14 in the algorithmoperationalize Theorem 1 by Samarghandi and Eshghi (2010).

ALGORITHM DIV-2

Input: An SRFLP instance with n facilities, P_Size, and H_Dist .Output: The initial population with P_Size permutations.Code

1. begin2. arrange the facilities in non-increasing order of their lengths;

/* with 〈i〉 denoting the i-th facility in this order. */3. for i from 1 to �n/2 do4. set Π [i] = 2× 〈n − i + 1〉;5. for i from 1 to n/2� do6. set Π [�n/2 + i] = 〈2i − 1〉;7. set P_List ← {Π}; PCount ← 1;8. while PCount < P_Size do begin9. set Pi∗ ← Π ;

10. for j from 1 to �n/2 do begin11. r ← random number chosen from [0, 1];12. if (r ≤ 0.5) then interchange Pi∗[ j] and Pi∗[n − j];13. end;14. for all Πi ∈ P_List do15. set M Di ← deviation distance between Pi∗ and Πi ;16. if (mini {M Di } > H_dist) then begin17. set P_List ← P_List ∪ {Pi∗}; PCount ← PCount + 1;18. end;19. end;20. output P_List ;21. end.

2.2 Improvement method

The improvement method tries to improve upon the permutations in the initial popula-tion and the permutations obtained by the solution combination method at later stagesof the algorithm. In our scatter search algorithms we use local search iterations withan insertion neighborhood to improve the permutations. An insertion neighbor of apermutation is obtained by removing a facility from its position and introducing it atanother position in the permutation. A local search iteration searches all the insertionneighbors of a permutation and returns the best insertion neighbor of the permutation.

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Detailed descriptions of methods to search the insertion neighborhood of a SRFLPsolution are provided in Kothari and Ghosh (2013b,c).

2.3 Reference set update method

The reference set update method is used to build and update a set of permutationsknown as the reference set. The reference set consists of B_Size good permutationswhich are best and diverse among those obtained during the run of the algorithm.The value of B_Size is typically not more than 20. At the start of the algorithm theuser inputs two parameters B1 and B2 such that B1 + B2 = B_Size. The referenceset is created from the set of permutations as follows. First the improvement methodis applied on each permutation in the initial population. The improved permutationsare then sorted in non-decreasing order of their costs and the first B1 and last B2permutations in the sorted list are selected to be the members of the reference set. HenceB1 permutations have low objective function values and B2 permutations improve thediversity in the reference set.

In the later stages of the scatter search algorithm, the permutations in the referenceset are combined using the subset generation method and solution combination method,and then improved using the improvement method to generate new permutations whichmay replace some of the existing members of the reference set. A new permutationreplaces the highest cost permutation in the reference set if its cost is lower thanit. When no new permutations get added to the reference set during a reference setupdate, we say that the reference set has converged and the lowest cost permutationin the reference set is output as a solution to the problem.

2.4 Subset generation method

The subset generation method produces subsets of permutations in the reference set.These subsets are used to generate new permutations using the solution combinationmethod. The outlines of various subset generation methods is suggested by Glover(1998). In our algorithms we generate all the subsets of the reference set with cardi-nality 2. We restrict ourselves to subsets of size 2 since scatter search algorithms whichuse subsets of higher cardinalities are computationally impractical for large SRFLPinstances.

2.5 Solution combination method

The solution combination method combines the permutations in each subset gener-ated by the subset generation method into a new permutation. The new permutationis then subjected to the improvement method till it converges to a local optimum. Thelocally optimal permutation is finally either added to the reference set or discardedbased on its cost. In our scatter search algorithms we use two combination methods,one of which is purely deterministic in implementation and the other one which has arandom component.

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The first combination method is the alternating combination method (see e.g.,Larrañaga et al. 1999). It creates a new permutation by alternately selecting facili-ties from the first permutation and the second permutation in the subset, omitting thefacilities that have already been located in the new permutation. The second combi-nation method is the partially matched combination method. It is similar to the PMXcrossover (see e.g., Larrañaga et al. 1999), and is the most widely used combinationoperator for permutation problems.

Having described the components of scatter search we are now in a position todescribe a generic scatter search algorithm for the SRFLP. It starts by generating aninitial population of permutations using a diversification generation method. It thenuses an improvement method to improve the permutations in the initial populationand builds a reference set using the reference set update method. The algorithm thenperforms iterations consisting of three steps till the reference set converges. In the firststep it creates subsets of the reference set using a subset generation method. In thesecond step, for each of the subsets, it combines the permutations in the subset using asolution combination method to create one permutation corresponding to each subset.All of these permutations are then improved using the improvement method. In thethird step it tries to update the reference set using these improved permutations. Animproved permutation replaces the highest cost permutation in the reference set if itscost is lower than that of the latter. If none of the improved permutations enter thereference set in the third step, then the reference set is said to have converged and thescatter search algorithm terminates after reporting the lowest cost permutation in thereference set. The pseudocode for scatter search algorithm is given below.

ALGORITHM GENERIC-SS

Input: An SRFLP instance with n facilities, B1, B2, B, H_Dist , U_Size,E_Size and P_Size.

Output: The best permutation which has the lowest cost among all thepermutations encountered in the algorithm.

Code

1. begin2. generate an initial population with P_Size permutations using the

diversification generation method DIV-1/DIV-2;3. improve each permutation in population using the improvement method;4. sort the improved permutations in the population in non-decreasing

order of their costs;5. choose first B1 and last B2 permutations in the sorted list to

construct the reference set RefSet;6. set flag←TRUE;7. while flag = TRUE do8. create all subsets of size 2 using the permutations in the RefSet;9. for i from 1 to

(B2

)do begin

10. flag←TRUE;11. generate a permutation using the solution combination method

on the i th subset generated from the RefSet;

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12. use the improvement method to obtain a local optima Π of thepermutation generated in the last step;

13. sort the permutations in the RefSet in increasing order of costs;14. if (cost (Π) < cost of the last permutation in RefSet) then begin15. replace the last permutation in RefSet by Π ;16. sort the permutations in the RefSet in increasing order of

their costs;17. flag←FALSE;18. end;19. end;20. if flag = FALSE then go to Step 9; else go to Step 21;21. end while;22. output the lowest cost permutation in RefSet;23. end.

The computational complexity of the GENERIC-SS algorithm depends on the rela-tive magnitudes of B and n. In scatter search algorithms, the size B of the reference setis kept small, so that each scatter search iteration (Steps 7 through 21 in the pseudocode)does not take too long to execute. The critical step in each iteration is Step 12 whichtakes an amortized O(n3) time (see Kothari and Ghosh 2013b,c). Assuming that B ismuch smaller than n, each scatter search iteration requires O(B2n3) time.

Note that a scatter search algorithm called SS was presented for the SRFLP inKumar et al. (2008). The algorithm presented here differs from SS in several ways.First, the diversification generation method in SS is one presented in Glover (1998).The algorithm presented here uses one of two different diversification generationmethods (DIV-1 and DIV-2) based on the concept of deviation distance proposed inSörensen (2007). Second, the SS algorithm uses a left and right insertion neighborhoodsearch to improve solutions. Our algorithm on the other hand performs an exhaustivesearch in the insertion neighborhood to obtain the best insertion neighbor. Third, theSS algorithm generates four subset types in their subset generation method while thealgorithm presented here only generates one subset type, corresponding to Subset-Type 1 in Kumar et al. (2008). Finally, the solution combination methods used in thetwo algorithms are completely different.

In the next section we present a computational comparison of our algorithm withthe best available in the published literature.

3 Computational experiments

We performed computational experiments with scatter search algorithms to comparetheir performance with other methods available in the literature to solve large sizeSRFLP instances. We developed four versions of scatter search. All the versions usedthe same improvement method, reference set update method, and subset generationmethod but differed in diversification generation methods and subset generation meth-ods. The details of the diversification generation methods and solution combinationmethods for the four algorithms are given in Table 1.

We coded these algorithms in C and performed our experiments on a personalcomputer with four Intel Core i5–2,500 3.30 GHz processors, 4GB RAM, running

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Table 1 The four scatter searchalgorithms used in experiments

Algorithm Diversification generation Solution combination

Method Method

SS-1A DIV-1 Alternating combination

SS-1P DIV-1 Partially matchedcombination

SS-2A DIV-2 Alternating combination

SS-2P DIV-2 Partially matchedcombination

Fig. 1 Comparison of the scatter search algorithms on an instance of size 100

Ubuntu Linux 11.10. Based on initial experiments we set U_Size = 1, 000, E_Size =500, and P_Size = 100. For DIV-2 we set the value of H_Dist to n2/4 where n isthe size of the instance. Following usual practice we set the size of the reference setB_Size at 20, with B1 = B2 = 10.

Since the four algorithms are very similar in their frameworks, we tested theirperformance on randomly generated instances of various sizes. From our experimentswe observed that these algorithms output very similar results. However, SS-1P wasfound to be marginally superior to the others for our randomly generated instances. Itoften outputs the best solution among the four variants on any given instance. It alsogenerates its final solution faster than the other variants. For example, Fig. 1 presentsthe variation of costs of the solutions obtained by the four algorithms on a randomlygenerated instance of size 100.

The solid line in the figure represents the trajectory for SS-1P and the dotted linesrepresent the trajectories of the other algorithms. Notice that for this instance, SS-1Aand SS-2A terminated with solutions of larger costs than those output by SS-1P andSS-2P. Also note that SS-1P obtained this solution after about 320 s of executiontime while SS-2P obtained the solution after about 390 s of execution time. Hence inour detailed computational experiments with benchmark instances, we used only theSS-1P algorithm.

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We experimented with three sets of benchmark SRFLP instances. The first set is dueto Anjos et al. (2005) and consists of four groups of five instances each of size 60, 70,75, and 80. We refer to these as the Anjos instances. These instances have been widelyexperimented with in the published literature (see e.g., Datta et al. 2011; Hungerländerand Rendl 2013; Ozcelik 2011; Kothari and Ghosh 2013b). The second set of instancesare QAP-based sko instances. This set consists of seven groups of five instances eachof size 42, 49, 56, 64, 72, 81, and 100. They have been experimented with in theliterature by Anjos and Yen (2009), Amaral and Letchford (2012), Hungerländer andRendl (2013), and Kothari and Ghosh (2013a). We refer to them as the sko instances.The third set is due to Letchford and Amaral (2011) and consists of three instances ofsize 110 each. They have been experimented with by Letchford and Amaral (2011),Ozcelik (2011) and Hungerländer and Rendl (2013) and we refer to them as the Amaralinstances.

3.1 Testing with 20 instances from Anjos et al. (2005)

There are several published studies on the SRFLP which have reported results on theAnjos instances. Among them Samarghandi and Eshghi (2010) and Kothari and Ghosh(2013c) use tabu search, Datta et al. (2011), Ozcelik (2011) and Kothari and Ghosh(2013a) use genetic algorithms, Kothari and Ghosh (2013b) use a Lin–Kernighanneighborhood based local search and Hungerländer and Rendl (2013) uses a SDP-relaxation based approach to provide upper bounds on the cost of optimal solutions.Among these, the results reported by Ozcelik (2011), Kothari and Ghosh (2013a,b,c))are the best known results in the literature.

In Table 2 we present the costs of the best permutations obtained by the SS-1Palgorithm and compare them with the results available in the literature. The first andsecond columns of the table provides the name of the instance and its size. The lasttwo columns of the table presents the costs of the best permutations known in liter-ature and those output by SS-1P. The results show that the SS-1P algorithm outputssolutions whose costs equal the best costs in the published literature for all the 20instances.

We next compare the execution times required by the algorithm presented by Kothariand Ghosh (2013b) and the SS-1P algorithm on the Anjos instances. We compare SS-1P with the Lin–Kernighan type neighborhood based local search algorithm in Kothariand Ghosh (2013b) since it was the fastest among all previous algorithms that outputthe same costs. (The times reported in Ozcelik (2011), Kothari and Ghosh (2013a)are the average times per genetic algorithm run, and to obtain a fair comparison withthe times reported in Kothari and Ghosh (2013b) we multiplied the average timesreported with the number of genetic algorithm runs executed per instance.) To obtaina fair comparison we ran the code of the Lin–Kernighan type neighborhood basedlocal search algorithm in Kothari and Ghosh (2013b) and the code for SS-1P on thesame machine and report our findings in Table 3. The first two columns in the tablepresent details about the instances, the third column presents times required by thealgorithm in Kothari and Ghosh (2013b) on an instance, and the fourth gives the timesrequired by the SS-1P algorithm.

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Table 2 Comparison of solution costs for Anjos instances

Instance Size Best costs in literaturea SS-1P

Anjos-60-01 60 1477834.0 1477834.0

Anjos-60-02 60 841776.0 841776.0

Anjos-60-03 60 648337.5 648337.5

Anjos-60-04 60 398406.0 398406.0

Anjos-60-05 60 318805.0 318805.0

Anjos-70-01 70 1528537.0 1528537.0

Anjos-70-02 70 1441028.0 1441028.0

Anjos-70-03 70 1518993.5 1518993.5

Anjos-70-04 70 968796.0 968796.0

Anjos-70-05 70 4218002.5 4218002.5

Anjos-75-01 75 2393456.5 2393456.5

Anjos-75-02 75 4321190.0 4321190.0

Anjos-75-03 75 1248423.0 1248423.0

Anjos-75-04 75 3941816.5 3941816.5

Anjos-75-05 75 1791408.0 1791408.0

Anjos-80-01 80 2069097.5 2069097.5

Anjos-80-02 80 1921136.0 1921136.0

Anjos-80-03 80 3251368.0 3251368.0

Anjos-80-04 80 3746515.0 3746515.0

Anjos-80-05 80 1588885.0 1588885.0

Average solution cost 1981690.6 1981690.6a Refers to the costs reported by Ozcelik (2011), Kothari and Ghosh (2013a,b,c)

From the table we see that on average, SS-1P requires 56.57 % of the time requiredby the algorithm in Kothari and Ghosh (2013b) on these problems and is thus signifi-cantly faster.

3.2 Testing with sko instances

In the published literature, results of computational experiments with sko instanceshave been reported by Amaral and Letchford (2012), Anjos and Yen (2009), Hunger-länder and Rendl (2013), Ozcelik (2011), Kothari and Ghosh (2013a,b,c) ). Not allthe studies experiment with all the sko instances. Amaral and Letchford (2012) use apolyhedral approach, Anjos and Yen (2009) and Hungerländer and Rendl (2013) useSDP-relaxation based approaches, Ozcelik (2011) and Kothari and Ghosh (2013a) usegenetic algorithms, and Kothari and Ghosh (2013b) use a Lin–Kernighan based localsearch heuristic to experiment with these instances. In Table 4 we compare the costsof the permutations output by the SS-1P algorithm with the best costs from the litera-ture. The first and second columns of the table present details about the instances, thethird column presents the best costs known in the published literature, and the fourthcolumn presents the corresponding source. The fifth column presents the costs of thepermutations output by the SS-1P algorithm.

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Table 3 Comparison of execution times (in seconds) for Anjos instances

Instance Size Best time in literaturea SS-1P

Anjos-60-01 60 57.35 46.73

Anjos-60-02 60 57.00 41.54

Anjos-60-03 60 45.90 50.4

Anjos-60-04 60 55.52 53.52

Anjos-60-05 60 55.15 52.35

Anjos-70-01 70 118.57 82.61

Anjos-70-02 70 123.34 88.16

Anjos-70-03 70 155.67 83.17

Anjos-70-04 70 143.78 102.26

Anjos-70-05 70 114.40 81.59

Anjos-75-01 75 257.13 127.66

Anjos-75-02 75 226.60 118.01

Anjos-75-03 75 256.81 143.69

Anjos-75-04 75 246.16 125.37

Anjos-75-05 75 257.16 138.71

Anjos-80-01 80 336.25 162.59

Anjos-80-02 80 375.93 174.8

Anjos-80-03 80 312.32 181.84

Anjos-80-04 80 306.22 148.41

Anjos-80-05 80 392.86 199.79

Average execution time 194.72 110.16aRefers to the times required by Lin–Kernighan algorithm in Kothari and Ghosh (2013b)

The results in Table 4 show that for 30 of the 35 instances, the costs of the solu-tions output by the SS-1P algorithms match the best known for these instances in theliterature. For four of the instances, the costs output by SS-1P are marginally worsethan the best costs known in the literature, and in one of the instances, the cost outputof SS-1P is better. The difference in costs of solutions output by SS-1P and those ofthe previously known best solutions is never more than 0.004 % of the costs of thepreviously known best solutions.

We next compare the execution times required by the previously known best algo-rithms in the literature and the SS-1P algorithm on the sko instances. In some ofthe instances, several algorithms output the currently known best cost solutions. Forthese instances we consider the times corresponding to that algorithm which requiredthe least time. Allowing for differences in computational set ups, the algorithms pre-sented in Kothari and Ghosh (2013a,b) are the fastest among the algorithms in theliterature for these problems. For each of the instances, we ran the codes for thealgorithm among the ones reported in Kothari and Ghosh (2013a,b) which gen-erated the least cost solution to the instance and the code for SS-1P on the samemachine, and report their execution times in Table 5. The first two columns in thetable describe the instance, the third column reports the total time required on the

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Table 4 Comparison of solution costs for sko instances

Instance Size Best cost in literature Sourcea SS-1P

sko-42-01 42 25525.0 H&R, KGA 25525.0

sko-42-02 42 216120.5 H&R, KGA 216120.5

sko-42-03 42 173267.5 1 173267.5

sko-42-04 42 137615.0 H&R, KGA 137615.0

sko-42-05 42 248238.5 1 248238.5

sko-49-01 49 40967.0 KGA 40967.0

sko-49-02 49 416178.0 H&R, KGA 416178.0

sko-49-03 49 324512.0 H&R, KGA 324512.0

sko-49-04 49 236755.5 H&R, KGA 236755.5

sko-49-05 49 666143.0 H&R, KGA 666143.0

sko-56-01 56 64024.0 KGA 64024.0

sko-56-02 56 496561.0 H&R, KGA 496561.0

sko-56-03 56 170449.0 KGA 170449.0

sko-56-04 56 313388.0 KGA 313388.0

sko-56-05 56 592294.5 KGA 592316.5

sko-64-01 64 96881.0 KGA 96883.0

sko-64-02 64 634332.5 1 634332.5

sko-64-03 64 414323.5 LKI, KGA 414323.5

sko-64-04 64 297129.0 KGA 297129.0

sko-64-05 64 501922.5 A&L,2 501922.5

sko-72-01 72 139150.0 LKI, KGA 139150.0

sko-72-02 72 711998.0 KGA 711998.0

sko-72-03 72 1054110.5 2 1054110.5

sko-72-04 72 919586.5 KGA 919586.5

sko-72-05 72 428226.5 KGA 428226.5

sko-81-01 81 205108.5 KGA 205106.0

sko-81-02 81 521391.5 LKI, KGA 521391.5

sko-81-03 81 970796.0 KGA 970796.0

sko-81-04 81 2031803.0 KGA 2031803.0

sko-81-05 81 1302711.0 KGA 1302711.0

sko-100-01 100 378234.0 OZ, KGA 378234.0

sko-100-02 100 2076008.5 OZ, KGA 2076008.5

sko-100-03 100 16145598.0 KGA 16145614.0

sko-100-04 100 3232522.0 OZ, KGA 3232531.0

sko-100-05 100 1033080.5 OZ, KGA 1033080.5

Average solution cost 1063341.5 1063342.8

Cost in boldface indicate the improvement of SS-1P over the published literaturea A&L, H&R, OZ, TS, LKI, and KGA refer to Amaral and Letchford (2012), Hungerländer and Rendl(2013), Ozcelik (2011), Kothari and Ghosh (2013a,b,c) ), respectively1 H&R, A&L and KGA2 TS, LKI and KGA

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Table 5 Execution times (in seconds) of scatter search algorithms for sko instances

Instance Size Best times in literature Sourcea SS-1P

sko-42-01 42 658.00 KGA 11.42

sko-42-02 42 680.00 KGA 16.60

sko-42-03 42 680.00 KGA 15.40

sko-42-04 42 668.00 KGA 12.10

sko-42-05 42 636.00 KGA 11.16

sko-49-01 49 1172.00 KGA 25.40

sko-49-02 49 1100.00 KGA 27.86

sko-49-03 49 1108.00 KGA 24.07

sko-49-04 49 1146.00 KGA 25.67

sko-49-05 49 1106.00 KGA 22.45

sko-56-01 56 1920.00 KGA 48.61

sko-56-02 56 1708.00 KGA 42.12

sko-56-03 56 1692.00 KGA 47.05

sko-56-04 56 1780.00 KGA 43.50

sko-56-05 56 1662.00 KGA 36.94

sko-64-01 64 3074.00 KGA 82.80

sko-64-02 64 2796.00 KGA 68.31

sko-64-03 64 96.32 LKI 80.33

sko-64-04 64 2856.00 KGA 91.05

sko-64-05 64 87.90 LKI 75.86

sko-72-01 72 194.06 LKI 115.21

sko-72-02 72 4598.00 KGA 192.12

sko-72-03 72 179.94 LKI 114.16

sko-72-04 72 4460.00 KGA 135.46

sko-72-05 72 4656.00 KGA 116.50

sko-81-01 81 8148.00 KGA 311.63

sko-81-02 81 519.89 LKI 226.91

sko-81-03 81 7176.00 KGA 241.50

sko-81-04 81 7498.00 KGA 193.98

sko-81-05 81 7188.00 KGA 287.22

sko-100-01 100 19868.00 KGA 877.12

sko-100-02 100 17220.00 KGA 526.09

sko-100-03 100 16474.00 KGA 599.92

sko-100-04 100 16122.00 KGA 592.00

sko-100-05 100 16926.00 KGA 591.75

Average execution time 4510.12 169.44a KGA and LKI refer to Kothari and Ghosh (2013a,b) respectively

instance by the best algorithm previously known in the literature, the fourth columnreports the source of this algorithm. The fifth column reports the time required by theSS-1P algorithm.

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From the table we see that SS-1P requires on average, 3.76 % of the executiontimes required by the fastest algorithms known previously and is thus significantlyfaster. This, combined with the fact that SS-1P generates the lowest cost solutionsto all but one of these instances make it the algorithm of choice for solving the skoinstances. Among the other algorithms known in the literature, the Lin–Kernighanbased local search algorithm presented in Kothari and Ghosh (2013b) is the fastest forthese problems, but it does not output the best solutions to most of the sko instances.

3.3 Testing with instances from Letchford and Amaral (2011) with 110 facilities

Results for the Amaral instances have been reported by Letchford and Amaral (2011),Ozcelik (2011), Hungerländer and Rendl (2013), Kothari and Ghosh (2013a,b,c) ), inthe literature. Among these, the best results are reported by Ozcelik (2011). In Table 6we report the best costs output by SS-1P on Amaral instances. The structure of Table 6is the same as that of Table 4.

We see from the table that the costs reported by Ozcelik (2011) supersede thoseoutput by the SS-1P algorithm. In only one of the instances the cost of the SS-1Poutput matches the cost reported by Ozcelik (2011).

Table 7 compares the execution times required by the previously best known algo-rithms with those required by the SS-1P algorithm on the Amaral instances. To ensurefairness of comparison, we multiply the average execution time per run reported inOzcelik (2011) by 20 and that reported in Kothari and Ghosh (2013a) by 200 to obtainthe total execution times. The structure of the table is similar to that of Table 5 pre-sented earlier. We see that on average, the SS-1P algorithm required 0.90 % of thetimes required by the genetic algorithms in Ozcelik (2011) and Kothari and Ghosh

Table 6 Comparison on solution costs to Amaral instances

Instance Size Best costs inliterature

Sourcea SS-1P

Amaral-110-01 110 144296664.5 OZ 144297440.0

Amaral-110-02 110 86050037.0 OZ 86050208.0

Amaral-110-03 110 2234743.5 KGA, OZ 2234798.5

Average solution cost 77527148.3 77527482.2aOZ and KGA refer to Ozcelik (2011) and Kothari and Ghosh (2013a) respectively

Table 7 Comparison on execution times (in seconds) for Amaral instances

Instance Size Best times in literature Sourcea SS-1P

Amaral-110-01 110 124701.60 OZ 975.11

Amaral-110-02 110 124062.00 OZ 680.10

Amaral-110-03 110 23804.00 KGA 811.08

Average execution time 90855.87 822.10aOZ and KGA refer to Ozcelik (2011) and Kothari and Ghosh (2013a) respectively

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140 R. Kothari, D. Ghosh

(2013a) for these instances. Although the experimental set up in Ozcelik (2011) isdifferent from that in the current work and in Kothari and Ghosh (2013a), the differ-ence between the execution times suggest that SS-1P is significantly faster for theseproblems than existing algorithms.

Based on results from our computational experiments, we conclude that the SS-1Palgorithm presented in this paper is competitive for solving large size SRFLP instancesand is to be especially preferred in presence of time restrictions while solving theseinstances.

4 Summary of contribution and future research directions

In this paper we have developed four scatter search algorithms for the single rowfacility layout problem (SRFLP). Our algorithms inherit their basic structure from thescatter search template presented by Glover (1998). Each one of our algorithms usesa diversification generation method to generate an initial population containing goodquality and diverse permutations, an improvement method to improve the permutationsin the algorithm, a reference set update method to build and maintain a reference set ofgood quality permutations, and a subset generation method and a solution combinationmethod to update the reference set with new permutations. We present two diversi-fication generation methods called DIV-1 and DIV-2 and two solution combinationmethods, called the alternating combination method and partially matched combina-tion method, whose combinations yield four scatter search algorithms called SS-1A,SS-1P, SS-2A, and SS-2P. Our initial experiments showed that the SS-1P algorithm ismarginally more promising than the other three variants, and we used this algorithmto perform detailed computational experiments.

We compared the performance of the SS-1P algorithm with the best known algo-rithms in the literature for three sets of benchmark instances comprising of 58 large sizeSRFLP instances. Our computational experience indicated that the SS-1P algorithmis competitive with other algorithms known in the literature, and is to be preferred incase there is a constraint on the execution time available.

In the literature on combinatorial problems, scatter search is often combined withpath relinking to obtain good quality solutions (see e.g., Glover et al. 2000). Weimplemented interpolation based path relinking to work with all the four scatter searchvariants that we present in this paper (see Kothari and Ghosh 2012a, for details). Formost of the benchmark instances, path relinking was not able to better the solutionsobtained by the scatter search algorithms. In the few cases in which they did, thesolutions obtained after path relinking were those obtained by other variants of scattersearch.

One immediate direction for future research is to try extrapolation based pathrelinking strategies to obtain improved layouts for large SRFLP benchmark instances.Another possible direction of research is to evaluate different subset generation andsolution combination methods, or to use different neighborhoods for local searchin an effort to create better scatter search algorithms. Research can also focus ondeveloping different metaheuristics like GRASP for the SRFLP. One can also per-form experiments to evaluate the performance of scatter search algorithms on related

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facility layout problems such as the double row facility layout problem (see Chungand Tanchoco 2010) or the corridor allocation problem (see Amaral 2012).

5 Appendix

The permutation of facilities for the instance sko-81-01 for which the SS-1P algorithmimproved the best solution known in the literature is given below. The facilities arenumbered from 1 through 81.66 74 3 61 30 25 50 9 48 56 67 65 33 69 23 17 4 6 41 21 81 14 15 19 38 49 31 7 3936 26 70 35 42 22 55 11 57 78 44 2 45 52 32 68 13 62 1 5 27 58 20 18 73 43 59 2451 37 63 12 79 75 80 77 71 72 29 64 46 28 10 40 34 76 16 54 53 60 47 8

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