a detailed analysis of the population-based ant colony...

A Detailed Analysis of the

Population-Based Ant Colony

Optimization Algorithm for the TSP and

the QAP

Sabrina Oliveira, Mohamed Saifullah Hussin, ThomasStutzle, Andrea Roli, Marco Dorigo

IRIDIA – Technical Report Series

Technical Report No.

TR/IRIDIA/2011-006

February 2011

IRIDIA – Technical Report SeriesISSN 1781-3794

Published by:IRIDIA, Institut de Recherches Interdisciplinaires

et de Developpements en Intelligence Artificielle

Universite Libre de BruxellesAv F. D. Roosevelt 50, CP 194/61050 Bruxelles, Belgium

Technical report number TR/IRIDIA/2011-006

The information provided is the sole responsibility of the authors anddoes not necessarily reflect the opinion of the members of IRIDIA. Theauthors take full responsibility for any copyright breaches that mayresult from publication of this paper in the IRIDIA – Technical ReportSeries. IRIDIA is not responsible for any use that might be made ofdata appearing in this publication.

A Detailed Analysis of the Population-Based Ant

Colony Optimization Algorithm for the TSP and

the QAP

Sabrina Oliveira† [email protected]

Mohamed Saifullah Hussin† [email protected]

Thomas Stutzle† [email protected]

Andrea Roli‡ [email protected]

Marco Dorigo† [email protected]

† IRIDIA, CoDE, Universite Libre de Bruxelles, Brussels, Belgium.‡ DEIS, Universita di Bologna.

February 2011

Abstract

The population-based ant colony optimization algorithm (P-ACO)differs from other ACO algorithms through its implementation of thepheromone update management. P-ACO keeps track of a population ofsolutions, which serves as an archive of solutions generated by the ants’colony. Pheromone updates in P-ACO are only done based on solutionsthat enter or leave the solution archive. The population-based schemereduces considerably the computation time needed for the pheromone up-date when compared to ACO algorithms such as Ant System. In thiswork, we study P-ACO’s behavior for solving the traveling salesman prob-lem and the quadratic assignment problem. In particular, we investigatethe impact of a local search on P-ACO parameters and performance. Theresults clearly show that P-ACO is a very competitive tool in which itsparameters and behavior depend strongly on the problem tackled andwhether or not a local search is used.

1 Introduction

Ant Colony Optimization (ACO) [5] is a metaheuristic that is mainly usedfor tackling combinatorial optimization problems. In ACO, the ant’s searchis stochastically biased by heuristic and pheromone information. The waypheromone update is implemented differs across ACO variants, and the choiceof an appropriate pheromone update mechanism is essential to obtain effectiveACO algorithms. Pheromone update usually involves pheromone evaporation,which is implemented as the reduction of the pheromone trail strength, normallyby a fixed percentage, and the deposit of pheromone by one or several ants. A

1

IRIDIA – Technical Report Series: TR/IRIDIA/2011-006 2

detailed description of pheromone update procedures of ACO algorithms can befound in [5].

In the population-based ACO (P-ACO) algorithm, the pheromone matrixvalues change according to a population of solutions. This population of solu-tions has the role of an archive of solutions generated by the colony of ants.Every time a new solution enters this archive, a fixed amount of pheromoneis deposited on its corresponding components in the pheromone matrix. Thepheromone deposited by a solution is removed from its components when itleaves the solution archive.

The mechanism used to deposit and remove the pheromone in P-ACO re-sults in a faster pheromone update procedure than in usual ACO algorithmssuch as Ant System orMAX–MIN Ant System (MMAS). For example, whenapplied to the traveling salesman problem (TSP) and to the quadratic assign-ment problem (QAP) on an instance of size n, the pheromone update in theP-ACO algorithm can be computed in O(n) steps, while for Ant System basedalgorithms, it takes O(n2) steps. The criteria used to define which solution andwhen it should enter or leave the solution archive is important for the searchbehavior of P-ACO. Different strategies to define these criteria were proposedin [7].

In this paper, our main objective is to deepen the understanding of the P-ACO behavior for tackling the TSP and the QAP. Our motivation arose fromthe work done by Guntsch [7], in which the author introduces the P-ACO al-gorithm. In fact, P-ACO has mainly been studied by Guntsch and a few otherauthors only [1,10,14,11], despite its potential interest. In this paper, we inves-tigate the algorithm’s behavior with different settings for its specific parametersin dependence of the computation time invested, which has not been done previ-ously. We also study the effect of using different solution selection strategies andthe impact of local search on P-ACO’s parameters and performance. Further-more, we have improved the P-ACO with the implementation of a pheromonematrix restart procedure. With regard to how efficient P-ACO is, we providesome statistical analysis and comparison with MMAS, a state-of-the-art ACOalgorithm for the TSP and the QAP [12].

This paper is organized as follows. In Section 2, we describe the P-ACOpheromone update procedure and give its specific features. We present anddiscuss the experiments set up and results in Section 3. In Section 4, we presentthe statistical analysis of the improved P-ACO algorithm and MMAS. Finally,in Section 5 we draw the conclusions of this work.

2 Population Based ACO

P-ACO introduces a new memory and pheromone update [7, 8, 9] but it keepsthe same solution construction and the objective function evaluation mechanismas used in most ACO algorithms [5]. The pheromone update operates on twodata structures: the pheromone matrix and the solution archive. The algorithmstarts with an empty solution archive, denoted by P , and all the entries ofthe pheromone matrix have a initial value of τ0, which has the same effect asthe minimum pheromone trail limit in MMAS [12]. The maximum number ofsolutions in the solution archive is K. After each iteration, a solution π may beadded to the solution archive. The pheromone matrix is updated reflecting the


solution components contained in π, biased by a value ∆ and a weight wk.The general form of the PACO pheromone matrix is as follows:

τij = τ0 + ∆|P |∑k=1

wk · Ikij , (1)

where∆ =

τmax − τ0K

; (2)

τmax is a parameter and Ikij is an indicator function that is equal to one if the

solution component ij is present in ant k and zero otherwise. The pheromonematrix is updated incrementally. Each time a solution enters P , the amount∆ · wk is added to the pheromone matrix.

When the solution archive reaches |P |=K solutions, every new solution re-places another solution in the population matrix. When this replacement hap-pens, all the pheromone of solution components of the solution that is removedfrom P are updated by −∆ ·wk. The mechanism used to manage the populationof solutions, that is, how and when a solution should enter and leave the pop-ulation matrix, gives rise to the pheromone update strategy, which is anotherimportant factor that may determine the algorithm’s performance.

Several strategies were proposed in the literature in order to improve P-ACO. In this paper, we analyzed three of the five strategies proposed in [7]: theAge-based Strategy, the Quality-based Strategy and the Elitist-based Strategy.The remaining strategies are the Probabilistic Quality-based Strategy and theAge&Prob combination. Both are not studied in this paper because they aresimple variations of the strategies mentioned above. These two strategies weresuggested in order to tackle the problem of having a homogeneous population,especially for the Quality-based Strategy. Despite suggesting several strategiesto improve the performance of P-ACO, Guntsch focuses his experiments on theAge-based strategy. We will provide our results of using the three strategies asmentioned above in Section 3.3.

Age-based Strategy. This strategy follows a FIFO (first in first out) queuebehavior. After each iteration, the iteration best solution enters the solutionarchive. If the archive is full, the new solution replaces the one in P thatentered the archive longest ago. Pheromone deposit is done on the componentsof all solutions that enter the archive and the pheromone removal is done on thecomponents of the solutions that leave the archive.

Quality-based Strategy. After each iteration, the iteration best solutionis allowed to enter the solution archive only if its solution quality is better thanthe worst in P . Thus, if the new solution π is worse than all of the solutions inP , the solution archive remains unchanged. When P is full, the new solutionreplaces the worst one in P at that moment. Otherwise, the pheromone updatefollows the same mechanism used by the age-based strategy.

Elitist-based Strategy. This strategy keeps track of the best solutionsfound during a run. Each time a new global best solution e is found, the elitistupdate is applied. This means that, at weight we ∈ [0, 1], the elitist update isemployed on the global best solution with a weight we · τmax and the remainingsolutions of the population are updated with a weight ∆ · (1− we) / (K − 1).


3 Experiments

In an attempt to support our objectives in this paper, several experiments wereperformed with and without 3-opt local search on the TSP instances and withand without 2-opt local search on the QAP instances. The constant parame-ters used in P-ACO for the TSP are the same as in [7]: m = 10 ants, β = 2and α = 1. For MMAS without local search, we set m = n/4 where n is theinstance size, β = 2, α = 1, ρ = 0.02 for small size instances, ρ = 0.05 formedium size instances and ρ = 0.2 for large size instances. When local searchis applied, the following values are set in MMAS: m = 25 and ρ = 0.2. TheTSP instances used in this paper were taken from the TSPLIB: eil101, d198,kroA200, rd400, d657, u724, pcb1173, u1817,d2103, u2319 and are available athttp://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/tsp/. Wegroup the first three instances as small instances, the following three as mediumsize instances, and the remaining ones as large instances. All results reportedfor P-ACO and the other algorithms, when applied to the TSP, were imple-mented and executed on top of the ACOTSP software package available athttp://www.aco-metaheuristic.org/aco-code/. Hence, the P-ACO algorithmtested in this paper shares the same data structure and speed up techniquesas MMAS.

Unless stated otherwise, the parameter settings used for the algorithms onsolving the QAP are the same as those used in the TSP. For MMAS we set ρ= 0.4 regardless of whether or not local search is executed. The QAP instancesused are kra30a, wil50, tai80b, tai100b, taken from the QAPLIB, and are avail-able at http://www.opt.math.tu-graz.ac.at/qaplib/inst.html. A largerinstance was also studied, es300, taken from a set of benchmark instances athttp://iridia.ulb.ac.be/supp/IridiaSupp2010-009/.

Experiments conducted in this work have been run on Intel Xeon 2.4 Ghzquad-core CPUs with 6MB cache and 8 GB of RAM running under ClusterRocks Linux. The algorithms studied are coded in C and compiled with gccversion 4.1.2. For the TSP, we set the stopping criteria at 100 seconds forsmall instances, 1000 seconds for medium instances, and 2000 seconds for largeinstances. For the QAP, the stopping criteria is based on the time needed torun our re-implementation of Taillard’s Robust Tabu Search [13] to perform10000 · n iterations.

This Section is divided into 3 subsections: Subsection 3.1 is the one in whichwe compare the impact between the time spent by the pheromone update pro-cedure and the solution construction procedure on P-ACO, MMAS and ACS.In Subsection 3.2 we analyzes the P-ACO specific parameters. Finally, in Sub-section 3.3 we propose an improvement of P-ACO.

It is important to note that the plots chosen to be presented in this paperare just a sample set to represent our findings. Other plots can be found athttp://anonymous/supp/tobeupdated.

3.1 Impact of the Pheromone Update on ComputationTime

The pheromone update mechanism of P-ACO is typically faster than that ofother ACO algorithms. In order to measure the impact between the time spentby the pheromone update procedure and the solution construction procedure, we

http://www.opt.math.tu-graz.ac.at/qaplib/inst.html

http://iridia.ulb.ac.be/supp/IridiaSupp2010-009/

http://anonymous/supp/tobeupdated


P-ACO ACS MMAS P-ACO+ls ACS+ls MMAS+lsInstance %cst %ph %cst %ph %cst %ph %cst %ph %cst %ph %cst %phd198 95.64 4.09 81.45 18.02 43.32 55.81 98.53 0.61 93.90 6.05 97.08 1.89eil101 93.63 5.87 83.02 16.86 47.14 52.56 97.20 0.85 95.50 4.17 97.40 1.24kroA200 94.77 4.86 77.77 22.02 42.70 56.98 97.91 0.85 95.43 4.26 97.44 1.47rd400 94.06 5.06 79.67 20.09 28.71 70.03 98.41 0.60 94.17 5.62 95.76 3.09pcb1173 94.53 5.02 77.49 22.34 11.11 88.21 97.41 1.57 93.15 6.08 95.75 3.86u2319 92.45 4.96 76.22 23.64 9.04 88.47 96.01 3.09 89.76 8.17 90.93 6.36

P-ACO MMAS P-ACO+ls MMAS+lsInstance %cst %ph %cst %ph %cst %ph %cst %phkra30a 99.70 0.30 90.46 9.09 99.99 0.01 99.46 0.54wil50 99.61 0.38 90.08 9.92 99.99 0.01 99.73 0.27tai80b 99.74 0.26 90.51 9.49 99.99 0.01 99.80 0.20tai100b 99.76 0.24 90.52 9.46 99.99 0.01 99.83 0.16es300 99.69 0.31 88.92 11.06 99.99 0.01 99.91 0.09

Table 1: gprof results summary - Percentage of computation time used forsolution construction (including local search if executed), %cst and pheromoneupdate, %ph. Two variants of P-ACO, ACS, MMAS (without local search, andwith local search,+ls) were used. Note that in most cases the total computationtime does not sum up to 100% due to other functions used by the code.

run P-ACO and other ACO algorithms using a profiling tool.1 Each algorithmwas executed only once. P-ACO, MMAS and Ant Colony System (ACS) wereexecuted for the TSP. The comparison with ACS for the TSP is interestingbecause its pheromone update procedure also takes only O(n) steps as P-ACO,but with a larger constant hidden in the O(n) notation. For the QAP, the samealgorithms were considered except ACS, since ACS has not yet been applied tothe QAP.

The percentage of computation time for all functions related either to phero-mone update or solution construction (including local search if executed), weresummed and are presented in Table 1 for both the TSP and the QAP.

As expected, P-ACO spends much less time for the pheromone update whencompared to MMAS. The amount of computation time saved is particularlyimpressive when compared to MMAS on the TSP without a local search. Forlarge instances, up to almost 90% of MMAS’s computation time is spent bythe pheromone update (and not in generating solutions), due to the fact thatthe evaporation affects all pheromone entries, which are quadratically many.(The big impact of the pheromone update, is also due to the fact that thesolution construction is of almost linear complexity because of the exploitationof candidate sets.) For P-ACO, the computation time taken by the pheromoneupdate is always a small percentage (around 5% in the discussed case). P-ACO’s pheromone update is also much faster than ACS’s one. Once local searchis added, the advantage of the faster pheromone update by P-ACO is muchreduced, due to two reasons. First, simply because relatively the local searchtakes a significant amount of time; second, MMAS when used with local searchdoes not update the full pheromone matrix, but only to the candidate sets.P-ACO’s advantage with respect to computation time is nevertheless still clearin the TSP case, where tour length computation and construction is done in(almost) linear time. On the QAP, the situation changes quite a bit. Withoutlocal search, the speed advantage of P-ACO with respect to MMAS is much

1A profiler is a program used to check how much time is used by each function whileexecuting the code


reduced and with a local search, the percentage time spent for pheromone updateby P-ACO and MMAS is almost negligible. The difference is due to the factthat for the QAP, all computations (solution construction and objective functioncomputation) are quadratic (or cubic for the local search). Consequently therelative impact for the pheromone update becomes less important.

In summary, the speedup of the pheromone update through P-ACO is di-rectly dependent by two factors: (1) the use of a local search, and (2) theproblem to which the algorithm is going to be applied, in particular, the com-plexity of the computation of the objective function. From these experiments,P-ACO could be a good tool to be applied to problems in which local searchdoes not perform well like the longest common subsequence problem [3] and thefounder sequence reconstruction problem [2].

3.2 Analysis of P-ACO Specific Parameter Settings

The P-ACO age-based algorithm has three specific parameters K, τmax, andτmin. To understand how they influence the algorithm behavior, especially con-cerning the pheromone update procedure, we tested the algorithm with differentvalues of K and τmax. τmin is always set to 1/(n− 1), which is also the initialpheromone value τ0. Initially, the values we examine are the same as thoseproposed by Guntsch [7], that is, K = 1, 5, and 25, and τmax = 1, 3, and 10.However, some additional values are proposed throughout the paper, in order toclarify the observations taken from the results achieved by the initial parame-ters setting. For this analysis, we consider only the age based strategy, which isthe simplest method implemented to manage the solution archive. The impactof using different strategies to manage the solution archive and the algorithm’sperformance are studied later.

3.2.1 Analysis of K

The plots in Figure 1 show that, for both problems, without the use of localsearch, P-ACO performs the worst when K = 1, and the best when K = 25,followed closely by K = 5, with negligible differences. Similar behavior wasobserved across all instance sizes for both problems. This behavior can bejustified by the fact that P-ACO is an algorithm that focuses on exploitation.Thus, a larger solution archive enables the algorithm to better explore the searchspace.

P-ACO shows the opposite behavior when solving the TSP if local search isapplied. The best value of K for solving the TSP now becomes one. However,when it is applied to the QAP, P-ACO with a larger value of K performs better.For the TSP, we can conclude that the search intensification of a local searcharound one solution is enough and also important to guide the algorithm throughhigh-quality solutions. However, for the QAP, a certain diversity of solutions inthe solution archive is still needed.

Note that in Figure 1, τmax is 3 for the TSP while for the QAP it is also 3when local search is applied but 100 when it is not applied. The choice of theseparameter values is justified in the following.


1e!03 1e!01 1e+01

1.0

1.1

1.2

1.3

1.4

1.5

1.6

Computation time (log scale)

Co

st

pe

rce

nta

ge

devia

tio

n

K01

K05

K25

mmas

t_max = 03

1e!01 1e+00 1e+01 1e+02 1e+03

1.0

01

.02

1.0

41

.06

1.0

81

.10


Co

st

pe

rce

nta

ge

devia

tio

n

K01

K05

K25

mmas

t_max=03

0.01 0.05 0.50 5.00 50.00

1.0

01

.02

1.0

41

.06

1.0

81

.10


Co

st

perc

en

tag

e d

evia

tio

n

K01

K05

K25

mmas

t_max = 100

0.2 0.5 1.0 2.0 5.0 10.0 20.0 50.0 200.0

1.0

00

1.0

05

1.0

10

1.0

15

1.0

20

1.0

25


Co

st

perc

en

tag

e d

evia

tio

n

K01

K05

K25

mmas

t_max = 03

Figure 1: Solution quality over time of MMAS and P-ACO using di!erent values of K for the best setting of!max, without local search on instance d198 (for TSP, top-left plot) and on instance wil50 (QAP, bottom-leftplot), with local search on instance u1817 (TSP top-right plot) and on instance tai100b (QAP, bottom-rightplot).

without ls with ls

TSP

QAP

Figure 1: Solution quality over time of MMAS and P-ACO using different valuesof K for the best setting of τmax, without local search on instance d198 for TSPand on instance wil50 for QAP, with local search on instance u1817 for TSP,and on instance tai100b for QAP.

3.2.2 Analysis of τmax

The ratio between τmax and τmin for P-ACO is approximately τmax · n. Thus,the amount of pheromone used to deposit in the pheromone matrix increaseslinearly according to the instance size, the maximum pheromone trail beingτmax. Note that this behavior is similar to the one in MMAS. For more detailswe refer the reader to [12].

In Figure 2, we have an improvement on the final solution quality on the TSPwhen τmax is equal to 3 or 10. On the other hand, no substantial differences canbe seen with the use of a local search. The results show that τmax = 3 is able toimprove the algorithm’s solution quality for the best values of K in comparisonto τmax = 1. No improvement can be seen in the final solution quality for largervalues of τmax and when local search is applied. We tested also τmax = 50.

For the QAP, we have tested a wider range for τmax, where we set the valueto 1, 3, 10, 30 or 100. We found that increasing the value of τmax considerablyimprove the performance of P-ACO. τmax = 100 appears to be the best value forQAP without the use of local search. In case of P-ACO with local search, thebest value of τmax is much lower. We found that using τmax = 3 results in goodperformance. We also found that interdependency exists between the settingsof τmax and K. Without local search, P-ACO always obtains good results whenlarge τmax is paired with large K, while with local search, small τmax with largeK always gives good results.


1e!02 1e!01 1e+00 1e+01 1e+02

1.0

1.2

1.4

1.6

1.8


Cost perc

enta

ge d

evia

tion

mmas

tmax01

tmax03

tmax10

K=25

1e!02 1e+00 1e+02

1.0

01

.02

1.0

41

.06

1.0

8


Cost perc

enta

ge d

evia

tion

mmas

tmax01

tmax03

tmax10

K=01

0.01 0.05 0.50 5.00 50.00

1.0

01

.02

1.0

41

.06

1.0

81

.10

1.1

2


Cost perc

enta

ge d

evia

tion

mmas

taumax01

taumax10

taumax100

K = 25

0.5 1.0 2.0 5.0 10.0 20.0 50.0 100.0 200.0

1.0

00

1.0

05

1.0

10

1.0

15

1.0

20

1.0

25


Cost perc

enta

ge d

evia

tion

mmas

taumax01

taumax03

taumax10

K = 25

Figure 2: Solution quality over time of MMAS and P-ACO using di!erent values of !max for the best settingof K, without local search on instance kroA200 (TSP, top-left plot) and on instance wil50 (QAP, bottom-leftplot), with local search on instance u1817 (TSP, top-right plot) and on instance tai100b (QAP, bottom-rightplot).

without ls with ls

TSP

QAP

Figure 2: Solution quality over time of MMAS and P-ACO using different valuesof τmax for the best setting of K, without local search on instance kroA200 forTSP, and on instance wil50 for QAP, with local search on instance u1817 forTSP, and on instance tai100b for QAP.

Overall, the ratio between τmax and τmin influences, together with the settingof K, how strong the search intensification is. Given the the two factors we hadmentioned in the Subsection 3.1 about P-ACO speedup dependence, we cannow visualize (1) the speed advantage of P-ACO with respect to the QAP vs.the TSP is much reduced when local search is applied and (2) P-ACO showsdifferent behavior when applied to different problems.

3.3 Influence of using Different Strategies to Update theSolution Archive

Another particularity of P-ACO is the way it updates the solution archive. Toimprove the performance of P-ACO, different strategies to manage the solutionarchive have been proposed. In order to compare their influence on the algorithmperformance, we have tested three strategies from those presented in [7]. Allstrategies tested here were previously explained in Section 2. We have usedfor each strategy, the best values of K and τmax found in Subsections 3.2.1and 3.2.2.

For the TSP, τmax is set to 3, and K is set to 25 when local search is notapplied and to 1 when it is applied. For the QAP, τmax is set to 30 when localsearch search is not applied, and to 3 when local search is applied. The valueof K is always set to 25. The values chosen for τmax and K are the ones that


1e!02 1e+00 1e+02

1.0

1.2

1.4

1.6

1.8


Cost perc

enta

ge d

evia

tion

age

elitist

mmas

quality

1e!02 1e+00 1e+02

1.0

01

.01

1.0

21

.03

1.0

41

.05

1.0

6


Cost perc

enta

ge d

evia

tion

age

elitist

mmas

quality

1e!02 1e!01 1e+00 1e+01 1e+02

1.0

1.1

1.2

1.3

1.4


Cost perc

enta

ge d

evia

tion

age

elitist

mmas

quality

10 50 100 500 1000 5000

1.0

00

1.0

02

1.0

04

1.0

06

1.0

08

1.0

10

1.0

12


Cost perc

enta

ge d

evia

tion

age

elitist

mmas

quality

Figure 3: Solution quality over time of MMAS and P-ACO considering all the strategies tested, withoutlocal search on instance d198 (TSP, top-left plot) and on instance tai80b (QAP, bottom-left plot), with localsearch on instance pcb1173 (TSP, top-right plot) and on instance es300 (QAP, bottom-right plot).

without ls with ls

TSP

QAP

Figure 3: Solution quality over time of MMAS and P-ACO considering all thestrategies tested, without local search on instance d198 for TSP and on instancetai80b for QAP, with local search on instance pcb1173 for TSP, and on instancees300 for QAP.

provide the best tradeoff between the two values.To define the best weight for the elitist-based strategy, we tested three values

(we = 0.25, 0.5 and 0.75) for both problems. These weights are the same asthose proposed in [7].

In Figure 3, we present the solution development plot of P-ACO with thebest parameters among those tested for each population update strategy. For theTSP, the quality-based strategy shows the best performance. The elitist-basedstrategy with the right weight (in this case is 0.50 and 0.75), shows to obtaingood results too. From all instances tested, P-ACO often outperforms MMAS,independently of the strategy or whether local search is implemented, especiallywhen short computation time is used. We show the results for QAP in bothplots at the bottom of Figure 3. Without and with local search, MMAS obtainsbetter performance compared to P-ACO. Without local search, the performanceof the elitist-based strategy is almost as good as MMAS, while the other twostrategies are much worse. With the use of local search, P-ACO performs muchbetter, where all strategies performs almost as good as MMAS.

Using the pheromone update strategy as suggested by Guntsch [7], we man-age to obtain similar results. In many cases, P-ACO shows a better performancethan MMAS after a short computation time. A concrete comparison to [7] can-not be made because of the lack of actual data from the thesis.


1e!02 1e!01 1e+00 1e+01 1e+02

1.0

1.1

1.2

1.3

1.4

1.5

1.6


Cost perc

enta

ge d

evia

tion

mmas

qrestart

quality

1e!02 1e+00 1e+02

1.0

01.0

21.0

41.0

61.0

8


Cost perc

enta

ge d

evia

tion

mmas

qrestart

quality

0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 50.00

1.0

01.0

11.0

21.0

31.0

41.0

51.0

61.0

7


Cost perc

enta

ge d

evia

tion

mmas

qrestart

quality

1 2 5 10 20 50 100 200

1.0

00

1.0

05

1.0

10

1.0

15

1.0

20


Cost perc

enta

ge d

evia

tion

mmas

arestart

age

Figure 4: Solution quality over time of MMAS and P-ACO considering the quality strategy with the restartprocedure, without local search on instance d198 (TSP, top-left plot) and on instance wil50 (QAP, bottom-leftplot), with local search on instance pcb1173 (TSP, top-right plot) and on instance tai100b (QAP, bottom-rightplot).

without ls with ls

TSP

QAP

Figure 4: Solution quality over time of MMAS and P-ACO considering thequality strategy with the restart procedure, without local search on instanced198 for TSP, and on instance wil50 for QAP, with local search on instancepcb1173 for TSP, and on instance tai100b for QAP.

3.4 P-ACO with Restart

P-ACO is an algorithm that presents a strong exploitation capability and, hence,a fast convergence to a right quality solution. On the other hand, its explorationduring the search may be insufficient. Following other ACO algorithms such asMMAS we implemented a restart procedure.

It consists of re-initializing the pheromone matrix values to τ0 after a certainnumber of iterations r without improvement. We tested this procedure over arange values of r. For the TSP we tested the pheromone re-initialization after n,25n, 50n and 100n iterations when local search was not applied and after 100,250, 500 and 1000 iterations when it was applied. For the QAP, we examinedvalues of r of n, 25n, 50n and 100n without the use of local search, and of 10,25, 50 and 100 iterations, with the use of local search.

Plots in Figure 4 show the results achieved for solving the TSP, using r =50n when local search is not applied and r = 500 when local search is applied.For the QAP, r is set to 100n if no local search is applied, and r = 10 if localsearch is applied. We chose those limits as they were the ones in which thealgorithm shows the best overall results.

In Figure 4, an improvement in the performance of P-ACO can be seen whenwe compare the final solution quality achieved by the same algorithm with andwithout the restart procedure, especially when local search is applied. As shownin the previous section, the algorithm exploitation capability remains. However,


Inst. B.known P-ACO MMAS P-ACO+ls MMAS+lskroA100 21282 0.00 0.01 - -kroA150 26524 0.52 0.97 - -rat195 2323 0.52 0.59 - -pcb442 50778 0.92 2.05 - -d493 35002 1.92 2.40 - -rl1889 316536 - - 0.15 0.26u2152 64253 - - 0.18 0.13pr2392 378032 - - 0.16 0.13pcb3038 137694 - - 0.24 0.25fnl4461 182566 - - 0.33 0.36

Inst. B.known P-ACO MMAS P-ACO+ls MMAS+lssko42 15812 1.71 2.44 0.00 0.00sko72 66256 1.55 1.88 0.09 0.02tai100a 21125314 2.94 3.04 2.04 0.99wil100 273038 0.68 0.91 0.02 0.02bl100 9272 5.96 6.20 4.62 2.11ci100 523146366 2.36 2.92 0.48 0.27dre132 4380 30.44 32.03 24.56 8.41bl144 13472 7.11 7.56 5.51 2.35ci144 795009899 3.01 3.79 0.89 0.53tai175e 57540 73.97 66.75 3.67 1.49

Table 2: Deviation of the average solution quality obtained by P-ACO andMMAS across 30 trials from the best known solution to date. Instances withno publicly available best known solution (dre132 and tai175e) are comparedbased on the best solution found in our experiments. Entries in bold refer toentries with significant difference to the other algorithm.

the use of a restart procedure is able to improve P-ACO finally reached solutionquality.

4 Statistical Test

In this section, we compare the finally reached average solution quality betweenP-ACO and MMAS with and without local search when applied to the TSPand to the QAP. The percentage deviation is deemed significant if the p-valueaccording to pairwise Wilcoxon signed rank test with holm correction is lessthan 0.05. The P-ACO algorithms chosen to be tested are the same as theones presented in Section 3.4 as they are the P-ACO variants that achieve bestperformance. The instances that we used in this section are different from thoseused in the previous ones. For the TSP, they were also taken from TSPLIBand are shown in Table 2. For the QAP, the following instances are used:sko42, sko72, tai100a, and wil100 from QAPLIB; bl100, ci100, bl144, and ci144from the microarray instance set [4], and dre132 and tai175e from Drezner etal. [6]. The P-ACO and MMAS algorithm’s configuration remains the same aswe presented in Section 3.4.

The results presented in Table 2 show that for the TSP without local search,P-ACO performs better than MMAS except on instance kroA100. Nevertheless,the performance of MMAS and P-ACO are alike when applied to large instancestested with local search. When solving the QAP, P-ACO without local search


always obtains better results compared to MMAS, except on tai175e. In 9 out of10 instances, the differences are significant. With the use of local search, in mostcases, MMAS performs slightly better than P-ACO, except on instance dre132,where the difference is rather large. In 8 out of 10 instances, the differences aresignificant.

Overall, these results shows that P-ACO with the restart procedure appearsto be competitive to MMAS, independent of the local search application.

5 Conclusions

In this article, we have discussed and analyzed the P-ACO algorithm for the TSPand the QAP. Extensive experiments were conducted to investigate the behaviorof P-ACO when different parameter settings and pheromone update strategiesare used. A significant part of the comparison between P-ACO and MMASrevealed that the pheromone update of the former is much faster than the latter,for the TSP. For the QAP on the other hand, the speed improvement by P-ACOis negligible, since the computation time is utilized for the solution construction,which is much more expensive than for the TSP. We have considered differentpopulation update strategies for P-ACO to investigate the effect of using thestrategies for solving the TSP and the QAP. We found that the quality-basedstrategy is the best option for solving the TSP, while for the QAP, the strategyselection depends on the use of local search. Without local search, the qualitybased strategy appears to provide the best overall tradeoff, while with localsearch, the age-based strategy appears to be a better choice.

New are also the insights that the usage or not of a local search has strongimpact on parameters settings for P-ACO applied to the TSP. For example,for the case with local search, a population size of one is clearly the best forP-ACO. For the QAP without local search, a very large setting of τmax, muchlarger than considered previously, resulted in good performance.

Extensive analysis of the development of solution quality over time showsthat in many cases P-ACO outperforms MMAS for short computation times.P-ACO, however, shows an early stagnation behavior compared to MMAS. Toovercome this, we have introduced a restart mechanism, and this has improvedsignificantly the overall performance of P-ACO.

As a main result we can conclude that, with the restart procedure and theright configuration, P-ACO is competitive to the state-of-the-art ACO algo-rithms with the advantage of finding good solution quality in a shorter compu-tation time.

References

[1] D. Angus and C. Woodward. Multiple objective ant colony optimisation.Swarm Intelligence, 3(1):69–85, 2009.

[2] S. Benedettini, C. Blum, and A. Roli. A randomized iterated greedy al-gorithm for the founder sequence reconstruction problem. In LION, pages37–51, 2010.


[3] C. Blum, M. J. Blesa, and M. Lopez-Ibanez. Beam search for thelongest common subsequence problem. Computers & Operations Research,36(12):3178–3186, 2009.

[4] S. A. de Carvalho Jr. and S. Rahmann. Microarray layout as a quadraticassignment problem. In Proc. German Conference on Bioinformatics, vol-ume P-83 of Lecture Notes in Informatics, pages 11–20, 2006.

[5] M. Dorigo and T. Stutzle. Ant Colony Optimization. MIT Press, Cam-bridge, MA, 2004.

[6] Z. Drezner, P. Hahn, and E. D. Taillard. A study of quadratic assignmentproblem instances that are difficult for meta-heuristic methods. Annals ofOperations Research, 2003.

[7] M. Guntsch. Ant Algorithms in Stochastic and Multi-Criteria Environ-ments. PhD thesis, Universitat Fridericiana zu Karlsruhe, 2004.

[8] M. Guntsch and M. Middendorf. Applying population based ACO to dy-namic optimization problems. In M. Dorigo et al., editors, ANTS 2002,volume 2463 of LNCS, pages 111–122. Springer, Heidelberg, 2002.

[9] M. Guntsch and M. Middendorf. A population based approach for ACO.In S. Cagnoni et al., editors, EvoWorkshops 2002, volume 2279 of LNCS,pages 71–80. Springer, Heidelberg, 2002.

[10] M. Guntsch, B. Scheuermann, H. Schmeck, M. Middendorf, and O. Diessel.Population based ant colony optimization on FPGA. In P. Leong andW. Luk, editors, FPT’02, pages 125–132. IEEE Press, Piscataway, NJ,2002.

[11] B. Scheuermann, K. So, M. Guntsch, M. Middendorf, O. Diessel, H. A.ElGindy, and H. Schmeck. FPGA implementation of population-based antcolony optimization. Appl. Soft Comput., 4(3):303–322, 2004.

[12] T. Stutzle and H. H. Hoos. MAX–MIN Ant System. Future GenerationComputer Systems, 16(8):889–914, 2000.

[13] E. D. Taillard. Robust taboo search for the quadratic assignment problem.Parallel Computing, 17:443–455, 1991.

[14] T. Thalheim, D. Merkle, and M. Middendorf. A hybrid population basedACO algorithm for protein folding. In S. I. A. et al., editor, IMECS ’08,Lecture Notes in Engineering and Computer Science, pages 200–205. Inter-national Association of Engineers, Newswood Limited, 2008.

a detailed analysis of the population-based ant colony...

Documents