An Analysis of Algorithmic Components for Multiobjective ...

An Analysis of Algorithmic Components for

Multiobjective Ant Colony Optimization: ACase Study on the Biobjective TSP

Manuel Lopez-Ibanez and Thomas Stutzle

IRIDIA, CoDE, Universite Libre de Bruxelles, Brussels, [email protected], [email protected]

Abstract. In many practical problems, several conflicting criteria existfor evaluating solutions. In recent years, strong research efforts have beenmade to develop efficient algorithmic techniques for tackling such multi-objective optimization problems. Many of these algorithms are exten-sions of well-known metaheuristics. In particular, over the last few years,several extensions of ant colony optimization (ACO) algorithms havebeen proposed for solving multi-objective problems. These extensionsoften propose multiple answers to algorithmic design questions arisingin a multi-objective ACO approach. However, the benefits of each one ofthese answers are rarely examined against alternative approaches. Thisarticle reports results of an empirical research effort aimed at analyzingthe components of ACO algorithms for tackling multi-objective combi-natorial problems. We use the bi-objective travelling salesman problemas a case study of the effect of algorithmic components and their possibleinteractions on performance. Examples of design choices are the use oflocal search, the use of one versus several pheromone matrices, and theuse of one or several ant colonies.

Keywords: Multiobjective Optimization, Ant Colony Optimization,Travelling Salesman Problem.

1 Introduction

Ant colony optimization (ACO) [6] is a general-purpose stochastic local search(SLS) method [9] inspired by the pheromone trail laying and following behaviorof some real ant species. The main application area of ACO is NP-hard com-binatorial optimization problems. Due to the practical relevance of this class ofproblems and the high performance reached by ACO algorithms, the number ofapplications of ACO algorithms has risen strongly over the recent years [6]. Inmany practically relevant problems, the quality of candidate solutions is evalu-ated with respect to various, often conflicting objectives. It is therefore not sur-prising that several extensions of ACO algorithms have been proposed to tacklesuch multi-objective combinatorial optimization problems (MCOPs). Two re-cent reviews [3,7] describe a few tens of papers that deal with multi-objectiveACO (MOACO) algorithms for MCOPs defined in terms of Pareto optimality.

P. Collet et al. (Eds.): EA 2009, LNCS 5975, pp. 134–145, 2010.c© Springer-Verlag Berlin Heidelberg 2010

An Analysis of Algorithmic Components for MOACO 135

Nevertheless, surprisingly little research on MOACO is targeted towards under-standing the contribution to performance of the specific design choices made inthese algorithms. Such design choices concern the definition of pheromone andheuristic information, the aggregation of specific information for each objective,different ways of selecting solutions for the pheromone update, the use of multi-ple ant colonies to specialize on specific areas of the Pareto front, and so on. Infact, most articles propose one specific way of tackling MCOPs by an ACO al-gorithm [5,4]; rare are comparisons of several MOACO algorithms [7] or studiesthat compare few design alternatives [1,2]. In fact, we believe that our previousarticle on an experimental analysis of MOACO algorithms for the biobjectivequadratic assignment problem (BQAP) [12] is one of the most complete con-cerning the systematic experimental comparison of specific design alternativesof MOACO algorithms.

In this article, we build and extend upon our earlier work in the area of theexperimental analysis of MOACO algorithms. As in our previous work, we use acomponent-wise view of the design of MOACO algorithms, but here we extendthe analysis in various ways. First, we use more advanced tools for the empiri-cal analysis of the behavior of multi-objective optimizers. In particular, we baseour analysis on the attainment function methodology [8], and we use graphicalillustrations to examine where in the objective space the various MOACO algo-rithms differ in performance [11,13]. This gives a detailed view of the impact ofspecific MOACO components on performance. Second, we tackle another prob-lem: the bi-objective traveling salesman problem (BTSP). The BTSP differs verystrongly from the BQAP in search space characteristics [16], and the solutionevaluation in the BTSP is much faster than in the BQAP. We also apply to theBTSP the usual algorithmic speed-up techniques for the single-objective TSPsuch as candidate lists [14]. Third, some of the algorithm details studied havenot been considered previously. In fact, the results we present here are part of alarger research effort that aims at a detailed analysis of the impact of MOACOcomponents on algorithm performance.

The article is structured as follows. In Section 2 we give basic definitions andintroduce the BTSP. Section 3 reviews concisely available ACO algorithms forMCOPs tackled in the Pareto sense and introduces the algorithmic componentswe study. We detail the experimental setup in Section 4 and present results inSection 5. We end with some concluding remarks in Section 6.

2 Multiobjective Combinatorial Optimization

In this article, we focus on MCOPs, where candidate solutions are evaluated byan objective function vector f = (f1, . . . , fd) with d objectives. MCOPs are oftensolved without a priori assumptions on the preferences of the decision maker. Insuch a case, the goal is to determine a set of feasible solutions that “minimizes”the objective function vector f according to the Pareto optimality criteria. Letu and v be vectors in Rd. We say that u dominates v (u ≺ v) iff u �= v andui ≤ vi, i = 1, . . . , d. Furthermore, u and v are nondominated iff u ⊀ v and

136 M. Lopez-Ibanez and T. Stutzle

v ⊀ u. To simplify the notation, we also say that a feasible solution s dominatesanother solution s′ iff f(s) ≺ f (s′). A solution s is a Pareto optimum iff noother feasible solution s′ exists such that f(s′) ≺ f (s). The goal in MCOPs thentypically is to determine the set of all Pareto-optimal solutions. However, thistask is often computationally intractable, and therefore it is usually preferableto approximate the Pareto set as well as possible in a given amount of time. Suchan approximation is always a set of solutions that are mutually nondominated.

In this paper, we deal with the BTSP, which is a direct extension of the widelystudied single-objective TSP with two cost values between each pair of distinctcities. A BTSP instance is defined by a complete graph G = (V, A) with n = |V |nodes {v1, . . . , vn}, and a set of arcs A that fully connects the graph. Each arc hasan associated cost vector whose components are c1(vi, vj) and c2(vi, vj), i �= j.Here we assume that instances are symmetric, that is cq(vi, vj) = cq(vj , vi),i �= j, q = 1, 2. The goal in the BTSP is to find the set of Hamiltonian toursp = (p1, . . . , pn) “minimizing” in terms of Pareto optimality the total tour cost,which is given by

fq(p) = cq

(vp(n), vp(1)

)+

n−1∑

i=1

cq

(vp(i), vp(i+1)

)q = 1, 2.

3 Multi-Objective Ant Colony Optimization

The first extensions of ACO for MCOPs were proposed at the end of the 90s.While various early approaches targeted problems where the objectives can beordered lexicographically or preferences of a decision maker are known, recentoverviews [3,7] show that most proposals of MOACO algorithms tackle multi-objective problems in terms of Pareto optimization.

3.1 Available MOACO Algorithms

All proposed MOACO algorithms somehow try to modify underlying ACO com-ponents so as to allow the algorithm to direct the search towards the differentregions of the Pareto front simultaneously. A main question is how to representthe solution components of different areas of the front in the form of pheromones.Essentially, the alternatives are to use one pheromone matrix [4,10] or severalpheromone matrices [5,1,10]. In the latter case, typically each objective has as-sociated one pheromone matrix. Related to this decision is the choice of the antsthat deposit pheromone. Typically, if one pheromone matrix is used, some or allnondominated solutions are selected for update [10], whereas when using severalmatrices, some elitist choices w.r.t. the objectives represented by the pheromonematrices are done [5,1]. During the solution construction, multiple pheromoneor heuristic matrices are commonly combined by means of weights [5,10].

Most papers concerning MOACO approaches do not study the various al-ternative algorithmic components, and, hence, little insight about the impact ofspecific choices is available. The following articles are, to some extent, exceptions.


Iredi et al. [10] explored some few design options regarding multiple colonies fora specific scheduling problem. Alaya et al. [1] studied four combinations of thenumber of pheromone matrices and the number of colonies for a multi-objectiveknapsack problem. The review article by Garcıa-Martınez et al. [7] experimen-tally compared various complete MOACO algorithms, rather than individualalgorithmic components. They used the BTSP as a benchmark problem, butthey did not apply usual algorithmic techniques such as candidate lists or localsearch. We therefore decided to start a significant research effort to analyze inmore detail the impact various MOACO algorithm components may have onperformance, extending in a strong way beyond our initial efforts [12].

3.2 Studied Algorithm Components

Based on our earlier experience with MOACO algorithms and the various propos-als made in the literature, we decided to study the following main algorithmiccomponents for composing MOACO algorithms for the BTSP. Note that forsome components, only specific levels of other components make sense. This isindicated below where necessary.Local Search. It is well known that local search has a strong influence onperformance for single-objective ACO algorithms [6]. Hence, we study the im-pact of local search for the BTSP by means of an iterative improvement algo-rithm based on the 2–exchange neighborhood. Our local search algorithm usesa weighted sum combination of the two objectives and exploits standard TSPspeed-up techniques (candidate sets are reordered for each weight vector).Pheromone Information. We use two levels for this component, either onepheromone matrix or two pheromone matrices. If one pheromone matrix is used,the solution construction exploits this pheromone matrix as usual in single-objective ACO algorithms. However, two heuristic values η1

ij , η2ij are available

for each arc (i, j), one for each distance matrix. We decided to aggregate themby giving equal weight to each: ηij = 0.5·η1

ij +0.5·η2ij. When multiple pheromone

matrices are utilized, each matrix is associated to one objective, and they areaggregated by means of weights. The MOACO literature contains examples ofboth linear versus non-linear aggregations. Here, we use a linear aggregation,since in our underlying ACO algorithm we ensure that the pheromones are inthe same range. Hence, the probability of choosing city j directly after city i forant k is given by:

pkij =

((1 − λk)τ1

ij + λkτ2ij

)α · ((1 − λk)η1ij + λkη2

ij

)β

∑l∈Nk

i((1 − λk)τ1

il + λkτ2il)

α · ((1 − λk)η1il + λkη2

il)β

if j ∈ N ki ,

where N ki is the feasible neighborhood of ant k, that is, those cities not visited

yet by ant k, τ1ij , τ2

ij are the pheromone trails for arc (i, j) for either objective,η1

ij , η2ij are the corresponding heuristic values, and λk is a weight.

Weight Setting Strategies. Whenever weights are used, either for aggregatingmultiple pheromone matrices or for performing a single-objective local search


over a scalarization of the multi-objective problem, several strategies can beapplied for setting the weights used at each iteration of the algorithm. A possiblestrategy is to define as many weights as ants. Then, each ant may be assigned adifferent weight from the other ants at each iteration. Iredi et al. [10] proposedthis approach. Alternatively, we can assign to all ants the same weight but modifythe weight slightly at each iteration. This latter strategy is more similar to theapproach followed by two-phase local search [15]. If an ant is assigned a weightvector, the subsequent local search will use the same weight vector.Pheromone deposit. This algorithmic component is mostly defined by theprevious choice of pheromone information. In the case of one pheromone matrix,typically the nondominated solutions are allowed to deposit pheromones. In thecase of multiple pheromone matrices, some elitist strategy is usually followed.Here, we use the method of updating each pheromone matrix with the best solu-tion for each objective. As for whether the iteration-best or global-best solutionsare used for updating, we follow directly the strategy of the underlying ACOalgorithm, MAX -MIN Ant System (MMAS) [18].Multiple Colonies. Multiple colonies are typically used to better direct thesearch towards specific areas of the Pareto front. Each colony has its own pher-omone information, that is, either one or two pheromone matrices. The totalnumber of ants is divided equally between the colonies. Solutions are placed in acommon nondominated archive, hence, colonies influence each other by sharinginformation in order to discard dominated solutions. Then, the pheromone ofeach colony is updated with solutions generated by ants from the same colony(update by origin) [10]. The cooperation among the colonies can be made strongerby exchanging solutions. One such approach is update by region [10], where thecommon archive is divided into as many parts as colonies, and each part is usedto update the pheromone information of a different colony. In the bi-objectivecase, this can be implemented by sorting the common Pareto front according tothe first objective, and subdividing it into smaller fronts of equal size.

4 Experimental Setup

The multi-objective ACO algorithms are implemented in C based on ACOTSP[17], and compiled with gcc, version 3.4. Experiments are carried out on AMDOpteronTM 2216 dual-core processors running at 2.4 GHz with 2 MB L2-Cacheand 4 GB RAM under Rocks Cluster GNU/Linux. Due to the sequential imple-mentation of the code, only one core is used for running the executable.

We wish to focus on the multi-objective components. Hence, we use a commonunderlying ACO algorithm, MMAS [18], for the management of pheromones(evaporation, pheromone trail limits, etc.). Basic parameters are set to α = 1,β = 2, Δτ = 1, τ0 = τmax, where Δτ is the amount of pheromone deposited byan ant and τ0 is the initial value of the pheromone. The evaporation factor isρ = 0.2 if 2-opt local search is used, otherwise it is ρ = 0.05. The total numberof ants is m = 30. Since the use of candidate lists is essential for obtaining high-quality solutions for the single-objective TSP, we also use them for the BTSP


by sorting edges according to dominance ranking [14]. The size of the candidatelist is 40.

We consider the algorithmic components described in the previous section.When weights are needed, that is, when using multiple pheromone matrices orlocal search, we use m maximally dispersed weight vectors, where each com-ponent of the weight vectors is in the interval [0, 1] and the sum of the twocomponents is equal to one. We test the two weight setting strategies explainedabove. These weight setting strategies are only relevant for components utilizingweights. In fact, they do not apply if only one pheromone matrix is used andno local search is done. We test different number of colonies ∈ {1, 3, 5}, withants equally divided among the colonies (m/). We tested both update by originand update by region strategies explained earlier. We found a slight advantage infavor of update by region, and, hence, we focus on the results obtained by thisapproach.

Each experiment was run for a time limit of 300 seconds and repeated 25 timeswith different random seeds. We performed experiments on 3 Euclidean BTSPinstances of 500 cities. Two were generated taking a single-objective Euclideaninstance, and altering the coordinates of the cities by a correlated random noiseto create a new distance matrix. This way, we generated a positively correlatedinstance, with a correlation between the distance matrices of 0.77, and an esti-mated correlation between the objectives of 0.81, measured by generating 10 000random solutions. We also generated a zero correlated instance with a correla-tion between the distance matrices of 0.01, and an estimated correlation betweenthe objectives of −0.01. In addition, we tested the instance euclidAB500 withcorrelation 0.01 available from http://eden.dei.uc.pt/~paquete/tsp/.

In contrast to previous results on the BQAP [12], the different correlation doesnot lead to large differences in the behavior of MOACO components. Hence, wediscuss only results obtained for the zero-correlated instance generated by us,but the same overall behavior was observed for the other instances.

5 Analysis of Experiments

Our goal is to identify fundamental differences between alternative componentsof multi-objective ACO algorithms, rather than obtaining the best configura-tion possible. For this purpose, we analyze the results by means of a graphicaltechnique [11,13] based on the empirical attainment function (EAF) [8].

The attainment function gives the probability of a particular point in the ob-jective space vector being attained by (dominated by or equal to) the outcomeof a single run of an algorithm. This probability can be estimated from severalruns of an algorithm, in order to calculate the empirical attainment function(EAF) of an algorithm. The EAFs of two algorithms can be compared by calcu-lating the difference in value of the EAFs for each point in the objective space.Differences in favor of one algorithm indicate that those points are more likelyto be attained by that algorithm than by its competitor, and, hence, that theperformance of that algorithm is better (in that region of the objective space)

http://eden.dei.uc.pt/~paquete/tsp/


Fig. 1. Differences in EAFs with and without local search, either using one (top) ortwo pheromone matrices (bottom), and one colony

than the performance of its competitor. We display this information graphicallyby plotting side-by-side the differences between the EAFs of two algorithms. Anexample is the plot in Fig. 3. Each side of the plot displays points in the objec-tive space where the difference in the value of the two EAFs is in favour of thecorresponding algorithm mentioned at the bottom. The points are colored de-pending on the size of the difference (points where the difference is smaller than0.2 are not shown). For example, a black point at the left side of the plot meansthat the corresponding algorithm has attained that point in (at least) 80% moreruns than the algorithm at the right side. The plots also display, as a continuousline, the grand best and the grand worst attainment surfaces, which respectivelydelimit the regions never attained by any algorithm and always attained by anyrun of the two algorithms. The median attainment surface of each algorithm,which delimits the region of the objective space attained by 50% of the runs, isalso shown in each side of the plot as a dashed line.

Here we discuss the main effects of the components we have studied, in orderof decreasing impact. The most important component in this sense is the use of


Fig. 2. Differences in EAFs between the two weight setting strategies. The algorithmsuse one colony, two pheromone matrices, and local search (top) or not (bottom).

local search. For the other components, we always give results with and withoutlocal search, since sometimes there are quite strong interactions observable. Inthe following, we present the results only for the BTSP. We applied the samealgorithms to the BQAP using the same instances as in our previous work [12];below we also comment shortly on the main findings for the BQAP, and indicatepossible differences from the BTSP results.

Component: Local Search. The strongest difference is caused by the use oflocal search. The plots in Fig. 1 compare the same configuration of MOACO withand without local search. The results in the top plot were obtained by using onepheromone matrix, whereas the bottom plot shows results using two pheromonematrices. In both cases, all differences are in favour of the variants using localsearch, showing that local search clearly improves the results obtained by ACO.This was the result in all our tests. A similar conclusion was obtained for theBQAP [12]. If we focus on the median attainment surfaces (dashed lines) on thebottom plot, the version without local search obtains solutions that are quite


Fig. 3. Differences in EAFs between one or two pheromone matrices. The comparisonis done without local search; one colony.

Fig. 4. Differences in the EAFs between one and multiple colonies. Results are shownfor one pheromone matrix (top) and two pheromone matrices (bottom).


close to those of the version with local search at the extremes of the Pareto frontbut very poor results in the center. This result suggests that handling the trade-off between objectives is difficult for ACO when using two pheromone matrices.With only one pheromone matrix (top plot), the ACO algorithm tries to handleextreme and trade-off solutions at a same time, and, as a consequence, the resultsare rather poor along the whole Pareto front.

Component: Weight Setting Strategies. Another notable observation is thedifference in results obtained by the two weight setting strategies: All ants usethe same weight at each iteration (TPLS-like strategy) versus each ant uses adifferent weight). The plots in Fig. 2 compare these two weight setting strategieswith (top plot) and without local search (bottom plot). Although the benefit ofthe TPLS-like strategy is clearly stronger by a large margin if no local search isused, even in the case with local search there is an important difference alongthe whole Pareto front. By comparison, for the BQAP, there is no noticeabledifference between weight setting strategies. In our experiments, we consistentlynoticed that the aggregation of multiple pheromones by using weights, which isnecessary for computing the probabilities of adding specific solution components,is an expensive operation. If each ant uses a different weight, this aggregationhas to be performed for each of the m ants. On the other hand, if all ants usethe same weight, the probabilities can be pre-computed once at each iteration.For the BTSP, the latter approach allows to perform ten times more iterationsthan the former within the same computation time.

Component: One vs. Two Pheromone Matrices. There were strong differ-ences in the results obtained when using one pheromone matrix (plus update bynon-dominated solutions) or two pheromone matrices (plus update by the bestsolution for each objective). Figure 3 compares both approaches when not usinglocal search. The first approach with only one pheromone matrix tends to pro-duce better results in the center of the Pareto front than the approach using twopheromone matrices. In contrast, when using two pheromones matrices, the re-sulting Pareto front is much better at the extremes of the Pareto frontier. If localsearch is used (not shown here), these structural differences disappear becausethe weighted local search explores the whole Pareto front explicitly. Moreover,when using local search, the approach using two pheromone matrices is betteralong the whole Pareto frontier, and in particular in the center of the Paretofront. Hence, there is a strong interaction between the components concerningthe pheromone matrices and local search.

Component: Number of Colonies. We observed, as well, differences in thestructure of Pareto frontiers obtained when using one or more colonies. Figure 4compares the use of one or more colonies for the approach based on one phero-mone matrix (top) or two pheromone matrices (bottom). As illustrated by thetop plot, when using one pheromone matrix, the results obtained with threecolonies are better in the extremes than with one colony. The opposite behavioris observed in the bottom plot when using two pheromone matrices. In this case,six colonies improve over the results obtained by only one colony, except for thevery extremes of the Pareto front.


Taking into account that using one pheromone matrix is good at the centerof the Pareto front but very poor at the extremes, the effect of the multiplecolonies is to improve the results at the extremes. On the other hand, using twopheromone matrices is very good at the extremes of the Pareto front, but quitepoor at the center. In this case, the use of multiple colonies improves the resultsat the center. Hence, one may conclude that the effect of multiple colonies is toovercome the core weakness of each approach. However, the multiple coloniesapproach is not better along the whole Pareto frontier than a single colony, be-cause each additional colony introduces an overhead in computation time and theavailable time is limited in our experiments. The benefits of additional colonieswere even stronger in the case of the BQAP [12], since the overhead introducedby each additional colony was smaller.

6 Conclusions

In this paper, we have presented an analysis of several key algorithm componentsfor the design of MOACO algorithms for the BTSP. We also have identifiedspecific components, where different behavior is induced as when to comparedto the BQAP. There are a number of ways of how this study can be extended.As next steps we plan to investigate the contribution of components of otherconcrete MOACO algorithms on performance both for the BTSP and BQAP.We also may consider comparisons to ACO algorithms that are embedded intohigher-level guidance strategies such as two-phase local search [15].

We hope to have made a step forward towards an understanding of the impactof MOACO algorithm components on performance; ultimately we hope thatthis leads towards informed design choices when applying MOACO to practicalapplications.

Acknowledgments. This work was supported by the META-X project, anAction de Recherche Concertee funded by the Scientific Research Directorate ofthe French Community of Belgium. Thomas Stutzle acknowledges support fromthe Belgian F.R.S.-FNRS, of which he is a Research Associate.

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