Pareto-based multi-colony multi-objective ant

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Pareto-based multi-colony multi-objective ant colony optimization algorithms: An island model proposal A.M. Mora · P. Garc´ıa-S´anchez · J.J. Merelo · P.A. Castillo

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Abstract Multi-objective algorithms are aimed to obtain a set of solutions, called Pareto set (PS), covering the whole Pareto front (PF), i.e. the representation of the optimal set of solutions. To this end, the algorithms should yield a wide amount of near-optimal solutions with a good diversity or spread along this front. This work presents a study on different coarse-grained distribution schemes dealing with Multi-Objective Ant Colony Optimization Algorithms (MOACOs). Two of them are a variation of independent multi-colony structures, respectively having a fixed number of ants in every subset or distributing the whole amount of ants into small sub-colonies. We introduce in this paper a third method: an island-based model where the colonies communicate by migrating ants, following a neighbourhood topology which fits to the search space. All the methods are aimed to cover the whole PF, thus each sub-colony or island tries to search for solutions in a limited area, complemented by the rest of colonies, in order to obtain a more diverse high-quality set of solutions. The models have been tested by implementing them considering three different MOACOs: two well-known and CHAC, an algorithm previously proposed by the authors. Three different instances of the bi-Criteria travelling salesman problem have been considered. The experiments have been performed in a parallel environment (inside a cluster platform), in order to get a time improvement. Moreover, the system scaling factor with respect to the number of processors will be also analyzed. The results show that the proposed Pareto-island model and its novel neighbourhood topology performs better than the other models, yielding a more diverse and more optimized set of solutions. Moreover, from the algorithmic point A.M. Mora, J.J. Merelo, P. Garc´ıa-S´anchez, P.A. Castillo ATC Department. University of Granada (Spain) E-mail: {amorag,pgarcia,jmerelo,pedro}@geneura.ugr.es

of view, the proposed algorithm, named CHAC, yields the best results on average. Keywords Ant Colony Optimization · Multi-Objective · MOACO · Multi-colony · Distributed algorithms · Coarsegrained · Island model · Pareto-based topology · Bi-criteria TSP

1 Introduction Multi-objective optimization problems (MOPs) are those where more than one independent function must be optimized at once. These functions are called objectives and the improvement of one does not mean an improvement in the rest. There is a set of the so-called non-dominated solutions (those which better satisfy each of the functions) rather than just one, which constitute the Pareto set (PS). The graphical representation of this Pareto-optimal set of solutions is named the Pareto front (PF). Thus, the main issue of a good multi-objective (MO) algorithm [9] is to obtain the maximum number of non-dominated solutions, having a high diversity and spread, and being as closer to the optimal set as possible. Ant colony optimization algorithms (ACOs) [17, 18] are bioinspired metaheuristics based in the behaviour of natural ants when searching for food, and the way they build shortest paths between their nest and any interesting point. Ants are modelled as a set of artificial agents which search in a weighted graph for the optimal paths. These agents cooperate to explore the search space and to reach the solution for a problem. Initially, ACO algorithms were created for single objective problems, but lately ACOs have also been applied to problems with several objectives; these are known as multi-objective ant colony optimization algorithms or MOACOs (see [24] for a survey).

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Most ACOs, and specially most MOACOs follow a single processor model. When approaching problems of a certain complexity, the distribution or parallelization of an algorithm usually has as aim the improvement of the running time for yielding an objective solution (if known), or for obtaining a certain degree of quality in the solution, if it is not the optimal. In some metaheuristics this distribution also imply a different searching scheme, which can lead to a different searching area of the space of solutions and thus, to a different solution (or set of solutions) for an optimization problem. This is the case of the ACOs which can be directly distributed due to the relative independence of the ants (they work independently, but all of them cooperate to find the solution). Some MOACO are based on multi-colony approaches, with independent subsets of ants working to find the best set of solutions, they are suitable to be easily distributed or parallelized without restrictions, since they can perform completely independent searches, sharing, once the have finished, the obtained Pareto sets. Following this approaches, the present work proposes a distribution model, named Pareto-based island model, which will spread the ants, grouped in colonies, into several computational nodes (normally processors in PCs or clusters). Each one will focus on a different area of the search space. Such a structure promotes the exploration and contributes to yield a better set of results than a mono-colony standard approach. Thus, a bigger amount of non-dominated solutions, with higher quality (closer to the optimal values), and with a better spread or distribution over the ideal set of solutions would be obtained. This method is inspired by the genetic algorithms (GAs) island scheme [6], where the colonies interchange (migrate) some ants during the algorithm running, adding diversity to the receivers colonies, and thus increasing the explorative component which turns on getting a better performance. This model defines a neighbourhood topology based on the Pareto front form, and is proposed for yielding better PSs, i.e. including more diverse, spread and closer to the optimal set non-dominated solutions. The model will be compared with two other methods based in independent multi-colony structures. All these models have been implemented as substrates for three different MOACOs from the literature. Two of them are well known approaches: Iredi’s et al. BIANT [27], and Bar´an’s et al. MOACS [3]; along with an algorithm previously presented by the authors named CHAC [41,42]. The distribution of the ants into sub-colonies or islands is called coarse-grained parallelization, i.e. at colony level, and have been implemented on a cluster of computers to get a running time improvement and to study the scaling of solution quality with the number of processors. The experiments have been performed considering a different number of computational nodes (processors), each of

A.M. Mora et al.

them containing an associated sub-colony or island. All methods have been applied to solve the Symmetric Bi-criteria Travelling Salesman Problem (bTSP from now on), which is the transformation into a two-criteria problem of the classical TSP [33]. In the single-objective TSP the target is to minimize the distance of a Hamiltonian cycle, while in this version there is a set of different costs between each pair of connected cities, which could correspond, for example, to distance and travel time. The three models have been tested in a wide array of experiments, comparing with each other and with respect sequential implementations. Several metrics and indicators usually calculated in multi-objective optimization studies have been considered, in order to analyze the profits in the quality, diversity and spread of the solutions of every approach. Moreover, the speedups and tradeoffs between solutions and time consumed to reach them, have been also analyzed. Statistical tools, such as the Analysis of variance (ANOVA) [20] and post-hoc Tukey’s Honestly Significant Difference tests (Tukey HSD) [13] have been applied in order to better support the reached conclusions. The rest of the paper is structured as follows: Section 2 introduces some concepts which are dealt in the work. The background and state of the art in this research area is presented in Section 3. The MOACOs to study are detailed in Section 4, in order to introduce the key parameter used in the work (λ ). Then, Section 5 describes the parallelization models considered. The performed experiments are presented in Section 6, along with the analysis of the results. Finally, conclusions and future work are commented in Section 7.

2 Preliminary concepts In this section, some initial concepts which will be extensively cited along the paper are introduced: the ant colony optimization metaheuristics and multi-objective optimization concepts.

2.1 Ant colony optimization The ant colony optimization (ACO) is a metaheuristic inspired by the naturally observed fact that some species of ants are able to find the shortest path from nest to food sources after a short time of exploring different paths between them [12]. This behaviour has been explained through the concept of stigmergy [25], that is, communication between agents using the environment, so every ant, while walking, deposits on the ground a substance called pheromone which the others can smell and which is evaporated after some time. One ant tends to follow strong concentrations of pheromone caused by repeated passes of ants; a pheromone trail is then formed

Pareto-based multi-colony MOACOs: An island model

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from nest to food source, so in intersections between several trails an ant moves with high probability following the highest pheromone level. ACO algorithms, introduced by Dorigo et al. [18] in 1991, take this model of ant behaviour as inspiration to solve combinatorial optimization problems using a colony of “artificial ants”, which are computational agents that communicate with each other through the environment using pheromones. One problem to be solved using ACO must be transformed into a graph with weighted edges. In every iteration, each ant builds a complete path (solution), by travelling through the graph. At the end of this construction (and in some versions, during it), each ant leaves a trail in the visited edges depending on the fitness of the solution it has found. This is a measure of desirability for that edge and it will be considered by the following ants. In order to guide its movement, each ant uses two kinds of information that will be combined: pheromone trails, which correspond to “learnt information” changed during the algorithm run, denoted by τ ; and heuristic knowledge, which is a measure of the desirability of moving to the next node, based in previous knowledge about the problem (it does not change during the algorithm run), denoted by η . The ants usually choose edges with better values in both properties, but sometimes they may “explore” new zones in the graph because the algorithm has a stochastic component, that broadens the search space to regions not previously explored. Due to all these properties, all ants cooperate in order to find the best solution for the problem (the best path in the graph), resulting in a global emergent behaviour. ACOs initially took two different variations: Ant System (AS) (Section 2.1.1) and Ant Colony System (ACS) (Section 2.1.2). Nowadays there are lots of variants and new approaches, based on these original algorithms, but applying other factors or parameters.

2.1.1 Ant System This was the first approach proposed by Dorigo et al. [15], it was initially designed for solving the classical Travelling Salesman Problem (TSP) [33]. Ant System builds solutions leaning strongly on the state transition rule (STR), since every ant uses it to decide which node j is the next in the construction of a solution (path), when the ant is at the node i. This formula calculates the probability associated to every node in the neighbourhood of i, as follows:        P(i, j) =

     

τ (i, j)α · η (i, j)β ∑ τ (i, u)α · η (i, u)β

i f j ∈ Ni

u∈Ni

0

(1) otherwise

where α and β are weighting parameters to set the relative importance of pheromone and heuristic information respectively, and Ni is the current feasible neighbourhood for the node i. In AS, the pheromone update is performed once all the ants have built their solutions. This updating is made at a global level by every ant, which retraces its solution path. It consists of an evaporation (left term) and a contribution (right term): τ (i, j)t = (1 − ρ ) · τ (i, j)t−1 + ∆ τ (i, j)t−1

(2)

t marks the new pheromone value and t-1 the old one. ρ in the range [0,1] is the common evaporation factor, and ∆ τ is the deposited amount of pheromone. As it can be seen in Equation 2, there are two phases: first, all the pheromone trails are reduced by a constant factor (evaporation), after this, every ant deposits an amount of pheromone in its path (in the graph edges) which depends on the quality of its solution (the better solution the ant has found, the higher amount of pheromone is added). 2.1.2 Ant Colony System ACS was presented by the same authors as AS [16] as an improvement over it. The first approach had three differences with regard to AS. The first one is the application of a different state transition rule (called pseudo-random proportional state transition rule), defined as: If (q ≤ q0 )

{

∑ τ (i, u)

j = arg max j∈Ni

Else

P(i, j) =

            

} α

β

· η (i, u)

(3)

u∈Ni

τ (i, j)α · η (i, j)β ∑ τ (i, u)α · η (i, u)β

i f j ∈ Ni

u∈Ni

0

(4) otherwise

where q is a random number in the range [0,1] and q0 is a parameter which set the balance between exploration and exploitation. If q ≤ q0 , the best node is chosen as the next one (exploitation), otherwise one of the feasible neighbours is selected, considering different probabilities for each one (exploration). The rest of the parameters are the same as in Equation 1. The second difference is that there is a global pheromone updating, which is only performed for the edges included in the best global solution: τ (i, j)t = (1 − ρ ) · τ (i, j)t−1 + ρ · ∆ τ (i, j)G

Best

∀(i, j) in SG

Best

(5)

Finally, there is also a local pheromone updating, which is performed by every ant every time a node j is added to the path which it is building. This formula is: τ (i, j)t = (1 − φ ) · τ (i, j)t−1 + φ · τ0

(6)

being φ in the range [0,1] the local evaporation factor, and τ0 the initial amount of pheromone, which corresponds to a

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lower trail limit. This formula results in an additional exploration technique, because it makes the edges traversed by an ant less attractive to the following ants and helps to avoid many ants following the same path. These two approaches, but specially ACS, have been adapted to particular problems mainly through the heuristic functions, and possibly also by changing the number of colonies or pheromone matrices. In particular, they can be used to approach multi-objective optimization problems, which will be introduced below. ACOs dealing with multiple optimization criteria are known as multi-objective ant colony optimization algorithms (MOACOs) [24].

2.2 Multi-objective optimization Multi-Criteria or Multi-Objective Optimization Problems (MOP) [44] are those where several objectives have to be simultaneously optimized. These problems do not have a single best solution, one that is better than any other with respect to every objective. Frequently improving the solution for one objective implies a worsening for another. Solving a multi-objective problem implies maximizing or minimizing a function f(x) composed by k cost functions (one per objective) and considering n parameters (decision variables): f (x) = (C1 (x),C2 (x), ...,Ck (x)) x = (x1 , x2 , ..., xn ) ∈ X

(7)

In a multi-objective optimization problem (MOP), there is a set of solutions that are better than the rest considering all the objectives; this set is known as the Pareto set (PS) [45]. They try to approach the ideal set of solutions which is usually represented graphically as the Pareto front (PF). These are related to an important concept, the dominance, defined as follows (a dominates b): a ≺ b if : ∀i ∈ 1, 2, ..., k | Ci (a) ≤ Ci (b) ∧ ∃ j ∈ 1, 2, ...k | C j (a) < C j (b)

(8)

where a ∈ X and b ∈ X are two different decision vectors of n values, and every C is a cost function (one per objective). If it intends to minimize the cost and Equation 8 is true, then b is dominated by a. Hence, the solutions in the PS are known as non-dominated solutions, while the remainder are known as dominated solutions. Since none of the solutions in the PS is absolutely better than the other non-dominated solutions, all of them are equally acceptable as regards the satisfaction of all the objectives. The interested reader is directed to [9] for a deeper examination of these concepts.

3 Background and state of the art The parallelization of ACO algorithms has been an open research field from the very first years of their introduction, since they are an intrinsically parallel method, based on a

set of independent agents. Thus, several parallel ACO approaches have been presented since then [28, 46], most of them devoted to address specific problems [36, 14, 1]. All the approaches mainly distribute the ants into several computational nodes (usually one node corresponds to one physical processor in a PC or cluster) following a different parallelization grain: fine-grained or ant-level implementations [4, 47] (every ant corresponds to one computational node), and coarse-grained or colony-level implementations [5,49, 34, 52] (every node contains a set of ants), which are so far the most extended. Typically, these models are centralized, following a master/slave architecture [50], which means that there is one node, called master process, which collects the solutions or the pheromone information from all the other nodes. After this, it performs the pheromone updating and computes the new pheromone matrix, which is then sent to the rest of nodes, known as slaves processes. On the other hand, in the decentralized approaches every node has to compute the pheromone update by itself, using information which has received from other nodes (if they communicate with any of the rest). The main goal of these approaches is to improve the running time without changing the optimization behaviour of the algorithm. Multi-colony model [28, 39], is the most extended so far model included in the decentralized architecture. In it the complete population of ants is divided into several sub-colonies which search independently for the solution. These colonies may interchange some information (pheromone, solutions) during the execution process. Two classical examples of multi-colony methods can be found in the works by Bullnheimer et al. [5], where the colonies share the pheromone trails every k iterations, concluding that the bigger the k value is, the faster the algorithm reaches the solution; and by St¨utzle [49] who perform a study by dividing the global number of iterations that a colony does into several subgroups of iterations executed in a set of independent subcolonies (no information is shared), obtaining the solution faster. Several other approaches have been proposed in the recent years, such as [36, 51, 53, 48]. Some other models and algorithms have been proposed in this line, as the multi-colony island model [37, 29] in which the colonies share the best solutions between them and substitute their own solutions if the received is better. Our work is focused in a model similar to this one, but enclosed in the multi-objective optimization area, which presents different properties as will be described below. Coarse-grained approaches have proved to perform better than the fine grained ones, since the latter should share big amounts of data (the pheromone information, for instance) through the network, so the improvement in running time reached by means of the parallelization is decompensated due to the communications delay. However all the ap-

Pareto-based multi-colony MOACOs: An island model

proaches present some problems which limit their performance: searching in a common space of solutions and the use of pheromone matrixes means a huge amount of data should be shared in order to improve the search. In addition, when the transfered information is just a solution (usually the best), some synchronization points are required and so, the computational time is increased as the communication frequency increases. Another line of research has been the improvement through massive parallelization (in both grains), i.e. consider different topologies and structures for take advantage on the current computers’ high power. Following this idea some works have considered grid-based implementations [40, 52], or recently, GPU-based approaches [7, 2,22] among others. Our approaches will be implemented on a cluster platform, the most extended. However, this work deals with the parallelization of MOACO algorithms, which have been widely used in the literature, but mostly in sequential implementations. They can deal with several objectives by directly assigning one colony to one specific objective or area of the search space. This method has proved [27,23, 3] to yield excellent results [24]. In this topic, the specialization of a set of ants corresponds to a division of the search space which they will explore in different complementary areas. In the multi-objective scope, the exploration of an area corresponds to a different part of the ideal Pareto set of solutions (or Pareto front). Some variations of specialization guide the ants’ search according to a parameter, for instance BIANT [27] and MOACS [3] or, the proposed by the authors, CHAC [42]. The combination of both topics, MOACOs and parallelization, is a young/recent and open research field, and is what we started to study in [43], with a preliminary research on one multi-colony approach. That research is widely extended and improved in the present work. Thus, this paper proposes and analyzes the performance of different models for multi-colony MOACO approaches, including an islandbased model (inspired in the Genetic Algorithms island model [6]), but defining a neighbourhood topology based in the Pareto front (PF) shape. In addition, some previously cited algorithms: BIANT, MOACS and CHAC, have been adapted for solving the symmetric bi-criteria TSP (bTSP), considering these models. This can be considered as another contributions which we present. Some other studies of the literature involving MOACOs and bTSP will be considered as important references in this work, such as the previously cited [24], in which Garc´ıaMart´ınez et al. analyzed and compared the performance of different MOACOs; the work by L´opez-Ib´an˜ ez et al. [35] that studies the effects of different configurations in the results of some MOACOs; and the paper by Cheng et al. [8] which describes a decomposition method for solving quite large instances of the bTSP using multi-colony MOACOs,

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and comparing some of the methods that we also consider in the present paper. However all of them study sequential approaches.

4 MOACOs to study As previously commented in the introduction, the distribution models which will be described in the next section have been applied to three different MOACOs: BicriterionAnt [27] and Multi-Objective Ant Colony System (MOACS) [3], two of the best state-of-the-art MOACOs according to the conclusions reached by Garc´ıa-Mart´ınez et al. in [24]; and the MOACO proposed by the authors named CHAC [41]. Each method was initially designed for solving a different MOP, so the three all have been adapted in this work to solve the symmetric bi-criteria TSP (bTSP). This problem can be defined as: given an undirected weighted graph G(N, A), with a set of nodes (cities) N, and arcs (links) A joining them, having associated two different costs per link; find the permutation of cities (solution path), composing a Hamiltonian circuit (a set of closed tours visiting each of the nodes exactly once), which minimizes both the global sum of each cost. It can be formulated as: { }    min F1 (S p ) = ∑ [C1 (n − 1, n)]    n∈S p  f (S p ) =

{ }       min F2 (S p ) = ∑ [C2 (n − 1, n)]

(9)

n∈S p

where F1 and F2 are the objective functions to minimize, S p is the solution path (permutation of N), n is a node (city) in that path, and C1 and C2 are the cost functions associated to every arc (link between cities). Usually distances are considered as costs functions. In order to address this problem, all the methods will consider the same heuristic functions (one per objective), described as: η1 (i, j) =

1 d1 (i, j)

(10)

η2 (i, j) =

1 d2 (i, j)

(11)

where d1 (i, j) and d2 (i, j) are the correspondent distances between nodes (cities) i and j in each objective. They depend on the problem definition, and in this work both are Euclidean distances, which can be defined, considering 2dimensional Cartesian coordinates (x,y) per node as: √ d(i, j) =

( jx − ix )2 + ( jy − iy )2

(12)

The evaluation functions (one per objective) have been also common to all the approaches, and are defined as: F1 (S p ) =

∑ [d1 (n − 1, n)]

(13)

∑ [d2 (n − 1, n)]

(14)

n∈S p

F2 (S p ) =

n∈S p

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where S p is the solution path to evaluate and n is a node (city) in that path. Each one of the dx is again the Euclidean distance (in each objective). Moreover, in the three methods it has been used a key parameter in the search, inside the correspondent state transition rule (the main element in a ACO algorithm), λ ∈ [0, 1], which let us to focus in a specific area of the search space to explore as will be shown in the next STR formulas. The first two algorithms (BIANT and MOACS) were initially defined with a ‘variable’ λ policy, which consists in assign a different value for the parameter to each ant h, following the expression: λh =

h−1 m−1

∀h ∈ [1, m]

(15)

Considering that there are m ants, the parameter takes an increasing value that goes from 0 for the first ant to 1 for the last one. This way, the algorithms search in all the possible areas of the space of solutions (every ant is focused on a zone of the PF). As will be deeply commented in the next section, in this work λ has been used to determine the area of the search space that each colony has to explore, so it will be constant for all the ants in a colony (and different to the rest of the colonies). Finally, the three approaches have been improved by means of a local search (LS) method: the 2-OPT algorithm [10], since it has been widely proved in the literature that this hybridization mechanism yields a better performance both in sequential and parallel ACOs [51].

Since BIANT is an AS, it is just performed a global pheromone updating, including evaporation in all nodes and contribution just in the edges of the best paths to the moment (those included in the PS). The Equations for the evaporation are: τ1 (i, j)t = (1 − ρ ) · τ2 (i, j)t−1

(17)

τ2 (i, j)t = (1 − ρ ) · τ2 (i, j)t−1

(18)

and for the contribution (in the edges of the solution paths): τ1 (i, j)t = τ1 (i, j)t−1 + 1/F1

(19)

τ2 (i, j)t = τ2 (i, j)t−1 + 1/F2

(20)

as usual, ρ is the evaporation factor, t marks the new pheromone value and t − 1 the preceding one; F1 and F2 are the evaluation functions for the two objectives (Equations 13 and 14). As a detail, the initial amounts of pheromone are calculated following the expressions: τ0,1 =

1 L1

(21)

τ0,2 =

1 L2

(22)

being Lx the cost obtained using a greedy algorithm for solving the problem defined by each one of the objective functions. It is an usual initialization value in ACOs. In the same work by Iredi et al. [27], there were presented some multi-colony approaches (Unsorted BiAnt, BiCriterion MC) which will be commented in the next section.

4.1 BIANT

4.2 MOACS

The first considered and adapted algorithm is BIANT (BiCriterion Ant), which was proposed by Iredi et al. [27] as a solution for a multi-objective problem with two criteria (the Single Machine Total Tardiness Problem, SMTTP). It is an Ant System (AS) which uses just one colony, and two pheromone matrices and two heuristic functions (one per objective). The STR for choosing the next node j when an ant is in the node i is as follows:

The second algorithm to study is MOACS (Multi-Objective Ant Colony System)1 , which was proposed by Bar´an et al. [3], to solve the Vehicle Routing Problem with Time Windows (VRPTW). It uses a single pheromone matrix for both objectives. The STR is defined as follows:

    

P(i, j) =

   

τ1 (i, j)α ·λ · τ2 (i, j)α ·(1−λ ) · η1 (i, j)β ·λ · η2 (i, j)β ·(1−λ ) ∑ τ1 (i, u)α ·λ · τ2 (i, u)α ·(1−λ ) · η1 (i, u)β ·λ · η2 (i, u)β ·(1−λ )

u∈Ni

i f j ∈ Ni

If (q ≤ q0 )

} { j = arg max τ (i, j) · η1 (i, j)β ·λ · η2 (i, j)β ·(1−λ )

Else (16)

0 otherwise

where the terms and parameters are the named as in Equation 1, but there are one τ and one η per objective (those described in Equations 10 and 11). In addition, it can be seen that this rule considers the λ parameter to weight the objectives in the search. This expression calculates the probability P(i, j) associated to each of the feasible nodes; then, the algorithm uses a roulette wheel to choose the next node in the solution path that the ant is building.

(23)

j∈Ni

P(i, j) =

            

τ (i, j) · η1 (i, j)β ·λ · η2 (i, j)β ·(1−λ ) ∑ τ (i, u) · η1 (i, u)β ·λ · η2 (i, u)β ·(1−λ )

i f j ∈ Ni

u∈Ni

0

(24) otherwise

where the terms and parameters are the same as in Equation 3, but this time there are two heuristic functions: η1 and η2 1 We have chosen this acronym instead of the extended MACS in order to distinguish the algorithm of the preceding MACS by Gambardella et al. [23]

Pareto-based multi-colony MOACOs: An island model

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(defined in Equations 10 and 11). This rule applies again λ to balance the priority of the objectives in the search. This STR works as follows: when an ant is building a solution path and is placed at one node i, a random number q in the range [0,1] is generated, if q ≤ q0 the best neighbour j is selected as the next node in the path (Equation 23). Otherwise, the algorithm decides which node is the next by using a roulette wheel considering P(i,j) as probability for every feasible neighbour j (Equation 24). Since MOACS is an ACS, there are two levels of pheromone updating, local and global. The equation for local pheromone updating is: τ (i, j)t = (1 − ρ ) · τ (i, j)t−1 + ρ · τ0

(25)

considering: τ0 =

If (q ≤ q0 )

1 L1 · L2

(26)

{ } j = arg max τ1 (i, j)α ·λ · τ2 (i, j)α ·(1−λ ) · η1 (i, j)β ·λ · η2 (i, j)β ·(1−λ )

(29)

j∈Ni

as previously, ρ is the evaporation factor, t marks the new pheromone value and t − 1 the preceding one. Again L1 and L2 are the cost obtained in each objective using a Greedy algorithm for solving the problem. MOACS applies a pheromone reinitialization mechanism, so the τ0 value is not constant during the algorithm run, as usual in ACS, but it undergoes adaptation. Every time an ant h builds a complete solution, it is compared to the Pareto set P generated until the moment, to check if the former is a non-dominated solution. At the end of each iteration, τ0′ is calculated following the formula: 1 τ0′ = ¯ ¯ F1 · F2

(27)

where F¯1 and F¯2 are respectively the average costs in each objective for the solution paths currently included in the Pareto set. Then, if τ0′ > τ0 (the current initial pheromone value), a better PS has been found, and the pheromone trails are reinitialized considering the new value for τ0 ← τ0′ . Otherwise the global pheromone updating is performed for every solution in the PS: τ (i, j)t = (1 − ρ ) · τ (i, j)t−1 +

It uses two pheromone matrices and two heuristic functions (one per objective), and considered a single colony originally. Two different approaches for CHAC were defined, using two STRs: the (Combined State Transition Rule, CSTR), which combines the pheromone and heuristic information for each objective weighted using α , β and λ parameters; and the (Dominance State Transition Rule, DSTR), which ranks neighboring cells according to how many they dominate [41]. In the present adaptation, the elements of CHAC are common to those of the previously commented algorithms, applying the same probability term as BIANT in its STR, but considering ACS scheme (as MOACS does). So, the STR is defined as:

ρ F1 · F2

Else      P(i, j) =

   

τ1 (i, j)α ·λ · τ2 (i, j)α ·(1−λ ) · η1 (i, j)β ·λ · η2 (i, j)β ·(1−λ ) ∑ τ1 (i, u)α ·λ · τ2 (i, u)α ·(1−λ ) · η1 (i, u)β ·λ · η2 (i, u)β ·(1−λ )

u∈Ni

i f j ∈ Ni (30)

0 otherwise

where all the terms are the same as in Equations 16, 23 and 24. It works as described in MOACS algorithm section. The use of the parameter λ ∈ (0,1), was defined as constant during the algorithm for all ants, so CHAC always searched in the same zone of the space of solutions (the zone related to the chosen value for λ ). The DSTR was designed to perform a global exploration of the space of solutions (it is not guided by λ ), so, in principle, it is not useful in the present study. As stated, CHAC is an ACS and considers the same pheromone updating scheme as MOACS, so there is a local pheromone updating phase, defined in Equation 25, and a global pheromone updating process performed for every solution in the PS and formulated in Equation 28.

(28)

where, as in previous equations, F1 and F2 are the evaluation functions for each objective, ρ is the common evaporation factor; and t and t − 1 stands for actual and previous pheromone values.

4.3 CHAC The third adapted algorithm is the so-called CHAC (Compan˜ ´ıa de Hormigas ACorazadas), which was proposed by the authors in [41], to solve the Bicriteria Military Unit Pathfinding Problem (MUPFP-2C). It is an ACS adapted to deal with two objectives, and it was designed considering some features of both BIANT (STR) and MOACS (pheromone updating scheme, without reinitialization mechanism).

5 Distributed Pareto-based approaches As explained in Section 3, multi-colony ACOs are usually designed so that every colony searches in the whole space of solutions. If the colonies are not independent, at a certain time of the run, they interchange or share some solutions (normally the best), and update the pheromone matrixes considering them, for guiding the search of the whole community of ants to the same area. However, the multi-objective optimization scope offers a different idea (Section 2.2), since the aim is not yielding the best solution for a function, but a set of non-dominated solutions (PS). Moreover, obtaining the ‘widest’ (as populated as possible) PS along the ideal set of solutions (represented by PF) is the priority, i.e. a set containing the highest amount of non-dominated solutions as possible.

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Following this idea, most of multi-colony MOACOs in the literature have chosen a different colony (or colonies) for optimizing each of the objective functions, in a variation of objective-specialized colonies. Thus, each subset of ants explore a different area of the search space, yielding a different PS. The previously described MOACS algorithm is an example where two colonies where initially defined to optimize both, the number of vehicles and the time (in the VRPTW). A global PS is obtained my mixing the Pareto sub-sets (one per colony), just considering the global non-dominated solutions, since the solutions obtained by a colony may be dominated by the solutions yielded by another one. The problem with these approaches is the low explorative component, tending to present a higher exploitative factor which leads to yield a set of solutions concentrated in specific zones (one per objective). We propose the use of space specialized colonies, i.e. a group of independent colonies, each of them searching in a different area of the space of solutions. The space ‘splitting’ is made my means of the λ parameter (see the algorithms’ description in Section 4), which will weight the objectives in the search for the ants in every colony. This model is similar to other proposals, such as Iredi’s Unsorted BiAnt [27], but presents some differences which will be commented in the following section. As usual, a global PS will be computed considering the global non-dominated solutions among all those yielded by each colony. Thus a parallel scheme will be implemented, taking a coarse-grained parallelization approach, i.e. at colony level, so every computational node (processor) will contain a set of ants (colony). However the main objective in this work is to get one profit: improve the quality, diversity and spread of the solutions obtained in solving the bTSP, rather than just improve the running time as usual in parallel approaches. Following this distribution idea three different multi-colony models are presented in the next sections.

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The ideal scheme2 of the model is presented in Figure 1, considering the minimization of the objectives.

Fig. 1 Sub-colony model scheme example for six sub-colonies: the distribution of the colonies tries to explore the whole Pareto Front, but some gaps (without solutions) arise between them. The dashed line corresponds to the value λ = 0.5.

As it can be seen the sub-colonies have different searching zones along the PF, but the good covering depends on their number, since the gaps which appear between them are wider when the number decreases. Moreover, when the amount of sub-colonies grows, the number of ants in each of them is lower, and thus the explorative factor decreases. So, for a better distribution of solutions, a good balance between the number of sub-colonies and the number of ants in every one should be reached. As previously pointed out, the distribution of the subcolonies is performed by means of the λ parameter, which weights the objectives in the state transition rules (STRs) of the algorithms which implement the model. Thus, when λ > 0.5, the objective function 1 (F1) has a bigger weight in the search, and the other way round when λ < 0.5. The equation which defines the value for λ ∈ [0, 1] considered in a sub-colony c is: λc = 1 −

5.1 Sub-colony model This model proposes the division of the global amount of ants Na in a number of sub-sets or sub-colonies: if there are Nc colonies, then there will be Na /Nc ants in each of them. Every sub-colony will be assigned to a different area of the search space, trying to ‘cover’ all the Pareto front. The subcolonies are independent so there is no communication between them. Every colony just updates its own pheromone matrix (or matrixes, depending on the algorithm) to guide the search.

c−1 Nc − 1

(31)

This is not a novel approach in multi-colony MOACOs field, for instance Iredi et al. [27] proposed a similar one, but it presents some differences with our proposal. The algorithm was named Unsorted Bicriterion Ant and it divides the population of ants in some sub-colonies which consider different pheromone matrixes, but the ants inside one subcolony take a different value for λ (variable policy), trying to explore the whole search space with all the ants at once. 2 Obviously

front

the colonies does not find solutions below the Pareto

Pareto-based multi-colony MOACOs: An island model

The model proposed here considers independent subcolonies where all the ants in every one share the same value for λ , so all of them only cover a specific area of the search space (and thus, of the PF), having a higher exploitation factor. In Figure 1 it can be seen the values for the parameter for six different sub-colonies.

5.2 Pareto-based island model The main problem of the proposed sub-colony model is the arise of gaps without solutions between the colonies. In order to address it, we propose an island model which presents communication between colonies trying to yield some extra solutions in the less explored areas. 5.2.1 Model topology Following the sub-colony model scheme, the colonies, called islands this time, are distributed along the PF (searching in different areas of the space of solutions). However it is also defined a neighbourhood topology aimed to covering the inter-colony ‘gaps’. The proposal is shown in Figure 2, which presents the ideal scheme of the model when minimizing the objectives.

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The model follows a specific neighbourhood topology, based in the Pareto front shape. Every island migrates ants (solutions) just to the closer neighbour and just in one direction. This direction depends on the objective which has higher priority in the island optimization process, so the islands devoted to mainly optimize objective function 1 (λ > 0.5), migrate the ants to the immediate left neighbours. On the other hand, the islands optimizing objective function 2 (λ < 0.5) share the solutions with the immediate right neighbours. The two central islands (on the left and right sides of λ = 0.5 value) interchange ants between them. If there is an island with λ = 0.5, it receives ants from two sides neighbours and shares its owns with them. In addition, the bounding islands just send solutions and do not receive at all, since the gaps to cover appear between the colonies. We have named this approach Pareto-based island model. The main reason for choosing this topology is its low communication cost, following the reasons explained by some authors, such as Twomey et al. [51], who concluded that preventing high communication rates makes more likely that colonies focus on different regions of the search space, emphasizing the exploration and possibly improving the results. The unidirectional (but directed to the optimal) topology could improve the search in the gaps between colonies, meanwhile the exploitation factor is maintained. We refer to migrant ants as an analogy to migrant individuals in distributed GAs, but what is actually migrated is a solution itself, i.e., in the case of the bTSP, a list of cities which compose a valid Hamiltonian cycle. 5.2.2 Model factors In addition to the neighbourhood topology and the information which is shared (individual ants/solutions), there are some other factors to define in an island model:

Fig. 2 Pareto-based island model scheme example for six islands: the neighbourhood topology is aimed to explore the whole Pareto front. The arrows show the migration flows between islands, trying to get overlapped searching areas, i.e. covering the gaps between colonies that the sub-colony model presents. The dashed line represents the value λ = 0.5.

As it can be seen in the figure, each island (sub-colony) searches in a different area of the PF. They have been distributed considering (in the STR of the algorithm) a value for λ computed as in the previous approach (Equation 31).

– Migration policy (which ants are migrated?): Following the conclusions reached by several authors [38, 51] the circulation of the locally best solutions to the neighbour colony (island) yields better results than other possibilities, so this is the policy we have considered. But in this case, since we are dealing with a multi-objective problem, there is not an absolutely best solution, but a set of non-dominated ones. This way, the migrant will be the best solution with regard to the objective with higher priority in the island. This policy along with the proposed neighbouring topology tries to guide the search in the neighbours to the same area than the island which sends the migrants, and thus fill the inter-island gaps. – Migration rate (how often are the migrants sent?): According to previous researches [38, 51], it is important to set a balance between the communication costs that the migrations imply and the profit these migrations give. To this end, we have considered a migration rate around the

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5-10% of the total number of iterations. This way, there should be performed several migrations (10 to 20) during the execution, the colonies have enough time to increase the exploration factor being influenced by the new solutions, and the costs in communications are not critically increased for maintaining a good computational performance. – Replacement policy (which ants/solutions are replaced in the receiver island?): Since the model is designed for multi-objective optimization problems, there is a set of non-dominated solutions in every island (PS). Thus, the immigrant ant/solution is compared with those in the current PS, removing from it those dominated by the new one. – Migrants influence (what effect have the migrants in the receiver island?): The ant/solution firstly performs a pheromone contribution (in all the matrixes of the receiver colony) to those edges of the solution according the correspondent equation depending on the algorithm, i.e. Equations 19 and 20 in BIANT and 28 in MOACS and CHAC cases. This contribution tries to guide (a few) the search in the following iterations. Moreover, the changes in the PS will also globally update the pheromone matrixes in the future steps of the process.

5.2.3 Model justification As stated in Section 3, very few researches has been devoted to propose or study island-based models for ant colony optimization algorithms, following any of the common communication/migration topologies [29], such as: ring, fully connected, small-world or random. But those studies have been focused on ACOs for solving mono-objective problems. There are also some works which present multi-colony approaches with communication between the colonies, but they are different to our proposal. Iredi’s Bicriterion MC [27] for instance, present a model where the ants update the pheromone matrix of the colonies placed in the PF area where the solution has been found, regardless in which subcolony they are included. The aim was to increase the exploring area of the colonies, but it presented the problem that the central zone of the obtained PS is much more populated of non-dominated solutions than the bounds (as proved the work [24]). In our proposed Pareto-based island model (termed just island model from now on) the update is performed in all the pheromone matrixes (per island) so, in principle, all the areas are equally explored, even the bounds. Moreover the novel Pareto-based neighbourhood topology has as aim covering the inter-colonies (islands) gaps which could be the main problem of the sub-colony approach, aiding the ants in every colony to explore in other (but close) areas of the PF.

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Other example is described by Kr¨uger et al. [32] where the pheromone matrixes are migrated instead of just the best solutions as in our proposed method, but which is much more expensive in computational time.

5.3 Multi-colony model Finally a basic model is proposed. It consist on ‘replicating’ the colonies along the PF, i.e. each colony has the same number of ants independently of the number of colonies which are, so the more colonies we deploy, the higher searching/optimizing power we will have. Since colonies are independent, there is no communication between them and they update their own pheromone matrixes. It is a type of multi-start algorithm. In essence the scheme is similar to the sub-colony approach (see Figure 1), but with a higher number of ants per colony. The distribution of the colonies is again performed using the λ parameter (Equation 31). This model presents a higher exploration factor, since there are more ants in each area, along with higher exploitation by frequent positive reinforcement in the pheromone matrixes, so there should not be inter-colonies gaps (or at least not so wide) if the number of colonies is quite big (lower than in previous models). This approach will be taken as a baseline for comparing with the others, because the solutions should be better than those yielded using sub-colony or island models, due to the higher exploration and exploitation factors.

6 Experiments and results Several experiments have been conducted to test the proposed models and algorithms. Different symmetric TSP instances of the Krolak/Felts/Nelson problem, available from http://comopt.ifi.uni-heidelberg.de/software/ TSPLIB95/, have been considered as the problem to solve in

three sizes: 100, 150 and 200 cities. A and B instances have been taken in every case, considering each of them as one independent objective, and thus, defining an instance of a bicriteria TSP (bTSP). The three problems have been named respectively kroAB100, kroAB150 and kroAB200. These instances have a medium/high associated difficulty, and have been considered in several other studies which compare MOACO algorithms [24, 35, 8]. All the approaches (models + algorithms) will be run in a different number of processors: 1, 4, 8 and 16, in a cluster platform with 32 processors, being each of them a 2.6 GHz AMD Opteron (AMD64e) with 2GB of memory running Linux. The cluster has a shared-memory architecture which allows running large threaded jobs (eg OpenMP) as well as message-passing jobs. It has two back-end X4600

Pareto-based multi-colony MOACOs: An island model

SMP nodes, both containing 16 processors. The parallelization has been implemented using the Message Passing Interface [26], not using the shared memory, but the own memory of every processor. The mono-processor approach corresponds to the sequential implementation of the algorithms, which will be used as a comparative baseline to evaluate the performance of the distributed models. In the other cases (4, 8 and 16 processors), the three distributed approaches are designed to fit one colony, sub-colony or island in each processor, considering its own pheromone matrix (or matrixes), i.e. not profiting the shared memory in order to avoid communication (in the island model). The results will be analyzed in a number of ways, starting with the graphical representation of the Pareto sets yielded by the approaches. As a tool for plotting those PSs, we have used median attainment surfaces. This is a technique presented in 1996 [21] but which has been used in several and recent works (for instance [35, 8]). An attainment surface graphically shows the distribution of the solutions and is considered as a quality indicator for an algorithm. It summarizes in a graph the outcomes of several runs, by dividing the space of solutions in two parts: the attained solutions (in a percentage) and those not attained. A median attainment surface plots the graph which limits a region in the space attained by the 50% of the runs. As the authors in [8], we have used the software implemented by Knowles [30] which can be found at: http://dbkgroup.org/knowles/ plot_attainments/, considering resolution 100. In order to compare the approaches (models + algorithms), and since the evaluation of a multi-objective approach requires metrics and indicators in different scopes (such as diversity, distribution, convergence, general quality), we have computed: – Hypervolume (HV ∈ [0, 1]) [54]: this metric calculates the volume, in the objective space, covered by a set of non-dominated solutions (PS). A higher value means a better result, being 1 the maximum. – Spread (Spr ∈ [0, 1]) [11]: this indicator measures the extent of spread of a set of non-dominated solutions. It considers the Euclidean distance between consecutive solutions on average and extreme distances. A value 0 means an ideal spread. – Epsilon indicator (ε ) [31]: this indicator is a measure of the smallest distance it would be necessary to translate every solution in a PS so that it dominates the optimal PF of the problem. It depends on the solutions range of values, but smaller values are better. – Cardinality of the Pareto set (|PS|): number of non-dominated solutions in the obtained PS. There is one metric/indicator related to convergence (Epsilon), one to diversity (Spread), and one to general quality

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(Hypervolume). To compute these metrics and indicators we have used the jMetal software [19], an open source package available from: http://jmetal.sourceforge.net/. Finally, since the experiments have been conducted in a parallel environment (a cluster platform), the running time improvement and scalability will be also analyzed, along with a tradeoff analysis between solution quality, diversity and spread, and execution time. Then, and in order to support the conclusions reached, statistical tests have been applied: ANOVA [20], which allows us to determine whether a change in the results of every metric is due to a change in a factor or to a random effect. In this case, the factors to consider are the possible approaches (model + algorithm). After applying ANOVA, if the test shows significative differences, post-hoc tests have been applied to find which factors are causing these differences. Tukey’s Honestly Significant Difference (HSD)[13] has been used for this purpose. We would like to remark some considerations which should be taken into account in these experiments: – The sequential approach (or mono-processor) uses just one colony and applies the previously commented variable λ policy (originally proposed by the authors for BIANT and MOACS), that is, a different value for that parameter per ant in a single colony. It is justified because the aim is covering the whole search space (PF) with a set of ants. – In the island model the sequential approach is the same as in the sub-colony, since there is no other island to communicate with. – The migration rate in the island model has been set to 50, after a systematic experimentation process. – All the algorithms (BIANT, MOACS and CHAC) have been implemented following the three models, thus each implementation is named with the prefixes mc (multicolony), sc (sub-colony) and i (island), having: mcBIANT , mcMOACS, mcCHAC, scBIANT , scMOACS, scCHAC, iBIANT , iMOACS and iCHAC. – All the algorithms have been improved by applying 2OPT local search method [10]. – The greedy solutions for every instance has been computed in order to use those values in the τ0 expressions (Equations 21, 22 and 26). – Algorithms terminate after doing a number of iterations, common to all approaches and dependent on the problem difficulty. – 20 runs have been performed in every case in order to get a better quality indicator and more statistically significant results. – A pseudo-optimal Pareto set has been created per each problem, since some of the applied metrics and indicators (to compare the approaches) need an optimal Pareto set to be computed. This, which will be named PSG , has

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been built by combining all the solutions yielded by all the approaches (combinations of models, algorithms and number of processors) in the 20 runs; then the dominated solutions have been removed. Due to the multiple combinations (3 algorithms x 3 models x 3 problems x 4 different number of processors x 20 runs = 720 possibilities), the experiments produce a high amount of data. For the sake of clarity (and due to space limitations) we have decided to present results, i.e. attainment functions, metrics and indicators, tradeoff study and statistical tests for just the hardest problem (kroAB200). This instance is completely representative of the other two (kroAB100 and kroAB150), but its higher difficulty makes it a more attractive problem to analyze. The algorithmic behavior and performance is similar/equivalent for all the problems, as can be checked in the url with the additional material of the paper: http://geneura.ugr.es/˜amorag/papers/ i-moacos_soco. Thus, the conclusions reached for this instance, can be extended to all of them.

6.1 Experimental setup Experimental setup is described in the next Tables 1 and 2. Parameter values have been established starting from similar ones in related and previous works [24, 35,8,43], and tuning them by means of intensive experimentation. First, the general configuration with respect to the number of iterations and ants to consider in every processor (i.e. colony, sub-colony or island), in each approach, and problem instance is shown in Table 1. Table 1 Experiments configuration for each model for solving the three bTSPs instances, considering 1, 4, 8 and 16 processors. The values are used in every sub-colony and island (which corresponds to the number of processors).

multi-colony

sub-colony

island

1p 4p 8p 16 p 1p 4p 8p 16 p 1p 4p 8p 16 p

kroAB100 Its Ants 20 500 20 20 20 32 500 8 4 2 32 500 8 4 2

kroAB150 Its Ants 30 750 30 30 30 48 750 12 6 3 48 750 12 6 3

kroAB200 Its Ants 40 1000 40 40 40 64 1000 16 8 4 64 1000 16 8 4

As Table 1 shows, the multi-colony approaches (mcMOACS, mcBIANT and mcCHAC) have the same number of ants in each processor/colony. This value is different from the considered in the other approaches since its philosophy

is also different. It is a type of multi-start algorithm so, it is similar to a replication of the whole algorithm (including all the ants). So a standard number of ants, depending on the problem difficulty, but independently of the number of processors to use, has been set. We initially considered these values as a baseline (20, 30 and 40), then, when the subcolony and island models were defined, we set the number of ants in each of them to the minimum common multiple for 1, 4, 8 and 16 close to the value set for the multi-colony model. So 32, 48 and 64 were chosen. In the sub-colony and island models the total number of ants is divided by the number of processors (which corresponds to the sub-colonies and islands), as commented in Section 5. Since the migration rate is 50, there will be respectively 10, 15 and 20 migrations in the runs for solving kroAB100, kroAB150, and kroAB200 instances. It follows that the results of multi-colony approaches should outperform those yielded by the other methods when the number of processors grows, because the number of ants is clearly higher. In the same way as the number of ants, the number of iterations, which is the termination condition in this study, grows with the problem difficulty. This is different from some other works (for instance [24]), where a running time limit is set. We have considered that this is a fairer restriction, since the implementation of the algorithms makes the PS checking and updating (when a new non-dominated solution is found) computationally expensive. Thus, the approaches which found more solutions have a higher time consumption which would limit their performance in solving the problem, if time would be the termination condition. Table 2 shows the common parameters shared by the three algorithms (MOACS, BIANT and CHAC) across all the experiments. They have been set starting from standard values and performing a systematic experimentation process for fine-tuning some of them.

Table 2 Parameter values used in the three algorithms (in every colony, sub-colony and island).

α β q0 (used in MOACS and CHAC) ρ φ (used in MOACS and CHAC) Number of iterations in Local Search (2-OPT)

1 2 0.8 0.2 0.1 15

It is interesting to notice that the value for q0 is smaller than usual in ACSs, which is typically close to 1 [24, 8]. Moreover, the evaporation rate (ρ ) takes a value that is double the usual. The reason in both cases is the aim of having a higher exploration component which could be a key factor in

Pareto-based multi-colony MOACOs: An island model

the multi-colony and sub-colony models, where the colonies are independent and should explore a wider area. The first parameter, used in the pseudo-random proportional STR in ACS algorithms (see Section 2.1.2), tends to choose as next in a solution path a node which is not the best more often than usually. Meanwhile, the evaporation rate considered in the global pheromone updating is higher in order to ‘forget’ more of the feedback (traces) of the previous ants, tending to try different routes in next iterations.

6.2 Attainment Surfaces analysis This study tries to compare the different PSs obtained by the proposed approaches. To this end the median attainment surfaces have been obtained for every approach, considering the 20 runs solutions (PSs) in every case. Each graph show, in essence, the representation of the PF as a continuous function. It joins the gaps, but it could be noticed a discontinuity in the function. The solutions would have more quality if they are close to the minimum values in the axes. They would be well distributed if the function plots are ‘soft’ (i.e. presenting a small number of discontinuities). As pointed out in the introduction of experiments section, there has been a high amount of experimental data and thus, a high number of attainment surfaces graphs. For this reason we have just considered those related to the hardest problem in the study, i.e. the kroAB200 instance. However, the attainment surfaces for the three problems, can be seen (in higher size) from http://geneura.ugr.es/˜amorag/ papers/i-moacos_soco. The graphs will be analyzed in different ways in order to make comparisons at distinct levels. Firstly, in Figures 3, 4 and 5 a comparison by model has been performed, plotting the attainment surfaces for the PSs obtained using a different number of processors: 1, 4, 8 and 16. From this first set of graphs some preliminary conclusions can be reached: – It can be clearly seen that the approaches run in 4 processors yield worse PSs (more irregular function graphs) in all the cases. The reason is that the number of colonies, sub-colonies and islands is absolutely insufficient for covering the whole space of solutions (and the PF). – Following the covering issue, it can be noticed in Figure 5 (island model) that the graphs are ‘softer’, i.e. more continuous, above all the iBIANT case (Subfigure 5.a). This effect is present in all the algorithms and for all the number of processors in that model, since it is an expected consequence of the migration (exploring new areas around the islands or between them). – With respect to the results by number of processors, the attainment surfaces show that the 16 processors approaches are usually the best (according to the shape and dis-

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tance to the minimum point in the axes). These results are similar to the mono-processor results sometimes, but better in general. This is an interesting point since the mono-processor approaches consider a λ variable policy, while the multi-processor/multi-colony approaches consider a constant λ value for all the ants. Once the best number of processors has been set, the best approaches by models of the previous plots are shown in Figure 6, i.e. the correspondent to 16 processors runs. These graphs are devoted to compare the performance of each algorithm against the others. From the graphs it can be concluded that: – BIANT performs better than the other algorithms in the multi-colony model, although the set of solutions it yields is not well distributed on the bounds, where MOACS and CHAC outperform it. – In the sub-colony model, MOACS and CHAC performs similarly, but the latter obtains lightly better solutions in the central part of the PF. Both of them performs much better than BIANT in this model. However BIANT yields a more diverse PS (plotted as a more continuous PF) due to its higher inherent exploration factor, since it is an AS (the others are ACSs). – In the island model, a similar situation can be observed, having CHAC an apparently better PF. Figure 7 shows the best approaches by algorithm of the graphs in Figures 3, 4 and 5, i.e. the correspondent to 16 processors runs, in order to analyze the performance of the algorithms depending on the model which they implement. From these graphs, it can be extracted the following conclusions: – Multi-colony model presents in general the best results in all the algorithms (above all in BIANT case), as was expected, since it is similar to the sub-colony model but increasing the exploration/exploitation factors by greatly increasing the number of ants per colony (40 ants per processor against 4 in this problem). – The results yielded by sub-colony and island models against multi-colony are excellent in MOACS and CHAC, being very close but worse distributed. It is very interesting and an important achievement of these models, due to the high difference of ants in every colony. – BIANT case is different since the algorithm is in essence an AS, which performs worse in the exploitation task, since it does not consider the q0 parameter, which tends to chose the best nodes in the 80% (0.8 value) of the cases in CHAC and MOACS. In the multi-colony model (Subfigure 7.a) the exploitation factor is increased due to the high amount of ants and thus, the plotted PF outperforms the other two. – Differences between sub-colony and island models are minimal, showing the latter, in general, a better set of

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solutions (above all in the central part of the PF). As it can be seen the function plot is ‘softer’ in the island model cases, because the migration mechanism tends to reduce the inter-colony ‘gaps’ (lack of solutions). Finally, the three best approaches (model + algorithm) are plotted in Figure 8 in order to choose the best combination of all the experiments (for kroAB200). There is one algorithm per model chosen from Figure 6. Looking at the last set of attainment surfaces (Figure 6), it can be concluded that mcBIANT yields the best set of results in the central part of the PF, but it presents a high lack of solutions in the bounds. However, as pointed out in Section 5.3, multi-colony model has been included in the experiments as a baseline to test the value of the other models, because obviously increasing the number of ants (10 times more than the other models in kroAB200 problem) must yield a better set of solutions. Having this in mind and comparing the sub-colony and island models, both performs similarly, showing some areas with better solutions in the sub-colony model and others with better solutions in the island model PFs. What can be noticed is the ‘softer’ function graph in the island case due to the inter-colony additional exploration thanks to the migration performed following the proposed Pareto-based neighbourhood. In essence, the difference between both approaches that we can conclude looking at the attainment surfaces is that sub-colony model has a higher exploitation factor, because the ants in the sub-colony are devoted to explore always the same area of the PF. On the other hand, in the island model, the exploration factor is increased during the run, due to the received migrants which change the pheromone trails focus, directing a part of the search to areas closer to the neighbour islands. In order to reach stronger conclusions, in the next section we will perform an analysis of the metrics and indicators values for every approach.

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Fig. 5 Attainment Surfaces of Island model for kroAB200 problem. Each graph shows the results for every algorithm in a different number of processors: square for mono-processor, circle for 4 processors, black triangle for 8 processors and grey triangle for 16 processors.

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Fig. 7 Attainment Surfaces of the best approaches by algorithm, considering each model for kroAB200 problem. The graphs show the results considering: square for Multi-colony, circle for Sub-colony and triangle for Island.

Pareto-based multi-colony MOACOs: An island model

6.3 Metrics and indicators analysis In this section an analysis based on the metrics and indicators presented at the introduction of the experiments section is conducted. This study includes a different subsection for each of the metrics/indicators considered, firstly showing a set of boxplots with the results of the metrics applied to the 20 PSs (one per run) obtained by every approach (model + algorithm, in a number of processors). Then, a tradeoff analysis between the quality of the solutions and the time to reach them has been performed. Finally (in Subsection 6.3.7) a table summarizing the average results is presented for concluding the study. As in previous section, just the kroAB200 results are included in this paper (due to space limitations), but the rest of metrics and indicators results can be found at: http: //geneura.ugr.es/˜amorag/papers/i-moacos_soco. The metrics which require an ideal Pareto set to be computed have considered a pseudo-optimal global PS (PSG ) per problem, created by joining all the solutions yielded by all the approaches, in all the processors, in the 20 runs. 6.3.1 Hypervolume study Figure 9 shows the boxplots for the hypervolume metric (HV ) results for every problem. This metric can be seen as a numerical representation of the optimality of the solutions in the PS. As it can be seen, the best results (highest values) are those obtained by the 16 processors approaches, clearly outperforming the mono-processor ones in all the cases. In the model comparison, multi-colony performs better as expected, but sub-colony and island models get similar results, being lightly better (see Table 3) those of the latter. Moreover, the results by these two models are closer to the multi-colony ones when the problem gets harder (Subfigure 9.c). With respect to the algorithms, BIANT obtains the best results when the multi-colony approach is considered, however CHAC is the best when sub-colony or island models are applied, being iCHAC (in 16 processors) the best approach according to this metric. 6.3.2 Spread study Figure 10 shows the boxplots for the spread indicator (Spr) results for the three instances. The best results in this indicator (smallest values) are yielded, in general, by the mono-processor approaches. This means that the λ variable policy works better than the constant policy with respect to the distribution of solutions along the PF (i.e. the diversity). This is a logical conclusion, considering that the number of colonies is a discrete value so, there would be necessary a high amount of them to cover

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in a continuous form the whole PF. Looking at the models, the Pareto-based island seems to perform better than the others, followed by multi-colony and at last sub-colony. The first conclusion is a big achievement for the proposed island model, while the latter was expected due to the main subcolony flaw (the gaps between colonies). The best algorithm in general is CHAC, showing some close results to MOACS and far from BIANT in all the cases. iCHAC is again the best option, but scCHAC also obtains very good results in this metric. 6.3.3 Epsilon study Figure 11 shows the boxplots for the epsilon indicator (ε ) results for every problem. Looking at the figure, it can be noticed that the best results (lowest values) in this indicator are again obtained by the 16 processors approaches, outperforming the monoprocessor results in the multi-colony model, but not clearly in the sub-colony and island cases. Multi-colony model performs much more better than the others, meanwhile subcolony and island models get similar results. The best algorithm so far is mcBIANT with 16 processors, and also mcCHAC gets very good results. However if the baseline approaches are not considered, iCHAC with 16 processors yields again the best numbers. 6.3.4 Cardinality study Figure 12 shows the boxplots for the number of solutions in the PSs (cardinality), |PS|. The best values (the highest) are obtained so far by the 16 processors approaches, where mcBIANT clearly outperforms the rest of approaches (above all in the hardest problem), due to its higher exploration component (it is an AS). mcCHAC presents good results too, which means it has a good exploration/exploitation balance. This time, the subcolony model yields higher amount of solutions in the PSs than the island model, but iCHAC and iMOACS yields close numbers to scCHAC and scMOACS. The sub-colony approaches solutions are found due to the higher exploitation component, so these might be Pareto-optimal solutions in the central zones of the partial search space of every colony. 6.3.5 Time scaling study In this study the running time consumed by every approach is shown in a boxplot. But prior to analyzing the results, it is important to notice that we have considered as the algorithm running time (in the multi-processor approaches) the highest time across all processors. Results are plotted for kroAB200 problem instance in Figure 13. Looking at the boxplots, it can be clearly seen

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that the multi-colony approaches consume more time when the number of processors grows (they maintain the same number of ants in every processor). Even having been parallelized in a cluster platform, there is a synchronization time at the beginning and at the end of the processing which increases the running time and makes it higher than in a monoprocessor. In the sub-colony and islands models, the situation is the opposite (they divide the number of ants per processor), presenting very good scalability values. MOACS is the fastest algorithm, and the sub-colony and island models consume similar times, being a bit higher in the latter due to the communications (migrations) between islands, but not too much.

6.3.6 Quality vs running time analysis In this section, the tradeoff between the quality of the solutions and the running time has been analyzed. The quality indicators and metrics have been considered: the hypervolume (HV ), spread (Spr) and epsilon (ε ) indicators, and the number of solutions in the PSs (cardinality, |PS|). The results for the 20 runs in the three instances have been divided by the time consumed in every case (model +

algorithm). They are presented for the instance kroAB2003 in Figures 17 to 15, grouped by the indicator/metric. Firstly, the Hypervolume (HV ) metric tradeoff results are shown in Figure 14. Looking at the figure, three main facts can be clearly noticed: first, the multi-colony tradeoff results are very low, due to the high computational cost that the model requires; second, the tradeoff improves when the number of processors (sub-colonies and islands) grows, as should be expected in every parallel approach; and third, MOACS in the island model and above all in the sub-colony model outperforms the rest. The reason is that is the faster algorithm, and in the sub-colony model, it yields very good non-dominated solutions, due to its exploitation factor. CHAC also obtains good tradeoffs, both in sub-colony and in island models, being even better in the latter, which is a new achievement for the proposed model. Next, the Spread (Spr) indicator is compared against the time in the graphs plotted in Figure 15. These results are similar to the previous analysis, having the same reasons, but this time, the tradeoff in both scMOACS and iMOACS in 8 processors, even outperforms scCHAC and iCHAC in 16 processors, being quite surprising. 3 The

rest of the tradeoff study boxplots can be found at:

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Fig. 10 Boxplots showing the spread indicator (Spr) values of the 20 PSs (one per run) for the kroAB200 instance of the problem. The models are plotted in different colors, being white for multi-colony, light gray for sub-colony and dark gray for island. Each one of the algorithms is labelled with its initial letter and the number of processors where it was run.

The Epsilon (ε ) indicator is compared against the time in the graphs shown in Figure 16. The results show that MOACS outperforms the rest of approaches in any number of multi-processors. Finally, the results for the number of solutions, |PS|, are plotted in Figure 17. Looking at the figure, and considering 16 processors, MOACS seems to be the best algorithm again, considering both the sub-colony and the island model. However CHAC obtains a very good tradeoff in both models too. Multi-colony results are low as usual.

From this tradeoff study it can be concluded that scMOACS or iMOACS are the best options when the low time consuming is a requirement, since it gets an excellent performance even using a small number of processors. scCHAC and iCHAC turns in very good approaches too. 6.3.7 Metrics summary tables Table 3 shows the average values for each of the metrics/indicators in every approach (algorithm, model and number of processors) for the kroAB200 problem instance. The other

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Fig. 12 Boxplots showing the number of solutions (cardinality) |PS| of the 20 PSs (one per run) for the kroAB200 instance of the problem. The models are plotted in different colors, being white for multi-colony, light gray for sub-colony and dark gray for island. Each one of the algorithms is labelled with its initial letter and the number of processors where it was run.

two tables can be consulted from: http://geneura.ugr.es/˜amorag/papers/i-moacos_soco/ Metrics_Indicators-Summary_Tables.pdf.

As it can be seen in the table, the conclusions reached in the previous studies of the work are correct, as a summary: – 16 processors executions performs better than the rest of configurations with a smaller number of processors. – Multi-colony approaches outperforms in almost all the metrics to the other algorithms, but since it is a baseline, we just consider this results for a comparison.

– Island model yields better results than sub-colony model, in general, but sometimes are just lightly better. – CHAC is the best algorithm above all, obtaining much better results than BIANT, but again lightly better than MOACS sometimes. – The distribution of the ants population in sub-colonies or islands provides a high improvement in the computational time, showing a good scalability factor. – Finally, the time improvement obtained using sub-colonies is lightly better than the profit obtained following

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Fig. 14 Boxplots showing the tradeoff between the Hypervolume (HV ) metric results and the running time to get them in each one of the 20 runs for the kroAB200 instance. The models are plotted in different colors, being white for Multi-Colony, light gray for Sub-Colony and dark gray for Island. Each one of the algorithms is labelled with its initial letter and the number of processors where it was run.

the island model. However, due to the better general performance in the rest of metrics, the first can be considered as a more recommended model when the quality in the solutions is the most important issue, and the latter model when the diversity and spread is the key factor.

6.4 Statistical analysis In order to support the conclusions reached in the analysis of previous experiments, statistical tests have been conducted for all the instances and approaches. Again, due to the huge amount of data and figures generated, just kroAB200 instance will be presented and analyzed in this paper, since the results are similar to those of kroAB100 and kroAB150. However, the rest of mate-

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http://

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Firstly ANOVA tests [20] have been applied for evaluating every one of the metrics and indicators, considering each one of the approaches (model + algorithm) as factors. The time has been also analyzed. The R 4 tool has been used to obtain the ANOVA tables, which show, for each factor, the degrees of freedom(D f )5 , the value of the statistical F (F − value) and its associated P − value. If the output P is smaller than 0.05, then the factor effect is statistically significant at a 95% confidence level. In the present study this would indicate that using different approaches give significative differences on the metric results. 4 http://www.r-project.org 5 There

are 9 approaches, so there are 8 degrees of freedom or possible values con compare with a given one

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Fig. 16 Boxplots showing the tradeoff between the Epsilon (ε ) indicator results and the running time to get them in each one of the 20 runs for the kroAB200 instance. The models are plotted in different colors, being white for Multi-Colony, light gray for Sub-Colony and dark gray for Island. Each one of the algorithms is labelled with its initial letter and the number of processors where it was run.

Tables 4 to 8 show the results per metric or indicator for the ANOVA test applied to the kroAB200 problem.

Looking at the tables, it can be noticed that the approaches (models + algorithms) have significative influence on the values of every metric and indicator, since the P-value (Sig. Level) is always much smaller than 0.05 (is the standard minimum value 2.2e-16). Moreover, they have significative influence at a confidence level greater than 99%, as indicated by the asterisks.

Given this conclusion, post-hoc tests might be used to determine which approaches are significantly different from which other. In that sense, one of the most widely used posthoc test is Tukey’s Honestly Significant Difference test [13]. Tukey HSD is a versatile, easily calculated technique that might be used after ANOVA and allows determining exactly where the significant differences are. However, it can only be used when ANOVA has found a significant effect. Otherwise, if the F-value for a factor turns out non-significant, the further analysis is not needed.

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This test compares pairs of the factor values (i.e. model + algorithm combinations), showing a segment (confidence level) for each comparison. Additionally, it shows a vertical dotted line (distance equals zero) that intersects some segments. Significant differences can be found in those cases where the vertical line does not intersect a segment (those values compared are significantly different). Thus, Tukey HSD test has been applied, comparing all the approaches (model + algorithm) one by one per number of processor. Figures 18 to 22 show the results for each one of the metrics and indicators, along with the time, in the kroAB200 instance. Looking at the graphs for HV metric (Figure 18), it can be seen that the number of processors affects the confidence segments of the approaches, making them shorter, and separating them when the number of processors grows. It means the results are more different between models, but more similar between algorithms with a specific model. This conclusion could be seen in the attainment surfaces study and boxplots for this metric (Figure 9.c), where the results for 16 processors were more similar between algorithms in every model. Regarding the significative differences, BIANT in all the models yields the most different results with respect the other approaches, while MOACS and CHAC are quite similar. It can be also seen that sub-colony approaches are similar to island approaches results, as was noticed in the cited attainment functions and boxplots. The test results for the Spr indicator (Figure 19), are quite different, since the confidence segments are longer than in previous study and their long is not affected by the num-

ber of processors. The reason is the higher variability in the results of the metric between runs. Moreover, the segments follow a similar distribution with any number of processors because of the similarity in the results per approach when every number of processors is considered (see boxplots in Figure 10). In mono-processor comparisons, several approaches yield results non-significantly different (the segments are cut by the vertical line), due to the λ variable policy is used, which promotes the exploration. The main differences in this metric arise when the multi-colony model approaches are compared against others, since this model gets very good distributed PSs (as it can be seen in Figure 3), above all when 16 processors (colonies) are used. The results when the island model is compared are normally significantly different from the rest (including sub-colony), which means this is a differentiation indicator between those models. It is logical since the island model was designed for improving the spread in the PS. With respect ε indicator results (Figure 20), BIANT continues being the most different algorithm, while 16 processors results (Figure 20.d) show the highest amount of significantly different approaches. In that figure just scBIANTiBIANT, iMOACS-iCHAC, mcMOACS-mcCHAC, and scMOACS-scCHAC are similar, which better support the conclusion that CHAC and MOACS performs similarly in most cases. Regarding the cardinality (|PS|) study shown in Figure 21, BIANT algorithm is the common differentiation factor, because it yields the greatest amount of solutions due to its commented higher exploration factor (it is an AS).

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Table 3 Metrics average results for kroAB200 problem. Mean and standard deviation of 20 runs is shown for every approach.

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1 proc 4 procs 8 procs 16 procs 1 proc 4 procs 8 procs 16 procs 1 proc 4 procs 8 procs 16 procs 1 proc 4 procs 8 procs 16 procs 1 proc 4 procs 8 procs 16 procs 1 proc 4 procs 8 procs 16 procs 1 proc 4 procs 8 procs 16 procs 1 proc 4 procs 8 procs 16 procs 1 proc 4 procs 8 procs 16 procs

|PSG | 92 180 335 405 125 90 164 233 160 83 140 229 91 145 249 252 124 88 231 291 144 107 185 283 91 108 137 122 124 83 177 218 144 83 177 218

69.15 97.1 175.35 275.15 106.65 63.5 111.7 169.05 105.1 63.85 107.6 159.35 71.5 70.75 111.65 137.7 110.15 72.05 128.85 187.45 117.65 63.25 117.35 172.6 71.5 68.3 81.45 85.4 110.15 69.8 116.9 152.85 117.65 61.75 101.95 154.35

|PS| ±10.65 ±11.51 ±17.40 ±36.36 ±8.46 ±7.10 ±14.81 ±12.85 ±8.33 ±7.44 ±9.08 ±13.76 ±7.32 ±6.26 ±15.42 ±18.09 ±8.94 ±11.19 ±13.63 ±13.17 ±9.71 ±7.30 ±11.16 ±9.32 ±7.32 ±7.51 ±8.43 ±10.67 ±8.94 ±8.10 ±12.37 ±11.95 ±9.71 ±7.03 ±10.04 ±13.63

0.711 0.725 0.792 0.811 0.761 0.660 0.779 0.803 0.758 0.649 0.768 0.795 0.722 0.696 0.741 0.736 0.763 0.657 0.772 0.791 0.761 0.637 0.757 0.785 0.722 0.684 0.719 0.736 0.763 0.666 0.765 0.788 0.761 0.666 0.765 0.788

HV ±0.004 ±0.008 ±0.005 ±0.003 ±0.003 ±0.005 ±0.002 ±0.001 ±0.001 ±0.004 ±0.001 ±0.001 ±0.004 ±0.015 ±0.004 ±0.004 ±0.003 ±0.009 ±0.004 ±0.003 ±0.001 ±0.010 ±0.005 ±0.003 ±0.004 ±0.010 ±0.007 ±0.004 ±0.003 ±0.007 ±0.003 ±0.003 ±0.001 ±0.007 ±0.003 ±0.003

0.745 1.290 1.062 0.883 0.624 1.255 0.904 0.667 0.630 1.226 0.924 0.670 0.740 1.291 1.074 0.892 0.617 1.355 1.125 0.905 0.621 1.356 1.121 0.906 0.740 0.988 0.826 0.743 0.617 1.172 0.921 0.750 0.621 1.172 0.921 0.750

Spr ±0.032 ±0.034 ±0.031 ±0.042 ±0.032 ±0.043 ±0.030 ±0.033 ±0.026 ±0.037 ±0.050 ±0.032 ±0.034 ±0.045 ±0.036 ±0.048 ±0.033 ±0.044 ±0.045 ±0.046 ±0.043 ±0.039 ±0.030 ±0.038 ±0.034 ±0.052 ±0.043 ±0.052 ±0.033 ±0.056 ±0.041 ±0.052 ±0.043 ±0.056 ±0.041 ±0.052

ε 29129.35 58609.1 21461.95 12144.8 21446.65 82144.45 29111.6 14118.3 22361.35 83928.3 32138.1 15765 26524.05 64958.7 32349 27266.45 21016.4 84383.9 33202.3 21428.75 21628.45 89129.5 37754.75 21731.9 26524.05 56713.5 31875 26811.6 21016.4 78679.85 32719.1 18895.95 21628.45 78679.85 32719.1 18895.95

±1751.82 ±3956.12 ±1893.66 ±1479.22 ±911.87 ±1853.08 ±1579.41 ±763.18 ±690.75 ±1897.72 ±1481.31 ±656.95 ±1371.04 ±5663.40 ±2292.63 ±2641.44 ±825.55 ±2208.64 ±2624.87 ±1577.72 ±883.05 ±3004.59 ±2508.94 ±1556.30 ±1371.04 ±4575.69 ±1917.84 ±1175.06 ±825.55 ±2749.71 ±2455.42 ±2141.49 ±883.05 ±2749.71 ±2455.42 ±2141.49

time(ms) 678244 ±11893.55 704307 ±25710.81 693878.5 ±1866.24 725320 ±29034.20 439416.5 ±11057.55 455407 ±21407.82 471652.5 ±3133.70 489691.5 ±16514.38 266076 ±6448.21 270149 ±7942.34 267169.5 ±902.06 277091 ±10173.50 1024529 ±16436.27 285174.5 ±10209.98 139045 ±339.22 72622 ±2913.28 667055 ±17785.16 186662 ±6776.28 94337.5 ±4447.04 49252.5 ±1220.61 402508 ±11366.14 106921 ±815.73 53483 ±174.57 27944 ±967.30 1024529 ±16436.27 289131 ±12638.51 138983 ±424.60 73331 ±2790.26 667055 ±17785.16 184290 ±7354.27 ±519.78 89458.5 47319.5 ±1691.80 402508 ±11366.14 110238 ±3785.00 53699.5 ±154.46 28068.5 ±915.11

Table 4 ANOVA test results for HV metric in problem kroAB200. Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. Param. model + algorithm Residuals

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 1099 < 2.2e-16 *** 171, 0.001475, 0.0000086 1 processor

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 187.24 < 2.2e-16 *** 171, 0.013114, 0.0000767 4 processors

Df F P (Sig. Level) 8 530.55 < 2.2e-16 *** 171, 0.002942, 0.0000172 8 processors

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 1615.2 < 2.2e-16 *** 171, 0.001563, 0.0000091 16 processors

The last study concerns the running time, and it is presented in Figure 22. It not very useful, since the confidence segments are very small, meaning that the time consumed moves in a narrow interval. Multi-colony approaches are the most different due to its much more high time consumption (they consider 10 times more ants in the hardest instance). Sub-colony and island models performs similarly as concluded in the time scaling analysis previously done (Subsection 6.3.5).

As a summary, some general conclusions can be extracted from these analysis: First, ANOVA tests indicate that the approach (model + algorithm) is a factor which definitively affects the metrics results with a confidence level higher than 99%. Then, according to post-hoc Tukey HSD tests:

– The number of processors normally increases the differences between results for every approach, being 16 processors the value with the most differentiation factor.

Pareto-based multi-colony MOACOs: An island model

25

Table 5 ANOVA test results for Spr indicator in problem kroAB200. Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. Param. model + algorithm Residuals

Df F P (Sig. Level) 8 60.298 < 2.2e-16 *** 171, 0.20593, 0.001204 1 processor

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 124.65 < 2.2e-16 *** 171, 0.35912, 0.00210 4 processors

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 156.23 < 2.2e-16 *** 171, 0.2623, 0.001534 8 processors

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 101.44 < 2.2e-16 *** 171, 0.33949, 0.001985 16 processors

Table 6 ANOVA test results for ε indicator in problem kroAB200. Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. Param. model + algorithm Residuals

Df F P (Sig. Level) 8 166.87 < 2.2e-16 *** 171, 51434, 301 1 processor

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 246.83 < 2.2e-16 *** 171, 1.9898e+09, 11636071 4 processors

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 80.946 < 2.2e-16 *** 171, 806585269, 4716873 8 processors

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 96.863 < 2.2e-16 *** 171, 487223291, 2849259 16 processors

Table 7 ANOVA test results for |PS| (cardinality) in problem kroAB200. Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. Param. model + algorithm Residuals

Df F P (Sig. Level) 8 108.65 < 2.2e-16 *** 171, 13491, 78.9 1 processor

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 33.564 < 2.2e-16 *** 171, 11919, 69.7 4 processors

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 78.89 < 2.2e-16 *** 171, 28076, 164.2 8 processors

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 166.87 < 2.2e-16 *** 171, 51434, 301 16 processors

– Sub-colony and island models are similar for almost all the metrics/indicators, but for spread (Spr). In the study of this indicator, the island model is significantly different from the first. It is an expected conclusion, since the island model was designed to improve the diversion and spread in the PSs. – BIANT is the most different algorithm. MOACS and CHAC performs similarly in several cases.

7 Conclusions and Future Work In this work three different ACO algorithm multi-colony distribution models have been studied. The first one, called multi-colony model, has been taken as a baseline since it is a type of multi-start algorithm, where a complete colony is replicated several times; the second, named sub-colony model, divides the whole amount of ants in a colony into independent sub-colonies; finally the third is a novel Paretobased island model, introduced in this paper, which divides the ants into sub-colonies (islands) and proposes a neigh-

bourhood topology based on covering (exploring) the whole Pareto front (representation of the ideal set of solutions in the search space in multi-objective problems). The three models distribute the colonies or sub-colonies along the PF by means of a parameter named λ which weights the objectives in the search. In order to test the value of the proposed model, three multi-objective ant colony optimization algorithms (MOACOs) have been adapted to the three models; two are wellknown state of the art approaches: BIANT [27] and MOACS [3], and the last is our own algorithm, named CHAC and previously presented in [41]. Experiments for solving three instances of the symmetric bi-criteria TSP have been conducted. Moreover the colonies have been distributed into a different number of processors (1, 4, 8 and 16). Several multi-objective performance metrics (cardinality, hypervolume, spread and ε indicator) and graphs (attainment surfaces) have been computed for making a deep analysis of the results. A time scaling and a tradeoff study between the quality of the solutions and the time consumed to obtain them have been conducted. In addi-

26

A.M. Mora et al.

Table 8 ANOVA test results for time in problem kroAB200. Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. Param. model + algorithm Residuals

Df F P (Sig. Level) 8 7612.8 < 2.2e-16 *** 171, 3.2996e+10, 1.9296e+08 1 processor

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 4136.9 < 2.2e-16 *** 171, 2.9666e+10, 1.7349e+08 4 processors

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 255165 < 2.2e-16 *** 171, 6.5574e+08, 3.8347e+06 8 processors

Param. model + algorithm Residuals

Df F P (Sig. Level) 8 9120.4 < 2.2e-16 *** 171, 2.3591e+10, 1.3796e+08 16 processors

tion, statistical tests ANOVA and post-hoc Tukey HSD have been applied to support this analysis. From this study we can conclude that the proposed island model can be considered the best of the three, since it combines a good quality level in the solutions (hypervolume metric) with a very good diversity and spread along the PF. Sub-colony model is also an excellent option, but it is focused on yielding very good solutions in specific areas of the searching space (and thus, of the PF), which means the arise of ‘gaps’ (without solutions) between the colonies, i.e. a lower spread value. Finall, the multi-colony approach is better considering the solutions quality (and spread in some cases) but is a greater time consumer, since it is just based in using a higher amount of ants (replication of colonies). Even so, the results obtained in several cases with ten times less ants in the island model are similar to those obtained in multi-colony approaches, which makes it worthwhile using the first model, given the excellent running time improvement this model offers, very close to the sub-colony running time, but performing some communication traffic between the islands (20 migrations per island in the hardest problem). In addition, CHAC algorithm performs, in general, better than the others, just being beaten by MOACS or BIANT in some punctual cases; this can be considered as another conclusion (and contribution) of the present paper. However, MOACS obtains the best tradeoff between solutions quality and running time, because it is the fastest algorithm. The study of different algorithms over these models yields the conclusion that the CHAC algorithm performs, in general, better than the others, being beaten by MOACS or BIANT only in some particular cases; this can be considered as another conclusion (and contribution) of the present paper. However, MOACS obtains the best tradeoff between solutions quality and running time, since it is the fastest algorithm. The approaches executed in 16 processors obtain better results than the mono-processor, i.e. the sequential approach, and get a higher time improvement due to the division of the number of ants. From these three studies we can conclude that the best approach is CHAC - island model 16 processors. Since this work is an initial research (for the authors) in these topics, the promising results and conclusions open

several future lines of work. Firstly a complete study on the island model possibilities should be conducted, testing different neighbourhood topologies (bi-directional, random, broadcast, etc), migration rates or migration policies. Another line could perform a deeper study on the best approach (CHAC with the island model) could be conducted, solving other multi-objective problems. The comparison with others MOACO multi-colony models is another interesting line of research. There are several models proposed in the literature such as the one by Iredi at al. [27]; it would be also interesting the comparison with state-of-the-art multiobjective evolutionary algorithms following the island model. Following a different branch on the study of the algorithm, the promising results invite us to test CHAC in other multi-objective problems, comparing it with other state-ofthe-art approaches. After this, and focusing to the parallelization scope, the next objective could be to implement a fine-grained parallelization approach (at the ant level), in order to improve the performance in time. The aim could be deal with very large instances of multi-objective problems. Acknowledgements This work has been supported in part by HPCEuropa 2 project (with the support of the European Commission Capacities Area - Research Infrastructures), and the P08-TIC-03903 project awarded by the Andalusian Regional Government, the FPU Grant 2009-2942, the TIN2011-28627-C04-02 project, the University of Granada PR-PP-2011-5 project and, in part, by the MUSES project (number 318508 - FP7).

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(a) 1 processor

(b) 4 processors

(c) 8 processors

(d) 16 processors Fig. 18 Tukey HSD test graph for the HV metric in every number of processors (kroAB200 problem). The lines which are cut by the vertical line in 0, correspond to approaches that are not significantly different.

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(a) 1 processor

(a) 1 processor

(b) 4 processors

(b) 4 processors

(c) 8 processors

(c) 8 processors

(d) 16 processors

(d) 16 processors

Fig. 19 Tukey HSD test graph for the Spr indicator in every number of processors (kroAB200 problem). The lines which are cut by the vertical line in 0, correspond to approaches that are not significantly different.

Fig. 20 Tukey HSD test graph for the ε indicator in every number of processors (kroAB200 problem). The lines which are cut by the vertical line in 0, correspond to approaches that are not significantly different.

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(a) 1 processor

(a) 1 processor

(b) 4 processors

(b) 4 processors

(c) 8 processors

(c) 8 processors

(d) 16 processors

(d) 16 processors

Fig. 21 Tukey HSD test graph for the cardinality (|PS|) in every number of processors (kroAB200 problem). The lines which are cut by the vertical line in 0, correspond to approaches that are not significantly different.

Fig. 22 Tukey HSD test graph for the time measure in every number of processors (kroAB200 problem). The lines which are cut by the vertical line in 0, correspond to approaches that are not significantly different.