New Inspirations In Swarm Intelligence

Int. J. Bio-Inspired Computation, Vol. 3, No. 1, 2011 1 New inspirations in swarm intelligence: a survey R.S. Parpinelli* Bioinformatics Laboratory, Federal University of Technology – Paraná (UTFPR), Curitiba (PR), 80230-901, Brazil and Applied Cognitive Computing Group, Santa Catarina State University (UDESC), Joinville (SC), 89223-100, Brazil E-mail: [email protected] *Corresponding author H.S. Lopes Bioinformatics Laboratory, Federal University of Technology – Paraná (UTFPR), Curitiba (PR), 80230-901, Brazil E-mail: [email protected] Abstract: The growing complexity of real-world problems has motivated computer scientists to search for efficient problem-solving methods. Evolutionary computation and swarm intelligence meta-heuristics are outstanding examples that nature has been an unending source of inspiration. The behaviour of bees, bacteria, glow-worms, fireflies, slime moulds, cockroaches, mosquitoes and other organisms have inspired swarm intelligence researchers to devise new optimisation algorithms. This tutorial highlights the most recent nature-based inspirations as metaphors for swarm intelligence meta-heuristics. We describe the biological behaviours from which a number of computational algorithms were developed. Also, the most recent and important applications and the main features of such meta-heuristics are reported. Keywords: swarm intelligence; new meta-heuristics; bio-inspired algorithms; problem solving methods; biological behaviours; social living beings. Reference to this paper should be made as follows: Parpinelli, R.S. and Lopes, H.S. (2011) ‘New inspirations in swarm intelligence: a survey’, Int. J. Bio-Inspired Computation, Vol. 3, No. 1, pp.1–16. Biographical notes: R.S. Parpinelli received his BSc in Computer Science (2000) from the Maringá State University (UEM), and MSc in Computer Science (2001) from the Federal University of Technology – Paraná (UTFPR). Currently, he is a PhD candidate in Computer Science at UTFPR. He is also a Professor in the Computer Science Department of the Santa Catarina State University (UDESC). His main areas of interest are data mining, bioinformatics and all kinds of bio-inspired algorithms. H.S. Lopes received his degrees in Electrical Engineering (1984) and MSc in Biomedical Engineering (1990) from the Federal University of Technology – Paraná – UTFPR, and PhD in Electrical Engineering (1996) from the Federal University of Santa Catarina. He is presently working as an Associate Professor in the Department of Electronics of UTFPR. He is the Head of the Bioinformatics Laboratory of UTFPR, since its foundation in 1997. He is also a Researcher of the Brazilian National Research Council. His current research interests are evolutionary computation applications, data mining and bioinformatics. 1 Introduction Swarm-based systems are inspired by the behaviour of some social living beings, such as ants, termites, birds, and fishes. Self-organisation and decentralised control are remarkable features of swarm-based systems that, such as in nature, leads to an emergent behaviour. Emergent behaviour is a Copyright © 2011 Inderscience Enterprises Ltd. property that emerges through local interactions among system components and it is not possible to be achieved by any of the components of the system acting alone (Bonabeau et al., 1999; Garnier et al., 2007). In the beginning, the two mainstreams of the swarm intelligence area were: ant colony optimisation (Dorigo and 2 R.S. Parpinelli and H.S. Lopes Stützle, 2004) and particle swarm optimisation (Kennedy and Eberhart, 2001; Poli et al., 2007). The ACO meta-heuristics is inspired by the foraging behaviour of ants. The ants’ goal is to find the shortest path between a food source and the nest. Each path constructed by the ants represents a potential solution to the problem being solved. When foraging for food, ants lay down a chemical substance called pheromone. Ants can locally communicate to each other by means of the pheromone trails deposited in the environment. This indirect communication system is called stigmergy. When an ant finds a path from a food source to the nest, it deposits certain amount of pheromone in the path biasing other ants to follow that path. This is known as positive feedback and it is the result of successive deposits of pheromone on the same path: as more ants use a path, more pheromone will be present, and consequently, more ants will be attracted to it. As a chemical substance, the pheromone evaporates along time, therefore, reducing attractiveness strength of the trail. From the combination of stigmergy, positive feedback and evaporation an emergent behaviour takes place in the ant colony, leading them to find the shortest path between a food source and the colony.1 The PSO meta-heuristics2 is motivated by the coordinate movement of fish schools and bird flocks. The PSO is compounded by a swarm of particles. Each particle represents a potential solution to the problem being solved and the position of a particle is determined by the solution it currently represents. In PSO, particles are ‘flown’ through hyperdimensional search space. Changes to the position of the particles within the search space are based on the socio-cognitive tendency of individuals to emulate the success of other individuals. Each individual of a population has its own life experience and is able to evaluate the quality of its experience. As social individuals they also have knowledge about how well their neighbours have behaved. These two kind of information corresponds to the cognitive component (individual learning) and social component (cultural transmission), respectively. Hence, an individual decision is taken considering both the cognitive and the social components, thus, leading the population to an emergent behaviour of forage for food or escape from a predator. Both methods above cited have been applied successfully in a vast range of problems (Clerc, 2006). In recent years, new swarm intelligence algorithms have appeared, inspired by bacterial foraging (BFO) (Passino, 2002), fireflies bioluminescense (Krishnanand and Ghose, 2009; Yang, 2008), slime moulds life cycle (Monismith and Mayfield, 2008), cockroaches infestation (Havens et al., 2008), mosquitoes host-seeking (Feng et al., 2009), bats echolocation (Yang, 2010a), and various bees algorithms (BAs), i.e., inspired by bees foraging (Karaboga, 2005; Pham et al., 2006a) and bees mating (Haddad and Afshar, 2004) (for a comprehensive review about algorithms inspired by only bee swarms (see Karaboga and Akay, 2009b). In spite of the swarm inspiration common to these approaches they have their own particular way to exploit and explore the search space of the problem. This work aims at surveying the most recent inspirations in the field of swarm intelligence, reporting them in a concise way. The way each approach search the space of solutions and other features (i.e., biological inspiration, communication model) are presented in the paper. The following section describes these new approaches, from the inspirative nature phenomenon to the corresponding meta-heuristics. Section 3 shows applications of these algorithms in the most different domains. Some important features concerning all algorithms are summarised and discussed in Section 4. Later, some general conclusions are presented. 2 New inspirations and meta-heuristics The careful observation of the behaviour of some living beings can give us insights on how to map their natural behaviour into algorithmic routines. That is why the new meta-heuristics discussed in this work are nature-inspired algorithms. These new approaches are global optimisation meta-heuristics and they are basically composed by a selection of the best scheme and by a randomisation scheme. The former guides, the algorithm convergence to the optimality (exploitation) and the later avoids both the loss of diversity and the algorithm to get trapped in local optima (exploration). A good balance between exploitation and exploration may lead to the global optimality achievement. Sections 2.1 to 2.8 present some natural or behavioural phenomena in specific living beings that inspired computational models for problem solving. 2.1 Bee foraging Many social insects, such as ants and bees, spend most of their life in foraging for food. Honey bee colonies have a decentralised system to collect the food and can adjust the searching pattern precisely in order to enhance the collection of nectar (Seeley, 1995). Bees can estimate the distance from the hive to food sources by measuring the amount of energy consumed when they fly, besides the direction and the quality of the food source. This information is shared with their nestmates by performing a waggle dance and trophallaxis (direct contact). The dance floor is the place in the hive where the coming back forager bees perform the waggle dance to recruit more foragers. Bees that decide foraging without any guidance from other bees are called scouts. Bees that attend to the waggle dance at the dance floor can decide which food source to go based on his its quality. The quality of a food source is proportional to the quantity of nectar found there, and this information is transmitted by changing the intensity of the waggle dance and through antennae contacts. The better the food source, the most intense is the dance and the contacts (Reinhard and Srinivasan, 2009). New inspirations in swarm intelligence: a survey Each forager bee can behave in three different ways after unloading the food: it can perform the waggle dance to recruit more foragers to the same food source; it can abandon the food source due to loss of available resources; or it can directly return to foraging. The basic idea concerning the algorithms based on the bee foraging behaviour is that foraging bees have a potential solution to an optimisation problem in their memory (i.e., a configuration for the problem decision variables). This potential solution corresponds to the location of a food source and has an aggregated quality measure (i.e., value of the objective function). The food source quality information is exchanged through the waggle dance that probabilistically biases other bees to exploit food sources with higher quality. Some algorithms inspired by bee foraging behaviour have been found in literature including bee system (Sato and Hagiwara, 1997), honey bee algorithm (Nakrani and Tovey, 2003), BeeHive (Wedde et al., 2004), virtual bee algorithm (Yang, 2005), bee colony optimisation (Teodorovic and Dell’Orco, 2005), bees swarm optimisation (Drias et al., 2005), artificial bee colony (ABC) algorithm (Karaboga, 2005), BA (Pham et al., 2006a), honey bee foraging (Baig and Rashid, 2007). The two most widely used bee foraging inspired algorithms are described next. 2.1.1 Bees algorithm The BA was first introduced by Pham et al. (2005) applied to a benchmark of mathematical functions. In this seminal work, the comparison with other meta-heuristics (simplex method, stochastic simulated annealing, genetic algorithm and ant colony system) showed that BA outperformed them, regarding processing speed and accuracy of results, thus, suggesting that BA is a powerful optimisation approach. In this algorithm, a bee is a d-dimensional vector containing the problem variables and represents a possible solution to an optimisation problem. Moreover, a solution represents a visited site (i.e., food source) and has a fitness value assigned. The fitness is computed according to the objective function being optimised. The algorithm balances exploration and exploitation by using scout bees that randomly search for new sites and use recruitment for neighbourhood search in sites with the higher fitness, respectively (Pham et al., 2006a). The algorithm starts with n scout bees randomly placed in the search space of dimension d. Each solution f xi = ⎣⎡ xi1 , xi 2 ,..., xid ⎦⎤ is evaluated by a fitness function f f ( xi ) , i = 1,..., n. Bees that have the highest fitnesses are chosen as ‘selected bees’ and sites visited by them (elite sites) are chosen for neighbourhood search. The algorithm conducts searches in the neighbourhood of the selected sites, assigning more bees to search near to the best sites (recruitment). The BA parameters are: number of scout bees (n); number of selected sites (m), out of the n bees; number of elite sites (e), out from the m selected sites; number of bees recruited for the best e elite sites (nep); number of bees recruited for the other (m – e) selected sites (nsp); and the 3 radius for neighbourhood search (ngh). The BA is shown in Algorithm 1. Further information about BA can be found in its repository.3 Algorithm 1 1 2 3 4 Bees algorithm (BA) Parameters: n, m, e, nep, nsp, ngh f Initialise the bees population xi randomly f Evaluate fitness f ( xi ) of the population while stop condition not met do 5 Select m sites from n 6 for each m do 7 Select nsp sites from m 8 for each nsp do 9 Perform neighbourhood search with radius ngh 10 Update bee position according to f() 11 end for 12 end for 13 Select e elite sites from m 14 for each e do for each nep do 15 16 Perform neighbourhood search with radius ngh 17 Update bee position according to f() 18 end for 19 end for 20 Assign remaining (m – e) bees to search randomly and evaluate their fitness 21 Rank all the bees and find the current best 22 end while 23 Postprocess results and visualisation 2.1.2 ABC algorithm The ABC algorithm was first proposed by Karaboga (2005) for solving multidimensional and multimodal optimisation problems. A recent work (Karaboga and Akay, 2009a) compared the ABC algorithm performance against other population-based algorithms (genetic algorithm, particle swarm optimisation, differential evolution and evolution strategies) upon several benchmark functions. Results showed that the performance of the ABC was better than or similar to those of the other algorithms. Another relevant work concerning the ABC algorithm analysed the tuning of control parameters (Akay and Karaboga, 2009a). Algorithm 2 1 2 3 4 Artificial bee colony (ABC) algorithm Parameters: n, limit f Initialise the food sources xi randomly f Evaluate fitness f ( xi ) of the population while stop condition not met do 4 R.S. Parpinelli and H.S. Lopes 5 6 7 8 9 10 for i = 1 to n / 2 do {Employed phase} Select k, j and r at random such that k ∈ {1, 2, …, n}, j ∈ {1, 2, …, d}, r ∈ [0, 1] f v = xij + r ⋅ xij − xkj ( 11 12 Greedy selection else counti = counti + 1 13 14 ) f f Evaluate solutions v and xi f f if f ( v ) is better than f ( x ) then end if 15 end for 16 for i = n / 2 + 1 to n do {Onlooker phase} 17 18 19 20 21 22 Calculate selection probability f f f (x ) P ( xk ) = n k f ∑ Greedy selection else counti = counti + 1 25 26 f ( xk ) Select a bee using the selection probability f Produce a new solution v from the selected bee f f Evaluate solutions v and xi f f if f ( v ) is better than f ( x ) then 23 24 k −i end if 27 end for 28 for i = 1 to n do {Scout phase} 30 if counti > limit then f xi = random 31 end if 29 32 end for 33 Memorise the best solution achieved so far 34 end while 35 Postprocess results and visualisation The ABC algorithm begins with n solutions (food sources) of dimension d that are modified by the artificial bees. f Each solution xi = ⎡⎣ xi1 , xi 2 ,..., xid ⎤⎦ is evaluated by a fitness f function f ( xi ) , i = 1,..., n. The bees aim at discovering places of food sources (regions in the search space) with high amount of nectar (good fitness). There are three types of bees: the scout bees that randomly fly in the search space without guidance; the employed bees that exploit the neighbourhood of their locations selecting a random solution to be perturbed; and the onlooker bees that use the population fitness to select probabilistically a guiding solution to exploit its neighbourhood. If the nectar amount of a new source is higher than that of the previous one in their memory, they update the new position and forget the previous one (greedy selection). If a solution is not improved by a predetermined number of trials, controlled by the parameter limit, then the food source is abandoned by the corresponding employed bee and it becomes a scout bee. The ABC is shown in Algorithm 2. More about the ABC algorithm can be found in the repository.4 The ABC algorithm attempts to balance exploration and exploitation using the employed and onlooker bees to perform local search, and the scout bees to perform global search, respectively. 2.2 Bee mating The honey bee mating process is started when the queen flies in a journey called mating flight. The queen is the only sexually productive female in the colony. Hence, it is the mother of all future queens, drones and workers. The lifetime of the queen is around one to three years, in average (Winston, 1991). The drones follow the queen and the mating takes place in the air during seven or more days. The sperm of the drones are stored into a small organ called spermatheca in the queen’s abdomen. The queen uses this random mix of accumulated sperm to fertilise its eggs during its whole live (Winston, 1991). The basic idea concerning the algorithms based on bee mating behaviour is that the queen is considered the best solution to an optimisation problem and during the mating flight, it selects drones probabilistically for reproduction so as to form the spermatheca. The spermatheca is, then, a pool of selected solutions. New broods are created by crossovering the genotypes of drones and the queen. Natural selection takes place by replacing weaker queens by fitter broods. The algorithm inspired by the bee mating behaviour is described in details next. 2.2.1 Marriage in MBO algorithm The first bee mating algorithm, called marriage in honey-bees optimisation (MBO) algorithm, was presented by (Abbass, 2001), and was applied to propositional satisfiability problems, known as 3-SAT problems. In the MBO the mating flight can be seen as a set of transitions in a state space (fitness landscape), where queens (solutions) moves between different states and mates probabilistically with the drone encountered at each state. The probability of mating depends on the queen’s energy and speed, and the fitness of the drone. The workers are heuristics, such as local search, used to improve the f solutions. Each queen qi = ⎡⎣qi1 , qi 2 ,..., qid ⎤⎦ is characterised by a genotype of dimension d, a speed at instant t(Si(t)), an energy at instant t(Ei(t)), and a spermatheca with a defined capacity. The MBO algorithm has some user-defined parameters: the number of queens (Q); the queen’s spermatheca size (M), representing the maximum number of matings in a single mating flight; the number of broods (B) that will be born from the queen; and a speed reduction factor (α). The MBO is shown in Algorithm 3 and it can be summarised in five main steps: New inspirations in swarm intelligence: a survey 1 Mating flight, where a queen selects drones probabilistically to form the spermatheca. A drone is randomly selected from the spermatheca for the creation of broods. 5 26 Generate B broods by crossover and mutation 27 Use workers to improve the broods 28 while the best brood is better than the worst queen do Creation of new broods (trial solutions) by crossoverring genotypes of drones and the queen. 29 Replace the least-fittest queen with the best brood 30 Remove the best brood from the brood list 3 Use of workers (heuristics) to conduct local search on broods (trial solutions). 31 32 end for 4 Adaptation of workers’ fitness based on the amount of improvement achieved on broods. 33 Postprocess results and visualisation 5 Replacement of weaker queens by fitter broods. 2 Marriage in honey-bees optimisation algorithm (MBO) Algorithm 3 1 Parameters: Q, M, B, α 2 f Initialise the population of queens qi randomly 3 f for each queen qi do 4 Use the workers to improve the queens’ genotype 5 Initialise spermathecai as empty 6 end for 7 for j = 1 to M do {mating-flight loop} 8 t=0 9 for i = 1 to Q do 10 Initialise Ei(t) and Si(t) at random 11 Initialise energy reduction step γ = 12 f Generate a drone D of dimension d at random 13 while E(t) > 0 do 14 15 0.5 Ei ( t ) M ( ) f Evaluate drone’s genotype f D and queen’s f genotype f ( qi ) ⎛ − if ⎜ rand < e ⎜ ⎝ then ( ) f f f ( qi ) − f D Si ( t ) ⎞ ⎟ and spermathecai < M ⎟ ⎠ Add drone’s sperm to spermathecai 16 17 end if 18 t=t+1 19 Update queen’s internal energy: Ei(t + 1) = Ei(t) – γ 20 Update queen’s speed: Si(t + 1) = αSi(t) 21 if rand < Si(t) then 22 Perturbate drone’s genotype 23 end if 24 end while 25 end for end while 2.3 Bacterial chemotaxis An Escherichia coli bacterium can move itself by rotating its flagella distributed around the cell body. When all flagella rotate counterclockwise they propel the bacterium along a trajectory, which is called run (or swim). When the flagella rotate clockwise, they pull on the bacterium in different directions and make the bacterium to tumble (Berg, 2003). The bacterium alternates between these two modes to search for nutrients in random directions. A proper combination of running and tumbling keeps the bacteria in places of higher concentration of nutrients. This foraging activity is called bacterial chemotaxis. This behaviour can be considered as an optimisation process that includes the exploitation of known resources and the exploration for new, potentially more valuable resources. The algorithm inspired by BFO is described next. 2.3.1 BFO algorithm The BFO algorithm was first reported by Passino (2002) that applied the algorithm to the optimisation of a benchmark function. Possible solutions to an optimisation problem are represented in the BFO algorithm by a colony of n bacteria f of dimension d. Each solution xi = ⎡⎣ xi1 , xi 2 ,..., xid ⎤⎦ is f evaluated by a fitness function f ( xi ) , i = 1,..., n. The BFO consists of three main routines: chemotaxis, reproduction, and elimination-dispersal. In chemotaxis, a bacterium with random direction represents a tumble and a bacterium with the same direction of the previous step indicates a run. In reproduction, the health of each bacterium represents its fitness value. All bacteria are sorted according to their health status and only the first half of population survives. The surviving bacteria are split into two identical ones in order to form a new population. Thus, the population of bacteria is kept constant. The elimination-dispersal process is responsible for increasing the diversity of the population. The dispersion happens after a certain number of reproduction steps, when some bacteria are chosen according to a preset probability. Such bacteria are killed and new ones are randomly generated in another position within the search space. The exploitation of the search space is accomplished by both chemotaxis and reproduction steps, while the exploration is done by the elimination-dispersal step. 6 R.S. Parpinelli and H.S. Lopes Bacterial foraging algorithm (BFO) Algorithm 4 2 Parameters: n, Nc, Ns, Nre, Ned, Ped, C(i) (i = 1, 2, …, n) f Initialise randomly the bacterial colony xi 3 for l = 1 to Ned do {Elimination-dispersal loop} 4 for k = 1 to Nre do {Reproduction loop} 5 for j = 1 to Nc do {Chemotaxis loop} 1 ( for i = 1 to n do 6 f Compute fitness f xi j , k ,l 7 ) Tumble: Generate random vector Δ(i) ∈ [–1, 1]d f f Δ (i ) Move: θ i j , k ,l = xi j , k ,l + C (i ) T 8 9 ( [ Δ ( i )] ⋅Δ ( i ) ) 10 Compute f θ i 11 m=0 12 while m < Ns do {Run loop} f f if f θ i j , k ,l < f xi j , k ,l then f j , k ,l ( 13 ) ( 14 Update solution 15 Move again 16 ) else 17 m = Ns 18 end if 19 end while 20 end for 21 end for 22 for i = 1 to n do 23 i = J health f ∑ ( f ( x )) N c +1 j =1 j , k ,l i 24 Sort bacteria by ascending values 25 Best half of the colony duplicates and replaces the worst part 26 end for 27 end for 28 for i = 1 to n do 29 if rand < Ped then 30 31 32 Generate a new random bacterium i end if end for 33 end for 34 Postprocess results and visualisation The parameters involved in the BFO algorithm are: the number of chemotactic steps (Nc); the number of run steps (Ns); the number of reproductive steps (Nre); the number of elimination-dispersal steps (Ned); the probability of elimination (Ped); and the size of the step taken in each run or tumble (C(i), for each bacterium i). The BFO is shown in Algorithm 4. 2.4 Lampyridae bioluminescense Lampyridae is a family of insects (order Coleoptera) that are capable to produce natural light (bioluminescense) to attract a mate or a prey. They are commonly called fireflies or lightning bugs. In the species Lampyris noctiluca the fireflies are also known as glow-worms and, despite of the name, they are not worms. In this species, it is always the female who glows, and only the male has wings. In other species, Luciola lusitanica, both male and female firefly may emit light and both have wings (Fraga, 2008; Shimomura, 2006). If a firefly is hungry or looks for a mate its light glows brighter in order to make the attraction of insects or mates more effective. The brightness of the bioluminescent light depends on the available quantity of a pigment called luciferin, and more pigment means more light (Tyler, 2002). Two optimisation algorithms were inspired by the bioluminescent behaviour and are described next. 2.4.1 Glow-worm swarm optimisation (GSO) algorithm The GSO algorithm was first presented by Krishnanand and Ghose (2005) as an application to collective robotics. In this algorithm, each glow-worm uses a probabilistic mechanism to select a neighbour that has a luciferin value associated with him and moves towards it. Glow-worms are attracted to neighbours that glow brighter. The movements are based only on local information and selective neighbour interactions. This enables the swarm to divide into disjoint subgroups that can converge to multiple optima of a given multimodal function. The GSO is shown in Algorithm 5. It starts by placing a population of n glow-worms of dimension d randomly in the f search space. Each solution xi = ⎡⎣ xi1 , xi 2 ,..., xid ⎤⎦ is f evaluated by a fitness function f ( xi ) , i = 1,..., n. At the beginning, all the glow-worms contain an equal quantity of luciferin l0 and the same neighbourhood range decision r0. Each iteration consists of a luciferin update phase followed by a movement phase based on a transition rule. Other involved parameters are the luciferin decay constant (ρ), the luciferin enhancement constant (γ), the step size (s), the number of neighbours (nt), the sensor range (rs) and a constant value (β). The authors observed that the only two parameters that influences the algorithm behaviour are n and rs. Algorithm 5 1 Glow-worm swarm optimisation algorithm (GSO) Parameters: n, l0, r0, ρ, γ, β, s, rs, nt 2 f Generate the glow-worms population xi randomly 3 for i = 1 to n do New inspirations in swarm intelligence: a survey 4 5 Initialise luciferin li(0) = l0 Initialise neighbourhood range rdi (0) 7 Algorithm 6 = r0 1 6 end for 2 7 t=1 3 8 while stop condition not met do 9 10 li(t + 1) = (1 – ρ) ⋅ li(t) + γ ⋅ f(xi(t)) 5 11 end for 12 for each glow-worm i do {movement phase} 13 14 4 for each glow-worm i do {update luciferin} Find neighbours Ni(t) for each glow-worm j ∈ Ni(t) do Find probability Pij (t ) = 15 ∑ k∈Ni ( t ) Parameters: n, β0, γ f Initialise the fireflies population xi randomly f Compute f ( x ) { } while stop condition not met do f i min = arg min i f ( xi ) { } 6 f f ximin = arg min xfi f ( xi ) 7 for i = 1 to n do 8 l j ( t ) − li ( t ) Firefly algorithm (FA) 9 lk ( t ) − li ( t ) ( ) for j = 1 to n do f f if f x j < f ( xi ) then {Move firefly i towards j} Calculate distance rj 16 end for 10 17 Select glow-worm j using Pij 11 18 Update glow-worm position with ⎛ x (t ) − xi (t ) ⎞ xi (t + 1) = xi (t ) + s ⎜ j ⎟ ⎝ x j (t ) − xi (t ) ⎠ 12 Obtain attractiveness: β ← β 0e f Generate a random solution ui 13 for k = 1 to d do 19 { Update decision range: 21 end for 22 t=t+1 23 24 { ( rdi (t + 1) = min rs ,max 0, rdi (t ) + β ⋅ nt − Ni (t ) 20 )}} xi,k = (1 – β)xi,k + βxj,k + ui,k 14 15 end for 16 17 18 − γ rj end if end for end while 19 end for f Generate a random solution u Postprocess results and visualisation 20 for k = 1 to d do {Best firefly moves randomly} 21 2.4.2 Firefly algorithm The firefly algorithm (FA) was proposed by Yang (2008) and the algorithm was applied to the optimisation of benchmark functions. The FA uses three main basic rules: 1 a firefly will be attracted by other fireflies regardless their sex 2 attractiveness is proportional to their brightness and decreases as the distance among them increases 3 the landscape of the objective function determines the brightness of a firefly. This algorithm assumes that a population of n candidate solutions for an optimisation problem are agents of type firefly. These agents are vectors of dimension d representing f the problem variables. Each solution xi = ⎣⎡ xi1 , xi 2 ,..., xid ⎦⎤ is f evaluated by a fitness function f ( xi ) , i = 1,..., n that represents his quality. Each agent glows proportionally to its quality which, together with his attractiveness (β), dictate how strong it attracts other members of the swarm. Two other user-defined parameters are the maximum attractiveness value (β0) and the absorption coefficient (γ) that determines the variation of attractiveness with increasing distance from communicated firefly. The FA is summarised in Algorithm 6. 22 23 24 xi min , k = xi min , k + uk f Compute f ( x ) end for Find the current best 25 end while 26 Postprocess results and visualisation 2.5 Slime mould life cycle The cells of a slime mould are known to biologists as the amoeba Dictyostelium discoideum – Dd. Amoebae perform a random search for food and move using their pseudopodia5 as sensors to detect nearby food sources. The pseudopod is not always completely accurate, but they work, most time, to direct the amoebae towards food when it is available (Kessin, 2001). Dd cells move independently until starvation, emitting a chemical substance known as cyclic adenosine monophosphate (cAMP). The last form that Dd undergo during starvation is a period of grouping together (aggregation). Thereafter, cells group together in streams and eventually form a mound. The mound forms a slime sheath about itself to protect from predatory multicellular organisms and to forage new regions. Once the mound is complete, cells orient themselves to form a head and a tail. At this point, the slug is not a multicellular organism, but a group of single-cell organisms working 8 R.S. Parpinelli and H.S. Lopes towards a common goal (find food). This process continues until the slug reaches a location with resources or until resources have been depleted. If all resources have been depleted, the amoeba dies. However, if a source is available, culmination occurs and a fruiting body is formed. During culmination and formation of the fruiting body, spores are created and are dispersed by wind, birds or invertebrates. When spores arrive at another location, they may lie dormant for some time. After a period of dormancy, spores form new amoebae, and the life cycle of Dd begins again. Some algorithms that are inspired by this behaviour have been found in literature including cell-based optimisation algorithm (Rothermich et al., 2003) that uses only a portion of the Dd life cycle and the slime mould optimisation algorithm (SMOA) (Monismith and Mayfield, 2008) that uses the life cycle as a whole. Next section describes the SMOA. 1 2 3 algorithm, amoebae may take a number of states: vegetative amoeba, aggregating amoeba, mound, slug, fruiting body, and dispersal. In the algorithm, amoebae begin in the vegetative state, and are assigned to random positions in the search space. They are given time to search for local optima (i.e., food). Based on their initial positions, a mesh is formed using the approximate nearest neighbour algorithm (ε-ANN, where ε is the number of neighbours being considered) (Arya et al., 1998). In the vegetative state, amoebae perform a semi-random search. This is accomplished by the ability of amoebae to extend their pseudopodia towards multiple directions and, so, perform a local search. A parameter k represents the number of pseudopodia (and the number of directions explored by each amoeba). A simulation of starvation is necessary to start an aggregative state for the amoebae. This change is controlled by two parameters: tunimproved iterations without improving the best solution of an amoeba; and tlifetime, the number of time steps since its last dispersal event. As the number of starving amoebae in one distinct lattice point increases above some preset threshold (A), the probability of forming a mound also increases. Once there is no more improvement in the slug movement, it must be dispersed to continue searching for new locations with better results. Through the algorithm, the communication is performed using cAMP trails. A high-level view of SMOA is shown in Algorithm 7. For a complete description (see Monismith and Mayfield, 2008). Parameters: n, k, ε, A f Generate the amoeba population xi randomly f Evaluate fitness f ( xi ) 4 To all amoeba set the state to VEGETATIVE 5 Archive the best objective function value from the amoebae 6 7 Create a mesh based on the results of ε-ANN 8 for each amoeba i do 9 2.5.1 Slime mould optimisation algorithm The SMOA was introduced by Monismith and Mayfield (2008), and was applied to the optimisation of benchmark functions. Although authors achieved good results compared to the known optimum values of the benchmarks, no comparisons were done with other algorithms. The SMOA consists of an amoebae population of size n representing possible solutions for an optimisation problem f of dimension d. Each solution xi = ⎣⎡ xi1 , xi 2 ,..., xid ⎦⎤ is f evaluated by a fitness function f ( xi ) , i = 1,..., n. In the Slime mould optimisation algorithm (SMOA) Algorithm 7 Input the locations of each amoeba to ε-ANN switch amoebae state 10 case VEGETATIVE: Vegetative movement 11 case AGGREGATIVE: Aggregation 12 case MOUD: Moud formation 13 case DISPERSAL: Dispersal 14 end switch 15 end for 16 Postprocess results and visualisation 2.6 Cockroaches infestation Cockroaches are insects of the order Blattodea and are one of the most ancient animals on earth, having appeared around for 350 million years. Communication between cockroaches occurs mainly through chemical trails in their feces as well as emitting airborne pheromone for swarming and mating. Using these signs other cockroaches can follow the trails to discover sources of food and water, and places where other cockroaches are hiding (Bell et al., 2007). Cockroaches are mainly nocturnal insects and will run away when exposed to light. To decide which path to follow cockroaches basically use two pieces of information: how dark is the environment, and how many other cockroaches are there (Halloy et al., 2007). In other words, in an infestation the cockroaches will prefer to group themselves in darker places with other cockroaches. Next section describes the cockroaches infestation algorithm. 2.6.1 Roach infestation optimisation The roach infestation optimisation (RIO) was introduced by Havens et al. (2008). They applied RIO to benchmark functions and achieved competitive results compared to a standard PSO. Actually, RIO has some elements that resemble the traditional PSO algorithm. In RIO algorithm, cockroaches agents are defined using three simple behaviours: • cockroaches search for the darkest location in the search space and the fitness value is directly proportional to the level of darkness (find darkness phase) New inspirations in swarm intelligence: a survey • cockroaches socialise with nearby cockroaches (find friend phase) • cockroaches periodically become hungry and leave the friendship to search for food (find food phase). Roach infestation optimisation algorithm (RIO) Algorithm 8 1 2 3 4 5 Parameters: n, tmax, C0, Cmax, A, thunger f f Initialise the roaches population xi and vi randomly Initialise hungeri randomly between {0, thunger – 1} end for 7 for t = 1 to tmax do 8 M = distances from one roach to each other 9 dg = average distance from M 10 for i = 1 to n do f f if f ( xi ) < f ( pi ) then {Find darkness phase : 11 12 Update best roach location} f f pi = xi 13 end if 14 Compute the neighbours of roach i: {j} = {k: 1 ≤ k ≤ n, k ≠ I, Mik < dg} 15 location pi found so far. The other tunable parameters involved are tmax that is the maximum number of iteration of the algorithm, C0 that ponderates the relative velocity importance, Cmax that ponderates the relative importance of both roach personal best and roach neighbour position, A that influences the neighbour choice, and thunger that defines the hunger interval. The RIO is shown in Algorithm 8. 2.7 Mosquito host-seeking for i = 1 to n do f f Set pi as xi 6 9 Mosquitoes are insects from the family Culicidae. Both male and female mosquitoes feed on nectar (or other sugar source). However, only the female of many species is also capable of taking blood. A blood meal is necessary to develop and nourish the eggs. To do so, female mosquitoes have to seek for animals or humans as possible sources of blood. This is known as host-seeking behaviour. To find its blood host the female mosquito receives external environmental information from its sensory receptors. These sensory receptors respond to varying concentrations of attractants such as carbon dioxide (CO2) and L-lactic acid in the air. A mosquito, upon a host, chooses a specific body region for feeding according to the skin temperature and humidity (Mehlhorn, 2001). A swarm of mosquitoes randomly searches a host to attack. The host-seeking behaviour of a mosquito can be summarised in three main steps: 16 Ni = number of neighbours | {j} | 1 it looks randomly for CO2 or a smelling substance 17 for q = 1 to Ni do {Choose a neighbour} 2 18 once identified the smell, it seeks towards a place of high-concentration of this smell 19 if rand < A then f f li = arg min k f ( pk ) , k = {i, q} 3 it lands when it feels the radiated heat of the host. 20 end if { } 2.7.1 Mosquito host-seeking algorithm 21 end for 22 if hungeri < thunger then {Find friend phase} f f f f f f vi = C0vi + Cmax R1 ( pi − xi ) + Cmax R2 li − xi 23 24 f f f xi = xi + vi 25 Increment hungeri 26 ( ) else {Find food phase} 27 xi = random food location. Random position. 28 hungeri = 0 29 30 end if end for 31 end for 32 Postprocess results and visualisation In fact, RIO is a cockroach-inspired PSO in which a population of n cockroaches (possible solutions) of dimension d use darkness sense (fitness function evaluation) and neighbourhood communication to move through a f f search space updating their positions xi and velocities vi . f Each solution xi = ⎡⎣ xi1 , xi 2 ,..., xid ⎤⎦ is evaluated by a fitness f function f ( xi ) , i = 1,..., n. Each roach keeps its best The MHSA treats every entry of the TSP matrix as an artificial mosquito mij. Hence, the n-city TSP is transformed into host-seeking behaviour of a swarm of n × n artificial mosquitoes. A host is considered as an edge between cities for the TSP. Each entry mij of the TSP matrix (a mosquito) is a triple composed by dij (distance between cities i and j), xij (mosquito sex: 0 for male and 1 for female), and rij (distance between mij and the host). Each rij ranges from 0 to 1 as the artificial mosquito moves, and rij = 1 represents that the artificial mosquito mij is attacking the host and the shortest path passes through this host. When all mosquitoes are in a state of equilibrium, all rij will be 1 or 0. The equilibrium state indicates that the swarm have found a possible solution for the problem. Hence, a solution for the TSP using the MHSA is the set of mosquitoes that have successfully attacked a host (rij = 1). For an extended description of the mosquito host-seeking algorithm (see Feng et al., 2009). 2.8 Bat echolocation Several animals such as dolphins, shrews, most bats, and most whales use echolocation (also called as biosonar) for 10 R.S. Parpinelli and H.S. Lopes navigation, communication and foraging. Most bats use echolocation at a certain degree and, among all the species, microbats use echolocation extensively. In microbats, echolocation is a type of sonar used to avoid close obstacles in the dark, detect prey, and locate roosting crevices (places to sleep over) (Altringham et al., 1998). During echolocation these microbats emit a series of short, high-frequency sounds and listen for the echo that bounces back from the surrounding objects. With this echo a bat can determine an object’s size, shape, direction, distance, and motion. When hunting for a prey, the rate of pulse emission can be speed up to about 200 pulses per second when they fly near their prey and every pulse has a constant frequency. Moreover, the wavelengths of a pulse are in the same order (is proportional) of their prey sizes. The loudness also varies from the loudest when searching for prey and to a quieter base when homing towards the prey (Altringham et al., 1998). Next section describes the bat echolocation algorithm. 7 1 Parameters: n, α,γ 2 f f Initialise the bats population xi and vi randomly 3 f f Define pulse frequency f i at xi 4 for i = 1 to n do 5 6 Initialise pulse rates ri and loudness Ai end for 9 while stop condition not met do for i = 1 to n do Generate new solutions by adjusting: ( 12 ( ) f f f f f Velocity: vit = vit −1 + x t − x∗t fi 14 f f f Location: x t = xit −1 + vit if rand > ri then 16 Select a solution among the best solutions 17 Generate a local solution around the selected best solution 18 end if 19 Generate a new solution by flying randomly f f if rand < Ai & f ( xi ) < f ( x∗ ) then 20 21 Accept the new solutions Increase ri : rit +1 = ri0 [1 − exp(−γt )] 22 Decrease: Ai : Ait +1 = α Ai 23 24 3 ) Frequency: f f f f fi = f min + f max − f min β , β ∈ [0,1] 13 15 (best solutions). Two other parameters are: the loudness decay factor (α) that acts in a similar role as the cooling schedule in the traditional simulated annealing optimisation method, and the pulse increase factor (γ) that regulates the Bat algorithm (BA) f Find the current best x∗ 11 The bat algorithm (BA) was first presented in Yang (2010b). They applied to benchmark functions, and achieved better results compared to genetic algorithms and PSO. To date, no other application to real-world problems was found using the BA. The basic idea behind the BA is that a population of n bats (possible solutions) of dimension d use echolocation to sense distance and fly randomly through a search space f f updating their positions xi and velocities vi . Each solution f xi = ⎣⎡ xi1 , xi 2 ,..., xid ⎦⎤ is evaluated by a fitness function f f ( xi ) , i = 1,..., n. The bats’ flight aims at finding food/prey Algorithm 9 8 10 2.8.1 Bat algorithm pulse frequency. The properly update for the pulse rate (ri) and the loudness (Ai) balances the exploitation and exploration behaviour of each bat, respectively. As the loudness usually decrease once a bat has found its prey/solution (in order to do not loss the prey), the rate of pulse emission increases in order to raise the attack accuracy. The BA pseudo-code is shown in Algorithm 9. The BA is shown in Algorithm 9. f Compute f ( xi ) end if 25 end for 26 Find the current best x* 27 end while 28 Postprocess results and visualisation Applications For all the algorithms mentioned in the previous sections, a search in the literature was done to find applications in the most different domains. Although this search is not exhaustive, it covers the most relevant applications, and emphasises the applicability of those algorithms. The BA (Section 2.1.1) was applied with success in some problems including training of multi-layered perceptron neural networks (Pham et al., 2006b), job shop scheduling optimisation (Pham et al., 2007a), data clustering (Pham et al., 2007b), multi-objective optimisation (Pham and Ghanbarzadeh, 2007), protein folding optimisation using the torsion angles model (Bahamish et al., 2008), optimisation of fuzzy logic controller parameters (Pham and Kalyoncu, 2009), peer-to-peer file sharing in mobile ad-hoc networks (Dhurandher et al., 2009), and interference suppression of linear antenna arrays (Guney and Onay, 2010). New inspirations in swarm intelligence: a survey Applications found in the literature using the ABC algorithm (Section 2.1.2) include: the generalised assignment problem optimisation (Baykasoğlu et al., 2007), energy distribution network configuration (Srinivasa et al., 2008; Linh and Anh, 2010), neural network training (Karaboga and Ozturk, 2009), multi-objective optimisation (Pawar et al., 2008), data clustering (Marinakis et al., 2009), solving integer programming benchmarks (Akay and Karaboga, 2009b), template matching in digital images (Chidambaram and Lopes, 2009), and signal model parameter extraction (Sabata et al., 2010). The Marriage in MBO algorithm (Section 2.2.1), was applied to several problems in the literature, for instance: three-SAT problem optimisation (Abbass, 2001; Teo and Abbass, 2003), MAX-SAT problem optimisation (Benatchba et al., 2005), water resources management (Haddad and Afshar, 2004), non-linear constrained and unconstrained optimisation (with an updated version called honey bee mating optimisation algorithm – HBMO) (Haddad et al., 2006), stochastic dynamic programming (Chang, 2006), continuous optimisation (with an improved version of HBMO) (Afshar et al., 2007), data clustering (Fathian et al., 2007), stepped spillway optimum design (Haddad et al., 2008), and reconfiguration of multiobjective distribution feeder (Taher, 2009). The BFO algorithm (Section 2.3.1) has been used for some applications such as multivariate PID controller tuning (Kim and Cho, 2005a; Luo and Chen, 2010), power systems harmonic estimation (Mishra, 2005), power transmission loss optimisation (Tripathy et al., 2006), machine learning (Kim and Cho, 2005b), multi-objective optimisation (Hazra and Sinha, 2008), prediction of stock market indexes (Majhi et al., 2009), identification of dynamic and non-linear systems (Majhi and Panda, 2008, 2009), and optimisation of fuzzy controller (Alavandar et al., 2010). Some other applications are summarised in Das et al. (2009). Recently, an updated version of the algorithm named self-adaptive bacterial foraging optimisation (SABFO) was proposed by Chen et al. (2008). Results obtained upon several benchmark functions showed a significant improvement in performance over the original BFO, whilst similar or even superior performance compared to PSO and GA. The GSO algorithm (Section 2.4.1) was applied in Krishnanand and Ghose (2009) to find multiple optima of multimodal benchmark functions. This work also conducted detailed parameter tuning experiments. The comparison with a Niched-PSO showed better results concerning the number of peaks found in almost all test functions. Another application using GSO is for hazard sensing in ubiquitous environments (Krishnanand and Ghose, 2008). For the FA (Section 2.4.2), a recent work done by Lukasik and Zak (2009) performed an extensive set of empirical tests for parameter tuning, and proposed some extensions to the algorithm. Some comparisons against PSO on benchmark functions showed competitive results, suggesting that the FA is a powerful optimisation approach. Some extensions in the algorithm was proposed in Yang 11 (2010a) combining Lévy flights with the FA search strategy. Since FA is a very recent meta-heuristic, to date no other application to real-world problems was found. Concerning the SMOA (Section 2.5.1) no other applications to problem solving were found to date, besides that of the original paper (Monismith and Mayfield, 2008). A single application of the RIO algorithm (Section 2.6.1) was found: it was used for pattern recognition in images, checking the posture and stability of elders while they are performing exercises (Havens et al., 2009). Concerning the mosquito host-seeking algorithm (MHSA, Section 2.7.1) and the BA (Section 2.8.1), to date, no other application to real world problems was found, besides the original works for solving the symmetric travelling salesman problem (TSP) (Feng et al., 2009), and for solving benchmark functions (Yang, 2010b), respectively. Most of the above-mentioned algorithms are very recent. Although few applications have appeared to date, we can notice a growing interest in bio-inspired computation. 4 Discussion Table 1 summarises all algorithms in terms of biological inspiration, optimisation domain, mechanisms of exploitation and exploration, and the communication model for each approach. The second column of Table 1 indicates the biological inspiration behind of each algorithm. All the inspiring behaviours have in common two facts: they are compounded by a distributed society/population of individuals where the control is also distributed among the individuals (there is no centralised control); and the individuals’ decision-making is stochastic and based only on local information, without knowledge of the global pattern/solution (communications among them are localised). Moreover, the society-level behaviour transcends the behaviour of a single individual, leading to an emergent behaviour through self-organisation. The third column of Table 1 classifies the algorithms according to the domain they were first applied, either continuous or discrete optimisation. This classification was done concerning the algorithms only in their first canonical publication. An approach inserted in the continuous domain is characterised by solving problems where the variables to be optimised in the objective function can assume only real values. On the other hand, the discrete domain is characterised by solving problems where the variables to be optimised in the objective function are restricted to assume only discrete values, such as integers. Moreover, some algorithms were further adapted to handle the other optimisation domain, different from the original. It is the case, e.g., of BA in Pham et al. (2007a), ABC in Akay and Karaboga (2009b), and MBO in Afshar et al. (2007). 12 R.S. Parpinelli and H.S. Lopes Table 1 Algorithm Meta-heuristics summary Inspiration First applied to… Mechanism of exploitation Mechanism of exploration Communication model BA Bee foraging Continuous optimisation Neighbourhood search in good food sources Random search of scout bees Broadcast-like ABC Bee foraging Continuous optimisation Neighbourhood search carried by employed and onlooker bees Random search of scout bees Broadcast-like MBO Bee mating Discrete optimisation Neighbourhood search in queens and broods carried by workers Spermatheca creation Direct BFO Bacterial foraging Continuous optimisation Chemotaxis and reproduction steps Elimination-dispersal step Direct GSO Firefly bioluminescense Continuous optimisation Glow-worm position update Find neighbour phase dictated by sensor range Broadcast-like FA Firefly bioluminescense Continuous optimisation Firefly movement according to attractiveness Random move of the best firefly Broadcast-like Amoebae foraging Continuous optimisation Vegetative state Dispersal state Stigmergic Cockroaches infestation Continuous optimisation Find friend phase Find food phase Broadcast-like Mosquito foraging Discrete optimisation Host attraction Mosquitoes interaction Broadcast-like Bat echolocation Continuous Optimisation Low loudness and high pulse rate values High loudness and low pulse rate values Broadcast-like SMOA RIO MHSA BA An interesting fact about the third column is that most approaches can be applied to continuous optimisation. This gives us an insight that they could be applied together to the same problem, without major modifications, in order to promote co-evolution. The co-evolution can occur when migrations of individuals from one population biases positively the evolution of another population that receives the individuals. Hence, each approach can be viewed as an island evolving with its own strategies upon a migration topology. The fourth and fifth columns of Table 1 show the exploitation and exploration mechanisms for each approach, respectively. All the mentioned algorithms use exploration and exploitation procedures in their own particular way to seek for the global optimum value of an optimisation problem. According to the no free-lunch theorem (Wolpert and Macready, 1997), it is not possible to point which is the best approach without considering a specific problem. At a higher level, currently it is not possible to point which algorithms are more efficient for generic classes of problems, such as continuous optimisation or discrete optimisation. Therefore, systematic studies comparing the performance of the swarm intelligence algorithms presented here are still missing and this will be a future research direction. The communication model (sixth column of Table 1) classifies the approaches regarding how their individuals communicate to each other. The communication can be: 1 Broadcast-like: The information propagates throughout the environment to some limited extent and/or is made available for a short time such as the bees waggle dance, and the fireflies glow intensity. 2 Direct: The communication is done through antennation, reproduction, trophallaxis (food or liquid exchange), and mandibular contact. 3 Stigmergic/indirect: Occurs when one individual modifies the environment and other individual responds asynchronously to the changes in the environment at a later time. The communication models for the well-known ACO and PSO meta-heuritics are indirect through pheromone trails in the environment and broadcast-like through the social component of particles, respectively. In the BA approach, best sites are broadcasted and recruitment takes place. The ABC algorithm uses the waggle dance to disseminate the information. The MBO algorithm uses the spermatheca formation as a direct communication strategy. The BFO algorithm uses the reproduction phase to propagate the information. Both GSO and FA approaches use the glow intensity as broadcast-like communication strategy. The SMOA uses indirect communication through cAMP trails in the environment. The RIO algorithm broadcasts the information using the find friend phase. In the MHSA, broadcast-like communication occurs through the radiated heat of the host. Finally, the BA approach uses the same PSO strategy that is based on a social component. New inspirations in swarm intelligence: a survey 5 Conclusions and future work This work presented a review of the most recent developments in the field of swarm intelligence. Going beyond the traditional ACO and PSO meta-heuristics, we focused on emergent works that take inspiration from the behaviour of social organisms, but are not much explored, to date. In the same way as in ACO and PSO, nature observation has lead to these new algorithms. This highlights the fact that the nature is an unending source of inspiration for computer scientists, and many other approaches will appear in the future. The computer science community have already learned about the importance of emergent behaviours for complex problem solving. As shown in this work, learning about the collective behaviour of living beings can provide interesting and useful swarm-based meta-heuristics. The studies that have been done to date show the potential of these new approaches to effectively find good solutions to many types of practical optimisation problems. In fact, there is no ‘best’ approach, independently of specific context (Wolpert and Macready, 1997). Different implementations will be more adequate for different problems, either leading to better solutions, or improved speed. Moreover, the convenience of a particular approach does not depend only on the problem: different methods will be more useful for different people, depending on their experience and expertise. A straightforward future research is the comparison of performance of these approaches upon a specific problem in a specific domain, such as numerical optimisation of mathematical functions (continuous optimisation), combinatorial optimisation (discrete optimisation), or multi-objective optimisation (either continuous or discrete optimisation). Such work will focus on unveiling the strengths and weakness of the several algorithms, as well as trying to point out their applicability for different classes of problems, from the user viewpoint. In evolutionary computation in general, and in swarm intelligence, in particular, the use of mechanisms for self-adaptation of parameters is scarce, but a subject of current research. Self-adaptation of parameters could be useful, e.g., for the BFO algorithm that has seven parameters, and for the GSO algorithm that has nine parameters to be tuned by the user. It is not a trivial task to set all these parameters to achieve the best performance for a specific problem. Hence, the design of strategies to fine-tune or to reduce the number of parameters of the swarm intelligence algorithms presented in this paper is another topic pointed for future research in this area. Another two trends of research could be either to explore the concept of co-evolution using these new approaches working as islands upon a migratory topology, or to explore the development of hybrid systems where some properties of one approach are combined with those of another one. Hopefully, plenty of other collective behaviours remain in the shadow waiting to be investigated, such as 13 dragonflies hunting, bees pollination, butterflies mating, ants nest building, locusts collective motion, and others. In the near future, these research opportunities will, possibly, lead to novel algorithms and problem-solving methodologies. Acknowledgements The authors would like to thank UDESC (Santa Catarina State University) and FUMDES program for the financial support to R.S. Parpinelli; as well as to the Brazilian National Research Council (CNPq) for the Research Grant No. 309262/2007-0 to H.S. Lopes. References Abbass, H. (2001) ‘MBO: marriage in honey bees optimization – a haplometrosis polygynous swarming approach’, in Proceedings of the 2001 Congress on Evolutionary Computation CEC2001, IEEE Press, pp.207–214. Afshar, A., Haddad, O., Marino, M. and Adams, B. 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ABC Repository: available at http://mf.erciyes.edu.tr/abc/. A pseudopod is an extension of the cytoplasm of unicellular organisms that imitates a foot and is used to move the cell.