A survey, taxonomy and progress evaluation of three decades of swarm optimisation

Abstract

While the concept of swarm intelligence was introduced in 1980s, the first swarm optimisation algorithm was introduced a decade later, in 1992. In this paper, nineteen representative original swarm optimisation algorithms are analysed to extract their common features and design a taxonomy for swarm optimisation. We use twenty-nine benchmark problems to compare the performance of these nineteen algorithms in the form they were first introduced in the literature against five state-of-the-art swarm algorithms. This comparison reveals the advancements made in this field over three decades. It reveals that, while the state-of-the-art swarm optimisation algorithms are indeed competitive in terms of the quality of solutions they find, their complexities have evolved to be more computationally demanding when compared to the nineteen original algorithms of swarm optimisation. The investigation suggests that there is an urge to continue to design swarm optimisation algorithms that are simpler, while maintaining their current competitive performance.

Appendix

1.1 Bacteria foraging optimisation

In 2002, Passino (2002) modelled the foraging activity of E.coli bacteria swarm as a distributed optimisation process and proposed bacterial foraging optimisation (BFO) algorithm, which consists of four main steps: chemotaxis, swarming, reproduction and elimination & dispersal. The E.coli bacteria swarm move by taking small steps to obtain maximum nutrients. During the chemotaxis step, bacteria have two ways of movement: swim and tumble. Each bacterium can keep swimming in a direction or tumble to another direction, and alternate between these two ways via flagella to search for the place with better nutrient concentration. It is called swarming when there are chemical signals released to help bacteria swarm together. Here we consider the case where no chemical signals released due to the small effect of swarming in BFO. Natural selection would eliminate the bacteria with poor healthy condition and sudden or long process of environmental change would kill a group of bacteria. This inspires the reproduction and elimination & dispersal steps.

The pseudo code of BFO is shown in Algorithm 12. The position of each bacterium $X_i$ is considered as a potential solution of the optimisation problem whose objective value is denoted as $f(X_i)$. After initialisation, BFO executes Chemotaxis step, where each bacterium i updates its position by tumbling to a random direction $\frac{\varDelta (i)}{\sqrt{\varDelta ^T(i)\varDelta (i)}}$ and moves the length of the step size Step(i). As long as it reaches a better area with a better $f(X_i)$, the bacterium continues to swim in this specific direction but with a maximum number of swim $N_s$. Otherwise the bacterium tumbles to a random direction again. After $N_c$ chemotaxis steps, a reproduction step is executed by reproducing the most healthy N/2 bacteria and replacing the worst healthy N/2 bacteria with the reproduced bacteria. When $N_{re}$ reproduction steps are exhausted, it is followed by the elimination-dispersal step where each bacterium may be eliminated according to the probability $p_{ed}$ and be placed at a new random location. Bacterial positions keep updating until $N_{ed}$ elimination-dispersal steps are executed for finding the best solutions.

Since the appearance of BFO, its improvement and application have become popular research fields for researchers from various areas like management and engineering. A hybrid algorithm involving PSO and BFO was presented by Biswas et al. (2007b) for the optimisation of multi-modal and high dimensional functions. Considering the chemotactic movement is one of the most important driving forces of BFO, Dasgupta et al. (2009) proposed adaptive computational chemotaxis schemes to accelerate the convergence rate and optimise the frequency-modulated sound wave synthesis problem successfully. Moreover, Niu et al. (2017) designed a coevolutionary strategy with convergence status evaluation to balance exploration and exploitation of BFO. Until now, BFO has been successfully applied to a wide range of real-world optimisation problems, including menu panning (Hernández-Ocaña et al. 2018), parameter extraction (Subudhi and Pradhan 2018), disease recognition (Al-Kheraif et al. 2018), power system optimisation (Kumar and Jayabarathi 2012; Kowsalya et al. 2014), stock market indices prediction (Majhi et al. 2009), among those.

1.2 Shuffled frog leaping algorithm

Based on the natural memetics for the optimisation of water distribution network design, Eusuff and Lansey (2003) first presented shuffled frog leaping algorithm (SFLA), which is also considered as an extension of PSO (Eusuff et al. 2006). It is assumed that frogs are leaping in a swamp which has a number of stones and are aiming at finding the stone with the maximum concentration of food. In SFLA, frogs are considered as hosts of memes and can be partitioned into different memeplexes which are introduced as groups of mutually supported memes. SFLA performs local search within the memeplex, where frogs interact with each other by getting information from the best frog of memeplex or the best of the entire population. This local search is similar to particle swarm optimisation in concept. After a certain times of local search, the global exploration happens periodically by shuffling the virtual frogs and reorganising them into new memeplexes. This shuffling strategy allows for the exchange of information between different memeplexes and benefits frogs to move toward a global optimum.

The pseudo code of SFLA is shown in Algorithm 13. The position of each frog $X_i$ is a candidate solution of the optimisation problem whose objective value is denoted as $f(X_i)$. After the initialisation, the fogs are ranked in decreasing order and are partitioned into $N_{mp}$ memeplexes according to the rank. Each memeplex evolves by local search according to the following frog-leaping algorithm. Within each memeplex, q frogs are selected to construct the submemeplex according to the probability $p_j$, and the worst frog’s position is improved by using the information of the best frog of the memeplex or the best of the whole population. If no improvement is made, a new frog will be generated randomly to replace the worst frog. Each memeplex is updated based on the updated worst frog. This evolutionary step executes $N_{es}$ times for each memeplex in the frog-leaping algorithm, followed by the shuffle of memeplexes where all the updated memeplexes are gathered to generate the new population X and the best position of the whole population Gbest is updated. This process starts again from partitioning the population into memeplexes until the termination conditions are met.

SFLA has been further improved and applied to various real-word problems, ranging from electrical power production (Ebrahimi et al. 2010; Niknam et al. 2011), job scheduling (Li et al. 2012) to clustering (Amiri et al. 2009). Elbeltagi et al. (2007) introduced a new search acceleration parameter into the original SFLA to balance the global and local search for solving project management problem. Fang and Wang (2012) presented a new SFLA for solving project scheduling problem with constrained resource. A permutation-based local research (PBLS) and forward-backward improvement (FBI) were combined to enhance the exploitation ability of SFLA. A modified SFLA (MSFLA) was proposed by Elattar (2019) by introducing the movement inertia equation from PSO and the crossover and mutation operators from GA to optimise the combined heat, emission and economic dispatch (CHEED) problem.

1.3 Cat swarm optimisation

Cat swarm optimisation (CTSO) was presented by Chu et al. in 2006 by inspecting and simulating the cat behaviour, consisting of two major models named seeking mode and tracking mode (Chu et al. 2006, 2007). Each cat has its position representing a candidate solution and a fitness value which is the objective value. The seeking model mimics the case of the cat in resting and looking around for seeking the next move position. The positions sought by the cat in the defined Seeking Range of the Dimension (SRD) are put in the Seeking Memory Pool (SMP) and one of them will be selected by the cat to move to according to the probabilities. In tracing model, the cat is tracing the cat with best fitness value by moving according to its own velocity.

The pseudo code of CTSO is shown in Algorithm 14. The Boolean variable SPC decides whether the current position of the cat will be put in the seeking memory pool. The Counts of Dimension to Change CDC decides how many dimensions will differ. After the initialisation, the Mixture Ratio MR is applied to decide whether a cat is in seeking model or tracing model. For each cat i, if it is in seeking mode, it will look around to find j candidate positions by randomly plus or minus SRD percent of the current value according to CDC. If SPC equals to 1, the current position of the cat will be retained as one of the candidate positions. After that, the cat will move to one candidate position selected randomly according to the probability $p_{it}$. If the cat is in tracing model, its move is decided by its own velocity $V_i$ which is guided by the cat with best fitness value Gbest. These two modes are applied to the cat alternatively until the termination conditions are met and the best position $X_*$ found by the cat swarm will be output as the optimal solution.

CSO has been extended and applied to many problems such as clustering (Santosa and Ningrum 2009) since its appearance. Pradhan and Panda (2012) presented a new multi-objective algorithm based on CTSO by adopting the concept of Pareto dominance and external archive. An enhanced parallel CTSO was presented by adding the Taguchi method into the tracing mode of CTSO to improve the accuracy and reduce the computational time (Tsai et al. 2012). Binary Cat Swarm Optimisation (BCSO) was proposed for dealing with discrete problems and was implemented on 0-1 knapsack problem (Sharafi et al. 2013). BCSO has also been used for the non-unicost set covering problem (Crawford et al. 2015), scheduling workflow applications in cloud environment (Kumar et al. 2017b) and manufacturing cell design problem (Soto et al. 2019).

1.4 Glowworm swarm optimisation

In 2006, Krishnanand and Ghose presented Glowworm Swarm Optimisation (GSO) for the optimisation of multimodal function problems (Krishnanand and Ghose 2006, 2009b). The agents in GSO are called glowworms that have luciferin to emit light and interact with other glowworms within the neighbourhood. Positions of glowworms represent candidate solutions and the fitness values of positions are evaluated by the objective function. The level of luciferin of each glowworm is associated with the fitness of its current location and updates with time. In the movement phase, glowworms are attracted by neighbours with higher luciferin value which means brighter glow.

The pseudo code of GSO is shown in Algorithm 15. $Dt_{ij}$ is the Euclidean distance between glowworms i and j. Each iteration in GSO consists of the luciferin-update phase and the movement phase. The algorithm initialises with random position $X_i$ and an equal level of luciferin $l_i$ for each glowworm i. After that, the luciferin $l_i$ updates based on its previous $l_i$ which decays with time and the fitness value $f(X_i)$ which enhances the luciferin of the current position. Then each glowworm moves towards a neighbour j selected based on the probability $p_{ij}$ from a set of neighbours $NB_i$ who should be within the neighbourhood range $r_d^i$ and have higher luciferin $l_j$. The neighbourhood range $r_d^i$ is updated adaptively to control the current number of neighbours $|NB_i|$. The positions of glowworms are updated through the movement phase to search for the optimal solution for the optimisation problem until the termination conditions are met.

GSO has been successfully applied to different problems and a number of variants of GSO have been presented to improve its performance. For example, Krishnanand and Ghose (2009a) implemented GSO in multi-robot system to locate multiple signal source such as light, heat, sound, etc. Ding et al. (2017) applied GSO to optimise the parameters in wavelet twin support vector machine for determining the parameter automatically before the training process. For dealing with Multi-Objective Environmental Economic Dispatch problem (MOEED), Jayakumar and Venkatesh (2014) presented a new GSO by incorporating an overall fitness ranking tool and a time varying step size. An improved GSO was proposed to enhance its accuracy and convergence rate by Wu et al. (2012) via using greedy acceptance criteria and new movement formulas in the movement phase.

1.5 Firefly algorithm

Yang (2009), Yang (2010a) proposed Firefly Algorithm (FA) by getting inspiration from the flashing characteristics of fireflies in 2009 and applied it to standard pressure design optimisation problem in 2010. The attractiveness of a firefly is decided by its brightness (light intensity) which is evaluated by the objective function and decays with the distance from it. A firefly is attracted to another brighter firefly and explores search space via the movement to other brighter fireflies.

The pseudo code of FA is shown in Algorithm 16. The firefly swarm are initialised randomly in the D dimensional search space and the light intensity of the firefly i is evaluated by the objective function $f(X_i)$. For each firefly, it moves towards each another firefly with higher light intensity in the swarm. The move distance is related to the attractiveness $\chi$ between the firefly i and the firefly j, and a random parameter $\psi$. The attractiveness $\chi$ updates after each movement via $e^{(-\varphi \cdot Dt_{ij}^2)}$ since it varies with the distance $Dt_{ij}$. The best position found during the iteration process is recorded as the Gbest and will be output when the algorithm stops.

Many variants of FA have been proposed to improve its performance in solving different problems. For example, Yang (2013) extended FA to solve multi-objective continuous problems. In order to accelerate convergence rate and enhance the global search ability, a random attracted model and a concept of Cauchy jump were cooperated into FA by Wang et al. (2016a). Wahid et al. (2019) combined the crossover operator of GA with the firefly position movement stage of GSO to enhance the exploitation capability. FA has also been used for a wide range of applications such as clustering (Senthilnath et al. 2011), load frequency control (Sekhar et al. 2016), privacy preserving (Langari et al. 2019) and blast-induced air overpressure forecasting (Zhou et al. 2019a).

1.6 Cuckoo search

Cuckoo search (CS) was proposed by Yang and Deb (2009) based on the cuckoo breeding behaviour and the L$\acute{e}$vy flight behaviour of some birds. Each egg in a nest represents a solution and a cuckoo egg represents a new solution. In this algorithm, it is assumed that each cuckoo lays one egg at a time and each nest has one egg. The cuckoo generates new egg (solution) via L$\acute{e}$vy flight and leave the egg in the host nest where the egg with higher quality is kept. The number of the available host nests is fixed. The host bird might abandon the old nest and build a new one with a probability. The updating way of solutions in CS is similar to that of PSO, but the L$\acute{e}$vy flight applied in CS is more beneficial to the exploration.

The pseudo code of CS is shown in Algorithm 17. The N host nests are initialised randomly in the D dimensional search space. Then perform L$\acute{e}$vy flights to generate new solutions which are compared with the old solutions. The moving step size Step is also related to the difference between $X_i$ and the best position Gbest. If the new solution is better, it is accepted as the new egg in the nest. After that, a fraction $P_{ab}$ of the worst nests are abandoned and replaced by the new nests with new eggs generated via L$\acute{e}$vy flights at new locations. The best position found during the evolution is recorded as the Gbest and will be output when the termination conditions are met.

CS has attracted much attention since its appearance due to its superior performance. Yang and Deb (2010) validated the effectiveness of CS on addressing engineering optimisation problems such as the design of springs and welded beam structure. Walton et al. (2011) proposed a modified CS by allowing information exchange among the top solutions. For training feedforward neural networks, an improved CS was proposed by employing a parameters tuning strategy (Valian et al. 2011). To solve the TSP problem, a discrete CS was presented by reconstructing its population and employing a new kind of cuckoos (Ouaarab et al. 2014).

1.7 Bat algorithm

Yang (2010b) presented bat algorithm (BA) in 2010 based on the echolocation behaviour of bats, which enables bats to find the prey and avoid obstacles in the dark environment. The bats sense distance and distinguish prey from background barriers via echolocation. To looking for prey, the bat at the position $X_i$ flies with velocity $V_i$, pulse frequency $Pf_i$, pulse emission rate $Pr_i$ and loudness $A_i$, which are automatically adjusted based on the proximity of the prey. BA can be considered as a combination of PSO and the local search controlled by the loudness and pulse rate (Yang and Gandomi 2012).

The pseudo code of BA is shown in Algorithm 18, where $\bar{A}$ is the average loudness of the bat swarm at the current iteration t. The position of the bat represents a candidate solution and the proximity of the prey represents the quality of the solution. After the initialisation, the velocity $V_i$ and position $X_i$ of each bat i are updated based on its pulse frequency $Pf_i$ and the position of the global best bat Gbest. The pulse rate $Pr_i$ is used as the probability of local search part where a new solution is generated via a local random walk based the best solution. If the new solution is better than the old one and a randomly generated value is less than the loudness Ai, the new solution will be accepted. $Pr_i$ will increase and Ai will decrease. After that, the best solution Gbest is recorded and this process loops until the algorithm stops.

Some variants of BA have been proposed for further improving its effectiveness and addressing various optimisation problems such as multi-objective problems (Yang 2011) and binary problems (Mirjalili et al. 2014b). Gandomi and Yang (2014) proposed a chaotic BA where different chaotic systems are used to replace the parameters in BA and validated its effectiveness on solving numerical benchmark functions. The chaotic BA was applied to address the constrained economic dispatch problem later (Adarsh et al. 2016). An improved BA was proposed by designing an optimal forage strategy to guide the search direction of the bat and employing a random disturbance strategy to increase the global search capability (Cai et al. 2016). Hybridising with other algorithms is also an efficient way to improve the search ability of the algorithm such as the hybridisation with DE (Yildizdan and Baykan 2020) and the hybridisation with CS (Shehab et al. 2019).

1.8 Fireworks algorithm

Fireworks algorithm (FWA) was proposed by Tan and Zhu (2010) based on the phenomenon of fireworks explosion. A number of sparks will fill the local area around the firework after the explosion, which is regarded as a local search in the space around a specific position in FWA. Two search mechanisms are employed in FWA by simulating two specific fireworks explosion. One is to generate numerous sparks within an amplitude in a well-manufactured explosion and another is to generate a few in a bad explosion. After this, N sparks are selected from the current sparks and set as fireworks for the next explosion.

The pseudo code of FA algorithm is shown in Algorithm 19, where $N_{sp}$ is used to control the number of sparks yielded by the firework, $\varepsilon$ is the smallest constant in computer. N fireworks are set off in the search space randomly in the initialisation. For each firework i, the number of sparks $\hat{N_{sf}^i}$ and the amplitude of explosion $A_i$ are calculated. $\hat{N_{sf}^i}$ sparks are generated randomly within the amplitude $A_i$ while Gaussian explosion is designed for generating ${\hat{m}}$ special sparks to keep the diversity. All the positions of new sparks are evaluated and the best position $X_*$ are kept for the next explosion. $N-1$ other positions are kept for next explosion, selected from all the sparks based on the probability $p(X_i)$ which is related to their distance to other sparks $R(X_i)$. This process repeats until the termination conditions are met and the best position will be output as the optimal solution.

FWA has been developed and applied to many problems. For example, Zheng et al. (2015) improved FWA by adding DE operators to guide the generation of new positions for improving the diversity and avoiding premature convergence. Considering the effect of information utilisation, a novel Guiding Spark (GS) was introduced by Li et al. (2016) for improving the exploration and exploitation ability of FWA. He et al. (2019) presented a Discrete Multi-Objective FWA (DMOFWA) to solve the Multi-Objective FlowShop Scheduling Problem with Sequence-Dependent Setup Times (MOFSP-SDST). Opposition-based learning and clustering analysis are adopted in DMOFWA to improve its exploration ability and cluster firework individuals respectively.

1.9 Fruit fly optimisation

Based on the foraging behaviour of fruit flies, Fruit fly Optimisation Algorithm (FOA) (Pan 2011, 2012) was proposed by Pan in 2011 with two search processes: osphresis search and vision search. Fruit flies locate the food source via smelling and fly to the corresponding positions in the osphresis search process while in the vision search process, fruit flies use vision to converge to the best fruit fly with the highest smell concentration which is evaluated by the fitness function.

The pseudo code of FOA algorithm is shown in Algorithm 20 . The initial position of fruit fly swam is generated randomly in the search space as $(x_{axis}, y_{axis})$. For each fruit fly i, it flies to the new position $(x_{i}, y_{i})$ around the swarm position with random directions and distances RandomValue for searching food in osphresis search process. The smell concentration judgement value $Sm_i$ is calculated as the reciprocal of its distance to the origin $Dist_i$. The best fruit fly Gbest with highest smell concentration Smell which is evaluated by the fitness function $f(Sm_i)$ is selected and set as the fruit fly swarm position $(x_{axis}, y_{axis})$ for next generation in vision search process. These two processes repeat until FOA stops.

A number of variants of FOA have been proposed and it has been applied to different problems. To solve the multidimensional knapsack problem (MKP), for instance, Wang et al. (2013) proposed a new binary FOA by designing a group generating probability vector for generating new solutions. A new control parameter was introduced to tune the search range in osphresis search process adaptively and a new solution generation way was designed to improve the accuracy of FOA by Pan et al. (2014). Mousavi et al. (2019) presented a hybrid intelligent FOA by combing it with a heuristic algorithm to generate and classify fuzzy rules and select the best rules.

1.10 Chicken swarm optimisation

Chicken swarm optimisation algorithm (CCSO) is a relatively new SOA proposed by Meng et al. (2014), simulating the hierarchical order and behaviour of the chicken swarm. The chicken swarm can be classified into different groups, each of which has one rooster, many hens and chicks. In CCSO, the several chicken with best fitness values are set as roosters while those with worst fitness values are acted as chicks. The rest are designated as hens and the mother-child relationship between hens and chicks is randomly established. Different kinds of chicken follow different rules to search for food, causing different updating rules for their positions.

The pseudo code of CCSO algorithm is shown in Algorithm 21, where $Gaussian(0,\sigma ^2)$ is a Gaussian distribution with mean 0 and standard deviation $\sigma ^2$, $\varepsilon$ is the smallest constant in computer, $X_r$ is a randomly selected rooster. The chicken swarm are randomly distributed in the search space in the initialisation, followed by the grouping of the swarm and the determination of the hierarchical order in each group, which only update every G iteration times. Different chicken in the swarm adopt different movement rules. For roosters with better fitness f(X), they can search in a wider area than those with worse fitness. The movement of hens are decided by the positions of the rooster in its group $X_{ri}$ and a randomly selected rooster or hen $X_{rh}$ while the chicks follow their mother $X_{mi}$ for foraging. The new positions are evaluated and chicken positions update if the new positions are better.

CSO has been developed and applied for solving a number of problems. For example, in order to solve 0-1 knapsack problem, Han and Liu (2017) proposed a binary improved CSO by integrating greedy strategy and mutated strategy. inertia weight and learning factor are adopted in CSO to avoid premature convergence when dealing with high dimensional optimisation problems (Wu et al. 2015). Elsawy et al. (2019) presented a new wrapper feature selection algorithm by combining CSO and GA.

1.11 Parameters tuning of PSO and ACO

Here we tuned two of the most famous algorithms (PSO and ACO) to demonstrate that fine tuning these algorithms in their original form does not help still when we compare their fine-tuned performance to the current state-of-the-art algorithms. As is known, the weighting coefficient w of the previous velocity V is the most important parameter in PSO and $\varPhi$ and q are the two most important parameters in ACO. The effects of different $w, \varPhi$ and q values on the performance of PSO/ACO when addressing 30-D problems comparison results are presented in Tables 5, 6, 7. The best results are shown in bold and ‘*’ means the statistical significance. We can see from these tables that different values of these parameters yielded best results on different problems, and even the best results cannot help the comparison to the results obtained by the state-of-the-art algorithms.

Table 5 The effects of w on the performance of PSO when addressing 30-D problems

Table 6 The effects of $\varPsi$ on the performance of ACO when addressing 30-D problems

Table 7 The effects of q on the performance of ACO when addressing 30-D problems

Table 8 Nomenclatures

Fig. 6

The taxonomy based on the proposed general framework where G is Guided, UG is Unguided, C is Continuous, E is Event-based (for linear fusion)

Fig. 7

The taxonomy based on the proposed general framework where G is Guided, UG is Unguided, C is Continuous, E is Event-based (for nonlinear fusion)

Table 9 Results for 10-D problems

Table 10 Results for 10-D problems (continued on Table 9)

Table 11 Results for 30-D problems

Table 12 Results for 30-D problems (continued on Table 11)

Table 13 Results for 50-D problems

Table 14 Results for 50-D problems (continued on Table 13)

Table 15 Results for 100-D problems

Table 16 Results for 100-D problems (continued from Table 15)

Table 17 Rank of all algorithms based on average fitness for 10-D problems

Table 18 Rank of all algorithms based on average fitness for 30-D problems

Table 19 Rank of all algorithms based on average fitness for 50-D problems

Table 20 Rank of all algorithms based on average fitness for 100-D problems

Table 21 Computational complexity