The firefly algorithm (FA) is a metaheuristic algorithm, inspired by the flashing behaviour of fireflies. The primary purpose for a firefly's flash is to act as a signal system to attract other fireflies. Xin-She Yang formulated this firefly algorithm by assuming:^[1]

1. All fireflies are unisexual, so that one firefly will be attracted to all other fireflies;
2. Attractiveness is proportional to their brightness, and for any two fireflies, the less brighter one will be attracted by (and thus move to) the brighter one; however, the brightness can decrease as their distance increases;
3. If there are no fireflies brighter than a given firefly, it will move randomly.

The brightness should be associated with the objective function.

Firefly algorithm is a nature-inspired metaheuristic optimization algorithm.

Algorithm description

The pseudo code can be summarized as:

Begin

1) Objective function: ;
2) Generate an initial population of fireflies ;.
3) Formulate light intensity I so that it is associated with 
   (for example, for maximization problems,  or simply ;
4) Define absorption coefficient γ
While (t<MaxGeneration)
   for i=1:n (all n fireflies)
      for j=1:i (n fireflies)
         if (Ij > Ii), 
          move firefly i towards j;
         end if 
      Vary attractiveness with distance r via ;
      Evaluate new solutions and update light intensity;
      end for j
   end for i
   Rank fireflies and find the current best;
end while
Post-processing the results and visualization;

end

The main update formula for any pair of two fireflies and is

where α_t is a parameter controlling the step size, while is a vector drawn from a Gaussian or other distribution.

It can be shown that the limiting case corresponds to the standard Particle Swarm Optimization (PSO). In fact, if the inner loop (for j) is removed and the brightness I_j is replaced by the current global best g ^* , then FA essentially becomes the standard PSO.

Implementation Guides

The γ should be related to the scales of design variables. Ideally, the β term should be order one, which requires that γ should be linked with scales. For example, one possible choice is to use where L is the average scale of the problem. In case of scales vary significantly, γ can be considered as a vector to suit different scales in different dimensions. Similarly, α_t should also be linked with scales. For example, . It is worth pointing out the above description does not include the randomness reduction. In fact, in actual implementation by most researchers, the motion of the fireflies is gradually reduced by an annealing-like randomness reduction via α = α₀δ^t where 0 < δ < 1(e.g.δ = 0.97).^[2] In some difficult problem, it may be helpful if you increase α_t at some stages, then reduce it when necessary. This non-monotonic variation of α_t will enable the algorithm to escape any local optima when in the unlikely case it might get stuck if randomness is reduced too quickly.

Parametric studies show that n (number of fireflies) should be about 15 to 40 for most problems.^[3]A python implementation is also available, though with limited functionalities.^[4]

Recent studies shows that the firefly algorithm is very efficient,^[5] and could outperform other metaheuristic algorithms including particle swarm optimization.^[6] Most metaheuristic algorithms may have difficulty in dealing with stochastic test functions, and it seems that firefly algorithm can deal with stochastic test functions^[7] very efficiently. In addition, FA is also better for dealing with noisy optimization problems with ease of implementation.^[8][9]

Variants of Firefly Algorithm

Discrete Firefly Algorithm (DFA)

A discrete version of Firefly Algorithm, namely, Discrete Firefly Algorithm (DFA) proposed recently by M. K. Sayadi, R. Ramezanian and N. Ghaffari-Nasab can efficiently solve NP-hard scheduling problems.^[10] DFA outperforms existing algorithms such as the ant colony algorithm.

For image segmentation, the FA-based method is far more efficient to Otsu's method and recursive Otsu.^[11] Meanwhile, a good implementation of a discrete firefly algorithm for QAP problems has been carried out by Durkota.^[12]

Multiobjective FA

An important study of FA was carried out by Apostolopoulos and Vlachos,^[13] which provides a detailed background and analysis over a wide range of test problems including multobjective load dispatch problem.

Lagrangian FA

An interesting, Lagrangian firefly algorithm is proposed to solve power system optimization unit commitment problems.^[14]

Chaotic FA

A chaotic firefly algorithm (CFA) was developed and found to outperform the previously best-known solutions available.^[15]

Hybrid Algorithms

A hybrid intelligent scheme has been developed by combining the firefly algorithm with the ant colony optimization.^[16]

Applications

Digital Image Compression

Very recently, an FF-LBG algorithm for vector quantization of digital image compression was based on the firefly algorithm, which proves to be faster than other algorithms such as PSO-LBG and HBMO-LBG (particle swarm optimization and honey-bee mating optimization; variations on the Linde–Buzo–Gray algorithm).^{[17] [18]} For minimum cross entropy thresholding, firefly-based algorithm uses the least computation time^[19]

Eigenvalue optimization

Eigenvalue optimization of isospectral systems has solved by FA and multiple optimum points have been found efficiently.^[20]

Feature selection and fault detection

Feature selection can be also carried out successfully using firefly algorithm.^[21] Real-time fault identification in large systems becomes viable, based on the recent work on fault identification with binary adaptive firefly optimization.^[22]

Antenna Design

Firefly algorithms outperforms ABC for optimal design of linear array of isotropic sources.^[23]

Structural Design

For mixed-variable problems, many optimization algorithms may struggle. However, firefly algorithm can efficiently sove optimization problems with mixed variables.^{[24] [25]}

Scheduling and TSP

Firefly-based algorithms for scheduling task graphs and job shop scheduling requires less computing than all other metaheuristics.^[26][27] A binary firefly algorithm has been developed to tackle the knapsack cryptosystem efficiently^[28] Recently, an evolutionary discrete FA has been developed for solving travelling salesman problems^[29] Further improvement in performance can be obtained by using preferential directions in firefly movements.

Semantic Web Composition

A hybrid FA has been developed by Pop et al. for selecting optimal solution in semantic web service composition.^[30]

Clustering

Performance study for clustering also suggested that firefly algorithm is very efficient.^[31]

Dynamic Problems

Firefly algorithm can solve optimization problems in dynamic environments very efficiently. ^[32][33]

See also

• Metaheuristic
• Glowworm swarm optimization (GSO)

References

External links

• Firefly Algorithm implemented in Python
• Firefly Algorithm in Matlab or Octave

Swarming Swarm algorithms Agent-based models Ant colony optimization Ant robotics Artificial Ants Bees algorithm Bee colony optimization Boids Crowd simulation Firefly algorithm Glowworm swarm optimization Particle swarm optimization Self-propelled particles Swarm intelligence Swarm (simulation) Biological swarming Agent-based model in biology Bait ball Collective animal behavior Feeding frenzy Flock Flocking Herd Herd behavior Mixed-species foraging flock Mobbing behavior Pack hunter Patterns of self-organization in ants Salmon run Sardine run Shoaling and schooling Sort sol Swarming behaviour Swarming (honey bee) Swarming motility Animal migration Animal migration Bird migration Bird migration flyways Diel vertical migration Fish migration Homing (biology) Insect migration Lessepsian migration Lepidoptera migration Reverse migration Tracking animal migration Swarm robotics I-Swarm Microbotics Swarm robotics Symbrion Related topics Allee effect Biomimicry Collective intelligence Eusociality Group size measures Microbial intelligence Mutualism Predator satiation Quorum sensing Spatial organization Stigmergy Swarming (military) Task allocation and partitioning of social insects

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