# Adaptive Mutation Dynamic Search Fireworks Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Search Fireworks Algorithm

#### 2.1. Explosion Operator

_{i}, its explosion sparks’ number is calculated as follows:

_{max}= max (f(X

_{i})), m is a constant to control the number of explosion sparks, and ε is the machine epsilon to avoid S

_{i}equal to 0.

_{i}is defined as.

_{min}= min f(X

_{i}), A is a constant to control the explosion amplitude, and ε is the machine epsilon to avoid A

_{i}equal to 0.

_{CF}(t) is the explosion amplitude of the CF in generation t. In the first generation, the CF is the best among all the randomly initialized fireworks, and its amplitude is preset to a constant number which is usually the diameter of the search space.

Algorithm 1. Generating Explosion Sparks |

Calculate the number of explosion sparks S_{i} |

Calculate the non-core fireworks of explosion amplitude A_{i} |

Calculate the core firework of explosion amplitude A_{CF} |

Set z = rand (1, d) |

For k = 1:d do |

If k ∈ z then |

If X_{j}^{k} is core firework then |

X_{j}^{k} = X_{j}^{k} + rand (0, A_{CF}) |

Else |

X_{j}^{k} = X_{j}^{k} + rand (0, A_{i}) |

If X_{j}^{k} out of bounds |

X_{j}^{k} = X_{min}^{k} + |X_{j}^{k}| % (X_{max}^{k} − X_{min}^{k}) |

End if |

End if |

End for |

_{min}

^{k}and X

_{max}

^{k}refer to the lower and upper bounds of the search space in dimension k.

#### 2.2. Selection Strategy

## 3. Adaptive Mutation Dynamic Search Fireworks Algorithm

_{CF}is the core firework in the current population, and the symbol $\otimes $ represents the dot product. Gaussian() is a random number generated by the normal distribution with mean parameter mu = 0 and standard deviation parameter sigma = 1, and Levy() is a random number generated by the Levy distribution, and it can be calculated with the parameter β = 1.5 [16]. The value of E varies dynamically with the evolution of the population, with reference to the annealing function of the simulated annealing algorithm, and the value of E is expected to change exponentially, and it is calculated as follows:

_{max}is the maximum number of function evaluations.

_{m}times, each time with the core firework X

_{CF}(N

_{m}is a constant to control the number of mutation sparks).

Algorithm 2. Generating Mutation Sparks |

Set the value of mutation probability p |

Find out the core firework X_{CF} in current population |

Calculate the value of E by Equation (8) |

Set z = rand (1, d) |

For k = 1:d do |

If k ∈ z then |

Produce mutation spark X_{CF}’ by Equation (7) |

If X_{CF}’ out of bounds |

X_{CF}’ = X_{min} + rand * (X_{max} − X_{min}) |

End if |

End if |

End for |

_{min}is the lower bound, and X

_{max}is the upper bound.

Algorithm 3. Pseudo-Code of AMdynFWA |

Randomly choosing m fireworks |

Assess their fitness |

Repeat |

Obtain A_{i} (except for A_{CF}) |

Obtain A_{CF} by Equation (6) |

Obtain S_{i} |

Produce explosion sparks |

Produce mutation sparks |

Assess all sparks’ fitness |

Retain the best spark as a firework |

Select other m−1 fireworks randomly |

Until termination condition is satisfied |

Return the best fitness and a firework location |

## 4. Simulation Results and Analysis

#### 4.1. Simulation Settings

#### 4.2. Simulation Results and Analysis

#### 4.2.1. Study on the Mutation Probability p

#### 4.2.2. Comparison of AMdynFWA with FWA-Based Algorithms

#### 4.2.3. Comparison of AMdynFWA with Other Swarm Intelligence Algorithms

- (1)
- Artificial bee colony (ABC) [18]: A powerful swarm intelligence algorithm.
- (2)
- Standard particle swarm optimization (SPSO2011) [19]: The most recent standard version of the famous swarm intelligence algorithm PSO.
- (3)
- Differential evolution (DE) [20]: One of the best evolutionary algorithms for optimization.
- (4)
- Covariance matrix adaptation evolution strategy (CMA-ES) [21]: A developed evolutionary algorithm.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**The EFWA, AFWA, dynFWA, and AMdynFWA searching curves. (

**a**) f1 function; (

**b**) f2 function; (

**c**) f3 function; (

**d**) f4 function; (

**e**) f5 function; (

**f**) f6 function; (

**g**) f7 function; (

**h**) f8 function; (

**i**) f9 function; (

**j**) f10 function; (

**k**) f11 function; (

**l**) f12 function; (

**m**) f13 function; (

**n**) f14 function; (

**o**) f15 function; (

**p**) f16 function; (

**q**) f17 function; (

**r**) f18 function; (

**s**) f19 function; (

**t**) f20 function; (

**u**) f21 function; (

**v**) f22 function; (

**w**) f23 function; (

**x**) f24 function; (

**y**) f25 function; (

**z**) f26 function; (

**A**) f27 function; (

**B**) f28 function.

Function Type | Function Number | Function Name | Optimal Value |
---|---|---|---|

Unimodal Functions | 1 | Sphere function | −1400 |

2 | Rotated high conditioned elliptic function | −1300 | |

3 | Rotated bent cigar function | −1200 | |

4 | Rotated discus function | −1100 | |

5 | Different powers function | −1000 | |

Basic Multimodal Functions | 6 | Rotated rosenbrock’s function | −900 |

7 | Rotated schaffers F7 function | −800 | |

8 | Rotated Ackley’s function | −700 | |

9 | Rotated weierstrass function | −600 | |

10 | Rotated griewank’s function | −500 | |

11 | Rastrigin’s function | −400 | |

12 | Rotated rastrigin’s function | −300 | |

13 | Non-continuous rotated rastrigin’s function | −200 | |

14 | Schewefel’s function | −100 | |

15 | Rotated schewefel’s function | 100 | |

16 | Rotated katsuura function | 200 | |

17 | Lunacek Bi_Rastrigin function | 300 | |

18 | Rotated Lunacek Bi_Rastrigin function | 400 | |

19 | Expanded griewank’s plus rosenbrock’s function | 500 | |

20 | Expanded scaffer’s F6 function | 600 | |

Composition Functions | 21 | Composition function 1 (N = 5) | 700 |

22 | Composition function 2 (N = 3) | 800 | |

23 | Composition function 3 (N = 3) | 900 | |

24 | Composition function 4 (N = 3) | 1000 | |

25 | Composition function 5 (N = 3) | 1100 | |

26 | Composition function 6 (N = 5) | 1200 | |

27 | Composition function 7 (N = 5) | 1300 | |

28 | Composition function 8 (N = 5) | 1400 |

**Table 2.**Mean value and average rankings achieved by AMdynFWA with different p, where the ‘mean’ indicates the mean best fitness value.

Functions | p = 0.1 | p = 0.3 | p = 0.5 | p = 0.7 | p = 0.9 |
---|---|---|---|---|---|

Mean | Mean | Mean | Mean | Mean | |

f1 | −1400 | −1400 | −1400 | −1400 | −1400 |

f2 | 3.76 × 10^{5} | 3.84 × 10^{5} | 4.56 × 10^{5} | 3.96 × 10^{5} | 4.13 × 10^{5} |

f3 | 1.01 × 10^{8} | 8.32 × 10^{7} | 5.56 × 10^{7} | 7.16 × 10^{7} | 6.69 × 10^{7} |

f4 | −1099.9872 | −1099.98 | −1099.988 | −1099.9870 | −1099.984 |

f5 | −1000 | −1000 | −1000 | −1000 | −1000 |

f6 | −870.38 | −876.05 | −875.5 | −875.29 | −874.71 |

f7 | −713.59 | −711.45 | −713.69 | −712.66 | −702.99 |

f8 | −679.069 | −679.057 | −679.052 | −679.063 | −679.067 |

f9 | −578.503 | −577.189 | −577.75 | −577.436 | −576.518 |

f10 | −499.976 | −499.968 | −499.968 | −499.972 | −499.974 |

f11 | −305.44 | −302.436 | −307.02 | −311.215 | −309.596 |

f12 | −164.688 | −174.843 | −163.865 | −173.722 | −154.561 |

f13 | −31.9988 | −36.4318 | −35.7453 | −30.6652 | −32.3421 |

f14 | 2616.647 | 2543.716 | 2676.641 | 2586.064 | 2704.535 |

f15 | 3664.113 | 3974.245 | 3888.197 | 3946.214 | 3723.16 |

f16 | 200.3942 | 200.3496 | 200.3884 | 200.3441 | 300.3698 |

f17 | 437.5601 | 425.8707 | 426.4633 | 424.32 | 428.1304 |

f18 | 583.18 | 577.8134 | 578.672 | 576.0805 | 573.5208 |

f19 | 506.931 | 506.5545 | 506.6363 | 507.0156 | 506.3289 |

f20 | 613.1458 | 613.154 | 613.113 | 613.594 | 613.423 |

f21 | 1047.089 | 1051.01 | 1016.475 | 1035.483 | 1049.556 |

f22 | 3871.804 | 3928.667 | 4109.614 | 4059.632 | 4032.769 |

f23 | 5402.42 | 5574.529 | 5524.135 | 5597.751 | 5338.983 |

f24 | 1264.25 | 1265.845 | 1265.61 | 1268.231 | 1264.214 |

f25 | 1390.105 | 1387.764 | 1387.808 | 1390.035 | 1391.654 |

f26 | 1408.752 | 1412.901 | 1424.752 | 1414.98 | 1412.238 |

f27 | 2203.579 | 2187.724 | 2192.054 | 2191.372 | 2181.232 |

f28 | 1812.154 | 1762.647 | 1707.262 | 1771.612 | 1830.575 |

Average Ranking | |||||

2.93 | 2.82 | 2.86 | 3.07 | 2.93 |

Functions | EFWA | AFWA | dynFWA | AMdynFWA |
---|---|---|---|---|

Mean Error | Mean Error | Mean Error | Mean Error | |

f1 | 7.82 × 10^{−2} | 0 | 0 | 0 |

f2 | 5.43 × 10^{5} | 8.93 × 10^{5} | 7.87 × 10^{5} | 3.84 × 10^{5} |

f3 | 1.26 × 10^{8} | 1.26 × 10^{8} | 1.57 × 10^{8} | 8.32 × 10^{7} |

f4 | 1.09 | 11.5 | 12.8 | 2.02 × 10^{−2} |

f5 | 7.9 × 10^{−2} | 6.04 × 10^{−4} | 5.42 × 10^{−4} | 1.86 × 10^{−4} |

f6 | 34.9 | 29.9 | 31.5 | 23.9 |

f7 | 1.33 × 10^{2} | 9.19 × 10^{1} | 1.03 × 10^{2} | 8.85 × 10^{1} |

f8 | 2.10 × 10^{1} | 2.09 × 10^{1} | 2.09 × 10^{1} | 2.09 × 10^{1} |

f9 | 3.19 × 10^{1} | 2.48 × 10^{1} | 2.56 × 10^{1} | 2.28 × 10^{1} |

f10 | 8.29 × 10^{−1} | 4.73 × 10^{−2} | 4.20 × 10^{−2} | 3.18 × 10^{−2} |

f11 | 4.22×10^{2} | 1.05 × 10^{2} | 1.07 × 10^{2} | 9.75 × 10^{1} |

f12 | 6.33 × 10^{2} | 1.52 × 10^{2} | 1.56 × 10^{2} | 1.25 × 10^{2} |

f13 | 4.51 × 10^{2} | 2.36 × 10^{2} | 2.44 × 10^{2} | 1.63 × 10^{2} |

f14 | 4.16 × 10^{3} | 2.97 × 10^{3} | 2.95 × 10^{3} | 2.64 × 10^{3} |

f15 | 4.13 × 10^{3} | 3.81 × 10^{3} | 3.71 × 10^{3} | 3.87 × 10^{3} |

f16 | 5.92 × 10^{−1} | 4.97 × 10^{−1} | 4.77 × 10^{−1} | 3.4 × 10^{−1} |

f17 | 3.10 × 10^{2} | 1.45 × 10^{2} | 1.48 × 10^{2} | 1.25 × 10^{2} |

f18 | 1.75 × 10^{2} | 1.75 × 10^{2} | 1.89 × 10^{2} | 1.77 × 10^{2} |

f19 | 12.3 | 6.92 | 6.87 | 6.55 |

f20 | 14.6 | 13 | 13 | 13 |

f21 | 3.24 × 10^{2} | 3.16 × 10^{2} | 2.92 × 10^{2} | 3.51 × 10^{2} |

f22 | 5.75 × 10^{3} | 3.45 × 10^{3} | 3.41 × 10^{3} | 3.12 × 10^{3} |

f23 | 5.74 × 10^{3} | 4.70 × 10^{3} | 4.55 × 10^{3} | 4.67 × 10^{3} |

f24 | 3.37 × 10^{2} | 2.70 × 10^{2} | 2.72 × 10^{2} | 2.65 × 10^{2} |

f25 | 3.56 × 10^{2} | 2.99 × 10^{2} | 2.97 × 10^{2} | 2.87 × 10^{2} |

f26 | 3.21 × 10^{2} | 2.73 × 10^{2} | 2.62 × 10^{2} | 2.12 × 10^{2} |

f27 | 1.28 × 10^{3} | 9.72 × 10^{2} | 9.92 × 10^{2} | 8.87 × 10^{2} |

f28 | 4.34 × 10^{2} | 4.37 × 10^{2} | 3.40 × 10^{2} | 3.62 × 10^{2} |

total number of rank 1 | ||||

1 | 4 | 7 | 23 |

Functions | p/Significance | EFWA | AFWA | dynFWA |
---|---|---|---|---|

f1 | p-value | 0 | NaN | NaN |

significance | + | - | - | |

f2 | p-value | 1.5080 × 10^{−}^{32} | 5.1525 × 10^{−50} | 2.6725 × 10^{−49} |

significance | + | + | + | |

f3 | p-value | 0.8004 | 0.4302 | 0.0778 |

significance | - | - | - | |

f4 | p-value | 1.5546 × 10^{−136} | 1.8922 × 10^{−246} | 8.8572 × 10^{−235} |

significance | + | + | + | |

f5 | p-value | 0 | NaN | NaN |

significance | + | - | - | |

f6 | p-value | 1.5957 × 10^{−14} | 0.7108 | 0.0139 |

significance | + | - | + | |

f7 | p-value | 1.8067 × 10^{−36} | 0.5665 | 0.0084 |

significance | + | - | + | |

f8 | p-value | 0.1562 | 0.0137 | 9.2522 × 10^{−6} |

significance | - | + | + | |

f9 | p-value | 7.0132 × 10^{−27} | 0.0278 | 6.6090 × 10^{−8} |

significance | + | + | + | |

f10 | p-value | 2.7171 × 10^{−134} | 7.3507 × 10^{−6} | 0.0364 |

significance | + | + | + | |

f11 | p-value | 2.2083 × 10^{−100} | 3.0290 × 10^{−10} | 0.0437 |

significance | + | + | + | |

f12 | p-value | 1.7319 × 10^{−101} | 1.3158 × 10^{−11} | 1.8212 × 10^{−7} |

significance | + | + | + | |

f13 | p-value | 2.3914 × 10^{−89} | 4.1645 × 10^{−36} | 8.6284 × 10^{−37} |

significance | + | + | + | |

f14 | p-value | 0.0424 | 0.0117 | 4.4964 × 10^{−5} |

significance | + | + | + | |

f15 | p-value | 1.1749 × 10^{−6} | 0.9976 | 0.6064 |

significance | + | - | - | |

f16 | p-value | 2.2725 × 10^{−17} | 8.9230×10^{−}^{12} | 2.3427 × 10^{−}^{13} |

significance | + | + | + | |

f17 | p-value | 1.5713 × 10^{−81} | 7.3257 × 10^{−}^{10} | 1.0099 × 10^{−}^{6} |

significance | + | + | + | |

f18 | p-value | 0.8510 | 0.2430 | 0.1204 |

significance | - | - | - | |

f19 | p-value | 3.6331 × 10^{−25} | 5.3309 × 10^{−}^{6} | 0.0086 |

significance | + | + | + | |

f20 | p-value | 3.5246 × 10^{−}^{14} | 0.2830 | 0.4615 |

significance | + | - | - | |

f21 | p-value | 2.2455 × 10^{−}^{6} | 0.0120 | 0.0028 |

significance | + | + | + | |

f22 | p-value | 3.2719 × 10^{−}^{46} | 0.0634 | 0.0344 |

significance | + | - | - | |

f23 | p-value | 2.1191 × 10^{−}^{33} | 0.1225 | 0.4819 |

significance | + | - | - | |

f24 | p-value | 8.9612 × 10^{−}^{69} | 9.0342 × 10^{−}^{5} | 6.0855 × 10^{−}^{4} |

significance | + | + | + | |

f25 | p-value | 1.2812 × 10^{−}^{59} | 1.0745 × 10^{−}^{6} | 1.6123 × 10^{−}^{8} |

significance | + | + | + | |

f26 | p-value | 4.6864 × 10^{−}^{39} | 2.5440 × 10^{−}^{16} | 1.1739 × 10^{−}^{11} |

significance | + | + | + | |

f27 | p-value | 2.3540 × 10^{−}^{46} | 4.8488 × 10^{−}^{6} | 2.1456 × 10^{−}^{7} |

significance | + | + | + | |

f28 | p-value | 6.4307 × 10^{−}^{92} | 0.4414 | 0.0831 |

significance | + | - | - |

Functions Type | EFWA | AFWA | dynFWA |
---|---|---|---|

Unimodal Functions (f1 − f5) | 4 | 2 | 2 |

Basic Multimodal Functions (f6 − f20) | 13 | 10 | 12 |

Composition Functions (f21 − f28) | 8 | 5 | 5 |

Total number of significance in EFWA, AFWA and dynFWA | |||

25 | 17 | 19 |

Functions | Mean Error/Rank | ABC | DE | CMS-ES | SPSO2011 | AMdynFWA |
---|---|---|---|---|---|---|

f1 | Mean error | 0 | 1.89 × 10^{−3} | 0 | 0 | 0 |

Rank | 1 | 2 | 1 | 1 | 1 | |

f2 | Mean error | 6.20 × 10^{6} | 5.52 × 10^{4} | 0 | 3.38 × 10^{5} | 3.84 × 10^{5} |

Rank | 5 | 2 | 1 | 3 | 4 | |

f3 | Mean error | 5.74 × 10^{8} | 2.16 × 10^{6} | 1.41 × 10^{1} | 2.88 × 10^{8} | 8.32 × 10^{7} |

Rank | 5 | 2 | 1 | 4 | 3 | |

f4 | Mean error | 8.75 × 10^{4} | 1.32 × 10^{−1} | 0 | 3.86 × 10^{4} | 2.02 × 10^{−2} |

Rank | 5 | 3 | 1 | 4 | 2 | |

f5 | Mean error | 0 | 2.48 × 10^{−3} | 0 | 5.42 × 10^{−4} | 1.86 × 10^{−4} |

Rank | 1 | 4 | 1 | 3 | 2 | |

f6 | Mean error | 1.46 × 10^{1} | 7.82 | 7.82 × 10^{−2} | 3.79 × 10^{1} | 2.39 × 10^{1} |

Rank | 3 | 2 | 1 | 5 | 4 | |

f7 | Mean error | 1.25 × 10^{2} | 4.89×10^{1} | 1.91 × 10^{1} | 8.79 × 10^{1} | 8.85 × 10^{1} |

Rank | 5 | 2 | 1 | 3 | 4 | |

f8 | Mean error | 2.09 × 10^{1} | 2.09 × 10^{1} | 2.14 × 10^{1} | 2.09 × 10^{1} | 2.09 × 10^{1} |

Rank | 1 | 1 | 2 | 1 | 1 | |

f9 | Mean error | 3.01 × 10^{1} | 1.59 ×10^{1} | 4.81 × 10^{1} | 2.88 × 10^{1} | 2.28 × 10^{1} |

Rank | 4 | 1 | 5 | 3 | 2 | |

f10 | Mean error | 2.27 × 10^{−1} | 3.42 × 10^{-2} | 1.78 × 10^{−2} | 3.40 × 10^{−1} | 3.18 × 10^{−2} |

Rank | 4 | 3 | 1 | 5 | 2 | |

f11 | Mean error | 0 | 7.88 × 10^{1} | 4.00 × 10^{2} | 1.05 × 10^{2} | 9.75 × 10^{1} |

Rank | 1 | 2 | 5 | 4 | 3 | |

f12 | Mean error | 3.19 × 10^{2} | 8.14 × 10^{1} | 9.42 × 10^{2} | 1.04 × 10^{2} | 1.25 × 10^{2} |

Rank | 4 | 1 | 5 | 2 | 3 | |

f13 | Mean error | 3.29 × 10^{2} | 1.61 × 10^{2} | 1.08 × 10^{3} | 1.94 × 10^{2} | 1.63 × 10^{2} |

Rank | 4 | 1 | 5 | 3 | 2 | |

f14 | Mean error | 3.58 ×10^{−1} | 2.38 × 10^{3} | 4.94 × 10^{3} | 3.99 × 10^{3} | 2.64 × 10^{3} |

Rank | 1 | 2 | 5 | 4 | 3 | |

f15 | Mean error | 3.88 × 10^{3} | 5.19 × 10^{3} | 5.02 × 10^{3} | 3.81 × 10^{3} | 3.87 × 10^{3} |

Rank | 3 | 5 | 4 | 1 | 2 | |

f16 | Mean error | 1.07 | 1.97 | 5.42 × 10^{−2} | 1.31 | 3.4 × 10^{−1} |

Rank | 3 | 5 | 1 | 4 | 2 | |

f17 | Mean error | 3.04 × 10^{1} | 9.29 × 10^{1} | 7.44 × 10^{2} | 1.16 × 10^{2} | 1.25 × 10^{2} |

Rank | 1 | 2 | 5 | 3 | 4 | |

f18 | Mean error | 3.04 × 10^{2} | 2.34 × 10^{2} | 5.17 × 10^{2} | 1.21 × 10^{2} | 1.77 × 10^{2} |

Rank | 4 | 3 | 5 | 1 | 2 | |

f19 | Mean error | 2.62 × 10^{−1} | 4.51 | 3.54 | 9.51 | 6.55 |

Rank | 1 | 3 | 2 | 5 | 4 | |

f20 | Mean error | 1.44 × 10^{1} | 1.43 × 10^{1} | 1.49 × 10^{1} | 1.35 × 10^{1} | 1.30 × 10^{1} |

Rank | 4 | 3 | 5 | 2 | 1 | |

f21 | Mean error | 1.65 × 10^{2} | 3.20 × 10^{2} | 3.44 × 10^{2} | 3.09 × 10^{2} | 3.51 × 10^{2} |

Rank | 1 | 3 | 4 | 2 | 5 | |

f22 | Mean error | 2.41 × 10^{1} | 1.72 × 10^{3} | 7.97 × 10^{3} | 4.30 × 10^{3} | 3.12 × 10^{3} |

Rank | 1 | 2 | 5 | 4 | 3 | |

f23 | Mean error | 4.95 × 10^{3} | 5.28 × 10^{3} | 6.95 × 10^{3} | 4.83 × 10^{3} | 4.67 × 10^{3} |

Rank | 3 | 4 | 5 | 2 | 1 | |

f24 | Mean error | 2.90 × 10^{2} | 2.47 × 10^{2} | 6.62 × 10^{2} | 2.67 × 10^{2} | 2.65 × 10^{2} |

Rank | 4 | 1 | 5 | 3 | 2 | |

f25 | Mean error | 3.06 × 10^{2} | 2.89 × 10^{2} | 4.41 × 10^{2} | 2.99 × 10^{2} | 2.87 × 10^{2} |

Rank | 4 | 2 | 5 | 3 | 1 | |

f26 | Mean error | 2.01 × 10^{2} | 2.52 × 10^{2} | 3.29 × 10^{2} | 2.86 × 10^{2} | 2.12 × 10^{2} |

Rank | 1 | 3 | 5 | 4 | 2 | |

f27 | Mean error | 4.16 × 10^{2} | 7.64 × 10^{2} | 5.39 × 10^{2} | 1.00 × 10^{3} | 8.87 × 10^{2} |

Rank | 1 | 4 | 2 | 5 | 3 | |

f28 | Mean error | 2.58 × 10^{2} | 4.02 × 10^{2} | 4.78 × 10^{3} | 4.01 × 10^{2} | 3.62 × 10^{2} |

Rank | 1 | 4 | 5 | 3 | 2 |

SR/AR | ABC | DE | CMS-ES | SPSO2011 | AMdynFWA |
---|---|---|---|---|---|

Total number of rank 1 | 12 | 5 | 9 | 4 | 5 |

Total number of rank 2 | 0 | 10 | 3 | 4 | 11 |

Total number of rank 3 | 4 | 7 | 0 | 9 | 6 |

Total number of rank 4 | 8 | 4 | 2 | 7 | 5 |

Total number of rank 5 | 4 | 2 | 14 | 4 | 1 |

Total number of rank | 76 | 72 | 93 | 87 | 70 |

Average ranking | 2.71 | 2.57 | 3.32 | 3.11 | 2.5 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, X.-G.; Han, S.-F.; Zhao, L.; Gong, C.-Q.; Liu, X.-J.
Adaptive Mutation Dynamic Search Fireworks Algorithm. *Algorithms* **2017**, *10*, 48.
https://doi.org/10.3390/a10020048

**AMA Style**

Li X-G, Han S-F, Zhao L, Gong C-Q, Liu X-J.
Adaptive Mutation Dynamic Search Fireworks Algorithm. *Algorithms*. 2017; 10(2):48.
https://doi.org/10.3390/a10020048

**Chicago/Turabian Style**

Li, Xi-Guang, Shou-Fei Han, Liang Zhao, Chang-Qing Gong, and Xiao-Jing Liu.
2017. "Adaptive Mutation Dynamic Search Fireworks Algorithm" *Algorithms* 10, no. 2: 48.
https://doi.org/10.3390/a10020048