# Mexican Axolotl Optimization: A Novel Bioinspired Heuristic

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}, x

_{2}, x

_{3}, …, x

_{n}that either minimize or maximize some function f(x

_{1}, x

_{2}, x

_{3}, …, x

_{n}) [1].

## 2. Related Works on Bioinspired Optimization

- Fireflies are unisexual, so a firefly will be attracted to other fireflies regardless of their gender.
- The attractiveness is proportional to the brightness and both diminish as your distance increases. Therefore, for two flashing fireflies, the less bright will move to the brighter. If there is no one brighter than a particular firefly, it will move randomly.
- The brightness of a firefly is determined by the landscape of the objective feature.

## 3. Mexican Axolotl Variable Optimization

#### 3.1. The Axolotl in Nature

#### 3.2. The Artificial Axolotl

- We divide the individuals into males and females.
- We consider the females more important, due to the fact that for each female we find the best male according to tournament selection, to obtain the offspring.
- We have an elitist replacement procedure to include new individuals in the population. In such a procedure, the best individual is considered to be a female, and the second-best to be a male. That is, our procedure has the possibility of converting a male into a female, if the male is best.

## 4. Results and Discussion

#### 4.1. Optimization Functions

#### 4.2. Optimization Results of the Compared Algorithms

^{®}Core™ i7-6700 CPU 3.40 GHz processor, 16 GB of RAM, and a Nvidia GeForce GTX 1070 graphics card.

#### 4.3. Statistical Tests

#### 4.4. Convergence Analysis

#### 4.5. Main differences of MAO with Respect Other Bioinspired Algorithms

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Convergence of the Mexican Axolotl Optimization for unimodal benchmark functions: (

**a**) F1 test function, (

**b**) F2 test function, (

**c**) F3 test function, (

**d**) F4 test function, (

**e**) F5 test function, (

**f**) F6 test function, and (

**g**) F7 test function.

**Figure A2.**Convergence of the Mexican Axolotl Optimization for multimodal benchmark functions: (

**a**) F8 test function, (

**b**) F9 test function, (

**c**) F10 test function, (

**d**) F11 test function, (

**e**) F12 test function, and (

**f**) F13 test function.

**Figure A3.**Convergence of the Mexican Axolotl Optimization for composite benchmark functions: (

**a**) F14 test function, (

**b**) F15 test function, (

**c**) F16 test function, (

**d**) F17 test function, (

**e**) F18 test function, and (

**f**) F19 test function.

**Figure A4.**Convergence of the Mexican Axolotl Optimization for CEC06 2019 “The 100-Digit Challenge” benchmark functions: (

**a**) CEC01 test function, (

**b**) CEC02 test function, (

**c**) CEC03 test function, (

**d**) CEC04 test function, (

**e**) CEC05 test function, (

**f**) CEC06 test function, (

**g**) CEC07 test function, (

**h**) CEC08 test function, (

**i**) CEC09 test function, and (

**j**) CEC10 test function.

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**Figure 1.**Pseudocode of the Transition procedure, corresponding to the Transition from larvae to adult state phase in the Mexican Axolotl Optimization (MAO) algorithm.

**Figure 2.**Pseudocode of the Accidents procedure, corresponding to the Injury and restoration state phase in the MAO algorithm.

**Figure 3.**Pseudocode of the NewLife procedure, corresponding to the Reproduction and Assortment phase in the MAO algorithm of the proposed Mexican Axolotl Optimization.

**Figure 4.**Reproduction in the MAO. (

**a**) Male parent, (

**b**) female parent, (

**c**) random numbers generated to uniformly distribute the parents’ information, and (

**d**) the resulting offspring

**Figure 6.**Convergence of the Mexican Axolotl Optimization for some benchmark functions (

**a**) F1 test function (

**b**) F8 test function (

**c**) F14 test function, and (

**d**) CEC01 test function.

Function | Range | Shift Position | Min ^{1} |
---|---|---|---|

$TF1\left(x\right)=\sum _{i=1}^{n}{x}_{i}^{2}$ | [−100, 100] | [−30, −30, …, −30] | 0 |

$TF2\left(x\right)=\sum _{i=1}^{n}{x}_{i}|\prod _{i=1}^{n}\left|{x}_{i}\right|$ | [−10, 10] | [−3, −3, …, −3] | 0 |

$TF3\left(x\right)={\displaystyle \sum}_{i=1}^{n}\left(\sum _{j-1}^{i}{x}_{j}{}^{2}\right)$ | [−100, 100] | [−30, −30, …, −30] | 0 |

$TF4\left(x\right)=max\left\{\left|x\right|,1\le i\le n\right\}$ | [−100, 100] | [−30, −30, …, −30] | 0 |

$TF5\left(x\right)=\sum _{i=1}^{n-1}\left[100{\left({x}_{i+1}-{x}_{1}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]$ | [−30, 30] | [−15, −15, …, −15] | 0 |

$TF6\left(x\right)=\sum _{i=1}^{n}{\left(\left[{x}_{i}+0.5\right]\right)}^{2}$ | [−100, 100] | [−750, …, −750] | 0 |

$TF7\left(x\right)=\sum _{i=1}^{n}i{x}_{i}^{4}+random\left[0,1\right]$ | [−1.28, 1.28] | [−0.25, …, −0.25] | 0 |

^{1}Minimum value of the function.

Function | Range | Shift Position | Min ^{1} |
---|---|---|---|

$TF8\left(x\right)=\sum _{i=1}^{n}-{x}^{2}\mathrm{sin}\left(\sqrt{\left|{x}_{i}\right|}\right)$ | [−500, 500] | [−300, …, −300] | −418.9829 |

$TF9\left(x\right)=\sum _{i=1}^{n}\left[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right]$ | [−5.12, 5.12] | [−2, −2, …, −2] | 0 |

$TF10\left(x\right)=-20\mathrm{exp}\left(-0.2\sqrt{\sum _{i=1}^{n}{x}_{i}^{2}}\right)-\mathrm{exp}\left(\frac{1}{n}\sum _{i=1}^{n}\mathrm{cos}\left(2\pi {x}_{i}\right)\right)+20+e$ | [−32, 32] | − | 0 |

$TF11\left(x\right)=\frac{1}{4000}\sum _{i=1}^{n}{x}_{i}^{2}-\prod _{i=1}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ | [−600, 600] | [−400, …, −400] | 0 |

$TF12\left(x\right)=\frac{\pi}{n}\left\{10\mathrm{sin}\left(\pi {y}_{1}\right)+\sum _{i=1}^{n-1}{\left({y}_{i}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}\left(\pi {y}_{i+1}\right)\right]+{\left({y}_{n}-1\right)}^{2}\right\}+\sum _{i=1}^{n}u\left({x}_{i},10,100,4\right)$$,\mathrm{where}{y}_{i}=1+\frac{x+1}{4}$$,\mathrm{and}u\left({x}_{i},a,k,m\right)=\left\{\begin{array}{c}k{\left({x}_{i}-a\right)}^{m}{x}_{i}a\\ 0-a{x}_{i}a\\ k{\left(-{x}_{i}-a\right)}^{m}{x}_{i}-a\end{array}\right\}$ | [−50, 50] | [−30, −30, …,−30] | 0 |

$TF13\left(x\right)=0.1\left\{{\mathrm{sin}}^{2}\left(3\pi x1\right)+\sum _{i=1}^{n}{\left({x}_{i}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(3\pi {x}_{i}+1\right)\right]+{\left({x}_{n}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(2\pi {x}_{n}\right)\right]\right\}+\sum _{i=1}^{n}u\left({x}_{i},5,100,4\right)$ | [−50, 50] | [−100, …, −100] | 0 |

^{1}Minimum value of the function.

Function | Range | Min ^{1} |
---|---|---|

$\mathit{T}\mathit{F}\mathbf{14}\mathbf{\left(}\mathit{C}\mathit{F}\mathbf{1}\mathbf{\right)}$ $\mathrm{f}1,\mathrm{f}2,\mathrm{f}3\dots \mathrm{f}10=\mathrm{Sphere}\mathrm{function}$; $\delta 1,\delta 2,\delta 3\dots \delta 10=\left[1,1,1\dots 1\right]$; $\lambda 1,\lambda 2,\lambda 3\dots \lambda 10=\left[\frac{5}{100},\frac{5}{100},\frac{5}{100},\dots \frac{5}{100}\right]$ | [−5, 5] | 0 |

$\mathit{T}\mathit{F}\mathbf{15}\mathbf{\left(}\mathit{C}\mathit{F}\mathbf{2}\mathbf{\right)}$ $\mathrm{f}1,\mathrm{f}2,\mathrm{f}3\dots \mathrm{f}10={\mathrm{Griewank}}^{\prime}\mathrm{s}\mathrm{function}$; $\delta 1,\delta 2,\delta 3\dots \delta 10=\left[1,1,1\dots 1\right]$; $\lambda 1,\lambda 2,\lambda 3\dots \lambda 10=\left[\frac{5}{100},\frac{5}{100},\frac{5}{100},\dots \frac{5}{100}\right]$ | [−5, 5] | 0 |

$\mathit{T}\mathit{F}\mathbf{16}\mathbf{\left(}\mathit{C}\mathit{F}\mathbf{3}\mathbf{\right)}$ $\mathrm{f}1,\mathrm{f}2,\mathrm{f}3\dots \mathrm{f}10={\mathrm{Griewank}}^{\prime}\mathrm{s}\mathrm{function}$; $\delta 1,\delta 2,\delta 3\dots \delta 10=\left[1,1,1\dots 1\right]$; $\lambda 1,\lambda 2,\lambda 3\dots \lambda 10=\left[1,1,1\dots 1\right]$ | [−5, 5] | 0 |

$\mathit{T}\mathit{F}\mathbf{17}\mathbf{\left(}\mathit{C}\mathit{F}\mathbf{4}\mathbf{\right)}$ $\mathrm{f}1,\mathrm{f}2={\mathrm{Ackley}}^{\prime}\mathrm{s}\mathrm{function}$$;\mathrm{f}3,\mathrm{f}4={\mathrm{Ackley}}^{\prime}\mathrm{s}\mathrm{function}$$;\mathrm{f}5,\mathrm{f}6={\mathrm{Ackley}}^{\prime}\mathrm{s}\mathrm{function}$$;\mathrm{f}7,\mathrm{f}8={\mathrm{Ackley}}^{\prime}\mathrm{s}\mathrm{function}$$;\mathrm{f}9,\mathrm{f}10={\mathrm{Ackley}}^{\prime}\mathrm{s}\mathrm{function}$$;\delta 1,\delta 2,\delta 3\dots \delta 10=\left[1,1,1\dots 1\right];$$\lambda 1,\lambda 2,\lambda 3\dots \lambda 10=\left[\frac{5}{32},\frac{5}{32},1,1,\frac{5}{0.5},\frac{5}{0.5},\frac{5}{0.5},\frac{5}{0.5},\frac{5}{0.5},\frac{5}{0.5}\right]$ | [−5, 5] | 0 |

$\mathit{T}\mathit{F}\mathbf{18}\mathbf{\left(}\mathit{C}\mathit{F}\mathbf{5}\mathbf{\right)}$ $\mathrm{f}1,\mathrm{f}2=\mathrm{Rastrigins}\mathrm{function}$$;\mathrm{f}3,\mathrm{f}4=\mathrm{Weierstrasss}\mathrm{function}$$;\mathrm{f}5,\mathrm{f}6={\mathrm{Griewank}}^{\prime}\mathrm{s}\mathrm{function}$$;\mathrm{f}7,\mathrm{f}8={\mathrm{Ackley}}^{\prime}\mathrm{s}\mathrm{function}$$;\mathrm{f}9,\mathrm{f}10=\mathrm{Sphere}\mathrm{function}$$;\delta 1,\delta 2,\delta 3\dots \delta 10=\left[1,1,1\dots 1\right]$$;\lambda 1,\lambda 2,\lambda 3\dots \lambda 10=\left[\frac{1}{5},\frac{1}{5},\frac{5}{0.5},\frac{5}{0.5},\frac{5}{100},\frac{5}{100},\frac{5}{32},\frac{5}{32},\frac{5}{100},\frac{5}{100}\right]$ | [−5, 5] | 0 |

$\mathit{T}\mathit{F}\mathbf{19}\mathbf{\left(}\mathit{C}\mathit{F}\mathbf{6}\mathbf{\right)}$ $\mathrm{f}1,\mathrm{f}2=\mathrm{Rastrigins}\mathrm{function}$$;\mathrm{f}3,\mathrm{f}4=\mathrm{Weierstrasss}\mathrm{function}$$;\mathrm{f}5,\mathrm{f}6={\mathrm{Griewank}}^{\prime}\mathrm{s}\mathrm{function}$$;\mathrm{f}7,\mathrm{f}8={\mathrm{Ackley}}^{\prime}\mathrm{s}\mathrm{function}$$;\mathrm{f}9,\mathrm{f}10=\mathrm{Sphere}\mathrm{function}$$;\delta 1,\delta 2,\delta 3\dots \delta 10=\left[0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1\right]$$;\lambda 1,\lambda 2,\lambda 3\dots \lambda 10=\left[0.1\ast \frac{1}{5},0.2\ast \frac{1}{5},0.3\ast \frac{5}{0.5},0.4\ast \frac{5}{0.5},0.5\ast \frac{5}{100},0.6\ast \frac{5}{100},0.7\ast \frac{5}{32},0.8\ast \frac{5}{32},0.9\ast \frac{5}{100},1\ast \frac{5}{100}\right]$ | [−5, 5] | 0 |

^{1}Minimum value of the function.

Function | Dimensions | Range | Min ^{1} |
---|---|---|---|

Storn’s Chebyshev polynomial fitting problem | 9 | [−8192, 8192] | 1 |

Inverse Hilbert matrix problem | 16 | [−16,384, 16,384] | 1 |

Lennard-Jones minimum energy cluster | 18 | [−4, 4] | 1 |

Rastrigin’s function | 10 | [−100, 100] | 1 |

Griewangk’s function | 10 | [−100, 100] | 1 |

Weierstrasss function | 10 | [−100, 100] | 1 |

Modified Schewefel’s function | 10 | [−100, 100] | 1 |

Expanded Schafeer’s F6 function | 10 | [−100, 100] | 1 |

Happy Cat function | 10 | [−100, 100] | 1 |

Ackley function | 10 | [−100, 100] | 1 |

^{1}Minimum value of the function.

Set | Function | ABC | CS | DE | FA | FDO | MBO | SMA | WOA | MAO |
---|---|---|---|---|---|---|---|---|---|---|

Unimodal | F1 | 5447.7037 | 9165.9901 | 8587.4237 | 2936.3110 | 2855.3899 | 2997.0703 | 13.0903 | 234.8699 | 321.0370 |

F2 | 2.8086 | 31.3204 | 35.0858 | 14.5341 | 13.8453 | 13.6921 | 0.3195 | 3.2792 | 4.1843 | |

F3 | 7228.5989 | 9879.7702 | 10,826.8313 | 4258.6261 | 5174.2327 | 6729.9519 | 5245.1086 | 13,758.8168 | 700.1304 | |

F4 | 57.9899 | 54.8784 | 59.7082 | 29.7369 | 25.2446 | 23.4809 | 0.3670 | 41.7093 | 12.3571 | |

F5 | 2.68 × 10^{6} | 1.10 × 10^{7} | 9.71 × 10^{6} | 1.23 × 10^{6} | 3.12 × 10^{6} | 4.40 × 10^{6} | 13.1924 | 3.51 × 10^{4} | 1.84 × 10^{4} | |

F6 | 5928.5177 | 9269.7192 | 7336.3097 | 3043.6261 | 2771.3385 | 3193.3915 | 3.0798 | 246.8890 | 266.5308 | |

F7 | 1.2196 | 2.3582 | 2.8935 | 0.6749 | 1.1486 | 2.5256 | 0.1911 | 0.2530 | 0.0484 | |

Multimodal | F8 | −2407.4569 | −1809.7610 | −1950.2114 | −1531.2946 | −1483.9961 | −3132.3800 | −3293.9270 | −2635.7968 | −2843.8943 |

F9 | 70.2922 | 91.3020 | 95.3271 | 67.2274 | 55.1742 | 44.5630 | 15.3040 | 56.0142 | 25.3499 | |

F10 | 17.9524 | 18.7192 | 19.4404 | 15.1343 | 10.1403 | 11.3944 | 0.5564 | 7.3438 | 7.1662 | |

F11 | 46.1170 | 83.3698 | 71.7901 | 27.2379 | 24.3820 | 22.7500 | 0.4801 | 2.5929 | 3.7582 | |

F12 | 2.36 × 10^{6} | 1.42 × 10^{7} | 1.62 × 10^{7} | 1.22 × 10^{5} | 1.56 × 10^{6} | 1.99 × 10^{6} | 2.5396 | 5.96 × 10^{3} | 5.95 | |

F13 | 7.54 × 10^{6} | 5.21 × 10^{7} | 5.64 × 10^{7} | 2.12 × 10^{6} | 9.22 × 10^{6} | 1.17 × 10^{6} | 1.0523 | 3.03 × 10^{4} | 3.28 × 10^{3} | |

Composite | F14 | 342.3150 | 354.0967 | 410.7029 | 758.6487 | 968.2830 | 393.5150 | 429.0229 | 395.7099 | 473.1349 |

F15 | 476.2411 | 463.9622 | 468.3889 | 821.8865 | 1048.0255 | 478.9110 | 513.0348 | 457.5810 | 507.2991 | |

F16 | 1072.7264 | 1052.7694 | 1056.7774 | 1491.2869 | 1393.0906 | 1087.9062 | 1062.6645 | 1114.6494 | 1147.8570 | |

F17 | 990.3510 | 1003.4222 | 1026.9885 | 1077.1571 | 1050.0625 | 1001.3681 | 904.2288 | 999.8535 | 952.0442 | |

F18 | 420.8507 | 424.6245 | 442.7791 | 894.0803 | 1136.6270 | 420.0096 | 508.7819 | 447.0013 | 524.1615 | |

F19 | 1003.9296 | 996.0488 | 978.5206 | 974.7888 | 951.7356 | 933.2716 | 874.2914 | 933.4341 | 905.2780 | |

Competition | CEC01 | 6.27 × 10^{11} | 1.13 × 10^{12} | 6.26 × 10^{11} | 1.04 × 10^{12} | 3.96 × 10^{11} | 8.13 × 10^{11} | 8.17 × 10^{11} | 1.01 × 10^{12} | 4.11 × 10^{10} |

CEC02 | 10,208.4315 | 8666.1661 | 4254.8599 | 4408.7645 | 4833.2511 | 7174.2428 | 41.9309 | 479.6424 | 424.2248 | |

CEC03 | 12.7058 | 12.7047 | 12.7039 | 12.7043 | 12.7037 | 12.7036 | 12.7035 | 12.7026 | 12.7026 | |

CEC04 | 8564.8124 | 16,171.5930 | 9055.5641 | 11,536.6751 | 4962.4534 | 7862.3602 | 17,307.1928 | 5700.3273 | 4460.3403 | |

CEC05 | 4.4837 | 5.4179 | 3.9627 | 3.8751 | 2.8207 | 3.4589 | 5.6967 | 3.1687 | 2.6745 | |

CEC06 | 11.8186 | 13.0760 | 13.3387 | 14.3070 | 13.5259 | 11.4701 | 13.0106 | 13.1277 | 12.5090 | |

CEC07 | 1016.2961 | 1318.4455 | 1425.1217 | 1635.1259 | 1506.3325 | 1057.3337 | 1204.1217 | 1275.5805 | 1184.9008 | |

CEC08 | 6.9177 | 7.2169 | 7.4573 | 7.5103 | 7.0885 | 6.9756 | 7.4529 | 7.1778 | 6.8954 | |

CEC09 | 2013.8642 | 3939.6893 | 2682.6881 | 1773.0340 | 1059.9142 | 1321.8686 | 4199.2309 | 1043.8402 | 431.0717 | |

CEC10 | 20.6068 | 20.7581 | 20.8089 | 20.8308 | 20.8116 | 20.5595 | 20.7914 | 20.7122 | 20.6511 |

Algorithm | Parameters ^{1} |

ABC | Number of food sources: 30; Maximum number of failures which lead to the elimination: Number of food sources * dimension |

CS | Number of nests: 30; Discovery rate of alien eggs/solutions: 10^{−5} |

DE | Population Size: 30; Crossover probability: 0.8; Scaling factor: 0.85 |

FA | Number of Fireflies: 30; Alpha: 0.5; Betamin: 0.2; Gamma: 1.0 |

FDO | Scout bee number: 30; Weight Factor: 0.0 |

MAO | Total population size: 30; damage probability dp = 0.5; regeneration probability rp = 0.1; tournament size k = 3; differentiation constant $\lambda =0.5$. |

MBO | Total population size: 30; The percentage of population for MBO: 5/12; Elitism parameter: 2.0; Max Step size: 1.0; 12 months in a year: 1.2 |

SMA | Number of search agent: 30; z: 0.03 |

WOA | Number of search agents |

^{1}As in the MATLAB code publicly available at www.mathworks.com (accessed on 31 March 2021).

Algorithm. | Ranking |
---|---|

MAO | 2.7069 |

SMA | 3.4483 |

WOA | 3.8448 |

MBO | 4.1724 |

ABC | 5.1724 |

FDO | 5.5517 |

FA | 6.4828 |

CS | 6.7931 |

DE | 6.8276 |

i | Algorithm | Z | p-Value | Holm |
---|---|---|---|---|

8 | DE | 5.729586 | 0.000000 | 0.006250 |

7 | CS | 5.68164 | 0.000000 | 0.007143 |

6 | FA | 5.250123 | 0.000000 | 0.008333 |

5 | FDO | 3.955572 | 0.000076 | 0.010000 |

4 | ABC | 3.428163 | 0.000608 | 0.012500 |

3 | MBO | 2.037719 | 0.041578 | 0.016667 |

2 | WOA | 1.582229 | 0.113597 | 0.025000 |

1 | SMA | 1.030846 | 0.302613 | 0.050000 |

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## Share and Cite

**MDPI and ACS Style**

Villuendas-Rey, Y.; Velázquez-Rodríguez, J.L.; Alanis-Tamez, M.D.; Moreno-Ibarra, M.-A.; Yáñez-Márquez, C.
Mexican Axolotl Optimization: A Novel Bioinspired Heuristic. *Mathematics* **2021**, *9*, 781.
https://doi.org/10.3390/math9070781

**AMA Style**

Villuendas-Rey Y, Velázquez-Rodríguez JL, Alanis-Tamez MD, Moreno-Ibarra M-A, Yáñez-Márquez C.
Mexican Axolotl Optimization: A Novel Bioinspired Heuristic. *Mathematics*. 2021; 9(7):781.
https://doi.org/10.3390/math9070781

**Chicago/Turabian Style**

Villuendas-Rey, Yenny, José L. Velázquez-Rodríguez, Mariana Dayanara Alanis-Tamez, Marco-Antonio Moreno-Ibarra, and Cornelio Yáñez-Márquez.
2021. "Mexican Axolotl Optimization: A Novel Bioinspired Heuristic" *Mathematics* 9, no. 7: 781.
https://doi.org/10.3390/math9070781