# Adaptive Gene Level Mutation

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^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Genetic Algorithm

Algorithm 1: Genetic algorithm main steps |

## 3. Locus Adaptive Genetic Algorithm

Algorithm 2: Genetic algorithm main steps |

## 4. Heuristically Partially Solvable Problems with Unknown Optimum

## 5. Results

#### 5.1. N-Queens Problem

#### 5.2. Traveling Salesman Problems

#### 5.3. Using Locus Mutation with Other Heuristic Algorithms

#### 5.4. Exploiting the Tuning of the Power Parameter

#### 5.5. Running Time Comparison

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Talbi, H.; Batouche, M.; Draa, A. A quantum-inspired evolutionary algorithm for multiobjective image segmentation. Int. J. Math. Phys. Eng. Sci.
**2007**, 1, 109–114. [Google Scholar] - Jin, Y.; Branke, J. Evolutionary optimization in uncertain environments-a survey. IEEE Trans. Evol. Comput.
**2005**, 9, 303–317. [Google Scholar] [CrossRef][Green Version] - Wang, S.; Wang, Y.; Du, W.; Sun, F.; Wang, X.; Zhou, C.; Liang, Y. A multi-approaches-guided genetic algorithm with application to operon prediction. Artif. Intell. Med.
**2007**, 41, 151–159. [Google Scholar] [CrossRef] [PubMed] - Krawiec, K.; Pawlak, M. Genetic programming with alternative search drivers for detection of retinal blood vessels. In Proceedings of the European Conference on the Applications of Evolutionary Computation, Copenhagen, Denmark, 8–10 April 2015; Springer: Cham, Switzerland, 2015; pp. 554–566. [Google Scholar]
- Buurman, J.; Zhang, S.; Babovic, V. Reducing risk through real options in systems design: The case of architecting a maritime domain protection system. Risk Anal. Int. J.
**2009**, 29, 366–379. [Google Scholar] [CrossRef] - Zhang, S.X.; Babovic, V. An evolutionary real options framework for the design and management of projects and systems with complex real options and exercising conditions. Decis. Support Syst.
**2011**, 51, 119–129. [Google Scholar] [CrossRef] - Milone, D.H.; Merelo, J.J.; Rufiner, H. Evolutionary algorithm for speech segmentation. In Proceedings of the 2002 Congress on Evolutionary Computation, CEC’02 (Cat. No. 02TH8600), Honolulu, HI, USA, 12–17 May 2002; Volume 2, pp. 1115–1120. [Google Scholar]
- Vadakkepat, P.; Tan, K.C.; Ming-Liang, W. Evolutionary artificial potential fields and their application in real time robot path planning. In Proceedings of the 2000 congress on evolutionary computation, CEC00 (Cat. No. 00TH8512), La Jolla, CA, USA, 16–19 July 2000; IEEE: Piscataway, NJ, USA, 2000; Volume 1, pp. 256–263. [Google Scholar]
- Pan, X.; Zhang, J.; Szeto, K.Y. Application of Mutation Only Genetic Algorithm for the Extraction of Investment Strategy in Financial Time Series. In Proceedings of the 2005 International Conference on Neural Networks and Brain, Beijing, China, 13–15 October 2005; Volume 3, pp. 1682–1686. [Google Scholar]
- Corus, D.; Oliveto, P.S. Standard Steady State Genetic Algorithms Can Hillclimb Faster than Mutation-only Evolutionary Algorithms. arXiv
**2017**, arXiv:1708.01571. [Google Scholar] [CrossRef][Green Version] - Berger-Tal, O.; Nathan, J.; Meron, E.; Saltz, D. The exploration-exploitation dilemma: A multidisciplinary framework. PLoS ONE
**2014**, 9, e95693. [Google Scholar] - Abdoun, O.; Abouchabaka, J.; Tajani, C. Analyzing the Performance of Mutation Operators to Solve the Travelling Salesman Problem. arXiv
**2012**, arXiv:1203.3099. [Google Scholar] - Eiben, A.; Michalewicz, Z.; Schoenauer, M.; Smith, J. Parameter Control in Evolutionary Algorithms. In Parameter Setting in Evolutionary Algorithms; Lobo, F.G., Lima, C.F., Michalewicz, Z., Eds.; Studies in Computational Intelligence Book Series; Springer: Berlin/Heidelberg, Germany, 2007; Volume 54, pp. 19–46. Available online: http://www.springerlink.com/content/978-3-540-69431-1/ (accessed on 9 January 2021). [CrossRef][Green Version]
- Case, B.; Lehre, P.K. Self-adaptation in non-Elitist Evolutionary Algorithms on Discrete Problems with Unknown Structure. arXiv
**2020**, arXiv:2004.00327. [Google Scholar] - Baldi, P. Gradient descent learning algorithm overview: A general dynamical systems perspective. IEEE Trans. Neural Netw.
**1995**, 6, 182–195. [Google Scholar] [CrossRef] - Ma, Y.A.; Chen, Y.; Jin, C.; Flammarion, N.; Jordan, M.I. Sampling Can Be Faster Than Optimization. arXiv
**2018**, arXiv:1811.08413. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bottou, L. Large-scale machine learning with stochastic gradient descent. In Proceedings of the COMPSTAT’2010, Paris, France, 22–27 August 2010; Springer: Cham, Switzerland, 2010; pp. 177–186. [Google Scholar]
- Kingma, D.P.; Ba, J. Adam: A method for stochastic optimization. arXiv
**2014**, arXiv:1412.6980. [Google Scholar] - Young, S.R.; Rose, D.C.; Karnowski, T.P.; Lim, S.H.; Patton, R.M. Optimizing deep learning hyper-parameters through an evolutionary algorithm. In Proceedings of the Workshop on Machine Learning in High-Performance Computing Environments, Austin, TX, USA, 15–20 November 2015; ACM: New York, NY, USA, 2015; p. 4. [Google Scholar]
- Such, F.P.; Madhavan, V.; Conti, E.; Lehman, J.; Stanley, K.O.; Clune, J. Deep neuroevolution: Genetic algorithms are a competitive alternative for training deep neural networks for reinforcement learning. arXiv
**2017**, arXiv:1712.06567. [Google Scholar] - Bezzel, M. Proposal of 8-queens problem. Berl. Schachzeitung
**1848**, 3, 1848. [Google Scholar] - Gupta, S.; Panwar, P. Solving Travelling Salesman Problem Using Genetic Algorithm. Int. J. Adv. Res. Comput. Sci. Softw. Eng.
**2013**, 3, 376–380. [Google Scholar] - Chu, P.C.; Beasley, J.E. A genetic algorithm for the multidimensional knapsack problem. J. Heuristics
**1998**, 4, 63–86. [Google Scholar] [CrossRef] - Korejo, I.; Yang, S. A Comparative Study of Adaptive Mutation Operators for Genetic Algorithms. In Proceedings of the 8th Metaheuristic International Conference, Hamburg, Germany, 13–16 July 2009. [Google Scholar]
- Jeong, I.K.; Lee, J.J. Adaptive Simulated Annealing Genetic Algorithm for System Identification. Eng. Appl. Artif. Intell.
**1996**, 9. [Google Scholar] [CrossRef] - Hinterding, R. Gaussian Mutation and Self-Adaptation for Numeric Genetic Algorithms. In Proceedings of the 1995 IEEE International Conference on Evolutionary Computation, Perth, WA, Australia, 29 November–1 December 1995; Volume 1, p. 384. [Google Scholar] [CrossRef]
- Lee, C.Y.; Yao, X. Evolutionary Programming Using Mutations Based on the LÉvy Probability Distribution. Evol. Comput. IEEE Trans.
**2004**, 8, 1–13. [Google Scholar] [CrossRef][Green Version] - Hong, T.P.; Wang, H.S.; Chen, W.C. Simultaneously Applying Multiple Mutation Operators in Genetic Algorithms. J. Heuristics
**2000**, 6, 439–455. [Google Scholar] [CrossRef] - Fan, Q.; Yan, X. Self-adaptive differential evolution algorithm with zoning evolution of control parameters and adaptive mutation strategies. IEEE Trans. Cybern.
**2015**, 46, 219–232. [Google Scholar] [CrossRef] - Li, C.; Yang, S.; Korejo, I. An Adaptive Mutation Operator for Particle Swarm Optimization. Available online: https://bura.brunel.ac.uk/handle/2438/5884 (accessed on 9 January 2021).
- Yang, S. Adaptive Mutation Using Statistics Mechanism for Genetic Algorithms. In Proceedings of the International Conference on Innovative Techniques and Applications of Artificial Intelligence, Cambridge, UK, 13–15 December 2004; pp. 19–32. [Google Scholar] [CrossRef][Green Version]
- Yang, S.; Etaner-Uyar, A. Adaptive mutation with fitness and allele distribution correlation for genetic algorithms. In Proceedings of the 2006 ACM Symposium on Applied Computing, Dijon, France, 23–27 April 2006; Volume 2, pp. 940–944. [Google Scholar] [CrossRef][Green Version]
- Sarkar, U.; Nag, S. An Adaptive Genetic Algorithm for Solving N-Queens Problem. arXiv
**2018**, arXiv:1802.02006. [Google Scholar] - Hussain, A.; Muhammad, Y.S.; Nauman Sajid, M.; Hussain, I.; Shoukry, A.; Gani, S. Genetic Algorithm for Traveling Salesman Problem with Modified Cycle Crossover Operator. Comput. Intell. Neurosci.
**2017**, 2017, 7430125. [Google Scholar] [CrossRef] [PubMed] - Patil, S.; Bhende, M. Comparison and analysis of different mutation strategies to improve the performance of genetic algorithm. Int. J. Comput. Sci. Inf. Technol.
**2014**, 5, 4669–4673. [Google Scholar] - Zhan, S.H.; Lin, J.; Zhang, Z.J.; Zhong, Y.W. List-based simulated annealing algorithm for traveling salesman problem. Comput. Intell. Neurosci.
**2016**, 2016, 1712630. [Google Scholar] [CrossRef][Green Version] - Hore, S.; Chatterjee, A.; Dewanji, A. Improving variable neighborhood search to solve the traveling salesman problem. Appl. Soft Comput.
**2018**, 68, 83–91. [Google Scholar] [CrossRef] - Xu, D.; Weise, T.; Wu, Y.; Lässig, J.; Chiong, R. An investigation of hybrid tabu search for the traveling salesman problem. In Proceedings of the Bio-Inspired Computing-Theories and Applications, Hefei, China, 25–28 September 2015; Springer: Cham, Switzerland, 2015; pp. 523–537. [Google Scholar]
- O’Neil, M.A.; Burtscher, M. Rethinking the parallelization of random-restart hill climbing: A case study in optimizing a 2-opt TSP solver for GPU execution. In Proceedings of the 8th Workshop on General Purpose Processing Using GPUs, San Francisco, CA, USA, 7–8 February 2015; pp. 99–108. [Google Scholar]
- Dawkins, R. The Selfish Gene; Oxford University Press: Oxford, UK, 1989. [Google Scholar]

**Figure 1.**Potential chromosome for 8-Queens Problem where queen 1 is hitting queen 8, and two queens (2, 3) are hitting each other and also hitting two other queens (4, 7) yielding a loss of 4 where the four hitting pairs are ([1, 8], [2, 3], [2, 4], [3, 7]).

**Figure 2.**Traveling salesman Problem with 10 cities, chromosomes. The vertices depict the cities where the first index refers to the position of the city inside the chromosome while the other index refers to the city label. An edge can be formed between each two sequential cities to show the path which the traveling salesman should take.

**Figure 3.**This figure depicts the pertinent distances of a specific gene (gene (9,8)) for a Traveling salesman Problem with 10 cities chromosome. The red vertex depicts the pertinent path between our gene and next gene. The green and blue vertices link the gene of interest with the farthest and closest gene in respect.

**Figure 4.**The figure shows the distribution of the fitness of the chromosomes in a generation in three different cases, initial generation, 20th generation using baseline mutation and the 20th generation using locus mutation. Locus mutation is not just attaining better chromosomes, smaller fitness, but also moving the entire distribution closer to zero.

**Figure 5.**It depicts a comparison between Locus mutation and traditional mutation with different sets of parameters. The figures depict N-queens problem with 32, 64, 128 and 256 queens where the results have been averaged out with two different generation size [200, 400], [400, 600], [600, 800] and [800, 1000] respectively. We only selected the best solution out of these five different values of mutation rate [0.01, 0.1, 0.6, 0.3, 0.9]. The center of the curve is the expected value while the range visualize the standard deviation. All the experiments have been repeated 50 times and then averaged out. We can notice that the number of hitting queens is escalating when we increase the number of queens.

**Figure 6.**The fitness value of 64-Queens problem as a function of the number of generations comparing locus mutation with traditional and adaptive mutation. For adaptive mutation, We have averaged out 30 different ranges and used the center of each range for traditional and locus mutation. All the experiments have been repeated 10 times with the same set of parameters. Although we have used the mean solution for locus mutation, we have selected the best solution for adaptive and traditional mutation.

**Figure 7.**Comparison between Locus mutation and traditional mutation with different sets of parameters. The figures depict TSP problem with 32, 64, 124 and 254 cites. We averaged out the runs with two different generation size [400,600], [400,600], [600,1000] and only 1000 for 254 cities constellation. We only selected the best solution out of these five different values of mutation rate [0.01, 0.1, 0.6, 0.3 and 0.9]. The center of each curve is the expected value while the range visualizes the standard deviation. All the experiments have been repeated 100 times and then averaged out

**Figure 8.**The fitness value of the best optimal solution for 64-Queens problem and TSP problem respectively as a function of Powers $Pow$ where we averaged out ten runs. It starts with uniform distribution then uses a logarithmic scale of $Pow$, and end up with L-infinite norm. We have achieved similar results with two separate problems.

**Figure 9.**This Figure depicts TSP Problem with 10 cities manifesting the speed and the ability of our approach to nearly reach the optimal solution in comparison with the traditional approach. All the experiments have been conducted with 200 population size, 200 generations and 0.5 mutation rate. The optimal solution has been obtained using brute force algorithm. The experiments were repeated 100 times.

**Figure 10.**This Figure depicts TSP Problem with 10 cities manifesting the ability of our approach to surpass the traditional approach consuming the same time. We did run the the traditional approach for two times more generations e.g., when axis x equals 25 generations for locus mutation (as in the figure), it equals 25 * 2 generations for traditional mutation. All the experiments have been conducted with 200 population size and 0.5 mutation rate. We run the traditional approach for 600 generations, while we run locus mutation only for 300 generations. The optimal solution has been obtained using brute force algorithm and the experiments were repeated 100 times.

**Table 1.**N-Queens optimal solution, minimum number of hitting queens, after 20 generations. Two different population size $PopSize$, 200 and 400, have been investigated with 50 different repetitions. All the runs have been averaged out.

IterNum | MutRate | Number of Hits | |
---|---|---|---|

Baseline | Locus | ||

32 | 0.01 | 1.78 | 0.01 |

32 | 0.1 | 2.06 | 0.27 |

32 | 0.3 | 1.7 | 0 |

32 | 0.6 | 1.52 | 0 |

32 | 0.9 | 1.64 | 0 |

64 | 0.01 | 6.6 | 1. |

64 | 0.1 | 7.16 | 2.33 |

64 | 0.3 | 6.64 | 0.71 |

64 | 0.6 | 6.38 | 0.35 |

64 | 0.9 | 6.52 | 0.24 |

128 | 0.01 | 19.82 | 7.98 |

128 | 0.1 | 20.28 | 11.48 |

128 | 0.3 | 19.57 | 7.17 |

128 | 0.6 | 19.24 | 6.19 |

128 | 0.9 | 19.89 | 5.91 |

256 | 0.01 | 51.04 | 33.33 |

256 | 0.1 | 51.8 | 38.8 |

256 | 0.3 | 50.37 | 32.07 |

256 | 0.6 | 50.18 | 30.75 |

256 | 0.9 | 51.17 | 31.06 |

**Table 2.**TSP optimal solution after 100 generation. Two different population size, 200 and 400, have been investigated with a 100 different repetition. All the runs have been averaged out. $IterNum$ is the number of cities (population number), while $MutRate$ is the mutation factor.

IterNum | MutRate | Minimum Distance | |
---|---|---|---|

Baseline | Locus | ||

32 | 0.01 | 8.486 | 8.287 |

32 | 0.1 | 8.574 | 8.341 |

32 | 0.3 | 8.62 | 8.30 |

32 | 0.6 | 8.997 | 7.93 |

32 | 0.9 | 8.915 | 8.36 |

64 | 0.01 | 22.281 | 21.470 |

64 | 0.1 | 20.293 | 21.265 |

64 | 0.3 | 21.038 | 20.859 |

64 | 0.6 | 21.627 | 21.002 |

64 | 0.9 | 22.579 | 20.714 |

124 | 0.01 | 48.797 | 45.380 |

124 | 0.1 | 46.679 | 46.256 |

124 | 0.3 | 47.830 | 46.338 |

124 | 0.6 | 47.381 | 46.320 |

124 | 0.9 | 49.481 | 45.774 |

254 | 0.01 | 110.328 | 102.464 |

254 | 0.1 | 108.453 | 107.808 |

254 | 0.3 | 109.147 | 107.704 |

254 | 0.6 | 110.102 | 106.701 |

254 | 0.9 | 110.946 | 105.642 |

**Table 3.**Comparison between baseline (Base) and locus (Loc) mutation for three different algorithm, genetic algorithm (GA), simulated annealing (SA) and Variable neighborhood search (VNS). We applied these algorithms on 16 different instances from TSPLIB dataset. In most cases, locus mutation is enhancing the performance of the algorithms.

Instance | VNS Loc | VNS Base | SA Loc | SA Base | GA Loc | GA Base | Optimal |
---|---|---|---|---|---|---|---|

att48 | 38,074.95 | 38,349.60 | 66,162.13 | 78,766.73 | 125,478.95 | 127,962.10 | 10,628 |

berlin52 | 8633.02 | 8718.89 | 15,005.75 | 16,852.92 | 24,670.25 | 25,154.85 | 7542 |

burma14 | 24.99 | 25.56 | 27.11 | 26.83 | 40.70 | 44.78 | 30.89 |

eil51 | 477.05 | 487.21 | 827.75 | 951.21 | 1375.58 | 1410.37 | 426 |

eil76 | 605.78 | 626.56 | 1387.55 | 1570.88 | 2167.07 | 2209.15 | 538 |

kroA100 | 25,923.15 | 26,478.45 | 90,299.24 | 103,019.28 | 148,699.96 | 150,347.27 | 21,282 |

kroB100 | 25,248.30 | 25,437.91 | 88,441.77 | 102,262.19 | 145,522.35 | 148,050.07 | 22,141 |

kroC100 | 24,983.77 | 25,042.63 | 88,651.16 | 102,050.32 | 146,813.87 | 147,521.41 | 20,749 |

kroD100 | 26,225.47 | 26,276.29 | 87,275.06 | 100,001.28 | 142,412.64 | 142,597.61 | 21,294 |

kroE100 | 24,608.56 | 24,779.56 | 90,021.99 | 104,828.39 | 148,325.58 | 150,791.90 | 22,068 |

pr76 | 129,505.66 | 132,538.60 | 315,049.50 | 354,149.35 | 491,017.72 | 500,005.56 | 108,159 |

rat99 | 1386.94 | 1388.55 | 4412.32 | 5126.53 | 7280.52 | 7344.15 | 1211 |

rd100 | 9580.81 | 9697.09 | 30,877.39 | 34,885.76 | 49,093.79 | 49,482.80 | 7910 |

st70 | 750.11 | 769.31 | 1902.85 | 2161.98 | 3091.04 | 3149.35 | 675 |

ulysses16 | 52.01 | 55.33 | 59.52 | 61.09 | 100.05 | 101.36 | 73.98 |

ulysses22 | 55.18 | 55.66 | 73.00 | 75.17 | 129.77 | 133.62 | 75.3 |

**Table 4.**Time comparison between locus and baseline mutation where PN is the number of cites, PS is population size, GN is the number of generations and MR is mutation rate. TSP timing gives us the time consumption for each algorithm using the specified parameters. Ratio gives us the speed rate, speed advantage, of the original approach.

PN | PS | GN | MR | TSP Timing | Ratio | |
---|---|---|---|---|---|---|

Baseline | Locus | |||||

32 | 200 | 25 | 0.01 | 0.6297 | 0.9822 | 1.5598 |

32 | 200 | 25 | 0.5 | 0.8174 | 1.243 | 1.5207 |

32 | 200 | 25 | 0.9 | 2.3387 | 3.3526 | 1.4335 |

32 | 200 | 50 | 0.01 | 1.2445 | 1.9482 | 1.5655 |

32 | 200 | 50 | 0.5 | 1.6238 | 2.4778 | 1.526 |

32 | 200 | 50 | 0.9 | 4.6735 | 6.72 | 1.4379 |

32 | 400 | 25 | 0.01 | 1.4315 | 2.1328 | 1.4899 |

32 | 400 | 25 | 0.5 | 1.8039 | 2.6621 | 1.4757 |

32 | 400 | 25 | 0.9 | 4.838 | 6.8533 | 1.4166 |

32 | 400 | 50 | 0.01 | 2.9027 | 4.3041 | 1.4828 |

32 | 400 | 50 | 0.5 | 3.6441 | 5.3463 | 1.4671 |

32 | 400 | 50 | 0.9 | 9.7452 | 13.8431 | 1.4205 |

64 | 200 | 25 | 0.01 | 1.0071 | 1.7144 | 1.7023 |

64 | 200 | 25 | 0.5 | 1.1947 | 1.9664 | 1.6459 |

64 | 200 | 25 | 0.9 | 2.7421 | 4.0509 | 1.4773 |

64 | 200 | 50 | 0.01 | 2.0139 | 3.3967 | 1.6866 |

64 | 200 | 50 | 0.5 | 2.385 | 3.9082 | 1.6387 |

64 | 200 | 50 | 0.9 | 5.4714 | 8.1485 | 1.4893 |

64 | 400 | 25 | 0.01 | 2.2131 | 3.5941 | 1.624 |

64 | 400 | 25 | 0.5 | 2.5717 | 4.1369 | 1.6086 |

64 | 400 | 25 | 0.9 | 5.7171 | 8.3777 | 1.4654 |

64 | 400 | 50 | 0.01 | 4.379 | 7.1742 | 1.6383 |

64 | 400 | 50 | 0.5 | 5.1617 | 8.1775 | 1.5843 |

64 | 400 | 50 | 0.9 | 11.3358 | 16.5914 | 1.4636 |

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Al-Afandi, J.; Horváth, A.
Adaptive Gene Level Mutation. *Algorithms* **2021**, *14*, 16.
https://doi.org/10.3390/a14010016

**AMA Style**

Al-Afandi J, Horváth A.
Adaptive Gene Level Mutation. *Algorithms*. 2021; 14(1):16.
https://doi.org/10.3390/a14010016

**Chicago/Turabian Style**

Al-Afandi, Jalal, and András Horváth.
2021. "Adaptive Gene Level Mutation" *Algorithms* 14, no. 1: 16.
https://doi.org/10.3390/a14010016