# 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

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**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