# On the Resilience of Ant Algorithms. Experiment with Adapted MMAS on TSP

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## Abstract

**:**

## 1. Introduction

## 2. The Concept of Resilience and Some of Its Instances

## 3. Adaptation of an Ant Algorithm to Allow Disturbances Simulation

#### 3.1. An Efficient Nature-Inspired Class of Algorithms

#### 3.2. TSP as Test Problem

#### 3.3. Implementations of Ant Algorithms for TSP

_{k}(i) of the nodes not visited yet) by applying the following probabilistic transition rule:

#### 3.4. New Parameters for MMAS

## 4. Experimental Settings

#### 4.1. TSP Test Instances

#### 4.2. Parameters Values and Ant Implementation

procedure MMAS(epoch, p_nod_ch, p_inten, ampl) set parameters m, α, β, ρ set the maximum pheromone trail ${\tau}_{max}$ to an estimate /* the maximum possible pheromone trail is asymptotically bound */ initialize pheromone trails to ${\tau}_{max}$ set epoch, p_nod_ch, p_inten, ampl set previous_best and global_best to an initial solution /* e.g., found with Nearest-Neighbor heuristic */ no_epochs = 0, better_sol = 0, epoch_previous_best = 0 iteration = 0 while (termination condition not met) do iteration = iteration + 1 ConstructSolutions /* each of the m ants builds its solution*/ ApplyLocalSearch /* 3-opt local search procedure is called */ set current_best to the best solution, delivered by best_ant if current_best < global_best then global_best = current_best set global_time_best to the duration (seconds) required by best_ant to deliver its tour iteration_best= iteration endif epoch_current_best = no_epochs UpdatePheromoneTrails /* only the arcs used by best_ant are updated */ /* according to (2), (3) and the rules of pheromone update in MMAS */ if iteration = epoch then no_epochs = no_epochs + 1 /* a new epoch starts */ N1 = the set of $\lfloor p\_nod\_ch\xb7\left|N\right|\rfloor $ nodes randomly chosen from N A1 = $\varnothing $ for every node in N1 do A1 = A1 ${{\displaystyle \cup}}^{\text{}}$ {$\lfloor p\_inten\xb7\left(\mathrm{no}\text{}\mathrm{of}\text{}\mathrm{arcs}\text{}\mathrm{incident}\text{}\mathrm{to}\text{}\mathrm{the}\text{}\mathrm{node}\right)\rfloor $ arcs, randomly chosen} end for /* alter the pheromone matrix */ for every arc in A1 do alter the pheromone load ${\tau}_{1}$ on the arc /* the new value is randomly chosen in [${\tau}_{1}-{\tau}_{1}$·ampl, ${\tau}_{1}+{\tau}_{1}$·ampl] */ end forif (current_best < previous_best) and (epoch_current_best > epoch_previous_best) then better_sol = better_sol + 1 /* a better approximation was found, although the pheromone matrix was altered */ end ifend ifend while recovery_speed = better_sol/no_epochs return current_best, better_sol, global_time_best, iteration_best, no_epochs, recovery_speedend |

#### 4.3. Recorded Data

- better_solutions: number of times when the application manages to find a better approximation of the optimum tour, although the information stored on arcs (the pheromone quantities) was disturbed (averaged over ten trials).
- number of epochs: how many times during the execution the information on the arcs was disturbed (on average, in ten trials). During an epoch, the pheromones evolve according to Equations (2) and (3).
- average-best: this is the main metric based on which we study the algorithm resilience. It represents the approximation of the TSP instance solution, as delivered by the modified ant algorithm MMAS (also averaged over ten trials).
- recovery speed: is the ratio between better_solutions and number of epochs. It gives a measure of the recovering capacity, showing how many times (on average, in ten trials) average-best improves during an epoch.
- average-iterations: is the average of the ten iteration numbers that found the best solution.
- stddev-best: standard deviation for the ten best solutions delivered by the ten trials.
- stddev-iterations: standard deviation for the ten iterations when the best solution was found.
- best try: the best solution in ten trials.
- worst try: the worst solution in ten trials.
- avg. time-best: average of the times for finding the best solution in ten trials.
- stddev.time best: standard deviation for the time durations needed to find the best solution, for the ten trials.
- distance-to-optimum: is the difference between the (average) solution found by the ant algorithm (average-best) and the optimum solution.
- relative-error: distance-to-optimum/optimum tour length ∗ 100.

- number of epochs = 516. We obtained 516 series of 20 iterations and, starting with the second one, the pheromone alteration takes place as described above.
- better_solutions = 246, that is, although the pheromones are altered at every 20 iterations, the algorithm improves the solution, getting closer to the optimum, 246 times.
- average-best = 80,555.5. This last approximation delivered by the algorithm is only at 105.5 = distance-to-optimum (units) far from the known optimum, which is 80,450. We notice that 105.5 represents 0.13% (= the value of the relative-error) from the optimum. The algorithm succeeded to provide a solution that is only 0.13% worse than the optimum, which is a very good outcome, given the repeated intervention into the solution construction mechanism. In terms of resilience, we can assess that this is high for MMAS(20, 20, 50, 10).
- stddev-best = 54.64. The solutions delivered by the algorithm are spread around the average-best = 80,555.5 with a standard deviation of 54.64. The best solution among the ten collected in ten trials is best try = 80,476 and the worst is worst try = 80,638.
- average-iterations = 633.5, that is, the average of the iteration numbers when the ten best solutions were delivered. The standard deviation of these ten values is stddev-iterations = 249.93, quite a big value, showing that the best solution can be found early in the iterative process, or after a significant number of iterations.
- avg. time-best = 36.81 s. This is the average value of the execution durations (measured in seconds) of the 10 trials. Given the computer system configuration that the experiment was processed on, the result is very good. The standard deviation for the execution times of the ten trials is stddev.time best = 13.

## 5. Results and Statistical Analysis

#### 5.1. Metrics for TSP Resilience Assessment with Adapted MMAS

- Time-performance metrics: better_solutions, number of epochs, average-iterations, stddev-iterations, avg. time-best, stdev.time best and recovery speed. All these data give us a feedback on how fast the altered applications work.
- Path-performance metrics: average-best (which is the objective function) and stddev-best, best try and worst try, three measures indicating how good the altered applications still perform.

- The way the time-performance metrics depend on the factors and how these are inter-correlated.
- The way the path-performance metrics depend on the factors and how these are inter-correlated.

#### 5.2. Discussion on Behaviour of Adapted MMAS with TSP Instance d2103

#### 5.3. Discussion on Behaviour of Adapted MMAS with TSP Instance fl3795

#### 5.4. Discussion on Behaviour of Adapted MMAS with TSP Instance rl1889

## 6. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Correlogram of the time-performance metrics: avg.time-best, stddev-best, stddev.time best, average-iterations, better_solutions and number of epochs for d2103.

**Figure 2.**Intercorrelations between the path-performance metricsaverage-best, stddev-best, best try and worst try for fl3795.

**Figure 3.**Correlogram of the time-performance metrics avg.time-best, stddev-best, stddev.time best, average-iterations, better_solutions and number of epochs for rl1889.

Line | p_nod_ch | p_inten | epoch | recovery speed | average-best | relative-error |
---|---|---|---|---|---|---|

1 | 20 | 50 | 20 | 0.4487 | 80,595.2 | 0.18 |

2 | 20 | 50 | 40 | 1.1479 | 80,554.3 | 0.13 |

3 | 20 | 50 | 60 | 1.0877 | 80,546.1 | 0.12 |

4 | 20 | 100 | 20 | 0.5082 | 80,521.6 | 0.09^{1} |

5 | 20 | 100 | 20 | 0.7103 | 80,593.8 | 0.18 |

6 | 20 | 100 | 40 | 0.8844 | 80,602.9 | 0.19 |

7 | 20 | 100 | 60 | 1.0838 | 80,596.2 | 0.18 |

8 | 40 | 50 | 20 | 0.4192 | 80,606.0 | 0.19 |

9 | 40 | 50 | 40 | 0.9000 | 80,590.1 | 0.17 |

10 | 40 | 50 | 60 | 1.0904 | 80,546.4 | 0.12 |

11 | 40 | 100 | 20 | 0.6965 | 80,616.7 | 0.21 |

12 | 40 | 100 | 40 | 0.8670 | 81,092.4 | 0.80^{2} |

13 | 40 | 100 | 40 | 0.9945 | 80,564.2 | 0.14 |

14 | 40 | 100 | 60 | 1.3642 | 80,529.4 | 0.10 |

15 | 100 | 50 | 20 | 0.5019 | 80,537.9 | 0.11 |

16 | 100 | 50 | 40 | 0.8445 | 80,587.0 | 0.17 |

17 | 100 | 50 | 60 | 1.2384 | 80,557.8 | 0.13 |

18 | 100 | 100 | 20 | 0.4173 | 80,583.8 | 0.17 |

19 | 100 | 100 | 40 | 1.1618 | 80,647.5 | 0.25 |

20 | 100 | 100 | 60 | 1.2051 | 80,561.8 | 0.14 |

^{1,2}Values for ampl=0.

Dependent Variable: average-best | ||||
---|---|---|---|---|

Method: Least Squares | ||||

Residuals: | ||||

Min | 1Q | Median | 3Q | Max |

−149.58 | −23.76 | −6.34 | 14.61 | 491.21 |

Coefficients: | ||||

Estimate | Std. Error | t value | Pr(>|t|) | |

(Intercept) | 8.058 × 10^{+04} | 3.199 × 10^{+01} | 2518.882 | <2 × 10^{−16} *** |

p_nod_ch | −2.363 × 10^{−02} | 1.986 × 10^{−01} | −0.119 | 0.906 |

p_inten | 1.508 × 10^{−01} | 2.700 × 10^{−01} | 0.558 | 0.578 |

epoch | 6.450 × 10^{−02} | 4.134 × 10^{−01} | 0.156 | 0.876 |

ampl10% | −1.956 × 10^{+01} | 2.135 × 10^{+01} | −0.916 | 0.362 |

ampl20% | −2.938 × 10^{+01} | 2.135 × 10^{+01} | −1.377 | 0.172 |

ampl30% | −2.251 × 10^{+01} | 2.135 × 10^{+01} | −1.054 | 0.295 |

ampl40% | −1.838 × 10^{+01} | 2.135 × 10^{+01} | −0.861 | 0.392 |

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 | ||||

Residual standard error: 64.04 on 82 degrees of freedom | ||||

Multiple R-squared: 0.02891, Adjusted R-squared: −0.05398 | ||||

F-statistic: 0.3488 on 7 and 82 DF, p-value: 0.9287 |

Dependent Variable: recovery speed | ||||
---|---|---|---|---|

Method: Least Squares | ||||

Residuals: | ||||

Min | 1Q | Median | 3Q | Max |

−0.31815 | −0.11957 | −0.01764 | 0.11520 | 0.54237 |

Coefficients: | ||||

Estimate | Std. Error | t value | Pr(>|t|) | |

(Intercept) | 0.0103022 | 0.0963029 | 0.107 | 0.91507 |

p_nod_ch | −0.0003021 | 0.0005978 | −0.505 | 0.61466 |

p_inten | −0.0010120 | 0.0008128 | −1.245 | 0.21665 |

epoch | 0.0235334 | 0.0012443 | 18.913 | < 2 × 10^{−16} *** |

ampl10% | 0.1944056 | 0.0642569 | 3.025 | 0.00332 ** |

ampl20% | 0.1923444 | 0.0642569 | 2.993 | 0.00365 ** |

ampl30% | 0.1871389 | 0.0642569 | 2.912 | 0.00462 ** |

ampl40% | 0.0401833 | 0.0642569 | 0.625 | 0.53348 |

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 | ||||

Residual standard error: 0.1928 on 82 degrees of freedom | ||||

Multiple R-squared: 0.8213, Adjusted R-squared: 0.8061 | ||||

F-statistic: 53.85 on 7 and 82 DF, p-value: < 2.2 × 10^{−16} |

Line | p_nod_ch | p_inten | epoch | recovery speed | average-best | relative-error |
---|---|---|---|---|---|---|

1 | 20 | 50 | 20 | 0.4651 | 29,853.4 | 3.94 |

2 | 20 | 50 | 40 | 0.5946 | 29,895.6 | 4.09 |

3 | 20 | 50 | 60 | 0.5500 | 29,969.7 | 4.34 |

4 | 20 | 100 | 20 | 0.4414 | 29,848.0 | 3.92 |

5 | 20 | 100 | 40 | 0.6176 | 29,904.7 | 4.12 |

6 | 20 | 100 | 60 | 0.6087 | 29,875.0 | 4.01 |

7 | 40 | 50 | 20 | 0.4538 | 29,868.3 | 3.99 |

8 | 40 | 50 | 40 | 0.7027 | 29,960.9 | 4.31 |

9 | 40 | 50 | 60 | 0.6750 | 29,705.0 | 3.42^{1} |

10 | 40 | 50 | 60 | 0.5667 | 29,904.5 | 4.12 |

11 | 40 | 100 | 20 | 0.5299 | 29,823.5 | 3.84 |

12 | 40 | 100 | 40 | 0.5581 | 29,897.3 | 4.09 |

13 | 40 | 100 | 60 | 0.4167 | 29,841.5 | 3.90 |

14 | 100 | 50 | 20 | 0.4206 | 29,828.7 | 3.85 |

15 | 100 | 50 | 40 | 0.6667 | 30,076.3 | 4.72 |

16 | 100 | 50 | 60 | 0.4333 | 29,837.4 | 3.88 |

17 | 100 | 100 | 20 | 0.4909 | 29,859.4 | 3.96 |

18 | 100 | 100 | 40 | 0.5333 | 29,888.2 | 4.06 |

19 | 100 | 100 | 60 | 0.5500 | 29,952.2 | 4.28 |

^{1}Value for ampl = 10.

Dependent Variable: average-best | ||||
---|---|---|---|---|

Method: Least Squares | ||||

Residuals: | ||||

Min | 1Q | Median | 3Q | Max |

−105.583 | −34.775 | −1.943 | 27.548 | 185.639 |

Coefficients: | ||||

Estimate | Std. Error | t value | Pr(>|t|) | |

(Intercept) | 2.976 × 10^{+04} | 2.666 × 10^{+01} | 1116.149 | <2 × 10^{−16} *** |

p_nod_ch | 2.951 × 10^{−02} | 1.655 × 10^{−01} | 0.178 | 0.859 |

p_inten | 1.722 × 10^{−01} | 2.250 × 10^{−01} | 0.765 | 0.446 |

epoch | 5.094 × 10^{−01} | 3.445 × 10^{−01} | 1.479 | 0.143 |

ampl10% | −1.551 × 10^{+01} | 1.779 × 10^{+01} | −0.872 | 0.386 |

ampl20% | 1.003 × 10^{+02} | 1.779 × 10^{+01} | 5.640 | 2.35 × 10^{−07} *** |

ampl30% | 1.143 × 10^{+02} | 1.779 × 10^{+01} | 6.426 | 8.13 × 10^{−09} *** |

ampl40% | 9.683 × 10^{+01} | 1.779 × 10^{+01} | 5.442 | 5.34 × 10^{−07} *** |

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 | ||||

Residual standard error: 53.38on 82 degrees of freedom | ||||

Multiple R-squared: 0.5471, Adjusted R-squared: 0.5084 | ||||

F-statistic: 14.15 on 7 and 82 DF, p-value: 6.583 × 10^{−12} |

Dependent Variable: recovery speed | ||||
---|---|---|---|---|

Method: Least Squares | ||||

Residuals: | ||||

Min | 1Q | Median | 3Q | Max |

−10.991 | −3.594 | −0.893 | 1.503 | 112.522 |

Coefficients: | ||||

Estimate | Std. Error | t value | Pr(>|t|) | |

(Intercept) | −2.52477 | 6.55924 | −0.385 | 0.701 |

p_nod_ch | 0.05625 | 0.04071 | 1.382 | 0.171 |

p_inten | 0.05549 | 0.05536 | 1.002 | 0.319 |

epoch | −0.10271 | 0.08475 | −1.212 | 0.229 |

ampl10% | −0.04463 | 4.37657 | −0.010 | 0.992 |

ampl20% | 0.01811 | 4.37657 | 0.004 | 0.997 |

ampl30% | 0.10873 | 4.37657 | 0.025 | 0.980 |

ampl40% | 6.98307 | 4.37657 | 1.596 | 0.114 |

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 | ||||

Residual standard error: 13.13 on 82 degrees of freedom | ||||

Multiple R-squared: 0.09325, Adjusted R-squared: 0.01584 | ||||

F-statistic: 1.205 on 7 and 82 DF, p-value: 0.3096 |

Line | p_nod_ch | p_inten | epoch | recovery speed | average-best | relative-error |
---|---|---|---|---|---|---|

1 | 20 | 50 | 20 | 0.8794 | 318,209.4 | 0.53 |

2 | 20 | 50 | 40 | 1.4754 | 318,245.4 | 0.54 |

3 | 20 | 50 | 60 | 1.5067 | 318,281.4 | 0.55 |

4 | 20 | 100 | 20 | 0.7461 | 318,173.4 | 0.52 |

5 | 20 | 100 | 40 | 1.4173 | 318,605.4 | 0.65 |

6 | 20 | 100 | 60 | 1.778 | 318,641.4 | 0.67 |

7 | 40 | 50 | 20 | 0.641 | 318,713.4 | 0.69 |

8 | 40 | 50 | 20 | 0.9652 | 318,720.6 | 0.69 |

9 | 40 | 50 | 40 | 1.8023 | 318,677.4 | 0.68 |

10 | 40 | 50 | 60 | 1.9559 | 318,317.4 | 0.56 |

11 | 40 | 100 | 20 | 0.8115 | 318,137.4 | 0.51 |

12 | 40 | 100 | 40 | 1.7883 | 318,353.4 | 0.57 |

13 | 40 | 100 | 60 | 2.3153 | 318,569.4 | 0.64 |

14 | 100 | 50 | 20 | 0.8474 | 318,461.4 | 0.61 |

15 | 100 | 50 | 40 | 1.3854 | 318,497.4 | 0.62 |

16 | 100 | 50 | 60 | 2.1518 | 318,533.4 | 0.63 |

17 | 100 | 100 | 20 | 0.9206 | 318,079.8 | 0.49^{1} |

18 | 100 | 100 | 20 | 0.9728 | 318,101.4 | 0.49 |

19 | 100 | 100 | 40 | 1.2885 | 318,389.4 | 0.59 |

20 | 100 | 100 | 60 | 2.4444 | 318,425.4 | 0.60 |

^{1}Value for ampl = 10.

Dependent Variable: average-best | ||||
---|---|---|---|---|

Method: Least Squares | ||||

Residuals: | ||||

Min | 1Q | Median | 3Q | Max |

−231.23 | −156.69 | −8.88 | 113.31 | 359.31 |

Coefficients: | ||||

Estimate | Std. Error | t value | Pr(>|t|) | |

(Intercept) | 3.183 × 10^{+05} | 9.001 × 10^{+01} | 3536.748 | <2 × 10^{−16} *** |

p_nod_ch | 1.731 × 10^{−01} | 5.587 × 10^{−01} | 0.310 | 0.7575 |

p_inten | −1.200 × 10^{+00} | 7.597 × 10^{−01} | −1.580 | 0.1180 |

epoch | 4.050 × 10^{+00} | 1.163 × 10^{+00} | 3.482 | 0.0008 *** |

ampl10% | −2.880 × 10^{+01} | 6.006 × 10^{+01} | −0.480 | 0.6328 |

ampl20% | −2.160 × 10^{+01} | 6.006 × 10^{+01} | −0.360 | 0.7200 |

ampl30% | −1.440 × 10^{+01} | 6.006 × 10^{+01} | −0.240 | 0.8111 |

ampl40% | −7.200 × 10^{+00} | 6.006 × 10^{+01} | −0.120 | 0.9049 |

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 | ||||

Residual standard error: 108.2on 82 degrees of freedom | ||||

Multiple R-squared: 0.1547, Adjusted R-squared: 0.08253 | ||||

F-statistic: 2.144 on 7 and 82 DF, p-value: 0.04789 |

Dependent Variable: recovery speed | ||||
---|---|---|---|---|

Method: Least Squares | ||||

Residuals: | ||||

Min | 1Q | Median | 3Q | Max |

−0.59491 | −0.12230 | −0.00215 | 0.12432 | 0.56298 |

Coefficients: | ||||

Estimate | Std. Error | t value | Pr(>|t|) | |

(Intercept) | 0.2704582 | 0.1120153 | 2.414 | 0.018 * |

p_nod_ch | 0.0010162 | 0.0006953 | 1.462 | 0.148 |

p_inten | 0.0007799 | 0.0009454 | 0.825 | 0.412 |

epoch | 0.0275353 | 0.0014473 | 19.025 | <2 × 10^{−16} *** |

ampl10% | −0.0380889 | 0.0747408 | −0.510 | 0.612 |

ampl20% | −0.0471611 | 0.0747408 | −0.631 | 0.530 |

ampl30% | −0.0201778 | 0.0747408 | −0.270 | 0.788 |

ampl40% | −0.0285889 | 0.0747408 | −0.383 | 0.703 |

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 | ||||

Residual standard error: 0.2242 on 82 degrees of freedom | ||||

Multiple R-squared: 0.8166, Adjusted R-squared: 0.801 | ||||

F-statistic: 52.17 on 7 and 82 DF, p-value: <2.2 × 10^{−16} |

© 2020 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**

Nechita, E.; Crişan, G.C.; Iantovics, L.B.; Huang, Y.
On the Resilience of Ant Algorithms. Experiment with Adapted MMAS on TSP. *Mathematics* **2020**, *8*, 752.
https://doi.org/10.3390/math8050752

**AMA Style**

Nechita E, Crişan GC, Iantovics LB, Huang Y.
On the Resilience of Ant Algorithms. Experiment with Adapted MMAS on TSP. *Mathematics*. 2020; 8(5):752.
https://doi.org/10.3390/math8050752

**Chicago/Turabian Style**

Nechita, Elena, Gloria Cerasela Crişan, Laszlo Barna Iantovics, and Yitong Huang.
2020. "On the Resilience of Ant Algorithms. Experiment with Adapted MMAS on TSP" *Mathematics* 8, no. 5: 752.
https://doi.org/10.3390/math8050752