# Visibility Adaptation in Ant Colony Optimization for Solving Traveling Salesman Problem

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. TSP and Recent Methods to Solve It

#### 2.1. TSP and Its Importance

#### 2.2. Solving TSP with ACO and Its Updated Models

#### 2.3. Solving TSP with Other Prominent Bio-Inspired Methods

## 3. ACO with Adaptive Visibility (ACOAV) for TSP

#### 3.1. Review of Conventional ACO

#### 3.2. Adaptive Visibility Integration to ACO for TSP

#### 3.2.1. Population Initialization

#### 3.2.2. Adaptive Visibility (AV) Heuristic and Formulation

_{21}) is the distance between C1 and C2. The tour cost will be different on the visiting priority between C2 and C3. There are two tour options: C1-C2-C3-C4 having the tour cost (${d}_{12}$ + ${d}_{23}$ + ${d}_{34}$) and C1-C3-C2-C4 having tour costs (${d}_{13}$ + ${d}_{23}$ + ${d}_{24}$), which are shown in Figure 1b,c, respectively. Their tour cost difference is

_{sle}= d

_{sl}− d

_{le}

_{124}− ad

_{134},

_{124}, ad

_{134}are the adaptive distances of C2 and C3, considering C1 as the current city and C4 as the destination city. According to Equation (9), ΔTC < 0, i.e., ad

_{124}< ad

_{134}means the tour choosing C2 first (i.e., C1-C2-C3-C4) is better than the second one choosing C3 first (i.e., C1-C3-C2-C4). On the other hand, the second tour (choosing C3 first) is better than the first one if ΔTC > 0, i.e., ad

_{124}> ad

_{134}. It concludes that choosing the city first with a lesser adaptive distance may lead to a shorter path. It is evident for producing the optimal tour that the next city should be selected considering an overall path distance regardless of the nearest issue. Thus, Equation (8) depicts an interesting hypothesis for choosing the next city, emphasizing the distance between the intended city and the destination city: prioritizing the city with a long distance from the destination city.

#### 3.2.3. Partial Solution Update with AV

#### 3.2.4. 3-Opt Algorithm Adaptation

^{r}-C, A-B-C

^{r}, A-B

^{r}-C

^{r}, A-C-B, A-C

^{r}-B, A-C-B

^{r}, A-C

^{r}-B

^{r}, where B

^{r}and C

^{r}are the reversed tours of B and C segments. The tour is updated according to the combination, which has the lowest cost [12]. Embedding the 3-Opt algorithm helps to overcome stagnation problems and increase searching capability.

#### 3.3. ACOAV Algorithm

Algorithm 1 ACOAV | |||

Algorithm 2 UpdateSolution(${Sol}^{i}$)//Partial Solution Update | |||

1. | TempSol$\leftarrow So{l}^{i}$ | ||

2. | r1 ← RandInt(1, NoOfCities), r2 ← RandInt(1, NoOfCities) | ||

3. | r ← r1 + 1 | ||

4. | CitiesToOrder ← Ø | ||

5. | while (r != r2) do | ||

6. | CitiesToOrder ← CitiesToOrder ∪ {TempSol.city[r]} | ||

7. | if (r < NoOfCities) | ||

8. | r ← r + 1 // Increase the index number by one | ||

9. | else | ||

10. | r ← 1//Reset the index number to start from the first visited city | ||

11. | end if | ||

12. | end while | ||

13. | r ← r1 + 1 | ||

14. | s ← TempSol.city[r1] | ||

15. | e ←TempSol.city[r2] | ||

16. | while (CitiesToOrder ≠ Ø or Null) | ||

17. | TempSol.city[r]$\leftarrow argma{x}_{l:l\in CitiesToOrder}{P}_{sle}$ //Equation (13) | ||

18. | s ← TempSol.city[r] | ||

19. | CitiesToOrder ← CitiesToOrder − {s} | ||

20. | if (r < NoOfCities) | ||

21. | r ← r + 1 | ||

22. | else | ||

23. | r ← 1 | ||

24. | end if | ||

25. | end while | ||

26. | if (TempSol.Cost < ${Sol}^{i}$.Cost) | ||

27. | $So{l}^{i}$ ← TempSol | ||

28. | end if | ||

29. | return $So{l}^{i}$ |

## 4. Experimental Studies

#### 4.1. Experimental Setup

#### 4.2. Experimental Results and Performance Comparison

#### 4.3. Statistical Analysis of Presented Results

#### 4.3.1. Friedman Test

- Observations are mutually independent. That is, the results within one row do not affect the results of other rows.
- For each row, results can be ranked based on their performance.

_{F}is

#### 4.3.2. Post-Hoc Test

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Adaptive visibility demonstration for optimal path selection from start city (C1) to destination city (C4) visiting C2 and C3. Tour option C1-C2-C3-C4 will be optimal when (d

_{12}− d

_{24}) < (d

_{13}− d

_{34}); otherwise, tour option C1-C3-C2-C4 will be optimal having (d

_{12}− d

_{24}) > (d

_{12}− d

_{34}).

**Figure 2.**Demonstration of TSP solving with the proposed adaptive visibility compared with ACO visibility. Tour cost emphasizing nearest city for ACO is 2000 and tour cost with proposed adaptive visibility is 1700.

**Figure 4.**Tour cost improvement over iterations for conventional ACO, with different methods with adaptive visibility in different modes. ACOAV (FSU) is ACO + AV with full solution update mode, ACOAV (PSU) is ACO + AV with partial solution update mode, and ACOAV is the proposed model as ACOAV (PSU) + 3-Opt.

**Figure 5.**Process time (in the second) comparison over iterations for conventional ACO, with different methods with adaptive visibility in different modes. ACOAV (FSU) is ACO + AV with full solution update mode, ACOAV (PSU) is ACO + AV with partial solution update mode, and ACOAV is the proposed model as ACOAV (PSU) + 3-Opt.

Sl. | TSP Instance | Optimal Tour Length | Best Tour Length (TL) and Error Rate (ER) Comparison | Average Tour Length (TL) and Standard Deviation (SD) Comparison | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ACO | ACOAV (FSU) | ACOAV (PSU) | ACOAV | ACO | ACOAV (FSU) | ACOAV (PSU) | ACOAV | |||||||||||

Best TL | ER (%) | Best TL | ER (%) | Best TL | ER (%) | Best TL | ER (%) | Avg. TL | SD | Avg. TL | SD | Avg. TL | SD | Avg. TL | SD | |||

1 | eil51 | 426 | 435 | 2.11 | 436 | 2.35 | 426 | 0 | 426 | 0 | 441.3 | 3.86 | 436.3 | 0.9 | 426 | 0 | 426 | 0 |

2 | berlin52 | 7542 | 7689 | 1.95 | 7677 | 1.79 | 7569 | 0.36 | 7542 | 0 | 7795.8 | 81.61 | 7677 | 0 | 7659.25 | 20.7 | 7542 | 0 |

3 | st70 | 675 | 696 | 3.11 | 710 | 5.19 | 683 | 1.19 | 675 | 0 | 712.75 | 6.11 | 716 | 3.46 | 683.9 | 0.3 | 675 | 0 |

4 | eil76 | 538 | 548 | 1.86 | 543 | 0.93 | 541 | 0.56 | 538 | 0 | 559 | 4.25 | 556.85 | 2.48 | 542.05 | 1.36 | 538 | 0 |

5 | pr76 | 108,159 | 119,176 | 10.19 | 115,930 | 7.18 | 110,465 | 2.13 | 108,159 | 0 | 126,121 | 2784.97 | 115,930 | 0 | 110,928 | 201.09 | 108,159 | 0 |

6 | rat99 | 1211 | 1264 | 4.38 | 1274 | 5.2 | 1244 | 2.73 | 1211 | 0 | 1297.4 | 16.9 | 1285.1 | 9.41 | 1244.5 | 0.97 | 1211 | 0 |

7 | kroA100 | 21,282 | 24,210 | 13.76 | 22,788 | 7.08 | 21,521 | 1.12 | 21,282 | 0 | 24,904.4 | 334.39 | 22,788 | 0 | 21,585.6 | 14.82 | 21,282 | 0 |

8 | kroB100 | 22,141 | 25,191 | 13.78 | 23,852 | 7.73 | 22,416 | 1.24 | 22,141 | 0 | 25,847.7 | 298.76 | 23,852 | 0 | 22,488.5 | 38.92 | 22,141 | 0 |

9 | rd100 | 7910 | 8547 | 8.05 | 8556 | 8.17 | 7943 | 0.42 | 7910 | 0 | 8880.95 | 148.21 | 8556 | 0 | 7974.65 | 22.54 | 7910 | 0 |

10 | eil101 | 629 | 682 | 8.43 | 652 | 3.66 | 636 | 1.11 | 629 | 0 | 692.95 | 7.05 | 653.65 | 1.31 | 637.2 | 0.51 | 629 | 0 |

11 | lin105 | 14,379 | 15,714 | 9.28 | 14,803 | 2.95 | 14,549 | 1.18 | 14,379 | 0 | 16310 | 197.1 | 14,803 | 0 | 14,601.6 | 16.56 | 14,379 | 0 |

12 | pr107 | 44,303 | 48,512 | 9.5 | 50,356 | 13.66 | 44,566 | 0.59 | 44,303 | 0 | 49,129.8 | 359.13 | 50,356 | 0 | 44,897.9 | 137.2 | 44,303 | 0 |

13 | pr124 | 59,030 | 66,702 | 13 | 62,977 | 6.69 | 59,990 | 1.63 | 59,030 | 0 | 68,970.2 | 910.83 | 62,977 | 0 | 60,049.8 | 22.11 | 59,030 | 0 |

14 | ch130 | 6110 | 6859 | 12.26 | 6581 | 7.71 | 6204 | 1.54 | 6110 | 0 | 7012.3 | 77.03 | 6581 | 0 | 6234.8 | 13.83 | 6110 | 0 |

15 | ch150 | 6528 | 7236 | 10.85 | 6941 | 6.33 | 6621 | 1.42 | 6528 | 0 | 7421.5 | 100.03 | 6962.45 | 29.34 | 6630.2 | 5.47 | 6528 | 0 |

16 | kroA150 | 26,524 | 31,857 | 20.11 | 29,330 | 10.58 | 27,092 | 2.14 | 26,524 | 0 | 33,233.9 | 503.85 | 29,330 | 0 | 27,168.5 | 37.65 | 26,524 | 0 |

17 | kroB150 | 26,130 | 31,262 | 19.64 | 28,386 | 8.63 | 26,537 | 1.56 | 26,130 | 0 | 33011.1 | 504.53 | 28,386 | 0 | 26,736.5 | 79.45 | 26,130 | 0 |

18 | rat195 | 2323 | 2524 | 8.65 | 2428 | 4.52 | 2362 | 1.68 | 2326 | 0.13 | 2582.2 | 32.64 | 2430.9 | 4.09 | 2371.4 | 4.19 | 2330.2 | 1.6 |

19 | d198 | 15,780 | 18,503 | 17.26 | 16,961 | 7.48 | 15,997 | 1.38 | 15,780 | 0 | 19,023.9 | 202.59 | 17,136.5 | 88.53 | 16,111.5 | 47.25 | 15,780 | 0 |

20 | kroA200 | 29,368 | 36,628 | 24.72 | 31,866 | 8.51 | 29,725 | 1.22 | 29,368 | 0 | 38,404.1 | 639.51 | 31,886.7 | 49.28 | 29,830.5 | 56.59 | 29,368 | 0 |

21 | kroB200 | 29,437 | 37,305 | 26.73 | 32,304 | 9.74 | 29,857 | 1.43 | 29,438 | 0.003 | 38,516.3 | 605.58 | 32,304 | 0 | 30,155.8 | 112.8 | 29,439.5 | 0.67 |

22 | tsp225 | 3861 | 4592 | 18.93 | 4208 | 8.99 | 3997 | 3.52 | 3923 | 1.61 | 4733.4 | 61.17 | 4208 | 0 | 4047.15 | 15.41 | 3956.9 | 11.62 |

23 | pr226 | 80,369 | 96,539 | 20.12 | 86,109 | 7.14 | 80,993 | 0.78 | 80,369 | 0 | 102,238 | 1712.21 | 86,127.4 | 9.2 | 81,289.2 | 71.16 | 80,369.6 | 1.43 |

24 | a280 | 2579 | 3089 | 19.78 | 2778 | 7.72 | 2656 | 2.99 | 2581 | 0.08 | 3241.75 | 46.44 | 2816.8 | 17.1 | 2666.8 | 8.53 | 2589.4 | 3.83 |

25 | pr299 | 48,191 | 63,759 | 32.3 | 54,470 | 13.03 | 50,159 | 4.08 | 48,215 | 0.05 | 67,450.6 | 1132.05 | 54,470.3 | 0.48 | 50,458.7 | 146.52 | 48,519.3 | 70.16 |

26 | lin318 | 42,029 | 57,586 | 37.01 | 44,892 | 6.81 | 43,079 | 2.5 | 42,203 | 0.41 | 58,481.8 | 514.22 | 44,892 | 0 | 43243.2 | 76.5 | 42,220.9 | 9.99 |

27 | rd400 | 15,281 | 20,664 | 35.23 | 16,899 | 10.59 | 15,973 | 4.53 | 15,467 | 1.22 | 21,210.2 | 199.2 | 16,899 | 0 | 16,055.5 | 42.85 | 15,576 | 44.23 |

28 | fl417 | 11,861 | 14,370 | 21.15 | 12,948 | 9.16 | 11,921 | 0.51 | 11,861 | 0 | 14,725.4 | 149.15 | 12,948 | 0 | 11,937 | 8.23 | 11,861.2 | 0.4 |

29 | pr439 | 107,217 | 135,080 | 25.99 | 121,360 | 13.19 | 111,024 | 3.55 | 107,613 | 0.37 | 139,745 | 1847.98 | 121,360 | 0 | 111,490 | 251.24 | 107,965 | 182.42 |

30 | pcb442 | 50,778 | 72,682 | 43.14 | 56,991 | 12.24 | 53,833 | 6.02 | 50,778 | 0 | 75,440.8 | 1069.75 | 56,991.8 | 3.49 | 54,150.7 | 171.18 | 50,945.2 | 54.98 |

31 | rat575 | 6773 | 9012 | 33.06 | 7344 | 8.43 | 7118 | 5.09 | 6935 | 2.39 | 9213.35 | 85.31 | 7345.85 | 3.6 | 7141.95 | 11.87 | 6972.25 | 15.73 |

32 | rat783 | 8806 | 12,286 | 39.52 | 9712 | 10.29 | 9407 | 6.82 | 9050 | 2.77 | 12,554.5 | 119.38 | 9712.8 | 0.98 | 9451.45 | 21.13 | 9101.1 | 20.77 |

33 | pr1002 | 259,045 | 371,087 | 43.25 | 300,757 | 16.1 | 284,985 | 10.01 | 266,155 | 2.74 | 374,401 | 2224.55 | 300,972 | 214.5 | 288,991 | 1397.71 | 268,111 | 678.67 |

34 | fl1400 | 20,127 | 29,486 | 46.5 | 22,673 | 12.65 | 21,157 | 5.12 | 20,215 | 0.44 | 30,313.1 | 395.21 | 22,799.7 | 128.91 | 21,271.4 | 92.34 | 20,226.7 | 5.39 |

35 | pr2392 | 378,032 | 581,878 | 53.92 | 439,849 | 16.35 | 428,523 | 13.36 | 392,461 | 3.82 | 588,396 | 3977.43 | 440,033 | 140.07 | 431,504 | 1610.47 | 397,871 | 1805.49 |

Optimal/Best Count | 0/0 | 0/0 | 1/1 | 22/35 | 0/0 | 0/0 | 1/1 | 19/35 | ||||||||||

Win-Draw-Loss over ACO | - | 30-0-5 | 35-0-0 | 35-0-0 | - | 33-0-2 | 35-0-0 | 35-0-0 | ||||||||||

Win-Draw-Loss over ACOAV (FSU) | 5-0-30 | - | 35-0-0 | 35-0-0 | 2-0-33 | - | 35-0-0 | 35-0-0 | ||||||||||

Win-Draw-Loss over ACOAV (PSU) | 0-0-35 | 0-0-35 | - | 34-1-0 | 0-0-35 | 0-0-35 | - | 34-1-0 |

**Table 2.**Best Tour Length (TL) and Error Rate (ER) Comparison of ACOAV with Existing State-of-the-Art Bio-inspired Methods.

Sl. | TSP Instance | Optimal Tour Length | GA-MARL + NICH-LS [35] | DSOS [34] | SSABC [31] | DSMO [33] | DLSO [32] | PSO-ACO [14] | PACO [11] | DEACO [13] | HAACO [10] | Proposed ACOAV | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Best TL | ER (%) | Best TL | ER (%) | Best TL | ER (%) | Best TL | ER (%) | Best TL | ER (%) | Best TL | ER (%) | Best TL | ER (%) | Best TL | ER (%) | Best TL | ER (%) | Best TL | ER (%) | |||

1 | eil51 | 426 | 426 | 0 | 427 | 0.23 | 427 | 0.23 | 428.86 | 0.67 | 428.87 | 0.67 | 426 | 0 | 426 | 0 | 426 | 0 | 426 | 0 | 426 | 0 |

2 | berlin52 | 7542 | 7542 | 0 | 7542 | 0 | 7542 | 0 | 7544.37 | 0.03 | 7544.37 | 0.03 | 7542 | 0 | 7542 | 0 | 7542 | 0 | 7542 | 0 | 7542 | 0 |

3 | st70 | 675 | 675 | 0 | 675 | 0 | 675 | 0 | 677.11 | 0.31 | 677.11 | 0.31 | 676 | 0.15 | 676 | 0.15 | 675 | 0 | 675 | 0 | 675 | 0 |

4 | eil76 | 538 | 538 | 0 | 542 | 0.74 | 538 | 0 | 558.68 | 3.84 | – | – | 538 | 0 | 538 | 0 | 541.6 | 0.67 | 538 | 0 | 538 | 0 |

5 | pr76 | 108,159 | 108,159 | 0 | 108,159 | 0 | 108,159.4 | 0.0004 | 108,159.43 | 0.01 | – | – | – | – | – | – | – | – | 108,159 | 0 | ||

6 | rat99 | 1211 | 1211 | 0 | 1224 | 1.07 | 1211 | 0 | 1225.56 | 1.2 | – | – | 1224 | 1.07 | 1213 | 0.17 | 1211 | 0 | 1211 | 0 | 1211 | 0 |

7 | kroA100 | 21,282 | 21,282 | 0 | 21,282 | 0 | 21,282 | 0 | 21,298.21 | 0.08 | 21,285.44 | 0.02 | 21,301 | 0.09 | 21,282 | 0 | 21,282 | 0 | 21,282 | 0 | 21,282 | 0 |

8 | kroB100 | 22,141 | 22,141 | 0 | 22,141 | 0 | – | – | 22,308 | 0.75 | 22,142.07 | 0.01 | – | – | – | – | 22,141 | 0 | – | – | 22,141 | 0 |

9 | rd100 | 7910 | – | – | – | – | – | – | 8041.3 | 1.66 | – | – | – | – | – | – | 7910 | 0 | – | – | 7910 | 0 |

10 | eil101 | 629 | 629 | 0 | 640 | 1.75 | 629 | 0 | 648.66 | 3.13 | 642.53 | 2.15 | 631 | 0.32 | 629 | 0 | 629 | 0 | 630 | 0.16 | 629 | 0 |

11 | lin105 | 14,379 | 14,379 | 0 | 14,381 | 0.01 | 14,379 | 0 | 14383 | 0.03 | 14,383.0 | 0.03 | 14,379 | 0 | 14,379 | 0 | 14,379 | 0 | 14,379 | 0 | 14,379 | 0 |

12 | pr107 | 44,303 | 44,303 | 0 | 44,314 | 0.02 | – | – | 44,385.86 | 0.19 | – | – | – | – | – | – | – | – | – | – | 44,303 | 0 |

13 | pr124 | 59,030 | 59,030 | 0 | 59,030 | 0 | – | – | 60,285.21 | 2.13 | – | – | – | – | – | – | 59,074 | 0.07 | – | – | 59,030 | 0 |

14 | ch130 | 6110 | 6132 | 0.36 | – | – | – | – | – | – | 6158.08 | 0.79 | – | – | – | – | 6110 | 0 | – | – | 6110 | 0 |

15 | ch150 | 6528 | 6528 | 0 | 6542 | 0.21 | – | – | – | – | 6530.90 | 0.04 | 6538 | 0.15 | 6570 | 0.64 | 6528 | 0 | 6566 | 0.58 | 6528 | 0 |

16 | kroA150 | 26,524 | 26,579 | 0.21 | – | – | – | – | 27,591.44 | 4.02 | – | – | – | – | – | – | 26,572 | 0.18 | – | – | 26,524 | 0 |

17 | kroB150 | 26,130 | 26,130 | 0 | – | – | – | – | 26,601.94 | 1.81 | – | – | – | – | – | – | 26,130 | 0 | – | – | 26,130 | 0 |

18 | rat195 | 2323 | – | – | – | – | – | – | 2372.89 | 2.15 | – | – | – | – | – | – | 2340 | 0.73 | – | – | 2326 | 0.13 |

19 | d198 | 15,780 | – | – | – | – | – | – | 15,978.13 | 1.26 | 15,808.93 | 0.18 | – | – | – | – | – | – | – | – | 15,780 | 0 |

20 | kroA200 | 29,368 | 29,435 | 0.23 | 29,477 | 0.37 | 29450 | 0.28 | 30,481.35 | 3.79 | 29,519.83 | 0.52 | 29,468 | 0.34 | 29,533 | 0.56 | 29,368 | 0 | 29,483 | 0.39 | 29,368 | 0 |

21 | kroB200 | 29,437 | – | – | – | – | – | 30,716.5 | 4.35 | 29,652.94 | 0.73 | – | – | – | – | – | – | – | – | 29,438 | 0.003 | |

22 | tsp225 | 3861 | 3865 | 0.1 | 3877 | 0.41 | – | – | 4013.68 | 3.95 | 3929.51 | 1.77 | – | – | – | – | – | – | – | – | 3923 | 1.61 |

23 | pr226 | 80,369 | 80,369 | 0 | 80,407 | 0.05 | – | – | 83,587.98 | 4.01 | – | – | – | – | – | – | – | – | – | – | 80,369 | 0 |

24 | a280 | 2579 | 2595 | 0.62 | – | – | – | – | – | 2609.54 | 1.18 | – | – | – | – | – | – | – | – | 2581 | 0.08 | |

25 | pr299 | 48,191 | 48,637 | 0.93 | 49162 | 2.01 | – | – | 50,579.82 | 4.96 | – | – | – | – | – | – | 48,455 | 0.55 | – | – | 48,215 | 0.05 |

26 | lin318 | 42,029 | 42,255 | 0.54 | 42,201 | 0.4 | – | – | 44,118.66 | 4.97 | 42,744.96 | 1.7 | – | – | – | – | – | – | – | – | 42,203 | 0.41 |

27 | rd400 | 15,281 | – | – | – | – | – | – | – | – | – | – | – | – | 15,578 | 1.94 | 15,323 | 0.27 | 15,603 | 2.11 | 15,467 | 1.22 |

28 | fl417 | 11,861 | – | – | – | – | – | – | 12,218.98 | 3.02 | – | – | – | – | 11,972 | 0.94 | 11,866 | 0.04 | 11,960 | 0.83 | 11,861 | 0 |

29 | pr439 | 107,217 | 107,833 | 0.57 | – | – | – | – | 112,105.2 | 4.56 | – | – | – | – | 108,482 | 1.18 | – | – | 108,730 | 1.41 | 107,613 | 0.37 |

30 | pcb442 | 50,778 | – | – | 51,418 | 1.26 | – | – | – | – | 52,330.24 | 3.06 | – | – | 51,962 | 2.33 | 50,964.5 | 0.37 | 51,780 | 1.97 | 50,778 | 0 |

31 | rat575 | 6773 | – | – | 7073 | 4.43 | – | – | – | – | – | – | – | – | 7003 | 3.4 | 6773 | 0 | – | – | 6935 | 2.39 |

32 | rat783 | 8806 | – | – | 9045 | 2.71 | – | – | – | – | – | – | – | – | 9111 | 3.46 | 8916.0 | 1.25 | – | – | 9050 | 2.77 |

33 | pr1002 | 259,045 | 266,886 | 3.03 | 272,381 | 5.15 | – | – | – | – | 273,696.03 | 5.66 | – | – | – | – | – | – | – | – | 266,155 | 2.74 |

34 | fl1400 | 20,127 | 20,304 | 0.88 | – | – | – | – | – | – | – | – | – | – | – | – | – | – | – | – | 20,215 | 0.44 |

35 | pr2392 | 378,032 | 397,314 | 5.1 | 419,246 | 10.9 | – | – | – | – | – | – | – | – | – | – | – | – | – | – | 392,461 | 3.82 |

Optimal/Best Count | 15/16 | 6/7 | 7/7 | 0/0 | 0/0 | 4/4 | 6/6 | 14/16 | 7/7 | 22/30 |

**Table 3.**Average Tour Length (TL) and Standard Deviation (SD) Comparison of ACOAV with Existing State-of-the-Art Bio-inspired Methods.

Sl. | TSP Instance | Optimal Tour Length | GA-MARL + NICH-LS [35] | DSOS [34] | SSABC [31] | DSMO [33] | DLSO [32] | PSO-ACO [14] | PACO [11] | DEACO [13] | HAACO [10] | Proposed ACOAV | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Avg. TL | SD | Avg. TL | SD | Avg. TL | SD | Avg. TL | SD | Avg. TL | SD | Avg. TL | SD | Avg. TL | SD | Avg. TL | SD | Avg. TL | SD | Avg. TL | SD | |||

1 | eil51 | 426 | 427.4 | – | 427.90 | 1.20 | 427.01 | 0.46 | 436.96 | 4.73 | 429.7 | 1.61 | 426.45 | 0.61 | 426.35 | 0.49 | 426 | 0 | 427.5 | – | 426 | 0 |

2 | berlin52 | 7542 | 7550.7 | – | 7542.60 | 0 | 7542 | 0 | 7633.6 | 85.4 | 7544.37 | 0 | 7543.2 | 2.37 | 7542 | 0 | 7542 | 0 | 7542 | – | 7542 | 0 |

3 | st70 | 675 | 679.43 | – | 679.2 | 2.8 | 675.77 | 1.17 | 702.64 | 15.04 | 678.78 | 3.38 | 678.2 | 1.47 | 677.85 | 0.99 | 675 | 0 | 676.5 | – | 675 | 0 |

4 | eil76 | 538 | 545.3 | – | 542,547.4 | 3.9 | 538.15 | 0.60 | 572.7 | 7.56 | – | – | 538.3 | 0.47 | 539.85 | 1.09 | 541.6 | 0.6 | 542 | – | 538 | 0 |

5 | pr76 | 108,159 | 109,556.57 | – | – | – | 111,299.3 | 2050.48 | 108,572.35 | 341.96 | – | – | – | – | – | – | – | – | 108,159 | 0 | ||

6 | rat99 | 1211 | 1223.3 | – | 1228.37 | 14.32 | 1211.50 | 0.67 | 1291.93 | 21.07 | – | – | 1227.4 | 1.98 | 1217.1 | 4.01 | 1211.7 | – | 1214.1 | – | 1211 | 0 |

7 | kroA100 | 21,282 | 21,354.4 | – | 21,409.50 | 149.15 | 21,287.19 | 8.10 | 22,024.27 | 508.89 | 21,370.09 | 44.66 | 21,445.1 | 78.24 | 21,326.8 | 33.72 | 21,282 | 0 | 21,364.2 | – | 21,282 | 0 |

8 | kroB100 | 22,141 | 22,283.4 | – | 22,339.20 | 230.18 | – | – | 23,022.37 | 277.32 | 22,270.58 | 95.52 | – | – | – | – | 22,141 | 0 | – | – | 22,141 | 0 |

9 | rd100 | 7910 | – | – | – | – | 8377.76 | 209.4 | – | – | – | – | – | – | 7910 | 0 | 7910 | 0 | ||||

10 | eil101 | 629 | 642.6 | – | 650.60 | 4.57 | 630.59 | 2.37 | 674.4 | 10.97 | 649.05 | – | 632.7 | 2.12 | 630.55 | 2.63 | 629 | 0 | 632.5 | – | 629 | 0 |

11 | lin105 | 14,379 | 14,385.63 | – | 14,431.73 | 14,379.10 | 1.30 | 15,114 | 500.76 | 14,433.33 | 34.23 | 14,379.15 | 0.48 | 14,393 | 19.76 | 14,379 | 0 | 14,411.8 | – | 14,379 | 0 | |

12 | pr107 | 44,303 | 44,424.73 | – | 44,445.10 | 181.35 | – | – | 45,666.99 | 1300.43 | – | – | – | – | – | – | – | – | – | 44,303 | 0 | |

13 | pr124 | 59,030 | 59,208.83 | – | 59,030 | 264.08 | – | – | 62,443.49 | 1644.93 | – | – | – | – | – | – | – | – | – | 59,030 | 0 | |

14 | ch130 | 6110 | 6204.17 | – | – | – | – | – | – | – | 6201.98 | 30.96 | – | – | – | – | 6110 | 0 | – | – | 6110 | 0 |

15 | ch150 | 6528 | 6547.67 | – | 6552.58 | – | – | – | – | 6597.83 | 38.83 | 6563.95 | 27.58 | 6601.4 | 15.01 | 6528 | 0 | 6578.8 | – | 6528 | 0 | |

16 | kroA150 | 26,524 | 26,891.83 | – | – | – | – | 28,354.09 | 524.91 | – | – | – | – | – | – | 26,524 | 0 | – | – | 26,524 | 0 | |

17 | kroB150 | 26,130 | 26,477.33 | – | – | – | – | 27,576.16 | 625.26 | – | – | – | – | – | – | 26,130 | 0 | – | – | 26,130 | 0 | |

18 | rat195 | 2323 | – | – | – | – | – | 2488.55 | 50.48 | – | – | – | – | – | – | – | – | – | – | 2330.2 | 1.6 | |

19 | d198 | 15,780 | – | – | – | – | – | 16,270.47 | 171.2 | 15,896.48 | 35.21 | – | – | – | – | – | – | – | – | 15,780 | 0 | |

20 | kroA200 | 29,368 | 29,621 | – | 29,651.23 | 29,469 | 20.03 | 31,828.64 | 652.32 | 29,766.27 | 118.37 | 29,646.05 | 114.71 | 29,644.5 | 53.43 | 29,368 | 0 | 29,633.2 | – | 29,368 | 0 | |

21 | kroB200 | 29,437 | – | – | – | – | – | – | 31,781.62 | 487.39 | 29,994.08 | 226.62 | – | – | – | – | 29,440 | 5.1 | – | – | 29,439.5 | 0.67 |

22 | tsp225 | 3861 | 3925.33 | – | – | – | – | – | 4162.79 | 66.08 | 3977.53 | 21.05 | – | – | – | – | – | – | – | – | 3956.9 | 11.62 |

23 | pr226 | 80,369 | 80,638.6 | – | – | – | – | – | 85,935.69 | 2105.13 | – | – | – | – | – | – | – | – | – | – | 80,369.6 | 1.43 |

24 | a280 | 2579 | 2655.47 | – | – | – | – | – | – | – | 2650.49 | 33.83 | – | – | – | – | – | – | – | – | 2589.4 | 3.83 |

25 | pr299 | 48,191 | 49,200.57 | – | 50,335.20 | 905.42 | – | – | 51,747.99 | 863.32 | – | – | – | – | – | – | – | – | – | – | 48,519.3 | 70.16 |

26 | lin318 | 42,029 | 42,996.63 | – | 42,972.42 | 2037.43 | – | – | 45,460.25 | 660.47 | 43,172.51 | 235.18 | – | – | – | – | 42,225 | 47 | – | – | 42,220.9 | 9.99 |

27 | rd400 | 15,281 | – | – | – | – | – | – | – | – | – | – | – | – | 15,613.9 | – | 15,385 | – | 15,644.2 | – | 15,576 | 44.23 |

28 | fl417 | 11,861 | – | – | – | – | – | – | 12,950.77 | 360.99 | – | – | – | – | 11,987.4 | – | 11,875 | – | 11,979.5 | – | 11,861.2 | 0.4 |

29 | pr439 | 107,217 | 109,577.87 | – | – | – | – | – | 116,379.2 | 2462.82 | – | – | – | – | 108,702 | – | – | – | 108,950.6 | – | 107,965 | 182.42 |

30 | pcb442 | 50,778 | – | – | – | – | – | – | – | – | 52,841.26 | 230.75 | – | – | 52,202.4 | – | 50,965 | – | 52,179.8 | – | 50,945.2 | 54.98 |

31 | rat575 | 6773 | – | – | 7117.32 | 171.65 | – | – | – | – | – | – | – | – | 7012.4 | – | 6804.0 | 10.3 | – | – | 6972.25 | 15.73 |

32 | rat783 | 8806 | – | – | 9102.67 | 37.28 | – | – | – | – | – | – | – | – | 9127.3 | – | 8935.9 | 12.44 | – | – | 9101.1 | 20.77 |

33 | pr1002 | 259,045 | 269,845.97 | – | 278,381.51 | 4328.62 | – | – | – | – | 275,825 | 1189.80 | – | – | – | – | – | – | – | – | 268,111 | 678.67 |

34 | fl1400 | 20,127 | 20,444.33 | – | – | – | – | – | – | – | – | – | – | – | – | – | 20,229 | 31.15 | – | – | 20,226.7 | 5.39 |

35 | pr2392 | 378,032 | 400,171.73 | – | 425,431.78 | 4352.75 | – | – | – | – | – | – | – | – | – | – | – | – | – | – | 397,871 | 1805.49 |

Optimal/Best Count | 0/1 | 1/1 | 1/1 | 0/0 | 0/0 | 0/0 | 1/1 | 13/16 | 1/1 | 19/31 |

**Table 4.**Ranking of the Friedman Test among proposed ACOAV and Existing State-of-the-Art Bio-inspired Methods.

n | Method | GA-MARL + NICH-LS [35] | DSOS [34] | SSABC [31] | DSMO [33] | DLSO [32] | PSO-ACO [14] | PACO [11] | DEACO [13] | HAACO [10] | Proposed ACOAV |
---|---|---|---|---|---|---|---|---|---|---|---|

Rank(R) | ${\mathit{R}}_{\mathit{i}1}$ | ${\mathit{R}}_{\mathit{i}2}$ | ${\mathit{R}}_{\mathit{i}3}$ | ${\mathit{R}}_{\mathit{i}4}$ | ${\mathit{R}}_{\mathit{i}5}$ | ${\mathit{R}}_{\mathit{i}6}$ | ${\mathit{R}}_{\mathit{i}7}$ | ${\mathit{R}}_{\mathit{i}8}$ | ${\mathit{R}}_{\mathit{i}9}$ | ${\mathit{R}}_{\mathit{i}10}$ | |

1 | eil51 | 6 | 8 | 5 | 10 | 9 | 4 | 3 | 1.5 | 7 | 1.5 |

2 | berlin52 | 9 | 6 | 3 | 10 | 8 | 7 | 3 | 3 | 3 | 3 |

3 | st70 | 9 | 8 | 3 | 10 | 7 | 6 | 5 | 1.5 | 4 | 1.5 |

4 | kroA100 | 5 | 8 | 3 | 10 | 7 | 9 | 4 | 1.5 | 6 | 1.5 |

5 | eil101 | 7 | 9 | 4 | 10 | 8 | 6 | 3 | 1.5 | 5 | 1.5 |

6 | lin105 | 5 | 8 | 3 | 10 | 9 | 4 | 6 | 1.5 | 7 | 1.5 |

7 | kroA200 | 4 | 8 | 3 | 10 | 9 | 7 | 6 | 1.5 | 5 | 1.5 |

${\mathit{R}}_{.\mathit{j}}={\displaystyle \sum}_{\mathit{i}=1}^{\mathit{n}}{\mathit{R}}_{\mathit{i}\mathit{j}}$for i = 1,2,3…, n | ${R}_{.1}=$ 45 | ${R}_{.2}=$ 55 | ${R}_{.3}=$ 24 | ${R}_{.4}=$ 70 | ${R}_{.5}=$ 57 | ${R}_{.6}=$ 43 | ${R}_{.7}=$ 30 | ${R}_{.8}=$ 12 | ${R}_{.9}=$ 37 | ${R}_{.10}=$ 12 | |

Average rank$\left(\frac{{\mathit{R}}_{.\mathit{j}}}{\mathit{n}}\right)$ | 6.43 | 7.86 | 3.43 | 10 | 8.14 | 6.14 | 4.29 | 1.71 | 5.29 | 1.71 |

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## Share and Cite

**MDPI and ACS Style**

Shahadat, A.S.B.; Akhand, M.A.H.; Kamal, M.A.S.
Visibility Adaptation in Ant Colony Optimization for Solving Traveling Salesman Problem. *Mathematics* **2022**, *10*, 2448.
https://doi.org/10.3390/math10142448

**AMA Style**

Shahadat ASB, Akhand MAH, Kamal MAS.
Visibility Adaptation in Ant Colony Optimization for Solving Traveling Salesman Problem. *Mathematics*. 2022; 10(14):2448.
https://doi.org/10.3390/math10142448

**Chicago/Turabian Style**

Shahadat, Abu Saleh Bin, M. A. H. Akhand, and Md Abdus Samad Kamal.
2022. "Visibility Adaptation in Ant Colony Optimization for Solving Traveling Salesman Problem" *Mathematics* 10, no. 14: 2448.
https://doi.org/10.3390/math10142448