# A Novel Discrete Group Teaching Optimization Algorithm for TSP Path Planning with Unmanned Surface Vehicles

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^{†}

## Abstract

**:**

## 1. Introduction

- This study presents the first application of the GTOA in the permutation-coded discrete form.
- The DGTOA is a novel and effective discrete optimization algorithm for solving the TSP, and the comparison shows that the solutions obtained by the DGTOA have comparable performance.
- The dynamic adaptive neighborhood radius can balance the exploration and exploitation for solving the TSP.
- The DGTOA has been successfully applied to USV path planning, and the simulation results indicate that the DGTOA can provide a competitive advantage in path planning for USVs.

## 2. Literature Survey

## 3. Background Work

#### 3.1. Group Teaching Optimization Algorithm

Algorithm 1: Pseudo-code of GTOA |

#### 3.2. Traveling Salesman Problem (TSP)

#### 3.3. Dynamic Adaptive Neighborhood Radius

## 4. Discrete Group Teaching Optimization Algorithm Detail Design

#### 4.1. Discrete Group Teaching Optimization Algorithm

Algorithm 2: Pseudo-code of DGTOA |

#### 4.2. New Greedy Crossover Algorithm

#### 4.3. Middle Student Algorithm

#### 4.4. Dynamic Neighborhood Shift Mutation Algorithm

#### 4.5. Dynamic Neighborhood Inversion Mutation Algorithm

#### 4.6. Dynamic Neighborhood 3-opt Mutation Algorithm

## 5. Results and Discussions

#### 5.1. Experiment 1: Comparisons with Random Initialization, Neighborhood Initialization, and Greedy Initialization

#### 5.2. Experiment 2: Comparisons with Adaptive Neighborhood Mutation and Dynamic Adaptive Neighborhood Mutation

#### 5.3. Experiment 3: Comparisons with the DJAYA, DTSA, ABC, PSO-ACO, and DSFLA

#### 5.4. Experiment 4: Case Study with USV Path Planning

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

USV | Unmanned surface vehicle |

DGTOA | Discrete group teaching optimization algorithm |

GTOA | Group teaching optimization algorithm |

DTSA | Discrete tree seed algorithm |

TSA | Tree seed algorithm |

DJAYA | Discrete Jaya algorithm |

JAYA | Jaya algorithm |

ABC | Artificial bee colony |

PSO-ACO | Particle swarm optimization-ant colony optimization |

DSFLA | Discrete shuffled frog-leaping algorithm |

GA | Genetic algorithm |

SA | Simulated annealing |

SCX | Sequential constructive crossover operator |

CX2 | Cycle crossover operator |

EAX | Edge assembly crossover |

GSTM | Greedy sub-tour mutation |

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**Figure 7.**Dynamic neighborhood inversion mutation. (

**a**) Reverse the order ${x}_{i}^{min}\sim {x}_{i}^{max-1}$. (

**b**) Reverse the order ${x}_{i}^{min+1}\sim {x}_{i}^{max-1}$. (

**c**) Reverse the order ${x}_{i}^{min}\sim {x}_{i}^{max}$. (

**d**) Reverse the order ${x}_{i}^{min+1}\sim {x}_{i}^{max}$.

**Figure 8.**RE curves with random initialization, neighborhood initialization, and greedy initialization for different iteration periods based on eil101.

**Figure 9.**RE curves with random initialization, neighborhood initialization, and greedy initialization for different iteration periods based on tsp225.

**Figure 10.**Box plot for the eil76 and ch150 benchmark problems with dynamic adaptive neighborhood mutation and adaptive neighborhood mutation.

**Figure 11.**Simulated USV path tracking results at 25 and 50 waypoints. (

**a**) Simulated USV path tracking results at 25 waypoints. (

**b**) Simulated USV path tracking results at 50 waypoints.

Problem | Number of Cities | Optimum Tour Length |
---|---|---|

eil51 | 51 | 428.87 |

berlin52 | 52 | 7544.37 |

st70 | 70 | 677.11 |

pr76 | 76 | 108,159.44 |

eil76 | 76 | 545.38 |

kroa100 | 100 | 21,285.44 |

krob100 | 100 | 22,141 |

kroc100 | 100 | 20,749 |

krod100 | 100 | 21,294 |

kroe100 | 100 | 22,068 |

eil101 | 101 | 642.31 |

ch150 | 150 | 6532.1 |

pr152 | 152 | 73,683.6 |

kroa200 | 200 | 29,460 |

tsp225 | 225 | 3859 |

**Table 2.**Comparisons with random initialization, neighborhood initialization, and greedy initialization on fifteen benchmark problems.

Problems | Random initialization | Neighborhood Initialization | Greedy Initialization | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Std Dev | RE (%) | Mean | Std Dev | RE (%) | Mean | Std Dev | RE (%) | |||

eil51 | 430.35 | 1.29 | 0.34 | 429.92 | 0.92 | 0.24 | 429.78 | 0.99 | 0.21 | ||

berlin52 | 7544.37 | 0 | 0 | 7544.37 | 0 | 0 | 7544.37 | 0 | 0 | ||

st70 | 681.3 | 3.38 | 0.62 | 677.11 | 0.02 | 0 | 677.11 | 0.02 | 0 | ||

pr76 | 110,499 | 1214.54 | 2.16 | 108,344 | 331.36 | 0.17 | 108,298.6 | 287.72 | 0.13 | ||

eil76 | 554.09 | 2.61 | 1.6 | 549.84 | 1.81 | 0.82 | 549.44 | 1.81 | 0.74 | ||

kroa100 | 21,466.9 | 97.44 | 0.85 | 21,285.8 | 1.75 | 0 | 21,286.16 | 2.43 | 0 | ||

krob100 | 22,410 | 130.91 | 1.22 | 22,235.4 | 32.65 | 0.43 | 22,216.7 | 48.05 | 0.34 | ||

kroc100 | 20,974.1 | 155.93 | 1.08 | 20,809 | 44.61 | 0.29 | 20,778.18 | 37.53 | 0.14 | ||

krod100 | 21,621.9 | 143.5 | 1.54 | 21,485.2 | 64.79 | 0.9 | 21,456.83 | 55.58 | 0.76 | ||

kroe100 | 22,330.1 | 97.78 | 1.19 | 22,169.9 | 48.77 | 0.46 | 22,152.85 | 33.32 | 0.38 | ||

eil101 | 651.89 | 2.88 | 1.49 | 647.08 | 2.92 | 0.74 | 646.75 | 2.62 | 0.69 | ||

ch150 | 6763.4 | 60.96 | 3.54 | 6554.55 | 5.28 | 0.34 | 6554.4 | 3.93 | 0.34 | ||

pr152 | 74,968.6 | 386.56 | 1.74 | 74,356.1 | 175.35 | 0.91 | 74,408.66 | 225.48 | 0.98 | ||

kroa200 | 30,995 | 213.99 | 5.21 | 29,631.1 | 54.81 | 0.58 | 29,606.35 | 70.2 | 0.5 | ||

tsp225 | 4018.67 | 25.94 | 4.14 | 3940.41 | 19.73 | 2.11 | 3938.95 | 16.76 | 2.07 |

**Table 3.**Comparisons with adaptive neighborhood mutation and dynamic adaptive neighborhood mutation.

Problems | Adaptive Neighborhood Mutation | Dynamic Adaptive Neighborhood Mutation | |||||||
---|---|---|---|---|---|---|---|---|---|

Mean | Std Dev | RE (%) | Best | Mean | Std Dev | RE (%) | Best | ||

eil51 | 429.49 | 0.75 | 0.15 | 428.98 | 429.78 | 0.99 | 0.21 | 428.87 | |

berlin52 | 7544.37 | 0 | 0 | 7544.37 | 7544.37 | 0 | 0 | 7544.37 | |

st70 | 677.12 | 0.02 | 0 | 677.11 | 677.11 | 0.02 | 0 | 677.11 | |

pr76 | 108,542 | 447.33 | 0.35 | 108,159 | 108,298.6 | 287.72 | 0.13 | 108,159.4 | |

eil76 | 549.33 | 2.01 | 0.72 | 545.39 | 549.44 | 1.81 | 0.74 | 545.97 | |

kroa100 | 21,286.9 | 7.12 | 0.01 | 21,285.4 | 21,286.16 | 2.43 | 0 | 21,285.4 | |

krob100 | 22,251.9 | 37.42 | 0.5 | 22,178.6 | 22,216.7 | 48.05 | 0.34 | 22,139.07 | |

kroc100 | 20,824.7 | 55.15 | 0.36 | 20,750.8 | 20,778.18 | 37.53 | 0.14 | 20,750.76 | |

krod100 | 21,463 | 52.66 | 0.79 | 21,345.5 | 21,456.83 | 55.58 | 0.76 | 21,323.48 | |

kroe100 | 22,157.1 | 34.25 | 0.4 | 22,119.9 | 22,152.85 | 33.32 | 0.38 | 22,115.61 | |

eil101 | 647.4 | 3.11 | 0.79 | 641.88 | 646.75 | 2.62 | 0.69 | 641.23 | |

ch150 | 6552.5 | 8.15 | 0.31 | 6530.9 | 6554.4 | 3.93 | 0.34 | 6545.16 | |

pr152 | 74,432.9 | 250.23 | 1.02 | 74,128.6 | 74,408.66 | 225.48 | 0.98 | 73,936.17 | |

kroa200 | 29,667.3 | 61.81 | 0.7 | 29,560 | 29,606.35 | 70.2 | 0.5 | 29,405.72 | |

tsp225 | 3950.47 | 18.52 | 2.37 | 3911.3 | 3938.95 | 16.76 | 2.07 | 3912.3 |

SL | Problem | RE (%) | |||||
---|---|---|---|---|---|---|---|

DJAYA | DTSA | ABC | PSO-ACO | DSFLA | DGTOA | ||

1 | eil51 | 2.64 | 3.51 | 0.28 | 0.11 | 0.1 | 0.21 |

2 | berlin52 | 0.48 | 0.02 | 0 | 0.02 | 0 | 0 |

3 | st70 | 3.72 | 4.66 | 0.66 | 0.47 | 0 | 0 |

4 | pr76 | 4.71 | 6.26 | 0.43 | - | - | 0.13 |

5 | eil76 | 5.1 | 6.09 | - | 0.06 | 0.2 | 0.74 |

6 | kroa100 | 1.99 | 1.06 | 0.98 | 0.77 | 0.14 | 0 |

7 | krob100 | 3.76 | 4.51 | - | - | - | 0.34 |

8 | kroc100 | 4.59 | 5.15 | - | - | 0.11 | 0.14 |

9 | krod100 | 6.28 | 7.88 | - | - | - | 0.76 |

10 | kroe100 | 2.33 | 1.83 | - | - | - | 0.38 |

11 | eil101 | 5.46 | 7.41 | 2.9 | 0.59 | 0.62 | 0.69 |

12 | ch150 | 1.63 | 3.32 | - | 0.55 | 0.53 | 0.34 |

13 | pr152 | - | - | - | - | 0.39 | 0.98 |

14 | kroa200 | - | - | - | 0.95 | 1.03 | 0.5 |

15 | tsp225 | 6.12 | 9.93 | 6.83 | - | - | 2.07 |

Optimal number/ratio (%) | 0/0 | 0/0 | 1/14.29 | 2/25 | 5/66.67 | 10/66.67 |

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**MDPI and ACS Style**

Yang, S.; Huang, J.; Li, W.; Xiang, X.
A Novel Discrete Group Teaching Optimization Algorithm for TSP Path Planning with Unmanned Surface Vehicles. *J. Mar. Sci. Eng.* **2022**, *10*, 1305.
https://doi.org/10.3390/jmse10091305

**AMA Style**

Yang S, Huang J, Li W, Xiang X.
A Novel Discrete Group Teaching Optimization Algorithm for TSP Path Planning with Unmanned Surface Vehicles. *Journal of Marine Science and Engineering*. 2022; 10(9):1305.
https://doi.org/10.3390/jmse10091305

**Chicago/Turabian Style**

Yang, Shaolong, Jin Huang, Weichao Li, and Xianbo Xiang.
2022. "A Novel Discrete Group Teaching Optimization Algorithm for TSP Path Planning with Unmanned Surface Vehicles" *Journal of Marine Science and Engineering* 10, no. 9: 1305.
https://doi.org/10.3390/jmse10091305