# An Improved NSGA-II Based on Multi-Task Optimization for Multi-UAV Maritime Search and Rescue under Severe Weather

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Knowledge

#### 2.1. Multi-Objective Optimization Problem

**x**= (x

_{1}, x

_{2}, …, x

_{D}) ∈Ω is a D-dimensional decision variable; $\Omega \subset {\mathbb{R}}^{D}$ is the decision space;

**f**(

**x**) is an objective vector with M objective functions; $f(\mathit{x})\subset {\mathbb{R}}^{M}$ is the objective vector.

**Definition 1.**

**u**= (u

_{1}, u

_{2}, …, u

_{D}),

**v**= (v

_{1}, v

_{2}, …, v

_{D}),

**u**is called to dominate

**v**(denoted as $\mathit{u}\succ \mathit{v}$) if and only if

**u**is no more than

**v**, that is:

**Definition 2.**

**x***∈Ω,

**x*** is called to be a Pareto optimal solution if and only if there are not any

**x**satisfy $\mathit{f}(\mathit{x})\succ \mathit{f}(\mathit{x}*)$. All of Pareto optimal solutions compose a Pareto optimal set (PS), marked as

**X**

^{∗}.

**Definition 3.**

#### 2.2. Non-Dominated Sorting Genetic Algorithm-II

_{all}, which is sized 2NP. Subsequently, the non-dominated sorting and crowding distance method [25] are used to select NP individuals from P

_{all}to obtain a new parent population. Repeat these steps until the termination conditions are met. The pseudocode of NSGA-II is given in Algorithm 1.

Algorithm 1: NSGA-II |

Input: Population size: NP; maximum generation: G |

1: Generate and evaluate the initial population P |

2: Set G = 1 |

3: while termination criterion not satisfied do |

4: OP ← Generate offspring by crossover and mutation strategies |

5: P_{all} ← OP ∪ P |

6: P ← Select NP individuals from P_{all} based on the non-dominated sorting and crowding distance |

7: G = G + 1 |

8: end while |

Output: The PS and PF |

## 3. Literature Review

## 4. Multi-Objective Maritime Search and Rescue Problem under Severe Weather

#### 4.1. Description of Multi-Objective MSR Problem under Severe Weather

#### 4.2. Mathematical Model of Multi-Objective MSR Problem under Severe Weather

_{P}, y

_{P}) and the number of UAVs is n. Moreover, there are q vessels in distress. The coordinate and contact range of the i-th vessel is represented as (x

_{i}, y

_{i}) and r

_{i}, respectively. The center position of the storm S is (x

_{s}, y

_{s}) and its influencing radius is r

_{S}. The j-th UAV is assigned to visit a set of vessels. Moreover, the nodes corresponding to these vessels are denoted as T(j), in which N

_{i}is denoted as the corresponding node of the i-th vessel. Therefore, all the targets visited by the j-th UAV is W(j) = {P, T(j), P}.

_{i}is used to determine whether the i-th vessel is within the influencing scope of the storm. It is calculated by the following formula:

_{iS}denotes the Euclidean distance between the i-th vessel and the center position of the storm. If E

_{i}= 1, the i-th vessel is within the influencing scope of the storm and should be visited as soon as possible. Otherwise, the i-th vessel is relatively safe.

_{uv}stands for the Euclidean distance between the two nodes.

_{1}is to minimize the total path length of multi-UAVs. The shorter the total path length of UAVs, the less energy they consume. The f

_{2}aims at minimizing the total task completion time, which is equivalent to minimizing the longest single path length. The f

_{3}aims at minimizing the completion time of urgent tasks. That is, UAVs are required to visit the vessels within the affected area of the storm as early as possible. Concretely, it is represented to minimize the path length from the MSR station to the vessels which need priority.

## 5. The Proposed Method

#### 5.1. Encoding and Decoding Method

**Individual encoding of the main task:**The individuals in the main task are all encoded by real numbers, where each individual is 3 × q. Moreover, the individual is divided into three segments in the current study. The first segment (i.e., Chromosome Segment I) represents the task allocation of UAVs. It consists of q integer numbers within the range of [1, n]. The second segment (Chromosome Segment II) contains a series of vessel numbers, whose order will be mapped to the visiting sequence in the UAV paths. It also consists of q integer numbers within the range of [1, q]. To ensure that the UAVs traverse all vessels without repetition and omission, all genes in Chromosome Segment II are different from each other. The locations of nodes are indicated in the third segment (Chromosome Segment III), in which genes are represented as angles between the vessels and nodes. The Chromosome Segment III is composed of q floating-point numbers within the range of [0, 360]. An example of a main task containing 5 vessels and 3 UAVs is shown in Figure 2.

**Individual encoding of the assistant task:**Unlike the encoding of the main task, a new encoding for the simplified MSR problem (i.e., the target is regarded as an ideal point) is used. Therefore, the length of each individual in the assistant task is 2 × q. All the individuals are encoded by integer numbers. Moreover, the individual is divided into two segments. The first segment (i.e., Chromosome Segment I) represents the task allocation of UAVs, and the second segment (i.e., Chromosome Segment II) denotes the visiting sequence of each UAV. An example of an assistant task containing 5 vessels and 3 UAVs is shown in Figure 3.

**Chromosome Segment I:**Each gene represents the number of a UAV, i.e., which UAV will visit the matching vessel. The same genes mean that these vessels are assigned to the same UAV.

**Chromosome Segment II:**Each gene represents the number of a vessel. For a given UAV, its visiting sequence is determined by the corresponding order in the Chromosome Segment II.

**Chromosome Segment III:**Each node location is computed by the following formulas:

#### 5.2. Population Initialization Method

**Step 1:**The total number of UAVs and population size is set to n and NP, respectively. Generally, NP is much greater than n. The number of individuals of each particular type is defined as Num = [NP/n]. Then Num individuals that dispatch one UAV and Num individuals that dispatch two UAVs are generated, respectively.

**Step 2:**For the remaining individuals in the initial population, they are generated randomly.

#### 5.3. Knowledge Transfer in the Multi-Task Optimization

**Knowledge transfer from main task to assistant task:**As mentioned in Section 5.1, the individual in the main task has three parts (i.e., Chromosome Segments I, II, and III), while there are two components of the individual in the assistant task. Therefore, if the knowledge of the main task is transferred to the assistant task, then the third part (i.e., Chromosome Segment III) should be deleted. It can be observed from Figure 7 that, to achieve knowledge transfer from the main task to the assistant task, the Chromosome Segment III of the individual in the main task is deleted to obtain a transferred individual, which can help the assistant task.

**Knowledge transfer from assistant task to main task:**The first two parts of individuals between two tasks are versatile. However, individuals in the assistant task do not contain the third part. Therefore, if the knowledge of the assistant task is transferred to the main task, the third part of the individual in the main task can be added to the transferred individual. It can be observed from Figure 7 that a Chromosome Segment III is randomly selected as a reference from the PS of the main task, which can provide high-quality node information. Then, combine this Chromosome Segment III with the assistant task individual to form a transferred individual.

#### 5.4. The Overall Process of the INSGA-II-MTO

_{1}and P

_{2}with NP individuals according to the proposed initialization method. In line 2, all individuals in P

_{1}are evaluated via the original multi-objective MSR problem; in line 3, all individuals in P

_{2}are evaluated via the simplified multi-objective MSR problem.

_{1}and P

_{2}via the binary tournament method, which are denoted as MP

_{1}and MP

_{2}. Then, the SBX and PM are adopted to generate an offspring population OP

_{1}sized NP/2 in line 7; the two-point crossover and multi-point mutation operator are adopted to generate an offspring population OP

_{2}sized NP/2 in line 8. After generating two offspring populations, in lines 9–10, OP

_{1}and OP

_{2}are evaluated via the original multi-objective and simplified multi-objective MSR problem, respectively. Subsequently, knowledge sharing between two tasks is achieved by transferring individuals. In line 11, generate OP

_{1new}as a transferred population by deleting the Chromosome Segment III in OP

_{1}; in line 12, generate OP

_{2new}as another transferred population by randomly adding the Chromosome Segment III to the individual in OP

_{2}. In lines 13–14, P

_{1}, OP

_{1}and OP

_{2new}are combined as P

_{1all}; P

_{2}, OP

_{2}and OP

_{1new}are combined as P

_{2all}. Next, in lines 15–16, NP individuals are selected from P

_{1all}and P

_{2all}for the next iteration according to the non-dominated sorting and crowding distance, respectively. Finally, the PS and PF of the main task is output when the termination condition is satisfied.

Algorithm 2: INSGA-II-MTO |

Input: Population size: NP; maximum generation: G_{max}; population of the main task: P_{1}; population of the assistant task: P_{2} |

1: Initialize P_{1} and P_{2} of size NP via the proposed initialization method in Section 5.2 |

2: Evaluate P_{1} on the original multi-objective MSR |

3: Evaluate P_{2} on the simplified multi-objective MSR |

4: while termination criterion not satisfied do |

5: MP_{1} ← Select NP/2 individuals from P_{1} using binary tournament selection method |

6: MP_{2} ← Select NP/2 individuals from P_{2} using binary tournament selection method |

7: OP_{1} ← Generate NP/2 offspring by MP_{1} according to SBX and PM |

8: OP_{2} ← Generate NP/2 offspring by MP_{2} according to two-point crossover and multi-point mutation operator |

9: Evaluate OP_{1} on the original multi-objective MSR problem |

10: Evaluate OP_{2} on the simplified multi-objective MSR problem |

11: OP_{1new} ← OP_{1} delete the Chromosome Segment III to generate a transferred population according to; |

12: OP_{2new} ← OP_{2} randomly add the Chromosome Segment III to generate a transferred population according to Section 5.3; |

13: P_{1all} ←P_{1} ∪ OP_{1} ∪ OP_{2new}; |

14: P_{2all} ←P_{2} ∪ OP_{2} ∪ OP_{1new}; |

15: P_{1} ← Select NP individuals from P_{1all} based on the non-dominated sorting and crowding distance |

16: P_{2} ← Select NP individuals from P_{2all} based on the non-dominated sorting and crowding distance |

17: end while |

Output: The PS and PF of the main task |

## 6. Experimental Results and Analysis

#### 6.1. Comparison Results with Other Algorithms

#### 6.2. Experimental Analysis

- (1)
- The effectiveness of the population initialization method

- (2)
- The effectiveness of the multi-task optimization

#### 6.3. Diversity of the Solutions

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**The scenes of the original and simplified MSR tasks. (

**a**) The scene of the original MSR task. (

**b**) The scene of the simplified MSR task.

**Figure 7.**The transfer process of multi-task optimization. The symbol “*” denotes the chromosome segment comes from a main task individual.

**Figure 9.**The NR values of three algorithms. (

**a**) The NR values obtained by three algorithms in scenario 1. (

**b**) The NR values obtained by three algorithms in scenario 2. (

**c**) The NR values obtained by three algorithms in scenario 3.

**Figure 10.**The number of various schemes obtained in experiments. (

**a**) The number of schemes obtained by two algorithms in scenario 1. (

**b**) The number of schemes obtained by two algorithms in scenario 2. (

**c**) The number of schemes obtained by two algorithms in scenario 3.

**Figure 12.**Two typical solutions in scenario 2. (

**a**) One UAV is used to complete the MSR task. (

**b**) Three UAVs are dispatched to complete the MSR task.

Name | Value | Algorithms |
---|---|---|

pc_{1} | 0.9 | INSGA-II-MTO, NSGA-II, NSGA-II-GLS |

pm_{1} | 0.6 | INSGA-II-MTO, NSGA-II, NSGA-II-GLS |

pc_{1} | 0.8 | INSGA-II-MTO |

pm_{1} | 1 | INSGA-II-MTO |

NP | 100 | INSGA-II-MTO, NSGA-II, NSGA-II-GLS |

Test Scenario 1 | Test Scenario 2 | Test Scenario 3 | |
---|---|---|---|

INSGA-II-MTO | 0.1929 (2.46 × 10 ^{−4}) | 0.2846 (1.1 × 10 ^{−2}) | 0.2567 (1.2 × 10 ^{−2}) |

NSGA-II | 0.1923 (5.8 × 10 ^{−4}) + | 0.2099 (2.5 × 10 ^{−2}) + | 0.1571 (4.1 × 10 ^{−2}) + |

NSGA-II-GLS | 0.1924 (5.98 × 10 ^{−4}) + | 0.2211 (3.6 × 10 ^{−2}) + | 0.1774 (3.9 × 10 ^{−2}) + |

Test Scenario 1 | Test Scenario 2 | Test Scenario 3 | |
---|---|---|---|

INSGA-II-MTO | 0.1929 (2.46 × 10 ^{−4}) | 0.2846 (1.1 × 10 ^{−2}) | 0.2567 (1.2 × 10 ^{−2}) |

INSGA-II-MTO-R | 0.1926 (3.7 × 10 ^{−4}) + | 0.2465 (3.0 × 10 ^{−2}) + | 0.1908 (3.1 × 10 ^{−2}) + |

Test Scenario 1 | Test Scenario 2 | Test Scenario 3 | |
---|---|---|---|

INSGA-II-MTO | 0.1929 (2.46 × 10 ^{−4}) | 0.2846 (1.1 × 10 ^{−2}) | 0.2567 (1.2 × 10 ^{−2}) |

INSGA-II-MTO-I | 0.1926 (2.8 × 10 ^{−4}) + | 0.2593 (1.2 × 10 ^{−2}) + | 0.2272 (1.8 × 10 ^{−2}) + |

Objective 1 | Objective 2 | Objective 3 | |
---|---|---|---|

(a) | 297.876 | 297.876 | 138.7735 |

(b) | 636.1917 | 228.4096 | 114.2048 |

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## Share and Cite

**MDPI and ACS Style**

Ma, Y.; Li, B.; Huang, W.; Fan, Q.
An Improved NSGA-II Based on Multi-Task Optimization for Multi-UAV Maritime Search and Rescue under Severe Weather. *J. Mar. Sci. Eng.* **2023**, *11*, 781.
https://doi.org/10.3390/jmse11040781

**AMA Style**

Ma Y, Li B, Huang W, Fan Q.
An Improved NSGA-II Based on Multi-Task Optimization for Multi-UAV Maritime Search and Rescue under Severe Weather. *Journal of Marine Science and Engineering*. 2023; 11(4):781.
https://doi.org/10.3390/jmse11040781

**Chicago/Turabian Style**

Ma, Yue, Bo Li, Wentao Huang, and Qinqin Fan.
2023. "An Improved NSGA-II Based on Multi-Task Optimization for Multi-UAV Maritime Search and Rescue under Severe Weather" *Journal of Marine Science and Engineering* 11, no. 4: 781.
https://doi.org/10.3390/jmse11040781