Optimization method of passive omnidirectional buoy array in on-call anti-submarine search based on improved NSGA-II - ScienceDirect
Conclusion
In
this paper, the optimization problem of buoy array deployment for
passive omnidirectional sonar buoys during on-call submarine search with
a known submarine speed and an approximate heading is studied. The main
contributions are as follows:
The model for the distribution of the initial submarine positions and the positions after movement is established, considering the distribution of submarine positions under a known submarine speed and approximate heading during on-call submarine search
CRediT authorship contribution statement
- Wenhao Bi: Methodology, Supervision.
- Jiuli Zhou: Formal analysis, Investigation, Validation, Writing – original draft. ResearchGate
- Junyi Shen: Software, Visualization, Writing – review & editing.
- An Zhang: Funding acquisition, Project administration, Resources.
Introduction
With advancements in stealth capabilities and offensive potential(Liu et al., 2019; Yeo et al., 2022), submarines pose an escalating threat to surface vessels, emerging as a formidable maritime strategic force that poses a severe danger to national security. Consequently, anti-submarine warfare has become a crucial area of development for naval forces worldwide(Redford, 2021). Among anti-submarine strategies, aviation-based anti-submarine operations are characterized by their high maneuverability, efficient submarine detection capabilities, secure concealment, and effective offensive measures, garnering substantial attention from naval forces globally as the most effective means for submarine detection, localization, tracking and engagement. The evolution of aerial unmanned platforms has further contributed to the prominence of anti-submarine unmanned aerial vehicles (UAVs) in aerial submarine detection, owing to their extended endurance, cost-effectiveness, extensive detection range, and high operational security. Therefore, UAVs are gradually becoming the mainstay of aerial submarine search operations, playing a more crucial role in anti-submarine warfare(Yue et al., 2022).
During anti-submarine search missions, UAVs often carry sonar buoys(Craparo and Karatas, 2020; Lee, 2018) as their primary search equipment, which are deployed in multiple formations to facilitate submarine detection and localization. Sonar buoys are characterized by their small size, light weight and excellent concealment capabilities and are widely used by major military powers worldwide as essential search equipment for underwater operations. When employed in conjunction with anti-submarine UAVs, sonar buoys can perform submarine search missions with low cost and high efficiency. The deployment of a sonar buoy array directly determines the effectiveness of the anti-submarine search and thereby affects the monitoring and control capabilities of the underwater operational space. Therefore, studying the optimal deployment of sonar buoy arrays is of great importance for rapidly and accurately completing underwater search missions and ensuring situational awareness in submarine warfare. The optimization of sonar buoy array deployment for sonar-based submarine search aims to optimize the buoy array configuration by selecting the delivery number and coordinates of deployment points, with the objective of maximizing the probability of submarine detection and minimizing the quantity of deployed buoys.
The optimization problem for sonar buoy deployment in anti-submarine search is a prominent research focus in aviation-based anti-submarine warfare. It is often based on certain assumptions to establish mathematical models and is solved using methods such as geometric techniques and intelligent algorithms. Karatas et al. (2018) formulated a mathematical model for multibase sonar problems using integral geometry and geometric probability. The model's validity was assessed through Monte Carlo simulations. Craparo et al. (2019) proposed an optimization method for multibase sonar buoy placement based on an enhanced algorithm that integrates integer linear programming, heuristic procedures, and iterative search methods. The feasibility of the proposed algorithm was demonstrated through simulation experiments conducted in three real-world scenarios. Fügenschuh et al. (2020) approached the multibase sonar localization problem by formulating a mixed-integer programming model that aimed to optimize the maximum coverage area while minimizing costs. Ma et al. (2020) addressed the problem of detecting underwater clustered targets by introducing the concept of buoy arrays and employing the adaptive fireworks algorithm to determine optimal sonar buoy deployment strategies. Kim (2022) introduced a target localization method based on multiple virtual measurement sets, highlighting its advantages in terms of temporal and spatial precision compared to alternative localization approaches. Taylor et al. (2023) employed a two-stage algorithm to determine an optimal deployment strategy. The initial placement scheme is obtained through a multiobjective evolutionary method, and then the buoy array deployment scheme is further refined using reinforcement learning. The optimization problem for sonar buoy deployment aims to adjust the positions of critical nodes to achieve the desired optimization objectives, similar to node deployment optimization problems in other domains. Sharp and DuPont (2018) approached the array optimization problem for wave energy converters using a genetic algorithm, laying the groundwork for future research on array optimization in more complex scenarios. Yakıcı and Karatas (2021) employed the nondominated sorting genetic algorithm II (NSGA-II) to solve the problem of deploying heterogeneous sensor networks for localization, highlighting the superior accuracy and efficiency of heuristic algorithms. In the context of reconnaissance UAV deployment in adversarial environments, Karatas et al. (2022) solved this problem by considering the objectives of maximizing search efficiency and minimizing threats. The study utilized a hybrid algorithm that combined integer programming and heuristic techniques and demonstrated the effectiveness of the proposed method.
In the process of solving node deployment optimization problems, such as buoy array deployment optimization problems, intelligent algorithms and their improved versions have been extensively applied. Intelligent algorithms such as the fireworks algorithm (Ma et al., 2020; REDDY et al., 2016), bacterial foraging optimization algorithm(Kim et al., 2022), multiobjective particle swarm optimization algorithm(Zhang et al., 2018), and nondominated sorting genetic algorithm-III(NSGA-III) (Adekoya and Aneiba, 2022; Hu et al., 2020) can solve multi-objective optimization problems well.
The genetic algorithm (GA) has a strong global search ability and can search in multiple directions, so it is often used to solve optimization problems. Deb et al. (2002) proposed the NSGA-II, which uses fast nondominated sorting to reduce the computational complexity. The crowding degree was used to replace the shared radius, and the diversity of the population was maintained. The elite retention strategy was introduced to prevent the loss of optimal individuals and improve the operation speed and robustness of the algorithm. The NSGA-III aimed to solve the problem that the NSGA-II cannot deal with a high-dimensional target space of more than three dimensions, and it maintains population diversity by introducing a wide range of reference points. Because the optimization goal of the buoy deployment optimization problem is to maximize the search efficiency and positioning accuracy and minimize the delivery amount of buoys, the NSGA-II has certain advantages in solving this problem.
The NSGA-II and its improved algorithm are widely used in engineering practice, and have good performance in fields such as path planning(Wu et al., 2022), supply chain design(Shabani-Naeeni and Ghasemy Yaghin, 2021), network transmission(Moniz et al., 2019; Pal et al., 2022), structural optimization design (Fu et al., 2018; Rasekh and Aliabadi, 2023), and resource scheduling (Xiong et al., 2020; Zhou, 2022). As an intelligent heuristic algorithm, the NSGA-II often falls into a local optimum and it converges prematurely in practical applications, which means that it is unable to obtain satisfactory results in a limited time. For specific application problems, scholars have made appropriate improvements to some aspects of the NSGA-II to obtain the global optimal solution.
Zhao et al. (2019) divided the original population into multiple populations in which different crossover operators were assigned. The introduction of the Pareto front to dynamically adjust the population sizes of different populations has enhanced the effectiveness and stability of the algorithm. Zhu et al. (2020) improved the crossover operator and mutation operator of the genetic algorithm based on the improved crossover model to avoid falling into a local optimum. Zhang et al. (2020) incorporated the Levy distribution into the NSGA-II to solve the problem that the tournament selection strategy generates many duplicate individuals, resulting in a decrease in population diversity. Gu et al. (2020) used the symmetric Latin hypercube design to generate the initial population, and introduced the mutation and crossover operators of the differential evolution algorithm to enhance the convergence of the algorithm. The control parameters of the mutation and crossover operators are adjusted adaptively to improve the diversity of the candidate solutions. Jiang et al. (2022) divided the population into two subpopulations and exchanged the optimal nondominated set to replace the worst nondominated set. The convergence speed is thereby increased and the algorithm avoids falling into a local optimum. Deng et al. (2022) used the congestion strategy to improve the selection strategy of the NSGA-II, and used the adaptive crossover operator to balance the convergence of the algorithm and the diversity of the population. Li et al. (2022) improved the population update method of the NSGA-II, established a new scoring standard to evaluate the dominance of individuals in the population, introduced an adaptive strategy to prevent falling into a local optimum, and improved the crossover and mutation operator. He and Zhang (2022) used the NSGA-II to optimize the seismic design of buildings, and the optimal scheme was selected based on the decision-making method of the Tchebycheff objective weight. Gandomi et al. (2023) combined the NSGA-II with machine learning to solve the optimization problem, and the fuzzy decision-making method, decision-making laboratory method and network analytic hierarchy process were combined to select the final scheme from the optimal solution set.
However, existing research on optimizing sonar buoy array deployment often involves inappropriate evaluations or only considers one or two specific metrics of maximum coverage area, positioning accuracy and cost consumption. Simplifying the calculation of the maximum coverage area of the buoy array may reduce the precision of these metrics, leading to decreased interpretability of the calculation method and decreased accuracy of the optimization results. Failing to consider positioning accuracy during the optimization process hinders further submarine localization, impeding subsequent anti-submarine operations. Neglecting cost consumption during the optimization process does not align with the requirements of modern warfare and can result in unnecessary loss of anti-submarine equipment. When solving the optimization problem of passive omni-directional sonar buoy deployment, intelligent algorithms are prone to falling into local optima and converging prematurely. Adapting intelligent algorithms to specific problems and improving their performance can help prevent them from becoming trapped in local optima.
Therefore, this study first considers the distribution pattern of submarine positions under the assumption of a known approximate heading and establishes models for the initial and postmaneuver dispersion of submarine positions. Second, based on the principles of passive omni-directional sonar buoys, the searching submarine efficiency models for a single sonar buoy and for a buoy array are established using the grid method. The LOFIX method is employed to establish an effective positioning probability model for the buoy array. Subsequently, for the optimization problem of buoy array deployment in a known approximate heading scenario, an objective optimization function is constructed to maximize the search probability, maximize the effective positioning probability, and minimize the quantity of buoys deployed. An improved NSGA–II–based buoy deployment optimization model is developed, and the algorithm performance is enhanced through probability selection operators, mixed crossover operators, and an improved elite preservation strategy. Finally, the effectiveness of the proposed method is verified through simulations.
Section snippets
Problem description
It is assumed that in an on-call submarine search, the target submarine position is determined to be point O in the target sea area Ro at time t1, and some prior information is obtained, such as the approximate heading and speed vA of the target submarine at time t1. This paper focuses exclusively on the deployment of passive omni-directional sonar buoys, assuming that they operate on the same horizontal plane. It also assumes that the UAVs are capable of simultaneously monitoring the entire
Submarine position distribution model under on-call submarine search
When the UAV performs an on-call submarine search, the initial position of the submarine and the speed and approximate heading are known as prior information. Although the initial position of the submarine is known, positioning errors remain in the on-call anti-submarine search mission. The submarine's approximate heading and speed vA are known. On this basis, the submarine position distribution model under the approximate heading of on-call submarine search is established.
Submarine search efficiency model for passive omnidirectional sonar buoys
Ensuring a rational and precise evaluation of the search effectiveness of passive omnidirectional sonar buoys is essential for providing a clear and efficient representation of the search performance of different buoy arrays against target submarines. This evaluation forms the foundation for optimizing array configurations and has a direct impact on the selection of array schemes.
Optimization model of buoy array deployment based on the improved NSGA-II
For the problem of optimal buoy deployment for anti-submarine UAVs, adaptive modifications were made to certain components of the NSGA-II, leading to the establishment of an optimization model of buoy array deployment based on the improved NSGA-II. The specific details are given below.
Experiments
With the development of submarine technology to achieve high speed, low noise, and other advancements, anti-submarine warfare, especially in the form of aviation-based operations with rapid response, long range, versatile equipment options, and reduced vulnerability to attacks, plays a crucial role in ensuring maritime security and protecting naval assets. As a result, it is very important to study the optimization problem of anti-submarine buoy deployment. In the experiments, the
Acknowledgement
The author greatly appreciates the reviewers' suggestions and the editor's encouragement. This research is supported by the National Natural Science Foundation of China (No. 62073267) and the Fundamental Research Funds for the Central Universities (HXGJXM202214).
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