An innovative coverage optimization method for smart information monitoring in agricultural IoT using the multi-strategy Pelican optimization algorithm

Abstract

With the rapid advancement of agricultural technology, Agricultural Wireless Sensor Network (AWSN) monitoring for crop growth in large-scale fields has become pivotal in smart agriculture. Optimizing AWSN coverage is crucial for enhancing production efficiency and resource utilization. However, traditional optimization algorithms struggle with local convergence and accuracy in large-scale sensor deployment, an NP-hard problem. To address this, we propose a novel Multi-Strategy Pelican Optimization Algorithm (MSPOA), integrating a good point set strategy, a 3D spiral Lévy flight strategy, and an adaptive T-distribution variation strategy. The good point set strategy expands the search range and enhances local search capability, while the 3D spiral Lévy flight strategy improves convergence speed and global search accuracy. The adaptive T-distribution variation strategy further boosts global search ability, and pelican-inspired movement and collaboration strategies enhance adaptability and robustness in diverse agricultural scenarios. Comparative experiments with Improved Artificial Bee Colony Algorithm (IABC), Chaotic Adaptive Firefly Optimization Algorithm (CAFA), Adaptive Particle Swarm Optimization (APSO), and Lévy Flight Strategy Chaotic Snake Optimization Algorithm (LCSO) demonstrate that MSPOA improves network coverage by 5.85%, 11.33%, 21.05%, and 20.66%, respectively. Additionally, MSPOA exhibits strong adaptability and stability in dynamic agricultural environments.

Introduction

In the realm of smart agriculture, the monitoring of crop growth information in large-scale agricultural fields has garnered significant attention due to the rapid advancements in agricultural technology. Agricultural Wireless sensor network (AWSN) technology plays a pivotal role in this domain, as it enables real-time acquisition and transmission of environmental data such as temperature, humidity, soil moisture, pest infestations, and disease occurrences to a centralized data processing node or user terminal^1,2. This real-time monitoring provides invaluable data support for precision agriculture management. Wireless sensor nodes, typically consisting of low-cost, low-power hardware devices equipped with sensing, communication, and computation capabilities³, are strategically distributed across the monitored area. These nodes collaboratively work through wireless communication to gather essential data^4,5. Among the key challenges in WSNs, wireless sensing network coverage stands out as a crucial issue. It involves the rational arrangement of sensor nodes to ensure complete or partial coverage within the target area, thereby fulfilling specific monitoring requirements⁶. Ensuring comprehensive monitoring of large-field crop growth necessitates optimizing the distribution and coverage of the AWSN. The ultimate goal of addressing the coverage problem is to maximize network coverage, i.e., to effectively cover the area of interest within the target region^7,8, while also minimizing the number of deployed sensors. This optimization directly impacts the cost, energy consumption, and overall performance of the network. Therefore, optimization algorithms play a crucial role in the distribution of AWSN and network self-organization optimization^9,10,11. It helps to reduce energy consumption and extend the lifespan of sensor nodes, which is crucial for sustainable and cost-effective smart agriculture practices^12,13,14,15.

In recent years, the WSNs coverage optimization problem has attracted extensive research interest, prompting scholars to propose a variety of optimization algorithms to address the problem^16,17. The common goal of these algorithms is to find an optimal sensor deployment strategy that maximizes the coverage area by minimizing redundancy^18,19. However, despite the theoretical advantages of these algorithms, there are still some challenges and limitations in practical applications. First, traditional optimization algorithms often suffer from the problem of premature convergence. Due to the complexity of the search space and the existence of local optimal solutions, traditional algorithms may converge early in the search process, resulting in the failure to find the global optimal solution. Secondly, the convergence speed of these algorithms is slow, especially in high-dimensional spaces, and the search efficiency decreases significantly. This not only increases the computation time of the algorithms, but also limits their feasibility in practical applications. In addition, due to the limitations of algorithm design and parameter settings, traditional optimization algorithms are prone to fall into local optimal solutions and cannot effectively explore the entire search space, resulting in the inability to achieve the optimal sensor deployment scheme.

In order to overcome these limitations, this paper proposes an MSPOA wireless sensor network coverage optimization method, MSPOA combines the good point set strategy, the adaptive distribution t-variance strategy, the Levy flight strategy, and the inverse learning strategy, and the algorithm is supposed to have a stronger global search capability, a faster convergence speed, and a better adaptation to different network environments and topologies. The adjustment and optimization of the algorithm parameters are also enhanced to improve the stability and reliability of the algorithm. The efficiency and effectiveness of sensor deployment in wireless sensor networks are improved. The core work of this article is shown in Fig. 1.

Fig. 1

Wireless sensor network coverage structure.

The main purpose of this study is to investigate the effectiveness of MSPOA in optimizing WSN coverage. Specifically, the contributions of this article are as follows:

• 1) This study introduces a novel MSPOA that effectively balances global optimization and convergence speed, as well as population exploration and development. By integrating a good point set strategy for population initialization, MSPOA introduces controlled perturbations to enhance stability, prevent local optima, and improve convergence performance.

• 2) A three-dimensional spiral Lévy flight strategy is proposed, combining the strengths of Lévy flight and spiral optimization search. This strategy enables MSPOA to explore a broader search space, escape local optima, and enhance global search capabilities. Additionally, an adaptive T-distribution variation strategy is fused with the Pelican algorithm to further improve search accuracy and balance local and global optimization.

• 3) Through extensive simulation experiments, MSPOA is compared with four other algorithms (IABC, CAFA, APSO, and LCSO) across various network sizes and topologies. The results demonstrate MSPOA’s superior performance in WSN coverage optimization, achieving significant improvements in coverage rate, network cost, energy efficiency, and overall performance, thereby validating its effectiveness and superiority in addressing WSN coverage challenges.

The findings of this research will contribute to the existing body of knowledge on coverage optimization for wireless sensor networks and provide valuable insights into the application of MSPOA in real-world wireless sensor network deployments. The optimization results obtained from MSPOA can help improve the efficiency, reliability, and cost-effectiveness of wireless sensor networks, making them more suitable for a wide range of applications.

In the following sections of this paper, we will first provide an overview of the related work on AWSN coverage optimization. Then, we will describe the proposed MSPOA approach in detail, including the mathematical model and the optimization process. Subsequently, the experimental setup, simulation results and comparative analysis will be presented. Finally, the paper will discuss the results, draw conclusions, and outline future research directions. Table 1 provides symbol abbreviations.

Table 1 Symbol abbreviation.

Related work

In the research of coverage issues in Agricultural Wireless Sensor Networks, traditional methods and heuristic methods have played pivotal roles, each showcasing distinct advantages and limitations within the context of smart agriculture. This chapter aims to summarize and critique these two approaches, ultimately emphasizing the necessity of the innovative methods proposed in this paper against the backdrop of the rapidly evolving smart agriculture landscape.

Traditional methods in AWSN coverage

In the research on the coverage problem of agricultural wireless sensor networks (WSN), some traditional methods were the first to be studied and solved^20,21,22. These methods are often based on geometric, Boolean logic, or probabilistic models, aimed at analyzing and optimizing the coverage performance of WSN through theoretical derivation and simplifying assumptions. The Boolean perception model, as one of the most fundamental coverage models, was widely adopted in early WSN research. This model assumes that the monitoring capability of sensor nodes is uniform within the sensing radius, meaning that the target is considered fully covered as long as it is within the sensing range of the node. Although this model simplifies problem analysis, it is often too idealized in practical applications and difficult to accurately reflect the coverage situation in real environments²⁰. On the other hand, the probabilistic perception model considers the impact of signal attenuation and environmental factors on coverage effectiveness, making coverage evaluation more realistic. In the probability aware model, the coverage probability of nodes to targets decreases with increasing distance, providing a more accurate modeling approach for WSN coverage problems²¹. Grid partitioning method is another traditional WSN coverage evaluation method. This method quantifies the network coverage by dividing the target area into several small grids and counting the number of grids covered by sensor nodes. This method is intuitive and easy to implement, suitable for evaluating the coverage performance of large-scale WSN²².

However, as smart agriculture pushes the boundaries of AWSN deployment, traditional methods encounter limitations. The complexity of real-world agricultural environments, with their varying soil conditions, climate patterns, and crop growth cycles, often exceeds the assumptions made by these models. Moreover, the computational demands of large-scale AWSNs can render traditional methods impractical for real-time coverage optimization.

Heuristic methods in AWSN coverage

Heuristic methods, such as genetic algorithms, particle swarm optimization, and ant colony optimization, have emerged as powerful tools for addressing the challenges posed by large-scale and complex AWSN coverage problems. These methods mimic natural processes to efficiently search for near-optimal solutions, balancing exploration and exploitation to avoid local optima. Heuristic approaches excel at handling NP-hard problems, which are prevalent in AWSN coverage optimization, and can adapt to the dynamic nature of smart agricultural environments.

Wang et al.¹⁶ proposed a novel mobile assisted coverage blind spot repair strategy. This strategy adopts a grid-based approach to partition the network and evaluates the coverage of each grid to determine which areas have insufficient coverage quality. Next, by applying the Particle Swarm Optimization (PSO) algorithm, the optimal sensor movement path is calculated to activate the dormant mobile sensors and effectively repair the coverage blind spots, thereby enhancing the network’s coverage capability. Meanwhile, Li et al.¹⁷ improved the classic multi-objective ant lion optimization algorithm (MOALO) with the goal of improving the service quality and extending the network lifespan of wireless sensor networks (WSNs). They introduced an improved mechanism based on fast non dominated sorting, forming a new NSIMOALO algorithm. This algorithm combines the idea of fast non dominated sorting algorithm with the elite strategy in NSGA-II algorithm, effectively avoiding premature convergence to local optimal solutions and improving the accuracy of the solution. In addition, by integrating the Levy flight mechanism and dynamically adjusting the weight coefficients between ant lions, elite ant lions, and Levy flights, the NSIMOALO algorithm further enhances its global search performance while maintaining population diversity. Tuba et al.¹⁸ proposed an improved Enhanced Fireworks Algorithm (FWA) for wireless sensor network coverage problems, which has good stability and can be easily extended to more complex coverage models. However, the convergence of this algorithm is poor and the complexity is high. Tuba et al.¹⁹ proposed a coverage optimization method for wireless sensor networks based on firefly algorithm (FA). This method has the advantages of strong global search capability, simplicity and low algorithm complexity. However, due to the slow convergence speed of FA and the algorithm’s dependence on initial solutions, parameter sensitivity is high, which may lead to the algorithm getting stuck in local optima or poor search results.

In paper²³, Arivudainambi et al. introduced a Genetic Algorithm (GA) to solve the connected coverage problem, considering both target coverage and sensor connectivity to the base station. This algorithm aims to maximize the sensor network’s lifetime and achieve effective coverage and connectivity. However, it has limited coding options and is prone to local optima. Hajizadeh et al.²⁴ proposed a controlled deployment algorithm based on the Multi-Objective Bee Swarm Optimization (MOBSO), inspired by honeybee foraging behavior. This method prioritizes network connectivity and coverage, offering better coverage and higher-quality solutions in simulations. Nevertheless, its high time and space complexity hinder its application to large areas. Ankit et al.²⁵ presented an optimal path routing algorithm for wireless networks using an ant clustering technique. This algorithm employs a multipath routing scheme with numerous routing metrics, enhancing previous algorithms to find optimal paths and accelerating wireless network coverage. However, its implementation is challenging due to its heavy reliance on a large number of path routing schemes, which can significantly increase algorithm complexity.

Kong et al.²⁶ proposed an Enhanced Particle Swarm Algorithm (IPSO) for improving coverage in wireless sensor networks, addressing the issue of irrational sensor deployment distribution. This algorithm enhances global convergence speed and particle diversity to avoid local convergence. However, it suffers from slow convergence speed and challenging parameter settings. Tian et al.²⁷ introduced the Elite Parallel Cuckoo Search Algorithm (EPCSA) for area coverage control in high-density wireless sensor networks. EPCSA efficiently optimizes network coverage and sensor node deployment. Despite incorporating elite selection and parallel operations, the algorithm still experiences slow convergence. Additionally, it is not suitable for combinatorial and discrete optimization problems. Inspired by army ants’ predatory behavior, Yao et al.²⁸ proposed the Army Ant Search Optimization Algorithm (AASO), a new swarm intelligence technique for solving the WSN coverage problem. AASO demonstrates better performance and robustness in various scenarios, adapting well to changes in node count, sensing radius, and viewing angle. However, it is prone to local optima and exhibits poor stability.

Liu et al.²⁹ proposed a novel clustering model based on work cycle (DCCM) to save energy by reducing the number of active nodes. This model achieves alternating node operation through the innovative design of the Coverage Relationship Matrix (CRM) and Coverage Sets (CSs), effectively slowing down the energy consumption rate. Zhu et al.³⁰ used the brainstorming optimization algorithm to address the full coverage problem in wireless sensor networks. However, the algorithm is highly dependent on weight coefficients, leading to decreased coverage and a reduced number of activated sensor nodes as these coefficients increase. Paper³¹ introduced a fruit fly algorithm to eliminate overlapping (redundant) nodes based on coverage probability, aiming to maximize the network period. The algorithm employs random deployment of sensor nodes with varying sensing radii to enhance sensing coverage quality and reduce sensor node usage. However, these algorithms tend to fall into local optima, and their runtime and computational overhead may significantly increase when dealing with large-scale problems.

Nevertheless, heuristic methods are not without their drawbacks. Their performance can vary significantly depending on the problem instance, initial conditions, and parameter settings. This uncertainty can make it difficult to predict and guarantee coverage quality in critical agricultural applications. Furthermore, the computational overhead of heuristic algorithms may hinder their deployment in resource-constrained AWSNs.

Motivation

In the booming field of intelligent agriculture, agricultural wireless sensor networks (AWSN) are increasingly receiving widespread attention as the core technology for capturing large-scale crop growth information in farmland. However, with the expansion of wireless sensor network scale and the complexity of application scenarios, traditional optimization algorithms have encountered multiple challenges in coverage deployment, such as computational complexity, model accuracy, and adaptability. Although traditional methods such as Boolean perception models, probabilistic perception models, and grid partitioning methods have demonstrated their unique value in small-scale network analysis and specific scenario applications, their limitations gradually become apparent when facing large-scale and complex scenarios.

Heuristic methods, such as genetic algorithms, particle swarm optimization algorithms, and ant colony algorithms, explore the solution space by simulating natural processes or biological behavior, bringing more flexible and efficient solutions to wireless sensor network coverage problems. These methods can approach the global optimal solution and demonstrate strong adaptability and robustness when dealing with wireless sensor network coverage problems in large-scale and complex scenarios. However, heuristic methods are often limited by specific parameter settings and initial conditions, and their calculation process is relatively complex, which, to some extent, restricts their widespread application in practical settings.

Based on the above background, this paper aims to develop a new optimization algorithm to improve the coverage performance of agricultural wireless sensor networks and promote the development of intelligent agriculture. Therefore, we are committed to proposing a Multi-Strategy Cormorant Optimization Algorithm (MSPOA) aimed at overcoming the limitations of traditional and heuristic methods, and enhancing the adaptability and stability of the algorithm in agricultural scenarios. We believe that by integrating a good point set strategy, a 3D spiral Lévy flight strategy, and an adaptive T-distribution mutation strategy, MSPOA can not only significantly improve network coverage performance, but also better cope with uncertainty and dynamic changes in agricultural environments, providing an innovative approach and method for optimizing wireless sensor networks in intelligent agriculture. In smart agriculture, where precision farming, resource optimization, and disease prevention are paramount, the coverage performance of AWSNs directly impacts crop yields, resource utilization, and environmental sustainability. The proposed methods, by addressing the gaps in existing approaches, have the potential to revolutionize AWSN deployment in smart agricultural systems, enabling farmers to make more informed decisions, optimize resource allocation, and ultimately enhance food security and environmental resilience.

Agriculture wireless sensor network coverage model

The agricultural Wireless Sensor Network (AWSN) coverage model constitutes a comprehensive theoretical framework designed to describe and analyze the spatial distribution of nodes within sensor networks, with a specific focus on their perception capabilities of the monitoring area in the context of AWSN and smart agriculture. Its primary objective is to ensure that the network can effectively monitor the agricultural environment or specific regions where it is deployed. This model takes into account a wide array of factors that can influence network performance in smart agricultural settings, including the strategic placement of sensor nodes, their sensing ranges, and potential obstacles that might hinder their communication or sensing abilities, such as varying crop growth patterns, weather conditions, and terrain features.

This section provides a detailed elaboration on the fundamental assumptions underlying the WSN coverage model, particularly emphasizing its applicability in AWSN and smart agriculture. It delves into the mathematical models used to represent coverage, illustrating how these models can be employed to calculate crucial metrics like coverage area, sensing overlap, and detection probabilities within agricultural contexts. By doing so, it offers insights into how the model can be optimized to enhance monitoring efficiency and accuracy in smart agricultural applications. Furthermore, this section discusses network connectivity in the context of AWSN and smart agriculture, examining how sensor nodes communicate with each other and with the base station. It explores the impact of factors like node density, transmission range, and interference on the overall network performance in smart agricultural settings. This analysis is crucial for understanding how to design and deploy WSNs that can effectively navigate the unique challenges of agricultural environments, such as varying signal strength due to crop growth and changing weather conditions.

To further illustrate the concepts discussed in this section, Fig. 2 is provided, showcasing the coverage structure of the wireless sensor network tailored for smart agriculture. This figure depicts the spatial arrangement of sensor nodes, their sensing ranges, and the resulting coverage areas specifically designed for monitoring agricultural environments. It offers a visual representation of the network’s monitoring capabilities and the relationships between individual nodes and the overall network architecture in the context of AWSN and smart agriculture, highlighting how the model can be applied to optimize monitoring and data collection in agricultural settings.

Fig. 2

Agriculture wireless sensor network coverage structure.

Model assumptions

The assumptions of the WSN coverage model serve as the theoretical foundation for constructing and analyzing network performance in the context of AWSN and smart agriculture. The following is a compilation of model assumptions tailored to WSNs deployed in farmland for smart agricultural applications:

(1) Isomorphism Assumption: In the realm of smart agriculture, all sensor nodes are presumed to be identical in both hardware and software configurations. This homogeneity ensures that each node possesses the same sensing radius and time synchronization capability, thereby guaranteeing consistency and uniformity in network monitoring across the agricultural landscape.

(2) Scale Assumption: Given the vastness of typical farmland monitoring areas in smart agriculture, the sensing radius and communication radius of individual sensor nodes are assumed to be much smaller in comparison. This assumption simplifies the coverage and connectivity analysis of the network, allowing for more efficient modeling and optimization of the sensor network’s deployment and performance within the agricultural environment.

(3) Initial State Assumption: At the onset of network deployment in a smart agricultural setting, all sensor nodes are presumed to have the same energy level and sensing radius. Additionally, their position information is determined by the physical characteristics of the nodes and the specific requirements of the agricultural monitoring task. This assumption ensures the uniformity and predictability of network deployment, facilitating effective planning and management of the sensor network in the context of AWSN and smart agriculture.

(4) Unique Identification and Distribution Assumption: Each sensor node in the network is assigned a unique ID, enabling individual identification and management within the agricultural monitoring system. These nodes are unevenly distributed across the monitoring area, reflecting the geographical and environmental conditions prevalent during actual deployment in farmland. This assumption acknowledges the heterogeneity of agricultural environments and allows for the design of sensor networks that can adapt to varying conditions, ultimately enhancing the overall effectiveness and efficiency of smart agricultural practices.

Coverage model

Figure 3 shows the area coverage model of AWSN (Advanced Wireless Sensor Network) in smart agriculture. This model depicts the distribution, coverage radius, and connectivity between sensor nodes within the monitoring area. The node distribution takes into account the complexity of the agricultural environment, such as crop layout and soil conditions. Each node has a limited coverage radius, and the design should ensure coverage of the target area and optimize resource utilization. In order to improve monitoring reliability, there may be a certain degree of overlap in the coverage areas of nodes. At the same time, the model also demonstrates the wireless communication links between nodes, emphasizing the importance of connectivity in forming a stable network. Finally, energy efficiency management strategies are also a key aspect considered in this model, aimed at extending network lifespan and improving overall performance.

Fig. 3

Area coverage model.

In the monitoring scenario of smart agriculture, it is assumed that N isomorphic sensor nodes are randomly deployed within the monitoring area of the farmland. Each sensor node has the same sensing radius Rs​ and communication radius Rc​. To ensure the connectivity of the entire network, the communication radius of a node is set to be no less than twice its sensing radius. Therefore, the set formed by these N nodes can be represented as U={u_i=(x_i, y_i): i = 1,2,3…,N }. The set of nodes in the farmland monitoring area V={v_j=(x_j, y_j): j = 1,2,3,…,m*n } The (x_i, y_i) and (x_j, y_j) distributions represent the two-dimensional spatial coordinates corresponding to v_i and u_j. In this paper, the binary sensing model is adopted as the node sensing model, and as long as the farmland monitoring area is within the sensing radius of the node, it is regarded as covering the node. Then the Euclidean distance between the sensor node and the node in the farmland monitoring area is shown in Eq. (1).

$$d\left( {{u_i},{v_j}} \right)=\sqrt {{{\left( {{x_i} - {x_j}} \right)}^2}+{{\left( {{y_i} - {y_j}} \right)}^2}}$$

(1)

The probability that the monitoring point v_j is sensed by the sensor node u_i is shown in Eq. (2).

$$p({u_i},{v_j})=\left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {1,}&{d({u_i},{v_j}) \leqslant {R_s}} \end{array}} \\ {\begin{array}{*{20}{c}} {0,}&{d({u_i},{v_j})>{R_s}} \end{array}} \end{array}} \right.$$

(2)

where, when P(u_i,v_j) is equal to 1, it means that the monitoring point v_j can be successfully monitored by the sensor u_i, and when P(u_i,v_j) is equal to 0, it means that the sensor u_j does not cover the monitoring point v_j.

If a specific point in a monitoring area can be detected by multiple sensor nodes at the same time, then the probability that the monitoring point is sensed by the set of nodes U is shown in Eq. (3).

$$P(U,{v_j})=1 - \prod\limits_{{i=1}}^{N} {[1 - p({u_i},{v_j})]}$$

(3)

The coverage of the node set U over the entire monitoring area is shown in Eq. (4).

$${R_{\operatorname{cov} }}=\frac{{\sum\limits_{{j=1}}^{{m \times n}} {p(U,{v_j})} }}{{m \times n}}$$

(4)

The node deployment optimization model in AWSNs aims to maximize the coverage of the monitored area while minimizing the number of sensor nodes. This model can be reduced to an engineering optimization problem with constraints in the form shown below:

$$F=\hbox{max} ({R_c})$$

(5)

$$s.t=\left\{ {\begin{array}{*{20}{c}} {{b_1}=\sum\limits_{{j=1}}^{{m \times n}} {p(U,{v_j}) \geqslant 0} } \\ {{b_2}=\sum\limits_{{j=1}}^{{m \times n}} {p(U,{v_j}) - m \times n \geqslant 0} } \\ {{b_3}=d({u_i},{v_j}) - R \geqslant 0} \end{array}} \right.$$

(6)

In this study, a multi-strategy pelican optimization algorithm is employed based on the established coverage model of AWSN in the context of smart agriculture. Within this model, Rc​, representing the communication radius of the sensor nodes, serves as a fitness function for the algorithm. The primary objective of this study is to maximize Rc​ in order to enhance node coverage within the model. This is achieved by combining various improvement strategies, ultimately aiming to optimize the overall performance of the WSN in terms of coverage and connectivity within the smart agriculture environment. The key purpose of the regional coverage model in smart agriculture is to effectively integrate these strategies to improve the monitoring and data collection capabilities of the sensor network.

Network connectivity

In AWSN for smart agriculture, network connectivity is the most fundamental requirement. While striving for maximum coverage, it is imperative to ensure that the entire network remains connected. Connectivity implies that every sensor node in the network can communicate and exchange information with each other, forming a seamless communication path. Interruptions between nodes can disrupt information transmission, leading to network failures or instability. Therefore, in the process of optimizing coverage, measures must be taken to maintain network connectivity for reliable data transmission and overall performance improvement in smart agriculture.

For ease of calculation, this paper assumes that Rc = 2R. By establishing an adjacency matrix V′of a directed graph, which records the connectivity information between any two nodes, it becomes feasible to determine whether the nodes are connected to each other. According to Eq. (7), a specific method is applied to ascertain node connectivity. If there is redundancy between two nodes, the corresponding element in the adjacency matrix is denoted as 1; conversely, if there is no connectivity, the element is 0. This approach enables the determination of node locations within the entire network, ensuring effective communication and information exchange in the smart agriculture environment.

$$V'_{{i,j}} = \left\{ {\begin{array}{*{20}c} {1{\text{ }}If{\text{ }}d\left( {u_{i} ,v_{j} } \right) \le R_{i}^{c} } \\ {0{\text{ }}otherwise} \\ \end{array} } \right.$$

(7)

where V’_{i, j}=1 means that the i_th node can pass information to the j_th node, and when its value is 0, it means that it is not connected. Finally, Eq. (8) is used to determine whether the whole network is connected or not.

$$S' = V' + V'^{2} + V'^{3} + \cdots + V'^{{N - 1}}$$

(8)

where, if S’ is 0 it means that the whole network is not connected and vice versa; N denotes the number of sensor nodes.

Solving AWSN coverage problem using MSPOA

Using the Multi-Strategy Pelican Optimization Algorithm (MSPOA) to solve the AWSN coverage problem in smart agriculture involves simulating the process of pelican predation in nature to search for an optimal solution. The core idea of MSPOA is to mimic the feeding behavior of pelicans and seek the optimal solution through collaborative and strategic actions. In the context of WSN coverage optimization for smart agriculture, particularly for large-scale field crops, the deployment scheme that maximizes the network coverage $\:{R}_{c}$ corresponds to the safest position in the population, akin to how pelicans secure their food sources.

The mathematical model of MSPOA represents the population of the algorithm using a population matrix $\:X$. Each row in this matrix represents a candidate solution for deploying sensor nodes in the agricultural field. Each column corresponds to a dimension of the problem variable, specifically the proposed values for sensor node positions or configurations. This approach is highly relevant to smart agriculture and large-scale field crops, as it allows for the optimization of sensor node deployment to ensure maximum coverage and connectivity, ultimately enhancing the monitoring and management of crops in the field.

$$X=\left[ {\begin{array}{*{20}{c}} {{X_1}} \\ {{X_2}} \\ \vdots \\ {{X_i}} \\ \vdots \\ {{X_M}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {{x_{1,1}}}&{{x_{1,2}}}& \cdots &{{x_{1,j}}}& \cdots &{{x_{1,d}}} \\ {{x_{2,1}}}&{{x_{2,2}}}& \cdots &{{x_{2,j}}}& \cdots &{{x_{2,d}}} \\ \vdots & \vdots &{}& \vdots &{}& \vdots \\ {{x_{i,1}}}&{{x_{i,2}}}& \cdots &{{x_{i,j}}}& \cdots &{{x_{i,d}}} \\ \vdots & \vdots &{}& \vdots &{}& \vdots \\ {{x_{M,1}}}&{{x_{M,2}}}& \cdots &{{x_{M,j}}}& \cdots &{{x_{M,d}}} \end{array}} \right]$$

(9)

where $\:X$ denotes the population matrix of pelicans, $\:{X}_{i}$ denotes the ith pelican, $\:M$ denotes the population size, and $\:d$ denotes the problem dimension.

In the MSPOA algorithm, each population member represents a pelican, i.e., a potential solution for a given problem. Therefore, this paper can utilize each candidate solution to evaluate the objective function of the problem. In this paper, a vector is used to determine the value of the objective function, which is called the objective function vector. Its expression is shown below:

$$F=\left[ {\begin{array}{*{20}{c}} {{F_1}} \\ {{F_2}} \\ \vdots \\ {{F_i}} \\ \vdots \\ {{F_M}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {F({X_1})} \\ {F({X_2})} \\ \vdots \\ {F({X_i})} \\ \vdots \\ {F({X_M})} \end{array}} \right]$$

(10)

where $\:F$ is the fitness function vector and Fi denotes the fitness function value of the ith candidate solution.

Good point set initialization populations

The theory of good point set was proposed by Mr. Hua Luogeng, and it has been well applied in swarm intelligence optimization. The good point set method is superior to other methods, mainly because a more uniform original population can be obtained by projecting the good points into the solution space. Compared with the traditional stochastic method, its initial allocation is more balanced and has better traversal performance, which also helps the optimization result more.

$$p_{n} (k) = \{ (\{ r_{1}^{{(n)}} \times k\} ,\{ r_{2}^{{(n)}} \times k\} ,...,\{ r_{i}^{{(n)}} \times k\} ){\text{ }},1 \le k \le n\}$$

(11)

Its deviation satisfies Eq. (12).

$$\varphi (n)={C^*}(r,\varepsilon ){n^{ - 1+\varepsilon }}$$

(12)

where r is the good point and C*(r,ε) denotes the constant associated only with r, ε (ε > 0).

$${r_i}=2\cos \frac{{2\pi i}}{p},1 \leqslant i \leqslant D$$

(13)

where p is the smallest prime number satisfying (p-3)/2 ≥ D, and D represents the dimension of the problem solving. Therefore, based on the theory of optimal point sets, the new initialization strategy is:

$${x_i}(k)=(u{b_i} - l{b_i})\{ {P_n}(k)\} +l{b_i}$$

(14)

where ub is the upper bound and lb is the lower bound.

3D spiral levy flight strategy

The proposed MSPOA simulates the behavior and strategy of pelicans when attacking and hunting prey, in order to update candidate solutions. Therefore, this hunting strategy is biomimetic simulated in two stages: moving towards prey (exploration phase) and flapping on the water surface (mining phase).

Exploration phase

In this paper, a three-dimensional spiral Levy flight strategy is designed to regulate the position update of individual pelicans during the exploration phase. During the exploration phase, individual pelicans move toward prey locations according to a spiral movement pattern and change the flight step length through the Levy flight strategy. This spiral movement method will constantly change the rotation angle to expand the neighborhood of the current local solution, thus increasing the probability of the pelican individual to discover other locations in early iterations, thus avoiding the problem of the MSPOA algorithm falling into a local optimal solution. Its updated position formula is shown below.

$$X_{{i,j}}^{{{P_1}}}=\left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{x_{i,j}}+Levy(pj - O{x_{i,j}}),}&{{F_p}<{F_i}} \end{array}} \\ {\begin{array}{*{20}{c}} {{x_{i,j}}+Levy(pj - O{p_j}),}&{{F_p} \geqslant {F_i}} \end{array}} \end{array}} \right.$$

(15)

where O denotes the three-dimensional helical position update coefficients for the x-, y-, and z-axes, where x = ρcosθ, y = ρsinθ, and z = ρθ denote the three-dimensional components of the coordinates (x, y,z) under the helical motion, respectively. Ρ = seθt, s, and t denote the diameters of logarithmic helixes, which are usually taken to be s = 0.05 and t = 0.05, respectively, and θ denotes the random number between (0,2π). The formula for the Levy flight is shown below.

$$Levy\sim \varepsilon {\text{ }} = {\text{ }}t^{{ - \lambda }} {\text{ }}\left( {1 < \lambda < 3} \right)$$

(16)

$$Levy\left( \lambda \right)\sim \frac{{\varepsilon \times \varphi }}{{{{\left| \eta \right|}^{\frac{1}{\lambda }}}}}$$

(17)

$$\varphi = \left[ {\frac{{\gamma \left( \lambda \right) \times \sin \left( {\pi \times \left( {\lambda - 1} \right)/2} \right)}}{{\gamma \left( {\lambda /2} \right) \times \left( {\lambda - 1} \right) \times 2^{{\left( {\lambda - 2} \right)/2}} }}} \right]^{{1/\left( {\lambda - 1} \right)}}$$

(18)

where α denotes the step factor, which typically takes values in the range [0, 1], λ denotes the stability parameter, which typically takes a value of 1.5, η and ε conform to the standard normal distribution, and γ is the standard gamma function.

In the MSPOA algorithm, the new position of the pelican is accepted if the objective function value is improved at the current position. This type of update is known as effective update. The algorithm does not move to a non-optimal region during this type of update. This process can be described by the Eq. (19).

$${X_i}=\left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {X_{i}^{{{P_1}}},}&{F_{i}^{{P1}}<{F_i}} \end{array}} \\ {\begin{array}{*{20}{c}} {{X_i},}&{else} \end{array}} \end{array}} \right.$$

(19)

Mining phase

The mining phase of MSPOA is the process by which an individual pelican reaches the surface of the water, spreads its wings, and captures its prey, which is modeled in a mathematical equation as shown below.

$$x_{{i,j}}^{{{P_2}}}={x_{i,j}}+R \times (1 - \frac{g}{G}) \times (2 \times rand - 1) \times {x_{i,j}}$$

(20)

where $x_{{i,j}}^{{{P_2}}}$ denotes the position of the $\:{i}_{th}$ pelican in the $\:{j}_{th}$ dimension after the second stage update, R is a constant equal to 0.2, and $1 - g/G$ denotes the neighborhood radius, where g denotes the number of current iterations, G denotes the maximum number of iterations, and rand denotes a random number in the range of [0,1].

In the initial iteration phase, the coefficient R(1-g/G) takes on a large value, resulting in a wide range of exploration around each member. As the iteration of the algorithm proceeds, this coefficient gradually decreases, resulting in a decrease in the radius of the neighborhood of each member. Such a mechanism of change allows us to explore the area around each member of the population in a finer way, and thus the concept-based search can gradually converge to a solution that is closer to the global optimum.

$${X_i}=\left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {X_{i}^{{{P_2}}}}&{,F_{i}^{{{P_2}}}<{F_i}} \end{array}} \\ {\begin{array}{*{20}{c}} {{X_i}}&{,F_{i}^{{{P_2}}} \geqslant {F_i}} \end{array}} \end{array}} \right.$$

(21)

where ${X_i}$ denotes the new individual position of the $\:{i}_{th}$ pelican after the second stage update, and $F_{i}^{{{P_2}}}$ denotes the fitness value corresponding to the new position of the $\:{i}_{th}$ pelican after the second stage update.

Adaptive T-distribution variation strategy

To further improve the optimality seeking performance of the pelican algorithm and effectively avoid the algorithm from falling into the local optimal solution prematurely at the late iteration stage, an adaptive t-distribution variation strategy is introduced in this study. The strategy adjusts the positions of the pelican individuals according to the current algorithm execution state and problem characteristics. Specifically, the strategy dynamically adjusts the mutation rate based on the current position of the individual and the solution space characteristics of the problem to ensure that the algorithm strikes a balance between exploration and exploitation. The introduction of this adaptive t-distribution mutation strategy helps to enhance the global search capability of the pelican algorithm and effectively promotes the algorithm to explore more extensively in the solution space, thus improving the quality and stability of the final solution.

$$X_{{i,j}}^{g}={X_{best}}+{X_{best}} \times \alpha \times t(iter)$$

(22)

where $X_{{i,j}}^{g}$ denotes the mutated pelican individual, ${X_{best}}$ denotes the current optimal position of the pelican individual, $\alpha$ denotes the control variance factor, and $t(iter)$ denotes the t-distribution of degrees of freedom parameterized by the number of algorithm iterations.

Proposed pseudo code and flowchart for MSPOA

After updating all members of the pelican population during the exploration and development phase based on the new state of the population and the value of the objective function, the best candidate solution to date is updated. Subsequently, the algorithm enters the next iteration and repeats the various steps of MSPOA based on Eqs. (15)–(22) until the entire process is completed. In the end, the best candidate solution obtained during the algorithm iteration process is considered as the quasi-optimal solution of the given problem. The position update of the population relies on the above three strategies to find the optimal solution. The flowchart of MSPOA is shown in Fig. 4. To provide a clearer description of the algorithm details, we present the pseudocode of MSPOA in Algorithm 1.

Fig. 4

Flowchart of MSPOA.

MSPOA can empower agriculture and field crops in multiple ways. By leveraging its optimization capabilities, MSPOA can help in optimizing crop planting patterns, improving irrigation systems, enhancing fertilization strategies, and optimizing harvest scheduling. These optimizations can lead to increased crop yields, reduced resource consumption, and improved overall efficiency in agricultural practices. Furthermore, MSPOA’s ability to handle complex optimization problems makes it a valuable tool for addressing challenges specific to large-scale field crop management, such as disease control, pest management, and precision agriculture. Ultimately, the integration of MSPOA into agricultural practices can contribute to sustainable and efficient farming, enhancing productivity and environmental sustainability.

Algorithm 1

Proposed pseudo code for MSPOA

The computational complexity of the proposed MSPOA

In this section, we conducted a detailed analysis of the computational complexity of the proposed MSPOA. The computational complexity of this algorithm is mainly based on the following four core steps: algorithm initialization, fitness function evaluation, prey generation, and solution update. Firstly, the computational complexity of the algorithm initialization process is O (M), where M represents the number of pelicans in the population. During each iteration, each pelican goes through two stages and evaluates the objective function once in each stage. Therefore, the computational complexity of fitness function evaluation is O (2 * G * M), where G represents the number of iterations. Next, considering that prey is generated and evaluated in each iteration, the computational complexity of prey generation can be expressed as O (G) + O (G * d), where d represents the dimension of the solution space. Finally, in each iteration, a two-stage update operation is required for M pelicans with d dimensions. Therefore, the computational complexity of the solution update is O (2 * G * M * d). In summary, the total computational complexity of the proposed MSPOA algorithm is O(M + G (1 + d)*(1 + 2*M)), which takes into account the computational costs of algorithm initialization, fitness evaluation, prey generation, and solution update.

Simulation experiment and result analysis

Experimental parameterization

In the context of Agricultural Wireless Sensor Networks (AWSNs), the arrangement of sensor nodes is pivotal to network performance, with a primary concern being the dual objective of maximizing target area coverage and minimizing energy consumption. To tackle this challenge, this paper introduces a novel MSPOA approach tailored for coverage optimization in wireless sensor networks pertinent to agricultural applications. This research undertaking encompasses a series of simulation experiments designed to assess the efficacy of MSPOA in addressing WSN coverage optimization problems specific to smart agriculture and large-scale field crop monitoring. The ensuing sections elaborate on the parameter configurations and detailed experimental protocols employed in these evaluations.

In this paper, all the simulation experiments are conducted in a unified environment to ensure the comparability and robustness of the experimental results. Specifically, a square area of 100 m × 100 m, 150 m × 150 m and 200 m × 200 m is selected as the experimental scenario in this paper, and the sensor nodes are randomly distributed in the whole area. In order to maintain the consistency of the experiments, the same scene layout and parameter settings are used in each experiment, and each set of parameter settings is run for several repetitions to obtain the statistical stability of the results. Therefore, the population size was set to 40, the problem latitude was set to 40, and the number of iterations was set to 1000. all algorithmic experiments were run independently for 100 times in the same environment, and the mean and standard deviation were used to evaluate the algorithm’s performance. Such an experimental design helps to minimize the influence of external factors on the experimental results and ensures the credibility and reproducibility of the results. Table 2 represents the parameter settings of the running environment.

Table 2 Operating environment parameter settings.

Coverage optimization analysis for AWSNs

Experimental scenario one

Figures 5a-e present the coverage simulations of five algorithms on a WSN characterized by a sensor radius of 10 m, an area of 100 m × 100 m, and 30 sensors, respectively. To thoroughly evaluate the performance of the MSPOA algorithm, this paper opts for comparative experiments involving four other algorithms: LCSO, IABC, APSO, and CAFA. These comparative algorithms were chosen due to their widespread application and proven performance in the optimization field, enabling a deep analysis of each algorithm’s optimization potential. As depicted in Fig. 5a, MSPOA exhibits the best coverage, with the largest and most uniform coverage area, fully harnessing the coverage capability of each sensor. Conversely, Fig. 5d reveals that the coverage areas of multiple sensors in APSO overlap, resulting in the poorest coverage effect. This is attributed to APSO’s premature convergence, which hinders the discovery of an approximate optimal solution in the later stages. By comparing MSPOA with these algorithms, we can gain a better understanding of its performance in different optimization tasks, thereby comprehensively demonstrating its performance advantages.

Fig. 5

(a) - (e). Representing WSN coverage simulations with a sensor radius of 15 m, an area of 100 m × 100 m, and 30 sensors respectively.

Figures 5c,d present the coverage simulation plots of the CAFA and APSO algorithms, respectively. From these figures, it is evident that the uncovered areas of CAFA and APSO are substantial, with APSO exhibiting a particularly high number of duplicated coverage areas. This duplication results in inadequate coverage of other areas, consequently failing to detect targets within those regions. Figures 5b,e display the coverage simulation graphs of the IABC and LCSO algorithms, respectively. These figures indicate that, while IABC and LCSO also exhibit some duplicate coverage, their uncovered areas are relatively smaller. A comprehensive analysis of Figs. 5a-e reveals that the MSPOA algorithm boasts the widest coverage area and exhibits minimal uncovered areas in the center, while also maintaining the least amount of duplicate coverage. This demonstrates that the MSPOA algorithm excels in enhancing the coverage area and minimizing duplicate coverage, ultimately leading to significant improvements in coverage efficiency and detection performance.

Fig 6 illustrates the coverage curves of the five algorithms: MSPOA, IABC, APSO, CAFA, and LCSO. It is observable that MSPOA converges approximately at the 300th generation, with its highest coverage rate reaching 87.2%, demonstrating rapid convergence and exceptional coverage performance. Conversely, the IABC and LCSO algorithms converge only near the 900th generation, exhibiting a slow and unstable convergence process. Their final coverage rates are 80.7% and 83%, respectively, which are lower than MSPOA’s, indicating deficiencies in optimization efficiency and coverage effect. Further analysis of the APSO algorithm reveals that it falls into local optimality at the 200th generation and fails to continue optimization, with a final coverage rate of only 62%, significantly lower than MSPOA’s. This suggests that the APSO algorithm has greater limitations in addressing complex coverage problems. The CAFA algorithm, despite its faster convergence speed, also falls into local optimality at the 200th generation, with a coverage rate of 76.5%, failing to achieve further improvement.

Fig. 6

Comparison of coverage rate of 5 algorithms.

Table 3 provides a detailed list of 10 experimental coverage data for five optimization algorithms (MSPOA, IABC, APSO, CAFA, and LCSO) under the same conditions, as well as the optimal, worst, average, and standard deviation values calculated based on these data. Through in-depth analysis of the data in Table 1, we can clearly observe the superiority demonstrated by MSPOA. Specifically, the average coverage rate of MSPOA is at a relatively high level among the five algorithms, and its 10 experimental results closely revolve around the mean, indicating its stable and reliable performance. Meanwhile, the standard deviation of MSPOA is much smaller than the other four algorithms, indicating that its experimental results have minimal fluctuations and high stability. In addition, the difference between the optimal and worst values of MSPOA is much smaller than other algorithms, further demonstrating its consistency and stability in performance. In contrast, the other four algorithms have their own shortcomings in the data performance in Table 2. Although IABC algorithm has achieved high coverage in some experiments, its average value and stability are not as good as MSPOA, and its standard deviation is relatively large, indicating that its performance fluctuates greatly. The average coverage of the APSO algorithm is at a moderate level, but the difference between its optimal and worst values is significant, indicating that its performance fluctuates greatly in different experiments. The average coverage of CAFA algorithm is low and the standard deviation is large, indicating that its performance and stability need to be improved. Although the LCSO algorithm has achieved low coverage in some experiments, its average and standard deviation are at a moderate level, indicating that its performance is relatively stable, but further improvement in coverage is still needed.

Table 3 The coverage data and statistical indicators of 10 experiments of five optimization algorithms are presented in scenario 1.

In the above analysis, MSPOA shows significant advantages in coverage rate and convergence speed, especially in avoiding falling into local optimality. In contrast, other algorithms are deficient in coverage rate and convergence stability, which further proves the superiority of MSPOA in solving the coverage problem.

Experimental scenario two

Figure 7a to e present a detailed depiction of the performance of various algorithms in wireless sensor network (WSN) coverage simulation under specific experimental conditions, characterized by a sensor sensing radius of 15 m, a monitoring area of 150 m by 150 m, and a deployment of 30 sensors. A thorough analysis of these graphs reveals that, despite minor uncovered gaps at certain edges or specific regions, the MSPOA demonstrates significant advantages in overall coverage compared to the other four comparative algorithms. This supremacy is manifested not only in terms of higher coverage but also in MSPOA’s exceptional ability to minimize duplicate coverage and optimize resource utilization, highlighting its effectiveness in achieving efficient network coverage. In the context of empowered agriculture, smart agriculture, and large-field crop monitoring, the implementation of MSPOA in WSNs can lead to substantial improvements in monitoring capabilities, enabling precise and real-time data collection for enhanced decision-making and resource allocation in agricultural practices.

Fig. 7

(a) - (e). Representing WSN coverage simulations with a sensor radius of 15 m, an area of 150 m × 150 m, and 30 sensors respectively.

Furthermore, the data analysis presented in Fig. 8 provides compelling evidence for the advantages of MSPOA. This figure unveils that MSPOA exhibits fast and stable convergence characteristics, indicating that the algorithm can swiftly approximate the optimal solution within fewer iterations and demonstrate robust stability during the optimization process, making it less prone to significant fluctuations caused by external factors. In contrast, the CAFA, APSO, and LCSO algorithms exhibit slower convergence speeds and are susceptible to being trapped in local optima during the optimization process, resulting in unsatisfactory coverage outcomes. Specifically, these algorithms have struggled to effectively mitigate redundant coverage when addressing coverage issues, leading to the wastage of sensor resources. Additionally, their instability during the optimization process severely hampers their ability to further enhance coverage. In summary, through an in-depth analysis of Figs. 5 and 8, it is evident that MSPOA has demonstrated substantial advantages in resolving the coverage problem of wireless sensor networks. It not only outperforms other algorithms in the two core evaluation metrics of coverage and convergence speed but also exhibits higher efficiency in terms of algorithm stability and avoiding local optima. These remarkable features position MSPOA as a highly promising and viable choice for tackling AWSN coverage optimization problems.

Fig. 8

Comparison of coverage rate of 5 algorithms.

Through in-depth analysis of Table 4, we can clearly observe the performance and stability of each algorithm. Firstly, from an average perspective, the coverage rate of MSPOA is 0.8348, which is at a relatively high level among the five algorithms, demonstrating the excellent performance of MSPOA in optimization problems. More importantly, the 10 experimental results of MSPOA closely revolve around this mean, indicating its stable and reliable performance. In contrast, the average coverage rates of IABC, APSO, CAFA, and LCSO optimization algorithms are 0.7987, 0.788, 0.8043, and 0.7562, respectively, all lower than MSPOA, indicating their relatively weak performance in optimization problems. Secondly, observing the standard deviation as an indicator, the standard deviation of MSPOA is only 0.0003, which is much smaller than the other four algorithms. This means that the experimental results of MSPOA have minimal fluctuations and extremely high stability. In practical applications, this stability is crucial for ensuring the continuity and predictability of algorithm performance. However, the standard deviations of IABC, APSO, CAFA, and LCSO are 0.0064, 0.0045, 0.0047, and 0.0045, respectively, which are relatively large, indicating that their experimental results fluctuate greatly and their stability is relatively poor. In practical applications, such large fluctuations may have adverse effects on the performance of the algorithm. Furthermore, from the comparison of the optimal and worst values, we can also see the performance fluctuations of each algorithm. The optimal value of MSPOA is 0.835, only slightly higher than its mean, while the worst value is 0.834, also only slightly lower than its mean, demonstrating its consistency and stability in performance. In contrast, the difference between the optimal and worst values of IABC, APSO, CAFA, and LCSO is significant, further demonstrating their instability in performance. Taking IABC as an example, its optimal value is 0.810, while its worst value is 0.790, indicating a relatively large difference. Similarly, there is a significant gap between the optimal and worst values of APSO, CAFA, and LCSO.

Table 4 The coverage data and statistical indicators of 10 experiments of five optimization algorithms are presented in scenario 2.

Experimental scenario three

From the results depicted in Figs. 9a to e, it is evident that the coverage area of the MSPOA algorithm visually surpasses that of the other comparative algorithms, while its redundant coverage area remains relatively smaller. This observation strongly indicates that the MSPOA algorithm can effectively enhance coverage efficiency while mitigating redundant coverage in the context of smart agriculture and smart farmland. Specifically, whether in dense, sparse, or mixed deployment scenarios of wireless sensor networks for precision agriculture, the MSPOA algorithm has demonstrated superior performance compared to other algorithms, highlighting its significant advantage in optimizing coverage areas for smart farming operations. Through the careful implementation of strategies such as the good point set strategy, 3D spiral Levy flight strategy, and adaptive distributed variation strategy, the MSPOA algorithm not only successfully reduces redundant coverage but also significantly improves resource utilization efficiency, thereby enhancing the overall performance and stability of the algorithm in empowering smart agricultural production. The effective combination of these strategies enables the MSPOA algorithm to converge faster and exhibit a stronger ability to escape from local optima during the process of seeking optimal solutions for smart farmland management.

Fig. 9

(a) - (e). Representing WSN coverage simulations with a sensor radius of 18 m, an area of 200 m × 200 m, and 50 sensors.

Furthermore, the coverage curve illustrated in Fig. 10 intuitively reveals that the coverage of the IABC algorithm closely follows the MSPOA algorithm, while the coverage of the CAFA algorithm is lower than that of the IABC. This ranking further confirms the exceptional performance of the MSPOA algorithm in terms of coverage efficiency for smart agriculture applications. However, it is noteworthy that the LCSO and APSO algorithms perform poorly in terms of coverage, exhibiting uneven coverage distribution and a substantial number of uncovered blank areas in the agricultural field. This unsatisfactory coverage effect is likely attributable to the failure of these two algorithms to effectively avoid redundant coverage when addressing coverage problems in smart farmland, as well as deficiencies in resource allocation and utilization for precision agriculture.

Fig. 10

Comparison of coverage rate of 5 algorithms.

In summary, through an in-depth analysis of the empirical results presented in Figs. 9 and 10, it is clearly evident that the MSPOA algorithm has demonstrated significant advantages in solving the coverage problem of wireless sensor networks for smart agriculture. It not only excels in coverage efficiency and reducing redundant coverage but also improves resource utilization efficiency and enhances the overall performance and stability of the algorithm through its unique strategy combination. These remarkable features position the MSPOA algorithm as a highly promising and viable solution for optimizing wireless sensor network coverage in smart farmland and empowering precision agriculture operations.

From Table 5, it can be seen that MSPOA exhibits significant advantages in coverage, stability, and consistency, with an average coverage of 0.9656 and a standard deviation of only 0.0004. The difference between the optimal and worst values is minimal, fully demonstrating its outstanding performance and high stability in optimization problems. In contrast, the average coverage rates of IABC, APSO, CAFA, and LCSO were 0.9122, 0.7994, 0.8675, and 0.8003, respectively, all significantly lower than MSPOA, indicating their relative inadequacy in optimization capabilities. Meanwhile, the standard deviations of these four algorithms are relatively large, namely 0.0058, 0.0050, 0.0050, and 0.0058, indicating that their experimental results fluctuate greatly and their stability needs to be improved. In addition, by comparing the optimal and worst values, it can be found that the performance fluctuation of MSPOA is minimal, while IABC, APSO, CAFA, and LCSO exhibit certain performance fluctuations and instability.

Table 5 The coverage data and statistical indicators of 10 experiments of five optimization algorithms are presented in scenario 3.

Algorithm runtime analysis

In Fig. 11, it is evident that the running time of MSPOA is consistently the shortest across all scenarios. This observation, validated under various experimental conditions and parameter configurations, underscores the notable advantage of MSPOA in terms of computational efficiency. Specifically, when subjected to identical experimental settings, MSPOA exhibits a significantly lower running time compared to other benchmark algorithms, including IABC, APSO, CAFA, and LCSO. This finding indicates that MSPOA not only achieves rapid convergence during the optimization process but also demands considerably fewer computational resources and time expenditure when addressing intricate coverage challenges. Such exceptional performance renders MSPOA more practical and reliable for real-world applications, particularly in meeting the heightened real-time requirements of wireless sensor network coverage optimization tasks within smart agriculture and precision farmland management. By diminishing runtime, MSPOA not only elevates the overall efficiency of the algorithm but also offers robust support for its deployment in large-scale agricultural networks and dynamic environmental settings, thereby empowering farming operations with advanced algorithmic capabilities.

Fig. 11

It represents the time required for five algorithms to perform 1000 iterations under different area sizes and different sensor radii.

Conclusion

This study proposes a novel agricultural wireless sensor network (AWSN) coverage optimization method based on the Multi-Strategy Pelican Optimization Algorithm (MSPOA). By integrating a good point set strategy, a 3D spiral Lévy flight strategy, and an adaptive T-distribution variation strategy, MSPOA significantly enhances coverage rate, convergence speed, and algorithmic stability. Simulation experiments demonstrate that MSPOA outperforms existing methods in terms of coverage optimization, effectively reducing duplicate coverage areas and improving resource utilization efficiency. The key advantages of MSPOA include: (1) High Coverage: The good point set strategy ensures superior sensor network coverage; (2) Rapid Convergence: The 3D spiral Lévy flight strategy accelerates convergence, saving computational time; (3) High Stability: The adaptive T-distribution variation strategy prevents local optima and ensures robust performance across diverse scenarios; and (4) Reduced Duplicate Coverage: The method minimizes overlapping coverage areas, enhancing resource efficiency.

In the context of smart agriculture, MSPOA offers significant potential to improve precision farming practices by optimizing sensor network deployment and resource allocation. This enables more accurate monitoring and management of agricultural environments, supporting data-driven decision-making and sustainable farming practices.

Discussion

Despite its strengths, MSPOA has certain limitations. In highly complex scenarios, some uncovered areas may persist, indicating a need for further refinement to address extreme conditions. Additionally, while the algorithm demonstrates strong overall performance, its high computational complexity may limit its applicability in large-scale networks and real-time applications. Future research should focus on: (1) enhancing MSPOA’s adaptability and robustness through advanced adaptive mechanisms and dynamic operators; (2) exploring its performance in diverse environmental conditions and larger agricultural networks; and (3) optimizing computational efficiency to broaden its practical applicability in smart agriculture and precision farming. These improvements will further solidify MSPOA’s role as a powerful tool for AWSN coverage optimization.