A novel swarm intelligence optimization method for efficient task allocation in industrial wireless sensor networks

Abstract

Industrial wireless sensor networks (IWSNs) play an increasingly important role in digital and intelligent industries; however, the communication, computing, and energy performance of their nodes are constrained by the demands and cost constraints of large-scale deployment. In this case, efficient task allocation plays a key role in improving the performance of the IWSNs. Task allocation in IWSNs is a nondeterministic polynomial (NP) hard problem whose complexity increases with an increase in node and task size. In this study, chaotic elite clone particle swarm optimization (CECPSO) was proposed. The algorithm first introduces the chaos theory to optimize the initial population. Subsequently, an elite cloning strategy was designed, which not only accelerated the exploration of the solution space and improved the accuracy of the solution but also avoided the problem of falling into the local optimal solution in the early stage through the dynamic adjustment strategy. In addition, the algorithm employs an exponential nonlinear decreasing inertia weight function that balances local and global search capabilities. By comparing the performance of CECPSO, Particle Swarm Optimization (PSO), Genetic Algorithm (GA), and Simulated Annealing (SA) in different experimental scenarios, we found that CECPSO is superior to PSO, GA, and SA in terms of the convergence rate and overall performance. Under the conditions of 40 sensors and 240 tasks, CECPSO’s performance improvement of CECPSO relative to PSO, GA, and SA reached 6.6%, 21.23%, and 17.01%, respectively. Experimental results show that the proposed algorithm can effectively improve the overall performance of IWSNs.

Introduction

With the rapid advancement of artificial intelligence (AI), industrial intelligence has emerged as a pivotal force in transforming traditional manufacturing systems. By integrating cutting-edge AI techniques with advanced information technologies, industrial intelligence has led to significant improvements in production efficiency, service quality, and management effectiveness¹. Among the core enablers of this transformation, Wireless Sensor Networks (WSNs) play a vital role in facilitating real-time monitoring, environmental awareness, and data-driven decision-making in smart manufacturing scenarios^2,3. An illustrative architecture of a typical WSNs deployment is shown in Fig. 1.

Fig. 1

Wireless sensor networks diagram.

In modern industrial applications, WSNs are widely deployed to collect and transmit diverse production and environmental data. These data not only support intelligent analysis and predictive maintenance but also enhance operational efficiency and reduce production costs^4,5. To fully exploit the potential of WSNs in industrial environments, Industrial Wireless Sensor Networks (IWSNs) must adopt effective task allocation strategies that optimize resource utilization and network longevity⁶.

Efficient task allocation in IWSNs ensures balanced workload distribution among sensor nodes, thereby improving data processing and transmission performance. Moreover, it significantly impacts the network’s lifetime, reliability, and scalability^7,8,9. By avoiding resource overuse on individual nodes, effective allocation strategies help reduce energy consumption, prevent communication bottlenecks, and enhance real-time responsiveness. In large-scale and long-term deployments, these strategies are essential for maintaining network robustness, reducing operational costs, and ensuring uninterrupted industrial monitoring and control.

However, the task allocation problem in Industrial Wireless Sensor Networks (IWSNs) is inherently NP-hard, and its complexity increases rapidly with the expansion of network size and the diversity of tasks¹⁰. This complexity arises from the exponential growth in task-node mapping combinations, along with the need to simultaneously optimize multiple conflicting objectives such as energy efficiency, latency reduction, and load balancing^11,12.

The problem is further exacerbated by the dynamic and often harsh conditions in industrial environments—such as signal interference, energy limitations, and hardware failures—which significantly constrain the computational and communication capabilities of sensor nodes^{13,14,15,16,17}. These challenges render traditional exact optimization approaches impractical in real-world applications due to their high computational demands.

Consequently, heuristic and metaheuristic algorithms—such as Genetic Algorithms (GA), Ant Colony Optimization (ACO), and Particle Swarm Optimization (PSO)—have been widely adopted to obtain near-optimal solutions within acceptable timeframes^18,19,20. However, these classical algorithms each have limitations in the context of IWSNs. GA often suffers from premature convergence due to the loss of population diversity in later generations, particularly in complex, multimodal search spaces. ACO, while effective in some routing and allocation tasks, can exhibit slow convergence and high sensitivity to parameter tuning, which limits its generalizability. PSO tends to rely excessively on the global best particle, which can cause the swarm to stagnate in local optima—especially when the initial population lacks diversity or when handling high-dimensional task allocation scenarios²¹. Therefore, developing more robust and adaptive algorithms that can simultaneously balance solution quality, convergence speed, and adaptability under these constraints remains a pressing challenge and the central motivation of this study.

This paper proposes a novel optimization algorithm, Chaotic Elite Clone Particle Swarm Optimization (CECPSO), specifically designed to address the complex and large-scale task allocation problem in IWSNs. Building upon the insights into the limitations of traditional metaheuristics discussed earlier, CECPSO incorporates three targeted improvements: chaotic initialization for enhanced population diversity, an exponential nonlinear decreasing inertia weight function for balanced global and local search, and elite cloning strategies for preserving high-quality solutions. These enhancements collectively reduce the risk of premature convergence, lower computational complexity, and improve adaptability in dynamic and large-scale industrial environments. The main contributions of this work are summarized as follows:

• 1. A task allocation model tailored for IWSNs is established, along with a customized evaluation function that addresses the absence of problem-specific performance metrics in existing research.

• 2. The proposed CECPSO algorithm incorporates multiple enhancements—chaotic initialization, nonlinear inertia weight adjustment, and elite-clone strategies—which collectively improve population diversity, balance exploration and exploitation, and enhance convergence efficiency in complex optimization environments.

• 3. Extensive simulation experiments are conducted against several benchmark algorithms, validating the superiority of CECPSO in terms of faster convergence speed, reduced dependence on environmental parameters, improved task distribution balance, and better energy efficiency across various IWSN scenarios.

The remainder of this paper is organized as follows. Section 2 reviews related work. Section 3 presents the task allocation model for IWSNs. Section 4 provides a detailed description of the proposed CECPSO algorithm. Section 5 validates the performance of CECPSO and discusses the results. Finally, Sect. 6 concludes the paper.

Related work

In recent years, task allocation strategies for Industrial Wireless Sensor Networks (IWSNs) have undergone rapid development, with increasing focus on energy efficiency, real-time responsiveness, fault tolerance, and distributed adaptability. Many studies have introduced artificial intelligence and machine learning techniques to enhance energy utilization and prolong network lifespan while maintaining robust and timely task allocation under dynamic industrial conditions. For example, reference²² proposed an artificial intelligence-based resource allocation strategy for wireless sensor IoT networks, leveraging deep learning architectures and multi-objective optimization to improve energy efficiency and data processing performance. However, the high implementation complexity and computational cost present practical challenges. Similarly, in²³, a combination of distributed multi-objective evolutionary algorithms and greedy heuristics was introduced, incorporating an NSGA3-based chromosome structure to improve local optimization capability. This hybrid method showed effectiveness in large-scale task allocation involving up to 250 agents and 1000 tasks, though its overall complexity remains a concern. Reference²⁴ addressed task reliability and real-time requirements in time-triggered IoT-WSNs by applying Discrete Particle Swarm Optimization (DPSO) enhanced with a Framework Replication and Elimination (FRER) method and a physical interference model. While this method improves reliability and system stability, its applicability depends on specific scenario adaptations due to computational overhead. A queue-theory-based scheduling model was also proposed in²⁵, which reduces task completion time and extends network lifetime, but its performance relies heavily on the accurate acquisition of network and task-specific parameters.

In the context of solving task allocation problems via heuristic algorithms, several approaches have demonstrated promising results but also revealed notable limitations. Reference²⁶ introduced an inter-cluster task allocation algorithm that improves energy efficiency and load balancing, yet suffers from high computational complexity and sensitivity to parameter tuning in large-scale networks. Other studies such as^27,28 proposed Social Network Optimization (SNO) and an improved PSO for WSN task scheduling, but both are prone to premature convergence and entrapment in local optima. An improved ant colony optimization algorithm in²⁹ aimed to reduce the communication and energy burden of central nodes while enhancing network stability, though its adaptability under real-world dynamic environments remains insufficient.

To address the limitations of single-strategy heuristics, hybrid metaheuristic algorithms have gained considerable attention for their ability to combine the complementary strengths of different techniques. For instance, a hybrid of the Arithmetic Optimization Algorithm (AOA) and Harris Hawks Optimizer (HHO) was proposed for multi-threshold image segmentation (MTIS), where AOA’s global exploration capabilities and HHO’s local exploitation strengths collectively improved optimization quality³⁰. The Competing Leaders Grey Wolf Optimizer (CGWO) introduced in³¹ employs a dynamic leadership mechanism with diversity-enhanced initialization, helping to prevent stagnation in local optima and accelerate convergence. Additionally, a hybrid Jellyfish Search and Particle Swarm Optimization (HJPSO) algorithm was developed in³² to optimize support vector machine parameters for large-scale financial data prediction, showcasing the scalability and effectiveness of hybrid designs in high-dimensional scenarios. These works reflect a broader trend in metaheuristic research: leveraging synergy between exploration and exploitation to improve convergence behavior, adaptability, and robustness.

Table 1 summarizes the key characteristics of representative algorithms, highlighting their strengths and limitations across core evaluation dimensions. Aligned with these design principles, the proposed CECPSO integrates chaotic initialization, elite cloning, and nonlinear inertia adaptation to enhance search efficiency and overall performance in IWSN task allocation.

Table 1 Comparison of representative task allocation algorithms in IWSNs.

System model

In this section, considering the computational capacity limitations in the complex environment of IWSNs, a mathematical model was designed to maximize the efficiency of task allocation execution. The model is applicable to large industrial plants and enhances task allocation efficiency by optimizing the scheduling of heterogeneous nodes in different groups under communication constraints.

Specifically, the model can be simplified to M wireless sensor nodes randomly deployed in an industrial plant and N monitoring tasks that need to be assigned. Each monitoring task can be assigned to different sensor nodes, and this allocation has different effects on overall performance. The signals emitted by these wireless sensor nodes are managed by a unified system, regardless of the type of sensor they come from. Additionally, there is no limit to the physical parameters monitored by these sensors, which can include various parameters such as temperature, humidity, vibration, pressure, etc.

In this model, the key problem is how to optimally assign monitoring tasks to sensor nodes. Effective allocation strategies can not only improve the overall performance of the system but also extend the service life of sensor nodes, reduce energy consumption, and enhance the reliability and accuracy of data transmission. Therefore, the aim of this study is to identify a task allocation scheme that maximizes the overall performance of the sensor network under these constraints.

Assuming that the advantages and priorities of task allocation have already been assessed and determined. Let $b$ represent the advantage of task allocation, where $b_{ij}$ denotes the advantage of assigning an $j_{th}$ monitoring task to the $i_{th}$ sensor. The value of $b$ is between 0 and 1. Let $u$ denote the urgency of allocating monitoring tasks, where $u{}_{j}$ represents the urgency of allocating an $j_{th}$ monitoring task. The value of $u$ is between 0 and 1. In Eq. (1),$p_{j}$ indicates the performance value of the sensor node executing the monitoring task. In Eq. (2), $P$ represents the total performance value generated by the allocation scheme.

$$p_{j} = u_{j} b_{ij}$$

(1)

$$P = \sum\limits_{j = 1}^{N} {p_{j} }$$

(2)

Assume that the model has only one particle, four sensors, and five monitoring tasks. The urgency matrix $U_{j}$ for all tasks is represented by (3). The advantage matrix $B_{ij}$ for task allocation is represented by Eq. (4).

$$U_{j} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {u_{1} } & {u_{2} } \\ \end{array} } & {u_{3} } & {u_{4} } \\ \end{array} } & {u_{5} } \\ \end{array} } \right](j \in [1,5])$$

(3)

$$B_{ij} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {b_{11} } & {b_{12} } & {b_{13} } \\ \end{array} } & {b_{14} } & {b_{15} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {b_{21} } & {b_{22} } & {b_{23} } \\ \end{array} } & {b_{24} } & {b_{25} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {b_{31} } & {b_{32} } & {b_{33} } \\ \end{array} } & {b_{34} } & {b_{35} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {b_{41} } & {b_{42} } & {b_{43} } \\ \end{array} } & {b_{44} } & {b_{45} } \\ \end{array} } \\ \end{array} } \right](i \in [1,4],j \in [1,5])$$

(4)

where $Z$ denotes a task allocation scheme with a randomly generated task allocation scheme represented by Eq. (5). The encoding method for generating the schemes is discussed in Sect. 4. In Eq. (5), $Z$ indicates the allocation of the first monitoring task to the third sensor, the second monitoring task to the first sensor, the third monitoring task to the second sensor, the fourth monitoring task to the fourth sensor, and the fifth monitoring task to the first sensor.

$$Z = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 3 & 1 & 2 \\ \end{array} } & 4 & 1 \\ \end{array} } \right]$$

(5)

Decimal encoding is converted to binary encoding with the resulting matrix presented in Eq. (6). The row vectors in the matrix represent sensors and the column vectors represent monitoring tasks, intuitively displaying the correspondence between the sensors and tasks.

$$Z^{*} = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \end{array} } \right]$$

(6)

$$B_{ij}^{*} = B_{ij} \circ Z^{*}$$

(7)

where $Z^{*}$ denotes the binary encoding matrix form of the allocation scheme. The $Z_{ij}^{*} = 1$ indicates that the $j_{th}$ monitoring task is allocated to the $i_{th}$ sensor. $Z_{ij}^{*} = 0$ indicates that the $j_{th}$ monitoring task was not allocated to the $i_{th}$ sensor. $B_{ij}^{*}$ represents the Hadamard product of $B_{ij}$ and $Z^{*}$, which can $B_{ij}^{*}$ obtained by allocating the $j_{th}$ monitoring task to the $i_{th}$ sensor.

$$B_{ij}^{*} = \left[ {\begin{array}{*{20}c} 0 & {b_{12} } & 0 & 0 & {b_{15} } \\ 0 & 0 & {b_{23} } & 0 & 0 \\ {b_{31} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {b_{44} } & 0 \\ \end{array} } \right]$$

(8)

Set a matrix $A$ consisting entirely of 1s, with the number of columns in $A$ equal to the number of sensors in the system model. By calculating Eq. (10), the advantageousness matrix $B$ for a set of allocation schemes can be obtained.

$$A = \left[ {\begin{array}{*{20}c} 1 & 1 & 1 & 1 \\ \end{array} } \right]$$

(9)

$$B = A \cdot B_{ij}^{*}$$

(10)

$$B = \left[ {\begin{array}{*{20}c} {b_{31} } & {b_{12} } & {b_{23} } & {b_{44} } & {b_{15} } \\ \end{array} } \right]$$

(11)

$$P = B \cdot U_{j}^{T}$$

(12)

where $P$ represents the total performance value of a set of allocation schemes, which can be calculated using Eq. (12).

Based on the established IWSNs task allocation model, this study utilizes the classic Particle Swarm Optimization algorithm, integrating various improvement strategies with $P$ from Eq. (12) as the evaluation function. To maximize the $P$ value, CECPSO effectively optimizes the task allocation problem in the IWSNs.

Proposed algorithm

We proposed an optimization algorithm based on CECPSO to address the task allocation problem in IWSNs. It is shown in Fig. 2. The algorithm flow is shown in Fig. 3. The algorithm is inspired by the collective foraging behavior of organisms, especially birds and fish. In the design of CECPSO, chaos theory, an elite clone strategy, and an exponential nonlinear decreasing inertia weight function are integrated. These strategies significantly improve the overall performance of the algorithm, so that the algorithm can solve the task allocation problem in the IWSN more effectively and quickly find the optimal task allocation scheme.

Fig. 2

The pseudo code of the CECPSO.

Fig. 3

The flow chart of CECPSO.

Compared with the traditional PSO, CECPSO performs better in dealing with complex multimodal optimization problems. Traditional PSO tends to become trapped in local optima during the search process, primarily because it may overly depend on the current best solution of the swarm, resulting in a limited search range. To overcome this flaw, CECPSO introduces a chaos operator in the initial population distribution to initialize the positions and velocities of the particles. This operator can effectively generate a wide and diverse set of initial solutions, thereby providing a more comprehensive search space for the algorithm. Moreover, the algorithm combines elitist and cloning strategies by replacing the worst individuals in the population with a certain number of global best individuals and dynamically increases the number of replacements as the iteration proceeds. This strategy helps the algorithm to break free from excessive dependence on the current best solution, thereby expanding the search range. Finally, CECPSO employs an exponential nonlinear decreasing inertia weight function, balancing the algorithm’s local and global search capabilities, thereby enhancing the overall performance of the algorithm. In summary, the design features of CECPSO enable it to solve task allocation problems in IWSNs, thereby demonstrating its application potential in the field of IWSNs.

The main execution steps of the CECPSO are as follows:

• Step 1: Population Initialization. An initial population was generated by chaos operators, and the initial parameters were set.

• Step 2: Performance Evaluation. The performance value of each particle is calculated, which is determined by the current position of the particle, and reflects its performance.

• Step 3: Update personal and global costs. Based on the performance evaluation, each particle’s personal best ($Pbest$) and global best ($Gbest$) of the entire population. The personal best is the best result found by the particle thus far, whereas the global best is the best result found by all particles so far.

• Step 4: Update Particle Position and Velocity. The updated personal and global bests were utilized along with an exponential nonlinear decreasing inertia weight function to update the velocity and position of each particle. This step is the core of the algorithm, and determines how the particles move and explore the search space.

• Step 5: Implementation of the elite cloning strategy. We re-evaluate the performance of each particle and replace the worst individuals with the global best individuals in the population, increasing the number of replacements as the number of iterations increases.

• Step 6: Termination conditions. Determine whether the algorithm has reached the maximum number of iterations; otherwise, repeat steps 2–5, otherwise, terminate the algorithm.

• Step 7: Terminate the algorithm and output the global best solution in the population, which is the optimal task-allocation scheme.

Coding scheme

An optimization algorithm based on CECPSO is proposed for the task allocation problem in IWSNs. This algorithm is inspired by the foraging behavior of groups in the animal kingdom, particularly the foraging patterns of birds and fish. In the design of CECPSO, chaos theory, elite cloning strategy, and an exponential nonlinear decreasing inertia weight function are integrated, significantly enhancing the overall performance of the algorithm. These improvements enable the algorithm to solve the task allocation problem in IWSNs more efficiently and quickly to find the optimal task allocation scheme.

In the task allocation problem of IWSNs, the core objective is to effectively distribute a series of distinct tasks among various sensors to maximize overall performance. Here, an effective coding strategy plays a crucial role, directly relating to the program’s execution efficiency, accuracy of the fitness calculation, and subsequent operations. Common coding strategies include binary, integer, and real number coding. In this study, an integer-coding scheme was adopted. Within CECPSO, choosing integer coding is highly effective for addressing discrete optimization problems, such as task allocation. This coding method can directly reflect tasks or nodes, simplifying the conversion from problem to algorithmic representation and aiding in increasing the diversity of the initial population through chaotic initialization. Integer coding simplifies the fitness computation, enabling the algorithm to assess each solution more directly and effectively while also facilitating the implementation of an elitist cloning strategy. Furthermore, its simplicity and intuitiveness are conducive to the adjustment, implementation, and debugging of the algorithm, making CECPSO more efficient and flexible in dealing with complex issues. Moreover, to comply with the integer encoding scheme, both CECPSO and PSO in this paper perform rounding operations when updating particle positions in the algorithm. Assume a population with $S$ particles, $N$ sensor nodes, and $M$ monitoring tasks. In a particle, all monitoring tasks are assigned to a sensor. In Eq. (13), assigning the $j_{th}$ task of the $i_{th}$ particle to the $n_{th}$ sensor node is represented as $c_{ij}$. The value of $c_{ij}$ is the serial number of the sensor.

$$C_{i,j} = \left[ {\begin{array}{*{20}c} {c_{11} } & {c_{12} } & \cdots & {c_{1j} } & \cdots & {c_{1M} } \\ {c_{21} } & {c_{22} } & \cdots & {c_{2j} } & \cdots & {c_{2M} } \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {c_{i1} } & {c_{i2} } & \cdots & {c_{ij} } & \cdots & {c_{iM} } \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {c_{S1} } & {c_{S2} } & \cdots & {c_{Sj} } & \cdots & {c_{SM} } \\ \end{array} } \right](i \in [1,S],j \in [1,M],c_{ij} \in [1,N])$$

(13)

Equation (14) represents the allocation scheme for five particles, six sensor nodes, and seven monitoring tasks. Where $c_{24} = 3$ indicates that the fourth task of the second particle is assigned to the third sensor.

$$C_{5,7} = \left[ {\begin{array}{*{20}c} 3 & 5 & 3 & 4 & 2 & 6 & 1 \\ 2 & 1 & 4 & 3 & 6 & 5 & 1 \\ 3 & 6 & 1 & 5 & 2 & 4 & 3 \\ 5 & 3 & 6 & 2 & 1 & 4 & 2 \\ 1 & 4 & 2 & 5 & 6 & 1 & 3 \\ \end{array} } \right]$$

(14)

Chaotic population initialization

Population initialization is a critical step in optimization algorithms. The quality of the initial population significantly impacts the performance of the algorithm, and the quality of the optimal solution is ultimately found. Incorporating chaos operators during population initialization can leverage the properties of the chaos theory to enhance the diversity of the initial population, thereby improving the search capability and efficiency of the entire algorithm. Traditional PSO typically employs random initialization to determine the initial distribution of the population’s positions, using computer-generated random numbers to randomly generate the initial positions of each particle according to Eq. (15).

$$Positions = rand*(ub - lb) + lb$$

(15)

where $Positions$ represents the generated particle positions; $rand$ denotes the generated random numbers with a range of [0, 1]; $ub$ and $lb$ are the upper and lower bounds of the solution space, respectively.

Although this type of random initialization can generate different initial populations each time, thereby increasing their diversity and applicability, it also has some obvious drawbacks. First, the distribution of initial particles in the solution space is often uneven, which can result in certain local areas being too densely populated with particles, whereas other areas are too sparse. This uneven distribution adversely affects the early convergence of the optimization algorithm, particularly for swarm optimization algorithms prone to local optima, potentially slowing down the convergence rate or even failing to converge to the global optimum in some cases.

To address this issue, chaotic initialization methods have been introduced. Chaotic initialization exhibits the following notable characteristics: randomness, exhaustiveness, and regularity. Unlike traditional random initialization, chaotic initialization iteratively explores the search space within a certain range, thereby ensuring that the same position is not revisited. Consequently, the distribution of the initial population is more uniform, with a stronger ability to cover the entire search space. Thus, the solution accuracy and convergence rate of the initial population can be significantly improved, thereby enhancing the overall performance of the optimization algorithm.

Various methods are available for chaotic mapping, including logistic chaotic mapping, tent chaotic mapping, Chebyshev chaotic mapping, and sine chaotic mapping. In this study, we chose logistic chaotic mapping owing to its simple mathematical form and good chaotic characteristics. Logistic chaotic mapping is defined by the following iterative formula:

$$x_{k + 1} = \lambda *x_{k} (1 - x_{k} ),\lambda \in (0,4)$$

(16)

where $\lambda$ is the mapping parameter of the chaos model, and the larger the value of $\lambda$, the higher the chaos. To make the initial population more random, here we set the $\lambda = 3.9$.

Performance value evaluation

In CECPSO, each particle represents a task-allocation scheme. The performance value represents the efficiency of the task allocation scheme. The performance value can be calculated using the performance evaluation function in (12). By comparing the performance values, a better optimization scheme can be selected, thereby optimizing the performance of the IWSNs.

Update position

This step is the core dynamic search mechanism of CECPSO, aimed at guiding particles to explore the space in search of the optimal solution. Each particle has its own position and velocity as well as a performance value determined by an evaluation function. The particles remember and follow the current optimal particle when searching for a solution space. The process of each iteration was not entirely random. If a better solution is found, it serves as the basis for finding the next solution. In CECPSO, during each iteration, particles update themselves by tracking two “extremes”: one is the best solution found by the particle itself, called the personal best extreme point $Pbest$; the other extreme point is the best solution currently found by the entire population, called the global best extreme point $Gbest$. After finding these two optimal solutions, the particles update their velocity and position according to Eqs. (17) and (18). It is assumed that there are $S$ particles in the population, $N$ sensor nodes, and $M$ tasks. The information of particle $i$ can be represented by an M-dimensional vector, where the position and velocity are denoted by $X_{i} = (x_{i1} ,x_{i2} , \cdots ,x_{iM} )^{T}$ and velocity by $V_{{\text{i}}} = v_{{{\text{i1}}}}, v_{{{\text{i}}2}}, \ldots, v_{iM} )^{T}$, respectively. The equations for updating the velocity and position are as follows:

$$v_{im}^{k + 1} = w(t)v_{im}^{k} + c_{1} rand_{1}^{k} (Pbest_{im}^{k} - x_{im}^{k} ) + c_{2} rand_{2}^{k} (Gbest_{m}^{k} - x_{im}^{k} )$$

(17)

$$x_{im}^{k + 1} = \left\lceil {x_{im}^{k} + v_{im}^{k + 1} } \right\rceil$$

(18)

where $v_{im}^{k}$ represents the velocity of particle $i$ in the $m_{th}$ dimension during the $k_{th}$ iteration (rounded to the nearest integer); $w(t)$ is an exponential, non-linearly decreasing inertia weight function; $c_{1}$ and $c_{2}$ are acceleration coefficients that adjust the maximum step size towards the direction of the global best particle and the personal best particle; $rand_{1,2}$ is a random number between [0, 1]; $x_{im}^{k}$ is the current position of particle $i$ in the $m_{th}$ dimension during the $k_{th}$ iteration; $Pbest_{im}^{k}$ is the position of the personal best extremum of particle $i$ in the $m_{th}$ dimension; $Gbest_{im}^{k}$ is the position of the global best extremum in the $m_{th}$ dimension for the entire population. To comply with the integer encoding scheme, rounding operations are performed in Eq. (18), with the rounding interval varying according to the number of sensors.

Exponential non-linearly decreasing inertia weight function

This function addresses the issue of traditional PSO easily getting stuck in local optima, which leads to a lower accuracy. In CECPSO, an exponential nonlinear decreasing inertia weight function is introduced, which can effectively balance the local and global search abilities of the algorithm. The value of the inertia weight decreases with an increase in the number of iterations, and it is a nonlinear form. It conforms to the search characteristics of the algorithm, which focuses on global exploration in the early stage and local development in the late stage, and is conducive to improving the convergence rate and accuracy of the algorithm. Figure 4 shows the change curve of the exponential nonlinear decreasing inertia weight function. The equation used is as follows:

$$w(t) = w_{0} \cdot e^{{ - \frac{t}{{T_{\max } }}}}$$

(19)

where $t$ is the current iteration number, $T_{\max }$ is the set maximum number of iterations, and $w_{0}$ is the initial inertia weight used to ensure the global search capability of the algorithm in its early stages.

Fig. 4

The change curve of the inertia weight function.

Elite cloning strategy

Elite and cloning strategies are important techniques in optimization algorithms that can enhance the performance and search capability of these algorithms. Elite strategy refers to preserving the best individual or the top few best individuals from the current population to the next generation during each evolutionary process. This ensures that the optimal solution is not lost owing to random operations such as crossover and mutation. An elite strategy improves the stability and convergence rate of the algorithm by retaining and passing on the optimal solution throughout the evolutionary process. The cloning strategy refers to the replication of some excellent individuals in some stages of the algorithm to enhance the influence and exploration ability of these individuals. Increasing the proportion of excellent individuals in the population accelerates the convergence rate of the algorithm, helps to explore the solution space around excellent individuals more carefully, and improves the precision of the solution. This paper combines the advantages of elite and clone strategies and proposes a new elite clone strategy.

Elite cloning strategy plays a key role in CECPSO and is dedicated to enhancing the preservation and improvement of excellent solutions within the population. The core of this strategy is to duplicate and appropriately mutate the global optimum individual of the current population, thereby accelerating the finding of optimized solutions while maintaining the effective exploration of the solution space by the algorithm. In the elite cloning strategy, the number of individuals participating in elite cloning is calculated according to a fixed proportion of the total population and is dynamically adjusted with the iteration of the algorithm.

The specific steps of the elite cloning strategy in a single iteration are as follows:

• 1. The number of individuals participating in elite cloning operation in this iteration is calculated by Eq. (20).

• 2. Replace the worst individual with the elite individual according to the value of $E$.

• 3. To ensure the diversity of the population, individuals undergoing elite cloning were subjected to a single-point mutation.

Figures 5 and 6 show the specific process of the elite cloning strategy when $E$ = 2. The formula for dynamically adjusting the number of individuals for elite cloning is as follows:

$$E = \left\lceil {\frac{\rho \cdot S \cdot t}{{T_{\max } }}} \right\rceil$$

(20)

where $E$ represents the number of individuals participating in elite cloning; $\rho$ is the participation coefficient for the elite cloning strategy, where $\rho = 0.1$ indicates that up to 10% of the population is selected for elite cloning operations; S is the population size; $t$ is the current iteration number; $T_{\max }$ is the set maximum number of iterations.

Fig. 5

When $E$ = 2, the process of cloning elite individuals.

Fig. 6

When $E$ = 2,the process of mutation.

As the number of iterations increases, the algorithm gradually shifts from the exploration phase to a more focused exploitation phase. In the early stages of the iteration, a smaller number of elite individuals is beneficial for exploring the solution space and preventing the algorithm from prematurely converging on local optima. As the iterations progress, gradually increasing the number of elite individuals can accelerate the convergence rate of the algorithm, ensuring that high-quality solutions already discovered are not lost because of random exploration. This mechanism of dynamically adjusting the number of elite individuals achieves a smooth transition of the algorithm from broad exploration to rapid convergence by gradually increasing the proportion of elite individuals during the iteration process. This strategy significantly enhances the performance of the algorithm, especially when dealing with complex optimization problems, as it enables the algorithm to escape local optima and accelerate the search for global optima.

Time complexity analysis

To ensure practical applicability in real-time industrial environments, any metaheuristic optimization algorithm must demonstrate reasonable computational complexity. In this section, we theoretically analyze the time complexity of the proposed CECPSO algorithm and compare it with that of the standard PSO.

Let the population size be $n$ and the maximum number of iterations be $T_{\max }$. In the worst case, the time complexity of each process in CECPSO is calculated as follows:

• 1. The time complexity required to initialize the parameter is $O(1)$.

• 2. The chaotic initialization of the population requires $O(n)$.

• 3. The time complexity required to calculate the performance values of all particles is $O(n)$.

• 4. The $Gbest$ and $Pbest$ individuals are updated. The time complexity of this operation at most is $2O(n)$.

• 5. Updating the inertia weight function requires $O(1)$.

• 6. Update the position and velocity of all particles, which requires $O(n)$.

• 7. Calculating the number of elite clones needs $O(1)$.

• 8. The time complexity to perform an elite clone operation requires $O(n)$.

Consequently, the total time complexity of CECPSO is calculated as follows:

$$\begin{gathered} \begin{array}{*{20}c} {} & {} \\ \end{array} O(1) + O(n) + T_{{{\text{max}}}} \cdot [O(n) + 2O(n) + O(1) + O(n) \hfill \\ + O(1) + O(n)] \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} = (5T_{\max } + 1) \cdot O(n) + (2T_{\max } + 1) \cdot O(1) \hfill \\ \end{gathered}$$

(21)

To be simplified, the total time complexity can be regarded as $O(T_{\max } \cdot n)$.

After calculating the time complexity of PSO using the same method, we found that the time complexity of PSO is also $O(T_{\max } \cdot n)$. The results indicate that both CECPSO and PSO have the same computational time complexity, suggesting that the three improvement strategies—chaotic strategy, elite cloning strategy, and the adoption of an exponential nonlinear decreasing inertia weight function—not only enhance the performance of CECPSO but also do not reduce its computational efficiency.

Simulation and analysis

Parameter settings

In this section, we validate the performance of CECPSO in task allocation for IWSNs using simulation experiments. To evaluate its effectiveness, CECPSO is compared with three widely adopted classical heuristic algorithms: Particle Swarm Optimization (PSO), Genetic Algorithm (GA), and Simulated Annealing (SA). These methods have been extensively used as benchmarks in optimization research and remain representative baselines for evaluating newly proposed algorithms. The experiments were performed under consistent conditions using a system model and evaluation function defined in this study. Each result represents the average of 100 independent runs to ensure statistical reliability. All simulations were conducted on a computer equipped with an Intel Core i5-14600KF 3.50GHz CPU, running MATLAB R2023a on a Windows 11 operating system. The results are presented using line graphs, bar charts, and data tables for comprehensive comparison.

In this simulation experiment, to better compare CECPSO, PSO, GA, and SA, the population size for all four algorithms was set to 100 with a maximum of 100 iterations. The experimental parameters of the four algorithms are listed in Table 2, Table 3 and Table 4. In CECPSO and PSO, the inertia weight $w_{0}$ was set to 0.6, and the acceleration factors $c_{1}$ and $c_{2}$ were both set to 1.5. In GA, the crossover probability was set to 0.8, and the mutation probability was set to 0.01. In the SA, the initial temperature was set to 200, and the annealing coefficient was set to 0.8.

Table 2 Key experimental parameters of CECPSO and PSO.

Table 3 Key experimental parameters of GA..

Table 4 Key experimental parameters of SA..

Results analysis and discussion

Table 5 reports the optimal performance values of CECPSO, PSO, GA, and SA under varying numbers of sensors and tasks in the IWSN task allocation scenario. As shown in the table, CECPSO consistently outperforms the other algorithms across most configurations. The only exception occurs in the case of 16 sensors and 30 tasks, where CECPSO and PSO achieve identical performance. This is due to the relatively small scale and low complexity of the problem setting, which reduces the difficulty of optimization and limits the performance gap between algorithms.

Table 5 Optimal performance values with different number of tasks and sensor nodes.

Figure 7a–d show the performance value versus iteration number curves of CECPSO, PSO, GA, and SA in IWSNs task allocation under various numbers of sensors and tasks, allowing us to clearly compare the convergence rate and optimal performance values of the four algorithms. Figure 7a demonstrates that, in comparison with GA and SA, CECPSO consistently outperforms both GA and SA. In the comparison between CECPSO and PSO, although the optimal performance values of CECPSO and PSO are consistent, CECPSO converges faster than PSO,making CECPSO’s overall performance still superior to PSO under the condition of 16 sensors and 30 tasks. In Fig. 7b–d, CECPSO continues to show superior overall performance in terms of both convergence rate and optimal performance value when compared to PSO, GA, and SA. This consistent superiority across various scenarios indicates that the integration of the chaotic strategy, elite cloning strategy, and an exponential non-linearly decreasing inertia function within CECPSO significantly enhances the algorithm’s efficiency and effectiveness.

Fig. 7

Comparison of performance and convergence rate of four algorithms: CECPSO, PSO, GA, and SA. (a) Comparison of performance and convergence rate of CECPSO, PSO, GA, and SA under conditions of 16 sensors and 30 tasks; (b) Comparison of performance and convergence rate of CECPSO, PSO, GA, and SA under conditions of 24 sensors and 60 tasks; (c) Comparison of performance and convergence rate of CECPSO, PSO, GA, and SA under conditions of 32 sensors and 120 tasks; (d) Comparison of performance and convergence rate of CECPSO, PSO, GA, and SA under conditions of 40 sensors and 240 tasks.

These results highlight the effectiveness of these improvement strategies in optimizing the performance of the algorithm. The chaotic strategy contributes to better initial search efficiency, the elite cloning strategy improves global search capability and stability, and the exponential nonlinear decreasing inertia function aids in balancing exploration and exploitation. Consequently, CECPSO not only converges faster but also achieves higher performance values compared to traditional PSO, GA, and SA, making it particularly effective for complex optimization problems in IWSNs task allocation.

Figure 8a–d clearly presents, via bar graphs, the optimal performance values in IWSNs task allocation after 100 iterations of the CECPSO, PSO, GA, and SA under conditions of varying numbers of sensors and tasks. Comparative calculations reveal that, with 16 sensors and 30 tasks, the CECPSO achieves a performance improvement of 2.29% and 0.39% over the GA and SA, respectively. When the number of sensors increases to 24 and the number of tasks to 60, CECPSO’s performance gain becomes even more significant, surpassing the PSO, GA, and SA by 1.41%, 7.88%, and 3.94%, respectively. Under the configuration of 32 sensors and 120 tasks, the performance improvement of CECPSO over PSO, GA, and SA is 5.48%, 15.49%, and 10.67%, respectively. Finally, under the conditions of 40 sensors and 240 tasks, CECPSO’s performance improvement relative to PSO, GA, and SA reaches 6.6%, 21.23%, and 17.01%, respectively. These data indicate that as the number of sensors and tasks increases, the performance advantage of CECPSO in IWSNs task allocation becomes significantly more pronounced compared to PSO, GA, and SA. The elite cloning strategy and the use of an exponential nonlinear decreasing inertia function endow CECPSO with enhanced search capabilities, particularly in the later stages of optimization, thereby effectively improving the algorithm’s optimal performance value.

Fig. 8

Comparison of the optimal performance values of the CECPSO, PSO, GA, and SA. (a) Comparison of the optimal performance values of CECPSO, PSO, GA, and SA with 16 sensors and 30 tasks; (b) Comparison of the optimal performance values of CECPSO, PSO, GA, and SA with 24 sensors and 60 tasks; (c) Comparison of the optimal performance values of CECPSO, PSO, GA, and SA with 32 sensors and 120 tasks; (d) Comparison of the optimal performance values of CECPSO, PSO, GA, and SA with 40 sensors and 240 tasks.

Overall, Fig. 8 demonstrates that CECPSO is particularly effective in handling larger-scale optimization problems in IWSNs task allocation. The synergistic combination of its improvement strategies ensures superior performance, making it a robust choice for complex and large-scale optimization tasks.

We set the number of iterations for the algorithm to 100 and analyzed the convergence rate within this timeframe. The convergence rate of the algorithm is determined by calculating the number of iterations required to achieve performance values within 1% of the optimal performance value. The fewer the iterations needed to reach this threshold, the faster the convergence rate. Through bar graphs, Fig. 9 illustrates the number of iterations required by CECPSO, PSO, GA, and SA to achieve performance values within 1% of the optimal value for various numbers of sensors and tasks. As shown in Fig. 9, for all configurations of sensors and tasks, CECPSO consistently requires fewer iterations to achieve performance values within 1% of the optimal value compared with PSO, GA, and SA. This demonstrates that CECPSO had the fastest convergence rate among the four algorithms. The superior convergence rate of CECPSO indicates that the integration of a chaos strategy and the use of an exponential nonlinear decreasing inertia function effectively optimizes the initial population distribution and enhances the global search capability early on in the algorithm. These strategies improve the ability of the algorithm to quickly converge to near-optimal solutions, thereby enhancing the overall performance.

Fig. 9

The number of iterations required for the CECPSO, PSO, GA, and SA to achieve performance values within 1% of the optimal performance value. (a) Comparison of the convergence rate of CECPSO, PSO, GA, and SA with 16 sensors and 30 tasks; (b) Comparison of the convergence rate of CECPSO, PSO, GA, and SA with 24 sensors and 60 tasks; (c) Comparison of the convergence rate of CECPSO, PSO, GA, and SA with 32 sensors and 120 tasks; (d) Comparison of the convergence rate of CECPSO, PSO, GA, and SA with 40 sensors and 240 tasks.

In summary, Fig. 9 highlights the effectiveness of CECPSO in rapidly converging to high-performance values across different tasks and sensor configurations. The combined use of the chaos strategy and exponential nonlinear decreasing inertia function not only accelerates the convergence rate but also ensures the stability of the algorithm, making it a highly efficient algorithm for complex IWSNs task allocation problems.

From a mechanistic perspective, PSO is characterized by its fast convergence and simplicity, yet it is prone to becoming trapped in local optima. On the other hand, GA exhibits robust global search capabilities and good adaptability, but its convergence speed is relatively slow. By mimicking the physical annealing process, the SA can effectively avoid local optima; however, its convergence speed is suboptimal and its computational efficiency is relatively low. Three improved strategies enable the CECPSO to overcome these limitations. CECPSO not only converges quickly, but also has high computational efficiency and is not easily trapped in local optima.

To demonstrate the superiority of CECPSO in addressing the task allocation problem in IWSNs more clearly, we selected two improved PSO algorithms for comparison. Reference³³ proposed an enhanced PSO that introduces exponentially decaying inertia weights and dynamically adjusted learning factors, thereby improving the overall performance of the algorithm. Here, we refer to this as IPSO. Reference³⁴ combined the global search capability of GA with the fast convergence of PSO to propose a hybrid heuristic algorithm called PGSAO, which effectively integrates the strengths of both approaches to enhance overall performance. Figure 10a–d show the performance value versus iteration number curves of CECPSO, IPSO, and PGSAO in IWSNs task allocation under various numbers of sensors and tasks, allowing us to clearly compare the convergence rate and optimal performance values of the three algorithms. In Fig. 10a, where the problem scale is relatively small with 16 sensors and 30 tasks, the final performances of all three algorithms are nearly identical, although PGSAO exhibits faster convergence speed. In Fig. 10b and c, although the early evolution speed of CECPSO is slightly slower than that of PGSAO, its optimal performance value is significantly higher than those of PGSAO and IPSO. Figure 10d demonstrates that under more complex problem scenarios, CECPSO outperformed the other two algorithms in terms of both convergence speed and optimal performance value. From Fig. 10a–d, it can be observed that as the problem scale increases, the advantage of CECPSO in handling large-scale wireless sensor network task allocation problems becomes increasingly prominent.

Fig. 10

Comparison of performance and convergence rate of four algorithms: CECPSO, IPSO, and PGSAO. (a) Comparison of performance and convergence rate of CECPSO, IPSO, and PGSAO under conditions of 16 sensors and 30 tasks; (b) Comparison of performance and convergence rate of CECPSO, IPSO, and PGSAO under conditions of 24 sensors and 60 tasks; (c) Comparison of performance and convergence rate of CECPSO, IPSO, and PGSAO under conditions of 32 sensors and 120 tasks; (d) Comparison of performance and convergence rate of CECPSO, IPSO, and PGSAO under conditions of 40 sensors and 240 tasks.

Figure 11a–d clearly presents, via bar graphs, the optimal performance values in IWSNs task allocation after 100 iterations of the CECPSO, IPSO, and PGSAO under conditions of varying numbers of sensors and tasks. Under the conditions of 16 sensors and 30 tasks, the optimal performance values of the three algorithms are very close, with CECPSO showing only a slight advantage. When the number of sensors increases to 24 and the number of tasks to 60, CECPSO’s optimal performance value is 1.97% and 1.41% higher than that of IPSO and PGSAO, respectively. As the number of sensors increases to 32 and tasks to 120, CECPSO outperforms IPSO and PGSAO by 4.13% and 2.51%, respectively. Finally, under the condition of 40 sensors and 240 tasks, CECPSO’s optimal performance value is 9.13% and 3.88% higher than that of IPSO and PGSAO, respectively. Figure 11a to d clearly show that CECPSO consistently achieves better optimal performance values than IPSO and PGSAO under all conditions, further demonstrating the superiority of CECPSO in addressing the task allocation problem in IWSNs.

Fig. 11

Comparison of the optimal performance values of the CECPSO, IPSO, and PGSAO. (a) Comparison of the optimal performance values of CECPSO, IPSO, and PGSAO with 16 sensors and 30 tasks; (b) comparison of the optimal performance values of CECPSO, IPSO, and PGSAO with 24 sensors and 60 tasks; (c) comparison of the optimal performance values of CECPSO, IPSO, and PGSAO with 32 sensors and 120 tasks; (d) comparison of the optimal performance values of CECPSO, IPSO, and PGSAO with 40 sensors and 240 tasks.

In conclusion, Figs. 10 and 11 clearly illustrate that CECPSO significantly outperforms IPSO and PGSAO in the task allocation problem for IWSNs, highlighting its distinct advantages in this specific application domain. However, it is essential to clarify that this does not imply that CECPSO universally outperforms IPSO and PGSAO in all scenarios. According to the No Free Lunch (NFL) theory, it has been theoretically proven that no single optimization algorithm can dominate across all possible problem domains³⁵. Therefore, this experiment only demonstrates that CECPSO exhibits high adaptability and outstanding performance in the task allocation problem within IWSNs.

In this section, to better validate the effectiveness of CECPSO, a comparative analysis is conducted between CECPSO and several other algorithms under varying conditions of sensor and task quantities. The experimental results indicate that in the task allocation optimization problem for IWSNs, CECPSO demonstrates better optimal performance values and faster convergence speed than several other algorithms. Moreover, CECPSO’s performance of CECPSO further improves as the scale of the problem increases. These results suggest that the integration of the chaos strategy, elite cloning strategy, and use of an exponential nonlinear decreasing inertia weight function in CECPSO significantly contributes to its performance enhancement. The chaos strategy optimizes the initial population distribution, the elite cloning strategy accelerates the exploration of understanding space and improves the accuracy of understanding, and the exponential nonlinear decreasing inertia weight function maintains a balance between exploration and exploitation throughout the optimization process. Collectively, these strategies enable CECPSO to efficiently solve complex and large-scale optimization problems, demonstrating its robustness and effectiveness in IWSNs task allocation. At the same time, the exceptional performance of CECPSO in optimizing task allocation for Industrial Wireless Sensor Networks (IWSNs) demonstrates its significant potential in industrial applications. It not only enhances operational efficiency and reduces industrial costs but also ensures scalability in large-scale deployments, providing strong support for industrial intelligence.

Ablation study

In this section, we evaluate the contribution of three key improvement strategies in CECPSO to the algorithm performance through ablation study with 40 sensors and 240 task conditions. Specifically, we treat each improvement strategy in CECPSO as a key component of the algorithm, and evaluate the contribution of each strategy to the algorithm by removing these components in turn and observing changes in algorithm performance.

We designed the following experimental versions:

• CECPSO: This is our original algorithm without any cuts, with all the improvement strategies.

• CECPSO-C: CECPSO, which removes chaotic strategies to evaluate their impact of chaotic strategies on the overall performance of the algorithm.

• CECPSO-EC: The CECPSO of the elite cloning strategy is removed to determine the effect of the elite cloning strategy on the overall performance of the algorithm.

• CECPSO-IW: CECPSO, which removes the exponential nonlinear decreasing inertia weight function, is used to evaluate the effect of this weight function on the overall performance of the algorithm.

• PSO: As a benchmark, we also include the traditional PSO algorithm for comparison with the above versions.

It can be observed from Fig. 12 that the complete CECPSO exhibited the best performance among all experimental versions, with various improvement strategies significantly enhancing the overall performance of the algorithm. Specifically, CECPSO, compared to CECPSO-C (which removes the chaos strategy), demonstrates a faster convergence rate, indicating that the chaos strategy plays a critical role in improving the initial search efficiency and the overall convergence rate of the algorithm. Compared with CECPSO-EC (which removes the elite cloning strategy) and CECPSO-IW (which removes the exponentially decreasing inertia weight function), the complete CECPSO converges faster and achieves higher final performance values. This indicates that the elite cloning strategy and inertia weight function significantly contribute to improving the global search capability and stability of the algorithm. The traditional PSO exhibits the lowest performance, with both its convergence rate and final performance values significantly lower than those of CECPSO and its variants. This further validates the effectiveness of the various improvement strategies within CECPSO, demonstrating their important role in enhancing algorithm performance. Through this comparative analysis, it is clear that the chaos strategy, elite cloning strategy, and exponentially decreasing inertia weight function contribute specifically to the performance enhancement of CECPSO. Their synergy enables CECPSO to solve complex optimization problems.

Fig. 12

Comparison of performance and convergence rate of CECPSO, CECPSO-C, CECPSO-EC ,CECPSO-IW and PSO with 40 sensors and 240 tasks.

Conclusion

This paper introduces a novel heuristic intelligent optimization algorithm aimed at optimizing task allocation in IWSNs. To evaluate the overall performance of the CECPSO, a task allocation system model and an evaluation function were designed based on the algorithm. In CECPSO, the chaos theory is incorporated to optimize the initial population and enhance population diversity. An exponential nonlinear decreasing inertia weight function is employed to balance the local and global search capabilities of the algorithm. Additionally, by combining the elite strategy with the cloning strategy and dynamically adjusting the number of elite clones, this approach not only accelerates the exploration of the solution space and improves the precision of the solution, but also effectively avoids the early stage tendency to fall into local optima. Simulation experiments demonstrate that CECPSO surpasses traditional PSO, GA, and SA in terms of both the convergence rate and optimization performance. Moreover, as the scale of nodes and tasks increases, the advantages of CECPSO become more pronounced, further proving its effectiveness in solving task-allocation problems in IWSNs. The ablation study further verified the positive effects of the three improved strategies on the algorithm. The introduction of chaos theory helps maintain a diverse and well-distributed initial population, which is crucial for effective exploration in the early stages of optimization. The exponential nonlinear decreasing inertia weight function ensures a smooth transition between global exploration and local exploitation, enhancing the robustness of the algorithm. The dynamic adjustment of elite clones prevents premature convergence, allowing CECPSO to maintain its search efficiency even in complex and large-scale scenarios. In conclusion, CECPSO integrates several effective strategies to achieve a superior performance in IWSNs task allocation. Its ability to adapt and optimize under varying conditions makes it a powerful tool for addressing challenges inherent in large-scale task allocation problems. The experimental results underscore the potential of CECPSO to significantly improve the efficiency and effectiveness of task allocation in IWSNs, thereby making it a valuable contribution to the field.

Although CECPSO has demonstrated significant advantages in solving the task allocation problem in IWSNs, it has certain limitations. The introduction of chaos optimization and elite cloning strategies enhances the global search capability and convergence performance of the algorithm; however, these improvements can also lead to increased computational complexity. Particularly, when dealing with large-scale problems, CECPSO may require more computational resources and time, which can pose challenges for applications with high real-time requirements. Additionally, the current CECPSO is primarily applied to the optimization of task allocation in static IWSNs. Our next step is to introduce the mobility of sensor nodes and more comprehensive evaluation metrics to study the optimization of task allocation by CECPSO in dynamic IWSNs. Simultaneously, we will delve deeper into the security of nodes and practical application scenarios, further enhancing the application value of the algorithm and providing robust support for achieving smart manufacturing and digital transformation.