An improved sparrow search algorithm and its application in the deployment of 3D wireless sensor nodes

Abstract

This manuscript proposes an Improved Sparrow Search Algorithm with Multi-Strategies (MSISSA) to address the shortcomings of the Sparrow Search Algorithm (SSA), and applies it to optimize the deployment of 3D Wireless Sensor Nodes (3D-WSNs). The improvements of MSISSA encompass three main components: firstly, it utilizes an adaptive weight operator for updating position to enhance local search accuracy and expedite convergence; secondly, it introduces the spiral flight mechanism of the Moth Flame Optimization (MFO) algorithm, which balances the algorithm’s local and global search abilities; finally, it uses the Levy flight mechanism to jump out the local optima. The experimental results demonstrate that MSISSA outperforms SSA, its four variants, and two classical intelligent algorithms. The simulation results indicate that when the number of nodes is 30 and 50, the optimal coverage rate of MSISSA reaches 91.89% and 99.11%, respectively, surpassing those achieved by SSA and its four variants, as well as two classical intelligent algorithms. Therefore, it can be concluded that MSISSA is well-suited for optimizing the deployment of 3D-WSNs.

Introduction

Metaheuristic algorithms, which are a class of algorithms designed to tackle complex optimization problems, progressively approximate the optimal solution by searching within the problem space without requiring detailed assumptions about the problem’s underlying model. In recent years, metaheuristic algorithms represented by Particle Swarm Optimization (PSO) Algorithm¹, Genetic Algorithm (GA)², Cuckoo Search Algorithm (CS)³, Moth-Flame Optimization (MFO) Algorithm⁴, Grasshopper Optimization Algorithm (GOA)⁵, Grey Wolf Optimizer (GWO) Algorithm⁶, and Egret Swarm Optimization Algorithm (ESOA)⁷, have been continuously improved and widely applied to solve complex engineering problems, such as path planning^8–10 and network optimizing^11–13. Wu et al.¹⁴ proposed a multi-attribute label-based ant colony system (LACS) algorithm to address vehicle heterogeneity and mixed backhaul loads. Liu et al.¹⁵ presented a new three-stage hybrid feature gene selection method. Huang et al.¹⁶ utilized a nonlinear activated beetle antennae search algorithm for fraud detection of public trading companies. Ye et al.¹⁷ explored a multi-objective Cuckoo Search Algorithm to optimize coordinated formation control for multiple drones, ensuring safe flight paths while aligning airspeed, yaw angle, altitude, and altitude rate within specified parameters. Ou et al.¹⁸ introduced an enhanced multi-strategy GWO algorithm for wireless sensor networks. Kang et al.¹⁹ proposed a harmony search algorithm with segmented search capabilities. Khan et al.²⁰ implemented a human-guided cooperative robot agent in smart homes based on the beetle antennae search. Shang et al.²¹ focused on multi-objective optimization for magnetic resistance magnetic gears using ESOA. S Sharma et al.²² solve multiple engineering design problems and two multi-gravity-assist spacecraft trajectory problems using an improved butterfly optimization algorithm. Sahoo et al.²³ segmentation of COVID-19 CT images using improved moth-flame optimization.

The Sparrow Search Algorithm (SSA)²⁴, proposed in 2020, is inspired by the foraging and anti-predation behaviors of sparrows. As a novel metaheuristic algorithm, SSA is renowned for its ability to achieve higher optimization accuracy than classical algorithms, and can be applied to solve a variety of engineering problems. However, SSA still faces challenges in solving high-dimensional problems, including a tendency to fall into local optima, slow convergence speed, and poor global search capability. Researchers have proposed some variants of SSA to address these shortcomings. Li et al.²⁵ introduced an adaptive T-distribution to enhance local search (T_SSA). Fu et al.²⁶ developed a multi-strategy SSA that combines chaotic reverse learning and Cauchy-Gaussian mutation to balance search capabilities (LSSA). Huang et al.²⁷ utilized t-distribution and Tent chaotic mapping to increase the diversity of the initial population (T_TentSSA). Zhang et al.²⁸ combined circular chaotic mapping with the butterfly optimization algorithm and dimension mutation to enhance global exploration (CASSA). The aforementioned variants demonstrate higher performance compared to the standard SSA, but still have various limitations in solving high-dimensional problems, such as slow convergence accuracy and convergence speed, poor global search capability, and susceptibility to local optima.

Aim to solve high-dimensional problems, this manuscript presents an improved Sparrow Search Algorithm with multi-strategy (MSISSA) from enhancing the convergence speed, balancing the ability of local and global search, and escaping the local optima. MSISSA integrates three strategies: adaptive weight operator, the MFO algorithm, and the Levy flight mechanism. To verify performance, ablation experiments are conducted to evaluate the individual contributions of MSISSA components. Subsequently, MSISSA is benchmarked against other seven algorithms: SSA²⁴, T_SSA²⁵, LSSA²⁶, T_TentSSA²⁷, CASSA²⁸, PSO¹, and MFO⁴ on ten benchmark functions. The results indicate that MSISSA outperforms these seven algorithms. Ultimately, MSISSA is implemented to optimize the deployment of Three-Dimensional Wireless Sensor Nodes (3D-WSNs). Simulation experiments with 30 and 50 nodes demonstrate MSISSA’s superior stability compared to its competitors.

The remainder of this manuscript is organized as follows: Section "SSA" introduces SSA; Section "Proposed MSISSA" explains the improvements and underlying rationale of the MSISSA; Section "Experiments and analyses" presents the analytical experiments; Section "Application in the deployment of 3D-WSNs" discusses the application of MSISSA in the deployment of 3D-WSNs; and Section "Conclusions" provides the conclusion.

SSA

SSA simulates the foraging and anti-predation behaviors of sparrows. Based on differences in individual foraging behaviors, the sparrow population is divided into discoverers, followers, and sentinels. The discoverers are responsible for searching for food sources, while the followers follow the discoverers to forage. Additionally, a portion of the sparrows are randomly selected from the population to serve as the sentinels, who notify the group to adopt anti-predation behaviors when danger is detected.

In SSA, 10–20% of the sparrows possessing the optimal fitness values are chosen to serve as discoverers. These discoverers prioritize food acquisition during the search process and are responsible for finding food. The position update formula for the discoverers is as follows (1):

1

where represents the position of the sparrow in the dimension. α is a random number in (0, 1).  and  denote the warning value in (0, 1) and safety value in (0.5, 1), respectively. is a random number drawn from a normal distribution. is a  matrix where each element equals 1. and represent the current iteration number and the maximum number of iterations, respectively. When , it indicates an absence of predators in the foraging environment, allowing discoverers to search more broadly. Conversely, when , it signifies that some sparrows have detected predators and alerted others, prompting all sparrows to quickly relocate to safer areas for foraging.

The position update formula for followers is defined by (2).

2

where  and  represent the current global worst position and the current optimal position of discoverers, respectively. A is a vector whose elements are randomly assigned values of −1 or 1, and . When , it indicates that the sparrow, with a lower fitness value, has not obtained food and is in a state of extreme hunger. In this scenario, it must search for energy by foraging in other locations.

In SSA, 10–20% of the sparrows are randomly selected as sentinels, whose task is to detect danger and alert the group to move to a safe foraging area. The position update formula is defined by (3).

3

represents current global optimal position. is a random number that follows a normal distribution. is a random variable in (− 1,1). , and represent the fitness values of the current individual, the current global worst, and the current global optima, respectively. is an infinitesimal constant to prevent division by zero.

Proposed MSISSA

MSISSA employs an adaptive weight operator to dynamically regulate convergence and facilitate automatic search tuning, integrates the spiral flight mechanism to enhance the balance between global and local search capabilities, and the incorporates the Levy flight mechanism to strengthen the algorithm’s ability to escape local optima.

Position update using an adaptive weight operator for the discoverers

In SSA, when , the discoverer leads followers in foraging. However, as the number of iterations increases, the discoverer’s search capability decreases, causing the algorithm to slow down its convergence speed. Meanwhile, when , the discoverer at this point will lead the followers to leave their original search position, causing the algorithm to struggle with convergence. To overcome it, MSISSA introduces an adaptive weight operator  with exponential decay^29–31 to the position update formula for the discoverers. Then, formula (1) is replaced with the following formula (4).

4

where  is defined by (5).

5

where  and  with an empirical value in (0.1, 0.7), and represent the minimum and maximum values of , respectively. At the beginning of the iterations, the large value of  ensures the algorithm’s global exploration, and as  decreases, the algorithm’s convergence speed is enhanced.

Position update integrating the spiral flight mechanism for the followers

During the process of converging towards a discoverer, followers tend to rapidly gather around the discoverer, which may lead the algorithm to get trapped in a local optimum, and weaken its global search capability. To overcome this issue, MSISSA integrates the spiral flight mechanism of the MFO algorithm into the position update formula for the followers.

MFO algorithm is an innovative optimization algorithm that mimics the flight mechanism of moths. In MFO, a population is first initialized within the feasible domain, and their corresponding fitness values are calculated. The moths’ spatial positions are then sorted in ascending order based on their fitness values, and these positions are assigned as the locations of the flames. The position update for the moths is calculated using Eq. (6).

6

where  represents the position of the moth, F_jrepresent the position of the flame, and  is a spiral function defined by (7).

7

where  defined by (8), represents the distance between the moth and the flame.  is a constant that defines the shape of the logarithmic spiral, and the path coefficient is a random value in (− 1, 1).

8

Seen from Eq. (8), the spiral flight mechanism in the MFO shows robust local and global search capabilities. Thereby, MSISSA balances the algorithm’s local and global search abilities by integrating the spiral flight mechanism. The position update for the followers is calculated using Formula (9).

9

where the coefficient is a random number in (− 1,1), and represents the distance between the current position and the current discoverers optimal position , as defined in Formula (10).

10

Position update combined with levy flight mechanism for the sentinels

When , the sparrows located at the edge of the population are vulnerable to attacks from predators, and they must converge towards the sparrow at the best position, which diminishes the algorithm’s global search capability and increases the likelihood of getting trapped in a local optimum. To overcome this issue, MSISSA combines the Levy flight mechanism with the position update for the sentinels.

Levy flight is a type of random walk where step lengths follow a heavy-tailed probability distribution. Its alternating short and long steps are essential for balancing the local and global search capabilities of optimization algorithms. Short steps enable local search, while occasional long leaps help the algorithm escape local optima. The expression for the Levy flight path formula is defined by (11).

11

where and are the parameters that obey the normal distribution:,. and is calculated by (12).

12

where is the Gamma function. is a random number in (0, 2), and the empirical value .

The random walk mechanism of Levy flight^32,33, due to the uncertainty of its step length, can enhance the algorithm’s ability to escape from local optima. Thereby, in MSISSA, the position update for the sentinels using Formula (13).

13

Pseudocode description

The flowchart of MSISSA is shown in Fig. 1, and the pseudocode is provided in Algorithm 1.

Fig. 1.

Flowchart of MSISSA.

Algorithm 1.

MSISSA.

Algorithm complexity analysis

Assuming the function has a dimension of and the population size is , the worst-case time complexity analysis of the algorithm is as follows.

1. The time complexity of population position initialization is .

2. The time complexity of calculating the objective function is .

3. In each iteration, the time complexity of updating the position of the discoverers with Formula (4) is , and the time complexity of calculating the objective function is .

4. In each iteration, the time complexity of updating the position of the followers with Formula (9) is , and the time complexity of calculating the objective function is .

5. In each iteration, the time complexity of updating the position of the sentinels with Formula (13) is , and the time complexity of calculating the objective function is . Where, is taken as .

In summary, the total worst-case time complexity of iterations are .

Experiments and analyses

Experimental environment

The experiments were conducted on desktop computer with Gen Intel(R) Core (TM) i5-12400F @2.50 GHz, 16 GB RAM and Window10 operating system, using Python 3.8 as the programming language.

The benchmark functions are usually used to verify performance of algorithm. According to the No Free Lunch Theorem³⁴, no single algorithm performs optimally across all tasks, necessitating specific analysis for each problem. An algorithm may excel in certain constrained problems but underperform in others. Consequently, a judicious selection of benchmark functions is essential for performance assessment. To evaluate the performance of MSISSA, ten benchmark functions selected from the IEEE CEC Benchmark Suite are shown in Table 1. F₁–F₃ are dimension-variable unimodal functions with a single extremum, used to evaluate the algorithm’s convergence speed, optimization accuracy, and local search capability. F₄–F₇ are multimodal functions with multiple local extrema, designed to test the algorithm’s ability to escape local optima and the global search capability. F₈–F₁₀ are fixed 2-dimension multimodal functions, chosen to further assess the algorithm’s convergence speed, stability, and accuracy.

Table 1.

The benchmark functions.

No	Name	Function	Dimension	Interval	fmin
F1	Sphere		30,50	(− 100,100)	0
F2	Schwefel 2.22		30,50	(− 10,10)	0
F3	Schwefel 1.2		30,50	(− 100,100)	0
F4	Rastrigin		30,50	(− 5.12,5.12)	0
F5	Ackley		30,50	(− 32, 32)	0
F6	Griewank		30,50	(− 600,600)	0
F7	Penalized		30,50	(− 50,50)	0
F8	Three Hump Camel		2	(− 5,5)	0
F9	Schaffer N.2		2	(− 100,100)	0
F10	Drop Wave		2	(− 5.12,5.12)	− 1

Ablation experiments and analysis

To affirm the viability of MSISSA, ablation studies on the three strategies were performed. The dimensions of functions F₁–F₇ are set to 30 and 50, while F₈–F₁₀ remain fixed at 2 dimensions. Each of the eight algorithms was run 30 times independently, with a maximum of 500 iterations per run. The eight algorithms involved in ablation experiments are shown in Table 2. The parameter settings for the ablation experiments are provided in Table 3.

Table 2.

Algorithms for ablation experiments.

SSA	Standard SSA
SSA1	Algorithm using an adaptive weight operator
SSA2	Algorithm combined with the spiral flight mechanism
SSA3	Algorithm combined with Levy flight mechanism
SSA4	Algorithm combined with an adaptive weight operator and the spiral flight mechanism
SSA5	Algorithm combined with an adaptive weight operator and Levy flight mechanism
SSA6	Algorithm using an adaptive weight operator
MSISSA	Proposed algorithm

Table 3.

Parameter setting for eight algorithms.

Algorithm	Parameters
SSA
SSA1
SSA2
SSA3
SSA4
SSA5
SSA6
MSISSA

From Tables 4, 5 and 6, key observations are:

1. On F₁ and F₂, MSISSA achieves the optimal performance in both convergence accuracy and stability.

2. On F₃ in 30 dimensions, SSA6 performs optimal in both convergence accuracy and stability, while on F₃ in 50 dimensions, MSISSA outperforms the others.

3. On F₄–F₆, MSISSA and SSA6 consistently outperform the other six algorithms in both convergence accuracy and stability.

4. On F₇, SSA4 achieves the optimal convergence accuracy and stability.

5. On F₈, MSISSA demonstrates superior convergence accuracy and stability.

6. On F₉ and F₁₀, all algorithms (SSA1–SSA6 and MSISSA) show identical convergence accuracy and stability.

Table 4.

The ablation experimental results (Dim = 30).

Function	Index	SSA	SSA1	SSA2	SSA3	SSA4	SSA5	SSA6	MSISSA
F1	Mean	9.79E − 17	1.01E − 21	7.58E − 18	3.06E − 20	1.87E − 26	4.87E − 24	2.83E − 37	5.40E − 44
	Std	4.22E − 16	3.40E − 21	4.08E − 17	1.59E − 19	1.00E − 25	2.46E − 23	1.52E − 36	2.85E − 43
F2	Mean	5.30E − 08	7.67E − 10	2.65E − 11	7.92E − 10	1.61E − 13	1.41E − 09	3.64E − 01	5.14E − 20
	Std	2.84E − 07	3.94E − 09	1.23E − 10	2.71E − 09	7.39E − 13	7.60E − 09	1.94E − 17	2.76E − 19
F3	Mean	3.07E − 15	2.85E − 16	1.20E − 17	6.20E − 21	3.61E − 15	1.63E − 20	3.40E − 40	1.67E − 40
	Std	1.13E − 14	1.39E − 15	6.24E − 17	2.81E − 20	1.94E − 14	8.78E − 20	1.50E − 39	6.51E − 40
F4	Mean	3.15E − 12	2.06E − 09	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00
	Std	1.70E − 11	3.37E − 13	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00
F5	Mean	9.50E − 10	1.51E − 12	1.84E − 12	1.97E − 12	1.03E − 12	3.88E − 12	4.44E − 16	4.44E − 16
	Std	3.62E − 09	6.92E − 12	6.92E − 12	1.81E − 10	4.54E − 12	2.01E − 11	0.00E + 00	0.00E + 00
F6	Mean	5.99E − 14	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00
	Std	3.22E − 13	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00
F7	Mean	1.44E − 08	3.24E − 08	2.12E − 09	3.33E − 08	6.19E − 10	6.70E − 08	1.04E − 08	7.47E − 09
	Std	1.91E − 08	4.27E − 08	5.27E − 09	3.85E − 08	1.41E − 09	1.16E − 07	2.42E − 08	3.21E − 08

Optimal values are in bold.

Table 5.

The ablation experimental results (Dim = 50).

Function	Index	SSA	SSA1	SSA2	SSA3	SSA4	SSA5	SSA6	MSISSA
F1	Mean	4.13E − 17	7.50E − 21	2.82E − 20	8.66E − 21	5.01E − 22	8.83E − 18	3.97E − 41	1.02E − 41
	Std	1.57E − 16	4.02E − 20	1.52E − 19	4.06E − 20	2.70E − 21	4.75E − 17	2.14E − 40	5.44E − 41
F2	Mean	5.29E − 08	3.98E − 11	4.80E − 10	2.10E − 11	1.16E − 11	2.25E − 10	1.29E − 21	1.63E − 21
	Std	1.98E − 07	1.70E − 10	1.80E − 09	9.04E − 11	6.09E − 11	1.21E − 09	6.01E − 21	5.60E − 21
F3	Mean	4.44E − 16	3.22E − 17	5.95E − 17	1.39E − 16	1.25E − 22	4.90E − 19	9.72E − 35	1.87E − 42
	Std	1.34E − 15	1.73E − 16	2.58E − 16	5.56E − 16	6.66E − 22	1.84E − 18	5.16E − 34	1.00E − 41
F4	Mean	5.68E − 13	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00
	Std	2.99E − 12	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00
F5	Mean	3.97E − 08	2.51E − 10	1.91E − 10	2.77E − 12	3.60E − 11	4.63E − 14	4.44E − 16	4.44E − 16
	Std	2.04E − 07	1.14E − 09	7.35E − 10	1.10E − 11	1.93E − 10	1.42E − 13	0.00E + 00	0.00E + 00
F6	Mean	1.75E − 13	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00
	Std	9.39E − 13	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00
F7	Mean	1.18E − 07	1.58E − 07	7.39E − 09	1.72E − 07	1.32E − 09	5.10E − 07	2.51E − 08	1.18E − 08
	Std	1.91E − 07	5.13E − 07	2.53E − 08	3.15E − 07	3.60E − 09	8.35E − 07	7.15E − 08	3.31E − 08

Optimal values are in bold.

Table 6.

The ablation experimental results (Dim = 2).

Function	Index	SSA	SSA1	SSA2	SSA3	SSA4	SSA5	SSA6	MSISSA
F8	Mean	7.76E − 14	7.96E − 24	1.11E − 19	3.66E − 22	1.87E − 26	1.05E − 26	8.64E − 44	1.87E − 46
	Std	4.17E − 13	4.29E − 23	5.95E − 19	1.97E − 21	6.84E − 26	5.63E − 26	3.82E − 43	8.84E − 46
F9	Mean	2.96E − 16	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00
	Std	1.39E − 15	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00
F10	Mean	− 1.00E + 00	− 1.00E + 00	− 1.00E + 00	− 1.00E + 00	− 1.00E + 00	− 1.00E + 00	− 1.00E + 00	− 1.00E + 00
	Std	1.44E − 11	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00

Optimal values are in bold.

Convergence curves on F₁–F₇ in 30 or 50 dimensions are shown as Figs. 2, 3, 4, 5, 6, 7 and 8. Convergence curves on F₈–F₁₀ in 2 dimensions are shown as Figs. 9, 10 and 11.

Fig. 2.

Convergence curves on F₁.

Fig. 3.

Convergence curves on F₂.

Fig. 4.

Convergence curves on F₃.

Fig. 5.

Convergence curves on F₄.

Fig. 6.

Convergence curves on F₅.

Fig. 7.

Convergence curves on F₆.

Fig. 8.

Convergence curves on F₇.

Fig. 9.

Convergence curves on F8.

Fig. 10.

Convergence curves on F9.

Fig. 11.

Convergence curves on F₁₀.

From Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11. MSISSA demonstrates the fastest convergence on the unimodal functions F₁–F₃ and the multimodal functions F₄–F₆. On F₇, MSISSA is slower than SSA2 and SSA4, but faster than the other algorithms. On the 2 dimensions functions F₈–F₉, MSISSA outperforms the others. On the 2 dimensions function F₁₀, MSISSA performs comparably to the other algorithms in convergence accuracy.

The ablation experimental results show that the main contributors to the improvement of convergence accuracy are the integration of the spiral flight mechanism and Levy flight mechanism, while the adaptive weight operator accelerates the convergence speed.

Performance comparison experiment and result analysis

To verify the performance of the algorithm, MSISSA was compared with the standard SSA²⁴, four high-performing SSA variants (T_SSA²⁵, LSSA²⁶, T_TentSSA²⁷, CASSA²⁸), and two classical algorithms (PSO¹, MFO⁴) on functions F₁–F₁₀. Each algorithm was run 30 times, with a maximum of 500 iterations per run. The parameter settings for algorithms are listed in Table 7.

Table 7.

Parameter settings for algorithms.

Algorithms	Parameter
MFO4
PSO1
SSA24
CASSA25
T_TentSSA26
LSSA27
T_SSA28
MSISSA

The performance comparison experimental results for eight algorithms on ten benchmark functions are detailed in Tables 8, 9 and 10.

Table 8.

The performance comparison experimental results (Dim = 30).

Function	Index	MFO	PSO	SSA	CASSA	T_TentSSA	LSSA	T_SSA	MSISSA
F1	Mean	6.04E + 04	6.45E + 04	2.19E − 14	9.92E − 22	1.57E − 18	4.69E − 16	1.42E − 14	1.97E − 40
	Std	8.64E + 03	5.94E + 03	1.09E − 13	3.72E − 21	7.89E − 18	1.79E − 15	7.62E − 14	1.06E − 39
F2	Mean	2.11E + 02	1.87E + 12	1.20E − 08	7.24E − 12	2.96E − 09	4.59E − 07	5.69E − 08	1.18E − 21
	Std	1.39E + 01	4.35E + 12	5.90E − 08	2.85E − 11	8.21E − 09	2.22E − 06	3.02E − 07	6.20E − 21
F3	Mean	9.86E + 04	1.31E + 05	4.53E − 10	9.97E − 22	1.22E − 14	7.43E − 14	3.04E − 12	3.28E − 43
	Std	1.25E + 04	3.16E + 04	2.41E − 09	3.67E − 21	6.53E − 14	2.51E − 13	1.63E − 11	1.25E − 42
F4	Mean	4.21E + 02	4.30E + 02	3.79E − 15	1.89E − 15	0.00E + 00	1.89E − 15	1.33E − 14	0.00E + 00
	Std	2.58E + 01	3.14E + 01	2.04E − 14	1.02E − 14	0.00E + 00	1.02E − 14	3.18E − 14	0.00E + 00
F5	Mean	2.00E + 01	2.00E + 01	6.82E − 08	6.50E − 11	4.49E − 08	6.98E − 08	1.13E − 08	4.44E − 16
	Std	5.10E − 03	1.09E − 03	2.66E − 07	3.48E − 10	1.78E − 07	3.68E − 07	4.30E − 08	0.00E + 00
F6	Mean	5.92E + 02	5.80E + 02	2.44E − 15	0.00E + 00	0.00E + 00	1.13E − 15	0.00E + 00	0.00E + 00
	Std	5.46E + 01	6.03E + 01	1.31E − 14	0.00E + 00	0.00E + 00	4.39E − 15	0.00E + 00	0.00E + 00
F7	Mean	9.68E + 08	1.03E + 09	4.51E − 08	1.29E − 10	1.87E − 07	8.34E − 08	1.26E − 07	1.69E − 09
	Std	1.78E + 08	1.64E + 08	9.64E − 08	2.16E − 10	3.13E − 07	1.55E − 07	1.96E − 07	4.38E − 09

Optimal values are in bold.

Table 9.

The performance comparison experimental results (Dim = 50).

Function	Index	MFO	PSO	SSA	CASSA	T_TentSSA	LSSA	T_SSA	MSISSA
F1	Mean	1.21E + 05	1.20E + 05	3.56E − 17	7.67E − 22	3.20E − 12	1.43E − 14	3.36E − 23	5.18E − 40
	Std	8.54E + 03	6.87E + 03	1.91E − 16	4.13E − 21	1.72E − 11	7.66E − 14	1.71E − 22	2.79E − 39
F2	Mean	1.47E + 03	2.93E + 22	5.39E − 08	9.90E − 12	2.52E − 07	3.75E − 08	1.85E − 10	8.63E − 22
	Std	4.58E + 03	6.85E + 22	2.87E − 07	4.42E − 11	1.35E − 06	9.98E − 08	8.95E − 10	4.40E − 21
F3	Mean	2.53E + 05	3.48E + 05	2.14E − 14	5.36E − 16	3.23E − 14	9.56E − 14	1.02E − 13	2.91E − 44
	Std	3.51E + 04	1.18E + 05	1.15E − 13	1.68E − 15	1.66E − 13	4.07E − 13	5.31E − 13	1.51E − 43
F4	Mean	7.49E + 02	7.58E + 02	1.81E − 11	2.27E − 14	1.06E − 13	0.00E + 00	1.14E − 14	0.00E + 00
	Std	3.52E + 01	2.72E + 01	9.76E − 11	1.22E − 13	4.69E − 13	0.00E + 00	4.50E − 14	0.00E + 00
F5	Mean	2.00E + 01	2.00E + 01	1.90E − 09	1.95E − 11	2.62E − 09	1.60E − 09	3.71E − 08	4.44E − 16
	Std	2.86E − 02	6.89E − 04	7.16E − 09	7.36E − 11	1.09E − 08	7.00E − 09	1.95E − 07	0.00E + 00
F6	Mean	1.07E + 03	1.10E + 03	0.00E + 00	0.00E + 00	0.00E + 00	3.63E − 16	0.00E + 00	0.00E + 00
	Std	6.85E + 01	7.16E + 01	0.00E + 00	0.00E + 00	0.00E + 00	1.70E − 15	0.00E + 00	0.00E + 00
F7	Mean	2.15E + 09	2.02E + 09	1.31E − 07	2.84E − 10	8.00E − 08	1.20E − 07	2.02E − 07	4.22E − 08
	Std	2.48E + 08	3.20E + 08	2.31E − 07	4.43E − 10	1.16E − 07	2.32E − 07	3.69E − 07	1.27E − 07

Optimal values are in bold.

Table 10.

The performance comparison experimental results (Dim = 2).

Function	Index	MFO	PSO	SSA	CASSA	T_TentSSA	LSSA	T_SSA	MSISSA
F8	Mean	6.60E − 03	5.13E − 01	4.99E − 12	1.48E − 20	6.30E − 13	1.21E − 15	4.30E − 14	7.49E − 44
	Std	2.88E − 02	3.00E − 01	2.05E − 11	7.82E − 20	3.29E − 12	5.22E − 15	2.32E − 13	4.03E − 43
F9	Mean	4.34E − 02	2.68E − 01	1.85E − 16	0.00E + 00	0.00E + 00	1.48E − 17	0.00E + 00	0.00E + 00
	Std	5.15E − 02	1.13E − 01	7.39E − 16	0.00E + 00	0.00E + 00	5.54E − 17	0.00E + 00	0.00E + 00
F10	Mean	− 9.39E − 01	− 6.97E − 01	− 1.00E + 00	− 1.00E + 00	− 1.00E + 00	− 1.00E + 00	− 1.00E + 00	− 1.00E + 00
	Std	2.50E − 02	1.47E − 01	8.61E − 10	0.00E + 00	4.75E − 10	5.89E − 15	7.48E − 13	0.00E + 00

Optimal values are in bold.

From Tables 8, 9 and 10. the following conclusions can be drawn:

1. On F₁–F₃, both in 30 and 50 dimensions, MSISSA demonstrates superior convergence accuracy and better stability compared to the other seven algorithms.

2. On F₄, both in 30 and 50 dimensions, MSISSA outperforms SSA, T_SSA, T_TentSSA, CASSA, PSO and MFO, achieves the same performance as LSSA, and can converge to the theoretical optimal value of 0.

3. On F₅, both in 30 and 50 dimensions, MSISSA outperforms the other seven algorithms in both convergence accuracy and stability.

4. On F₆, both in 30 and 50 dimensions, MSISSA surpasses MFO, PSO, and LSSA in both accuracy and stability, while matching the performance of the other four algorithms, and can converge to the theoretical optimal value of 0.

5. On F₇, both in 30 and 50 dimensions, MSISSA performs worse than CASSA, but better than the others.

6. On F₈–F₉, MSISSA excels in both convergence accuracy and stability compared to all other algorithms, and converges to the theoretical optimal value of 0.

7. On F₁₀, MSISSA matches the performance of SSA, CASSA, T_TentSSA, and LSSA in both accuracy and stability, outperforming MFO and PSO, and can converge to the theoretical optimal value of -1.

Convergence curves on F₁–F₇ in 30 or 50 dimensions are shown as Figs. 12, 13, 14, 15, 16, 17 and 18. Convergence curves on F₈–F₁₀ in 2 dimensions are shown as Figs. 19, 20 and 21.

Fig. 12.

Convergence curves on F₁.

Fig. 13.

Convergence curves on F₂.

Fig. 14.

Convergence curves on F₃.

Fig. 15.

Convergence curves on F₄.

Fig. 16.

Convergence curves on F₅.

Fig. 17.

Convergence curves on F₆.

Fig. 18.

Convergence curves on F₇.

Fig. 19.

Convergence curves on F₈.

Fig. 20.

Convergence curves on F₉.

Fig. 21.

Convergence curves on F₁₀.

From Figs. 12, 13, 14, 15, 16, 17, 18, 19, 20 and 21. on F₁–F₆ and F₈–F₉, MSISSA demonstrates faster convergence than the others. On F₇, MSISSA is superior to the other compared algorithms except CASSA. On F₁₀, MSISSA shows comparable convergence speed to SSA, CASSA, T_TentSSA, LSSA, and T_SSA, while outperforming MFO and PSO. MFO and PSO exhibit higher initial fitness values because they generate initial solutions through completely random initialization, causing these random solutions to fall far from the optimal solution region, which results in elevated initial objective function values.

Wilcoxon test

Based on the experimental data in Sections "Ablation experiments and analysis" and "Performance Comparison Experiment and Result Analysis", the Wilcoxon rank-sum test³⁵ is employed to further evaluate whether there are significant differences among MSISSA, MFO, PSO, SSA, CASSA, T_TentSSA, LSSA, and T_SSA. The results of the rank-sum test on F₁–F₁₀ are presented in Table 11. In the table, the “W”column shows the comparison outcomes, where the symbols ‘+’, ‘−’ and ‘=’ indicate that MSISSA outperforms, underperforms, or performs equally to the compared algorithm, respectively.From Table 11, the number of '+' significantly exceeds the number of '-' and '=', indicating that MSISSA is superior to the other seven algorithms in terms optimization stability and convergence accuracy.

Table 11.

The results of the Wilcoxon rank sum test.

Function	Dim	MFO	PSO	SSA	CASSA	T_TentSSA	LSSA	T_SSA
F1	30	+	+	+	+	+	+	+
	50	+	+	+	+	+	+	+
F2	30	+	+	+	+	+	+	+
	50	+	+	+	+	+	+	+
F3	30	+	+	+	+	+	+	+
	50	+	+	+	+	+	+	+
F4	30	+	+	+	+	=	+	+
	50	+	+	+	+	+	=	+
F5	30	+	+	+	+	+	+	+
	50	+	+	+	+	+	+	+
F6	30	+	+	+	=	=	+	=
	50	+	+	=	=	=	+	=
F7	30	+	+	+	−	+	+	+
	50	+	+	+	−	+	+	+
F8	2	+	+	+	+	+	+	+
F9	2	+	+	+	=	=	+	+
F10	2	+	+	=	=	=	=	=
+		17	17	15	11	12	15	14
−		0	0	0	2	0	0	0
=		0	0	2	4	5	2	3

.

Total number are in bold.

Application to classical engineering problems

The MSISSA was applied to three classical engineering problems. Experiments were conducted to compare its results with those of other optimization algorithms, further validating its feasibility and effectiveness. To objectively reflect the experimental results, the parameter settings for each algorithm were kept consistent with the aforementioned comparative tests. Each of the eight algorithms was independently run 30 times, with a maximum of 500 iterations per run.

Case 1: Spring Design.

The objective of the tension/compression spring design problem is to minimize the spring weight while satisfying constraints on minimum deflection, vibration frequency, and shear stress. This problem involves three continuous decision variables: wire diameter (), mean coil diameter (), and number of active coils (). The objective function for this problem is defined as,

14

Constraints:

15

16

17

18

where ,,.

Case 2: Three-Bar Truss Design.

In the three-bar truss design problem, variables , and  represent the cross-sectional areas of the three bars, respectively. Due to symmetry, . Thus, the objective of this design can be described as: minimizing the volume of the truss by adjusting the cross-sectional areas . The truss is subject to stress constraints on each structural member. The objective function for this problem is defined as,

19

Constraints:

20

21

22

where , , , , .

Case 3: Pressure Vessel Design.

The objective of the pressure vessel design is to minimize the total manufacturing cost (including material, forming, and welding costs). The cylindrical pressure vessel is sealed at both ends by heads, with one head being hemispherical. The four optimization variables for this problem are: length of the cylindrical section (excluding heads), Inner radius of the cylindrical shell, Thickness of the cylindrical wall, Thickness of the hemispherical head. represent  respectively. The objective function is defined as,

23

Constraints:

24

25

26

27

where ,,,.

The simulation results on the above three classical engineering problems are presented in Table 12. As shown in Table 12, MSISSA outperforms all other algorithms on three classical engineering applications. 

Table 12.

Experiments Results.

Case	Index	MFO	PSO	SSA	CASSA	T_TentSSA	LSSA	T_SSA	MSISSA
1	Maen	1.97E + 04	5.48E + 05	1.10E + 04	1.65E + 04	9.34E + 03	8.73E + 03	9.43E + 03	8.18E + 03
	std	1.64E + 04	3.37E + 05	1.02E + 03	1.20E + 03	7.23E + 02	3.42E + 02	8.17E + 02	5.32E + 01
	min	8.07E + 03	1.15E + 05	9.49E + 03	1.34E + 04	8.34E + 03	8.23E + 03	8.34E + 03	8.07E + 03
2	Maen	2.65E + 02	2.73E + 02	2.67E + 02	2.68E + 02	2.64E + 02	3.49E + 02	2.64E + 02	2.64E + 02
	std	9.89E-01	7.33E + 00	2.51E-02	2.82E + 00	5.59E-02	2.40E + 01	6.21E-02	4.47E-02
	min	2.64E + 02	2.64E + 02	2.64E + 02	2.64E + 02	2.64E + 02	3.81E + 02	2.64E + 02	2.64E + 02
3	Maen	1.97E + 04	5.48E + 05	1.10E + 04	1.65E + 04	9.34E + 03	8.73E + 03	9.43E + 03	8.18E + 03
	std	1.64E + 04	3.37E + 05	1.02E + 03	1.20E + 03	7.23E + 02	3.42E + 02	8.17E + 02	5.32E + 01
	min	8.07E + 03	1.15E + 05	9.49E + 03	1.34E + 04	8.34E + 03	8.23E + 03	8.34E + 03	8.07E + 03

Optimal values are in bold.

Application in the deployment of 3D-WSNs

3D-WSNs have shown wide application potential in various fields such as drone swarm flight³⁶, environmental monitoring³⁷, and 3D WSN localization^38,39. However, as the network scale continues to expand, the issue of node deployment has become increasingly prominent, posing a key technological challenge that restricts network coverage and performance improvement. This manuscript applies MSISSA to optimize the deployment of 3D-WSNs, focusing on improving the coverage of wireless sensor networks.

Model of the deployment of 3D-WSNs

When discussing the deployment of 3D-WSNs, it is assumed that each node is connected via a wireless network. The monitoring range of each sensor in 3D space is abstracted as a sphere, with the sensor’s position serving as the center and the monitoring distance as the radius.

Assume the deployment area is a 3D space with dimension , and need deploy sensor nodes. Any point in the 3D space is denoted as , and the position of sensor node is denoted as . Each node has the same sensing radius . If a target point is located within the sensing range of the node , it means g can be detected by . The Euclidean distance⁴⁰ between and is calculated as,

28

If point is within the monitoring range of the node , the coverage probability is 1; otherwise, it is 0. The coverage probability is calculated as,

29

If there are sensor nodes in the space, the coverage of point g by the sensors is calculated as,

30

The overall coverage of 3D-WSNs represents the average coverage of all target points in the space. If the coverage of the network is denoted as , the following Formula (31) can act as the objective function for optimizing the deployment of 3D-WSNs.

31

Simulation experiments

In the same experimental environment, the deployment of 3D-WSNs is optimized using the MSISSA and compared with standard SSA, CASSA, T_TentSSA, LSSA, T_SSA, PSO, MFO. The of the 3D-WSNs is to evaluate the superiority and applicability of MSISSA. Each algorithm was run independently 10 times, with a maximum of 100 iterations. The parameter settings for the 3D-WSNs coverage model are as follows: is 30 and 50, .

The experimental results are shown in Table 13.

Table 13.

The simulation experimental results.

Algorithm	Performance index	Number of nodes 30	Number of nodes 50
MFO	Mean	84.83%	94.83%
	Best	87.19%	95.18%
	Std	1.477E−02	3.161E−03
PSO	Mean	85.20%	94.16%
	Best	89.03%	94.87%
	Std	2.266E−02	4.879E−03
SSA	Mean	88.60%	96.85%
	Best	90.28%	97.16%
	Std	1.236E−02	2.940E−03
CASSA	Mean	88.11%	96.55%
	Best	88.82%	97.48%
	Std	6.068E−03	8.194E−03
T_TentSSA	Mean	80.11%	92.78%
	Best	84.57%	94.83%
	Std	4.756E−02	2.081E−02
LSSA	Mean	87.44%	96.47%
	Best	90.04%	97.39%
	Std	1.937E−02	1.029E−02
T_SSA	Mean	86.35%	96.28%
	Best	87.23%	97.15%
	Std	8.564E−03	8.040E−03
MSISSA	Mean	88.94%	96.86%
	Best	91.89%	99.11%
	Std	1.266E−02	1.081E−02

Optimal values are in bold.

From Table 13, when the number of sensor nodes is 30, MSISSA outperforms MFO, PSO, SSA, CASSA, T_TentSSA, LSSA, and T_SSA in terms of maximum coverage rate, achieving improvements of 4.69%, 2.86%, 1.60%, 3.07%, 7.32%, 1.85%, and 4.66%, respectively. Additionally, MSISSA’s average coverage rate is increased by 4.11%, 3.73%, 0.33%, 0.83%, 8.83%, 1.50%, and 2.59%. When the number of sensor nodes is 50, MSISSA also demonstrates superiority over MFO, PSO, SSA, CASSA, T_TentSSA, LSSA, and T_SSA in terms of maximum coverage rate, with improvements of 3.94%, 4.24%, 1.95%, 1.63%, 4.29%, 1.72%, and 1.96%, respectively. Furthermore, MSISSA’s average coverage rate is enhanced by 2.02%, 2.70%, 0.01%, 0.30%, 4.08%, 0.39%, and 0.58%. Obviously, MSISSA exhibits better performance in optimizing the deployment of 3D-WSNs than the other algorithms.

The convergence curves are depicted in Fig. 22.

Fig. 22.

Convergence curves.

From Fig. 22 Convergence curves, compared to the other seven algorithms, MSISSA exhibits superior convergence accuracy and faster convergence speed, has better ability to jump out of the local optimal and find the global optimal, can more effectively improve 3D-WSNs coverage, make node distribution more uniform, and reduce coverage redundancy.

In summary, MSISSA has obvious advantages to optimize the deployment of 3D-WSNs and reduce deployment costs.

Conclusions

To address the shortcomings of the standard SSA in solving high-dimensional problems, such as slow convergence accuracy and convergence speed, poor global search capability, and susceptibility to local optima, this manuscript proposes an improved algorithm named MSISSA, and then applied MSISSA to optimize the deployment of 3D-WSNs. MSISSA integrates three strategies: an adaptive weight operator, the spiral flight mechanism of MFO, and the Levy flight mechanism. First, MSISSA enhances the convergence speed by introducing an adaptive weight operator ω with exponential decay to the position update formula for the discoverers. Second, MSISSA balances the algorithm’s local and global search abilities by combing the spiral flight mechanism of the MFO into the position update formula for the followers. Third, MSISSA escapes local optima by using the Levy flight mechanism with position updates for the sentinels. The ablation experiments, performance comparison experiments, and validation on optimization of classical engineering problems have verified the superiority of MSISSA, and the simulation experiments have confirmed its applicability and stability.

Future research aims to further improve the applicability and robustness of MSISSA across diverse environments and under various constraints. Furthermore, plans include utilizing MSISSA to optimize deep learning models in subsequent work.

Author contributions

Shan Li conceived and directed the research, managed data, and wrote the manuscript draft; Liping Mo provided the methodology and participated in manuscript editing; Biao Yin processed the data; Wei Min conducted data validation. All authors reviewed and approved the manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 62266019) and the Natural Science Foundation of Hunan Province, China (Grant No. 2024JJ7394).

Data availability

Data are contained within the manuscript.

Declarations

Competing interests

The authors declare no competing interests.