SSMA: simplified slime mould algorithm for optimization wireless sensor network coverage problem

Wireless sensor network (WSN) coverage problem is to think about how to maximize the network coverage to obtain reliable monitoring and tracking services with guaranteed quality of service. In this paper, a simplified slime mould algorithm (SSMA) for solving the WSN coverage problem is proposed. In SSMA, we mainly conducted 13 groups of WSNs coverage optimization experiments and compared them with six well-known meta-heuristic optimization algorithms. The experimental results and Wilcoxon rank-sum test show that the proposed SSMA is generally competitive, outstanding performance and effectiveness. We proposed SSMA algorithm could be helpful to effectively control the network nodes energy, improve the perceived quality of services and extend the network survival time.


Introduction
In general, the wireless sensor network (WSN) is composed of a large number of densely distributed sensor networks node, each node has limited computing, storage and wireless communication capabilities and can sense the surrounding environment at close range. It has the characteristics of small size, low cost and low power consumption. It can assist in sensing, collecting and processing the information of the monitored object in real time, so with the rapid development of science and technology in various fields, WSNs have been widely used in military affairs, environmental monitoring, medical and health care, safety monitoring and other fields (Singh et al., 2021). In various applications of WSNs, coverage has always been a crucial issue, which determines the monitoring ability of the target area. An appropriate node deployment strategy can not only improve the service quality of WSNs but also effectively promote its energy utilization. Generally, sensor nodes are randomly scattered in the workspace. However, this method will lead to a high concentration of nodes, which will lead to a low coverage rate and affect the monitoring quality (Zhang & Fok, 2017). Therefore, it is necessary to adaptively adjust and deploy the sensor nodes in WSNs, so that they are distributed more evenly in the detection area and have CONTACT Yongquan Zhou yongquanzhou@126.com an effective coverage rate, so as to rationally allocate network space resources and better accomplish the tasks of environment awareness and information acquisition, which is of great significance to improve the network survivability, network reliability and network construction cost. Searching for the optimal node deployment scheme is a difficult task, especially for large-scale sensor networks (Adulyasas et al., 2015;Wang et al., 2010;Zhao et al., 2004). In this context, most scholars choose to focus on meta-heuristic algorithms (MAs), Because of its advantages of availability and simple concept and few parameters, some scholars have proposed new algorithms and used them to solve practical application problems in recent years. Such as, Deng et al. (2022) proposed an enhanced MSIQDE algorithm with novel multiple strategies for global optimization problems. Song et al. (2021) proposed a new algorithm MPPCEDE: multi-population parallel co-evolutionary differential evolution (DE) for parameter optimization; Deng et al. (2021) use the DE with wavelet basis function and optimal mutation strategy for complex optimization problems; Zhang et al. (2016) proposed a new Grey wolf optimizer for unmanned combat aerial vehicle path planning. Zhou et al. (2018) proposed a sensor deployment scheme based on a social spider optimization algorithm for WSNs.
MAs have made a good contribution to the coverage optimization of WSNs. Mendis et al. (2006) used PSO optimized sink node path location in WSNs a new method is proposed to deduce the optimal path of sink nodes in fixed sensor node networks with practical difficulties such as mobility constraints. The algorithm achieves ideal results, but it mainly solves the experiments of wireless sensor nodes with fixed dimensions, and the number of experiments is small. Song et al. (2018) a novel change step of the FOA algorithm was introduced to the coverage optimization, at the same time, the mathematical modelling of two network models is carried out respectively. In the experiment part, the number of experiments is less and the comparison algorithm is less. You can increase the number of experiments and comparison algorithms. Aziz et al. (2009) used PSO and Voronoi diagrams to optimize sensor coverage in WSNs. PSO is used to find the optimal deployment of the sensors that gives the best coverage while the Voronoi diagram is used to evaluate the fitness of the solution. But the comparison algorithm and the number of experiments are less. Kuila and Jana (2014) propose a novel DE-based clustering algorithm for WSNs to prolong the lifetime of the network by preventing faster death of the highly loaded CHs and incorporating a local improvement phase to the traditional DE for faster convergence and better performance of the proposed algorithm. Liao et al. (2011) used an ACObased sensor deployment protocol for WSNs. Based on the ACO algorithm, the author proposed a deployment scheme to prolong the network lifetime, while ensuring complete coverage of the service region. The simulations show that the algorithm can prolong the lifetime of the network. Ambareesh and Madheswari (2021) proposed a hybrid red deer salp swarm algorithm for optimal routeing protocol in wireless multimedia sensor networks. The work outlined in this paper is to minimize four different objectives namely packet loss, memory, delay and expected transmission cost. The main intention of the multi-objective function involves generating a diverse optimal solution set that is utilized to evaluate the trade-off among various objectives. Rajeswari et al. (2021) Powell Flower Pollination Algorithm (PFPA) propose to solve the MEB problem in WSNs. The proposed algorithm is compared with other heuristic approaches and the performance of the algorithm is assessed using benchmark instances with 50 and 100 nodes. Pakdel and Fotohi (2021) used FA for power management in WSNs. In this paper, a method is presented using the FA and using the four criteria of residual energy, noise rate, number of hops and distance. The proposed method called EM-FIREFLY is introduced which selects the best cluster head with high attractiveness and based on the fitness function and transfers the data packets through these cluster head to the sink. The proposed method is evaluated with an NS-2 simulator and compared with the Algorithm-PSO and Optimal clustering methods. Karaboga et al. (2010) propose a novel hierarchical clustering approach for WSNs to maintain energy depletion of the network at a minimum using the Artificial Bee Colony Algorithm which is a new swarm-based heuristic algorithm. They present a protocol using Artificial Bee Colony Algorithm, which tries to provide optimum cluster organization in order to minimize energy consumption.
The slime mould algorithm ) is a novel meta-heuristic optimization algorithm, which was proposed by . It is to simulate the positive and negative feedback generated by the propagation wave of biological oscillator during the foraging process of Slime mould in nature and to guide their behaviour and morphological changes. In SMA, the oscillation factor in the process of foraging is very important, which reflects the perception of slime mould to the food concentration in the search space. And it will adjust their foraging patterns following the food concentration during foraging. Slime mould can obtain multiple information of food source, during searching food. When the food concentration is high, the wave source generated by the oscillator is relatively strong, and the feedback given by slime mould will approach the food source and surround it. Otherwise, when the food concentration is not perceived, Slime mould will randomly search in the search space. The algorithm uses the propagation wave of oscillator to simulate the feedback when sensing food concentration. Through this oscillation mode, feedback information can be given to other slime mould. Through this feedback information transmission mechanism, the foraging mode of slime mould is effectively simulated, and the superiority of the algorithm is improved. Because of its simple concept and strong searching ability, it has been paid attention to and studied by many scholars since it was put forward. Abdel-Basset et al. (2020) used HSMA_WOA based on the thresholding technique to overcome ISP for COVID-19 chest X-ray images by integrating SMA and WOA to maximize the Kapur's entropy; SMA was presented for the solar cell estimation (Kumar et al., 2020); Gao et al. (2020) proposed an improved SMA with cosine controlling parameters; Zhao and Gao (2020b) proposed a chaotic slime mould algorithm with Chebyshev map. Naik et al. (2022) used a leader slime mould algorithm (LSMA) introducing (NSD) based multilevel thresholding technique for multispectral images; Zhao and Gao (2020a) used hybridized the SMA and HHO algorithms for global optimization; Zhao et al. (2020) proposed an improved SMA; Agarwal and Bharti (2021) and Huang (1999) have implemented a modified NISI meta-heuristic approach known as a slime mould optimization algorithm (SMOA) in this research for path planning and obstacle avoidance problem in mobile robots; Zubaidi et al. (2020) proposed hybrid ANN model with SMA for prediction of urban stochastic water demand; Mostafa et al. (2020) based on SMA proposed a new strategy to extract the optimal model parameters of solar PV panel; SMA for feature selection (Abdel-Basset et al., 2021); SMA-AGDE for solving various optimization problems (Houssein et al., 2021); mitigating the effects of magnetic coupling between HV transmission line and metallic pipeline using SMA (Djekidel et al., 2021), Wei et al. (2021) proposed an improved slime mould algorithm for optimal reactive power dispatch problem.
In this paper, to improve the search accuracy and population diversity of the con SMA in solving the coverage optimization problem of WSNs, we propose a simplified SMOA have four highlights as follows: (1) A simplified location updating formula is proposed, which improves the accuracy and speed of the algorithm in solving the coverage optimization problem of WSNs.
(2) An improved adaptive oscillation factor is proposed to improve the exploration ability of SSMA in the early stage of search and the convergence speed of the algorithm. The rest of this paper is structured as follows: Section 2 briefly introduces the WSN coverage problem. Section 3 introduces the SMA. The SSMA is proposed in Section 4. Section 5 carries out an experimental analysis and discussion of the SSMA for WSNs. Section 6 provides the conclusions and future work.

WSN coverage problem
Assumed that the monitoring area of WSNs is a twodimensional plane, which is digitized into L × M grids, and the size of each grid is set to unit 1. N homogeneous sensors are deployed in this area, and the node set can be expressed as Z = {Z 1 , Z 2 , . . . , Z N } having the same sensing radius R. In this paper, the Boolean model is used as the node sensing model, so long as the target is within the node sensing range, it can be successfully sensed. Assuming that the coordinates of a node z i in the detected area are (x i , y i ) and the position coordinates D j of the target point are (x j , y j ), the distance between the node and the target point is where p(z i , D j ) represents the perceptual quality of node z i to D j . When the position D j of node z i is within the circle of perceived range, the perceived quality is 1, otherwise, it is 0, and the mathematical expression is Generally, the sensor's perception probability of the target is less than 1. In order to improve the perception probability of the target, multiple sensors need to cooperate to detect, so the sensor's perception probability of a certain target is The coverage rate of the node set Z over the entire monitoring area is Equation (4) is the objective function of the WSN coverage optimization problem.

Slime mould algorithm
The slime mould algorithm  mainly simulates the behaviour and morphological changes of slime mould in the foraging process in nature. The algorithm effectively simulates the information transmission mechanism of feeding back the food concentration in the search space through the propagation wave generated by the oscillator and simulates the threat to the existence in the foraging process with the weight factor, which improves the effectiveness of the algorithm. Slime mould can judge the food source position according to the concentration of surrounding food during foraging, and adjust their foraging mode according to the influence of environment. It also can obtain multiple food sources and develop them at the same time. When obtaining information of multiple food sources, biological oscillators will generate different propagation waves according to the concentration of food sources. When the concentration is higher, the cytoplasm flow of veins connecting food will increase, the radius of veins will along with increase, and the channels will become more stable, thus forming the optimal path connecting food sources.
When slime mould obtains multiple food sources, they will adjust their foraging patterns according to food concentration, and when food concentration is high, slime mould will use local search; While the food concentration is low, Slime mould will use the multi-area search mode, more spontaneous to leave the low concentration area to search for food in another area.

Search for food
When slime mould is foraging, they can sense the concentration and position of food through the smell in the air and then approach it constantly. This process can be expressed by the following equation: where the parameter − → kb is a number between [−a, a] and − → kc is linearly decreasing from 1 to 0, t represents the current iteration times, − → S * represents the current optimal individual, − → S represents the current individual, − → S R1 and − → S R2 represent two individuals randomly selected in the slime mould population.
− → W represents weight. The q mathematical formula is as follows: where i ∈ 1, 2, . . . , n, f (i) indicated fitness of individual − → S , DF represents the best fitness value obtained in all iterations. The − → kb mathematical formula is as follows: where − → W is the mathematical formula as follows: where r is a random values interval of [0,1], T max indicates the maximum number of iterations, BF indicates the best fitness value in the current iteration, WF indicates the worst fitness value in the current iteration, smell represents the corresponding values sorted by fitness values. The search position of individual − → S is updated by the best individual − → S * obtained at present and two random individuals − → S R1 and − → S R2 in the search space, and the position of individual − → S is changed by fine tuning oscillation factor − → kb and − → kc weight factor − → W . Equation (5) simulates the uncertainty of venous contraction mode. Logarithmic log is used to reduce the numerical change rate reducing the rate of change of numerical value makes the numerical change of contraction frequency not too large, so that the searching individual can search in all possible directions near the optimal solution, thus simulating the fan-shaped structure of slime mould when it is close to food. This concept can also be applied to extend to hyper-dimensional space.

Wrapped food
In this section, we proposed a mathematical model for the second foraging mode of slime mould to wrapped food. In the process of approaching and then food encirclement, the organic matter in slime mould will secrete enzymes to digest food. If the concentration of food becomes higher, the cytoplasmic flow will also be greater, the greater the radius of the blood vessels, and therefore the greater the weight. On the contrary, when the food concentration in the current search area is low, the cytoplasmic flow becomes less, the vein radius becomes smaller, also the weight becomes smaller, and the slime mould will turn to other areas to search for food. According to the above principle, the second phase formula for updating the position of slime mould is as follows: where ub and lb represent the upper and lower boundaries of the search range, rand and r represent random values between [0,1], and pc is a constant, which is taken as 0.03 in this paper.

Oscillation
During the foraging process, slime mould control the cytoplasmic flow on the vein network structure according to the propagation wave generated by its oscillator, and the width of veins can be adjusted by the flow. This mechanism can make slime mould find a better food concentration position. The algorithm has three variables: − → W , − → kb and − → kc which better simulates the foraging mode of slime mould under the action of oscillation mode.
− → W simulate the oscillation frequency of slime mould close to 1 under different food concentrations, which makes slime mould approach food more quickly when finding high-quality food and more slowly when the food concentration is lower in individual positions, thus improving Algorithm 1. SMA pseudo-code 1. Initialize the parameter, population, ub and lb, Max_iter; 2. Initialize the position of slime mould S i (1, 2, . . . , n); 3. While (t < Max_iter) 4.
Calculate the fitness of all slime mould; 5.
For each search portion 8.
the efficiency of slime mould in selecting the best food source.
The value of − → kb oscillates randomly and approaches 0 with the increase of iteration times. The value of − → kc oscillates between −1-1, and finally tends to 0. The synergistic effect between and simulated the selective behaviour of slime mould. In order to find a better food source, even if slime mould has found a better food source, it will still secrete some organic matter to explore other fields and try to find a higher quality food source instead of investing everything in one source.
In addition, the oscillation process of − → kb simulates the moving state of slime mould and decides whether to approach food sources or seek other food sources. At the same time, the process of exploring food is not smooth. During this period, there may be various obstacles, such as a light and dry environment, which limit the spread of slime mould. However, it also improves the possibility of slime moulds looking for higher quality food and avoids falling into local optimum. The pseudo-code of the SMA is shown in Algorithm 1.

Our proposed SSMA
Slime mould algorithm ) is a newly swarm intelligence optimization algorithm, which is a stochastic optimizer to form the shortest path connecting food based on the oscillation mode generated by slime mould in nature. The algorithm has been concerned and studied by many scholars since it was put forward. The SMOA has the advantages of a simple concept and good results in solving various practical problems. However, like many meta-heuristic optimization algorithms have the disadvantages of premature convergence and easy to fall into local optimum. The stander SMA has obvious defects such as orientation local optimum, low coverage rate and insufficient convergence speed in solving the coverage optimization problem of WSNs. So, we propose an SSMA to solve the optimization problem of WSNs. The performance of the standard slime mould algorithm is not ideal when solving e coverage optimization in WSNs. The SMA for optimization problems mainly depends on the best leader (global best) and two randomly collected slime mould in the population, which leads to poor development and requires more iterations to converge this algorithm. Because slime mould will search in a limited area under the influence of food concentration in space, slime mould will search in the possible range around the best individual or the individual's own range when the food concentration is high, and the probability of random search of slime mould is relatively small when the food concentration is not felt, so the probability of searching in a limited area becomes larger, which leads to premature convergence of stander SMA when solving some optimization problems.
It can be obviously from the above that due to the deficiency of the SMA updating formula, the algorithm cannot achieve good results in solving the coverage optimization problem of WSNs. The exploration and development of the algorithm are not well balanced, which leads to local optimum. In the experiment, we found that the optimized slime mould algorithm mainly depends on the best leader and two randomly collected slime mould in the population. Updating the formula leads to slime mould easily falling into local optimum, which leads to poor development, needs more iterations to converge, and leads to an imbalance between exploration and development. Therefore, we have further improved the updating and oscillation factor. The updated position (Equation (12)) is removed from the development stage of the improved algorithm. In the second part, the standard oscillation factor of SMA is changed into the adaptive oscillation factor of the cosine function, which improves the exploration ability and convergence speed of the algorithm at the searching. The improvement of these two parts makes the exploration and development of SSMA get a good balance, which is also proved in the experimental part.
From the above analysis, a mathematical model of updating the formula can be obtained as follows: where ub and lb represent the upper and lower boundaries of the search range, r and rand represent random values between, and pc is a constant, which is 0.03, in this paper as same the stander SMA . Parameter − → kb is a number between [−a, a], − → kc is a linearly decreasing number from 1 to 0, t represents the current iteration times, − → S * (t) represents the current best individual, − → S represents the current individual, − → S R1 and − → S R2 represents two individuals randomly selected in the slime mould population.
− → W represents the weight. The oscillation factor of SMA has an important relationship with the foraging speed and accuracy of slime mould. We find that the oscillation factor is larger and the search range is wider at the initial stage of iteration, which leads to the slow convergence of the algorithm and the local optimum at the initial stage of search; However, when the oscillation factor range is small at the initial stage of searching, slime mould will accelerate its searching speed and precision. To solve this problem, we propose an improved oscillation factor, and the oscillation factor formula is as follows: where a is the value of the change range of γ , γ = 0.5; λ is the value step of the slime mould, λ = 1. The oscillation factor trend − → kb of SMA and SSMA is shown in Figure 1. During the search, because the range of search elements is not large, the accuracy and speed of search can be improved, and the optimization speed of the algorithm can be improved.
The formula of weight − → W is as follows: Calculate the W by Equation (16); 7.
For each search portion 8.

15.
End For where r is a random number between [0,1], T max represents the maximum iteration times, BF represents the best fitness value in the current iteration, WF represents the worst fitness value obtained by the current iteration times, and smell represents the corresponding values sorted by fitness values. The SSMA mainly depends on the best leader and two random slime mould individuals in the population. Equation (13.2) is the core of the SSMA. When z less than 0.03, slime mould searches randomly in the whole search space, and Equation (13.1) is executed; otherwise, Equation (13.2) is executed. The best individual in Equation (13.2) can lead the development stage of slime mould, while two random individuals in the population mainly dominate the exploration stage of the SSMA. This formula can balance the exploration and development of the algorithm. The pseudo-code of the SMA is shown in Algorithm 2. The intuitive and detailed process of SSMA is shown in Figure 2.

Problem definition
As mentioned in the second section, we will define it as follows: the coverage of sensor nodes is a circle with a fixed radius. Whether a certain position is in the coverage area of the sensor node can be determined by calculating the distance between the target point and the node. In order to evaluate the coverage of WSNs in a twodimensional area, the whole monitoring area is divided into grids. If WSNs can cover points, then the coverage rate is 1. In addition, we assume that all sensors are the same monitoring area, and ignore the boundary effect on the sensor network.
During initialization, all sensor nodes are randomly scattered in a given monitoring area, and the initial coordinates of these sensor nodes are the initial input values of the algorithm. Each search agent in the algorithm represents a placement scheme of sensor nodes. In the two-dimensional monitoring area, the dimension of search agent is twice the number of sensor nodes; In other words, the 2x − 1 dimensional data represents the abscissa of the xth sensor node, while the 2x dimensional data represents its ordinate. The algorithm takes WSN coverage as the fitness function that is Equation (4), as mentioned in the second section, and maximizes the fitness function as the optimization goal. Finally, the optimized coverage and the positions of all sensor nodes are output.

Parameter set
In this section, we compare the experimental results of SSMA with SMA , PSO (Kennedy & Eberhart, 2002), GWO (Mirjalili et al., 2014), WOA (Mirjalili & Lewis, 2016), MPA (Faramarzi et al., 2020) and FPA (Yang, 2012). Table 1 summarizes the experimental parameter settings for these algorithms. In order to eliminate the experimental error caused by chance, the average value  of 20 independent runs is used as the comparison result.
In this paper, we are mainly divided into two kinds of experiments: low dimension and high dimension. Other parameter settings: population size: N, maximum iteration: T; The monitoring radius is: R, the monitoring area is L × M, the sensor of node is: Node, and the dimension is: Dim.

Comparative analysis of experiments
In this sector, we compared the SSMA with some competitive MAs on WSN coverage optimization. Furthermore, all of the experimental series were performed on MATLAB  2017b and were run on a CPU Core i3-6100 v4 (3.70 GHz) with 8 GB RAM.
In the experimental part, we will set 13 groups of coverage optimization experiments, among which there are two groups of low-dimensional ones, which are divided into Case1-Case3 and C1-C4, the number of nodes is unequal between 5 and 80 nodes, besides, there are six groups of high-dimensional experiments, mainly from 100 to 1200 nodes.

Low-dimension experimental analysis
In this section, we mainly test and analyze the number of nodes in the low dimension. There are two types of cases in low dimension: Case1-Case3; C1-C4. The case Case1 to Case3 comes from the literature (Miao et al., 2020), C1-C4 are self-defined cases in this paper, and the parameter settings of the examples are shown in Tables 2 and 3. The experiments mainly compared the experimental results of SSMA with six state-of-the-art algorithms to analyze the robustness and convergence speed. In the previous section, we used the above two types with different ratios for experimental comparison. Case 1-Case 3 compares these three cases through Tables 4-7 and Figures 3-5,  moreover, Tables 8-10 and Figures 6-8 represent the data tables and graphs of C1-C4 cases. Respectively, best values, worst values, average values and STD represent the best value, the worst value and STD of coverage. Black bold numbers represent the best value, dark grey bold represents the second-best value, and light grey bold represents the third best. Tables 4-7 respectively show the best value, the worst value, the average value and the standard deviation of independent operation for 20 times.
It can be clearly seen from Table 4 (best value) that PSO, SMA, FPA, GWO, WOA and MPA rank first in the coverage rate of Case1, which is 8% higher than PSO and SMA rank second in the coverage rate; In Case2, SSMA ranked first, with a coverage rate of 99%, and MPA ranked second was 1% higher than PSO, FPA, GWO, WOA and SMA by 21%, 13%, 5%, 11%, 1% and 19%. The coverage rate of SSMA in Case3 is 94.22%, which is 27.11%, 16%, 5.78%, 13.33%, 3.11% and 21.78% higher than PSO, FPA, GWO, WOA, MPA and SMA, respectively. In Case1 in Table 5 (the worst value), SSMA ranks second In Case2 and Case3, SSMA ranked first, MPA ranked second and FPA ranked third. It can be seen from  ranks first, while FPA ranks second, GWO and MPA rank third in Case1; MPA ranks second and GWO ranks third in Case2 and Case3. In Table 7 (standard deviation), SSMA still ranks in the top three, among which in Case1, SSMA ranks first with a standard deviation of 0.0239, FPA ranks second with 0.0178, and GWO ranks third with 0.0275; In Case2, FPA ranked first with a standard deviation of 0.0081, SSMA ranked second with a standard deviation of 0.0139 and SMA ranked third with a standard deviation of 0.0157. In Case3, SSMA ranks third, with a value of 0.0193, FPA ranks first, and SMA ranks third. From Figures 3-5, it can be seen that the performance of SSMA is superior to several other algorithms compared with it. The convergence curve of Case1-Case3 in Figure 3 shows that the convergence curve of SSMA is much faster than the other six algorithms, and other algorithms are   easy to fall into the local optimum or are not suitable for this problem, there is no way to optimize. In Figure 4, we can see that the standard deviation of SSMA is more stable and higher than other algorithms. As can be seen from the coverage in Figure 5, the coverage of SSMA is superior than other algorithms. As the dimensionality increases, the coverage performance is outstanding. Tables 8-11 are comparative data of low-dimensional examples C1-C4, which respectively show the best value, the worst value, the average value and the standard deviation of independent operation for 20 times. It can be seen from Table 8 (Best values) that SSMA ranks first in C1-C4, with coverage rates of 0.6356, 0.8797, 0.9977 and 0.8662 respectively. The coverage rate of SSMA in C1 is 0.16%, 2.76%, 0.24%, 1%, 0.48% and 7.43% higher than that of PSO, FPA, GWO, WOA, MPA and SMA respectively. The coverage rate of SSMA in C2 is 8.36%, 13.37%, 1.03%, 9.59%, 6.41% and 17.8% higher than that of PSO, FPA, GWO, WOA, MPA and SMA respectively. In C3, the coverage rate is 7.46%, 8.13%, 1.22%, 4.39%, 1.73% and 10.83% higher than that of PSO, FPA, GWO, WOA, MPA and SMA respectively. The coverage rate of SSMA in C4 is 19.59%, 13.78%, 3.27%, 8.48%, 2.58% and 18.81% higher than that of PSO, FPA, GWO, WOA, MPA and SMA, respectively. The four examples of SSMA in Table 9 (Worst values) rank first, among which MPA ranks second in C1, GWO is ranked second in C2 and MPA is ranked third. MPA is ranked second in C3 and C4, and GWO is ranked third. It can be seen from Table 10 (Average) that the average coverage rate of SSMA is significantly higher than that of the other six algorithms. The average coverage rate of SSMA in C1 reached 63.05%, ranking first, which was 1.61%, 4.54%, 0.56%, 3.07%, 1.66% and 10.36% higher than PSO, FPA, GWO, WOA, MPA and SMA respectively. The average  coverage rate of SSMA in C2 reached 86.43%, which was 13.97%, 15.07%, 2.55%, 11.44%, 7.9% and 19.35% higher than PSO, FPA, GWO, WOA, MPA and SMA respectively. The average coverage rate of SSMA in C3 reaches 99.47%, which is 13.46%, 8.76%, 6.89%, 5.79%, 3.2% and 12.41% higher than PSO, FPA, GWO, WOA, MPA and SMA respectively. The average coverage rate of SSMA in C4 is 85.13%, which is 21.58%, 13%, 10.64%, 9.64%, 4.14% and 15.15% higher than PSO, FPA, GWO, WOA, MPA and SMA respectively. In Table 11 (Standard Deviation), SSMA ranks among C1-C3 three examples FPA ranks second and GWO ranks third in C2. FPA ranks second and SMA ranks third in C3. FPA ranks first and SMA ranks second in C4. From the above data display and analysis, it can be seen that SSMA has obvious advantages in solving the coverage optimization problem of WSNs, and has relatively strong robustness and fast convergence speed. Figures 6-8 show the C1 to C4 of each algorithm, Figure 6 shows the convergence curve, Figure 7 shows the variance diagram, and Figure 8 shows the sensor coverage of each algorithm. From Figures 6-8, it can be concluded that SSMA has superior performance in coverage optimization in WSNs on the whole, and MPA has available performance in this optimization problem, which is generally second only to SSMA.

Large-scale experimental analysis
In this section, we test the proposed SSMA on a largescale wireless sensor node and compare the results with other algorithms. The experimental parameters are the same as before. In experiments, the population size of the algorithm is set to 30, the maximum number of iterations is 100, and the average value of the parameters is taken after 20 independent operations of the algorithm. Table  12 has list the other parameters set.
Other algorithms may adapt in small-scale experiments, but they may lose performance in large-scale experiments. However, the SSMA is still applicable in the     Figure 9 shows the coverage convergence curve from C5 to C10. From the convergence graph, it can be seen that the convergence curve of SSMA is faster than other algorithms, and the optimization speed is stronger. With the increase of dimensions, the ability of optimization is less obvious and the difficulty of optimization increases. Figure 10 shows the graph of standard deviation, from which it can be seen that SSMA is the best, and Figure  11 shows the optimization of node distribution of randomly selected samples in each case. It can be observed that with the increase of iteration times, the network coverage gradually rises to the optimal value. However, the increase of the dimension of the optimization problem brings great challenges to the algorithm. It can be seen that with the increase of dimensions, the performance of optimization becomes a challenge, and the performance of optimization gradually weakens on high-dimensional tasks.

Analysis of statistical significance
Wilcoxon sum-rank test (WSRT), as a non-parameter test, can effectively evaluate the statistically significant difference between the two optimization algorithms.  Tables 14-16 show the P-values of the Wilcoxon test (Herrmann, 1984) obtained by different SSMA for 13 optimization questions, which are statistically significant when the significance level is 0.05. When we calculate the P-value higher than 0.05, it means that there is no significant difference between the two methods. On the contrary, when the P-value is less than 0.05, we can see that there are great differences between them. We use the Wilcoxon test to compare the performance difference between the two methods. Test results are shown in Table 12-14. '+' indicates that the SSMA algorithm is superior to the comparison algorithm, '−' indicates that the SSMA algorithm is second only to competitors, and ' = ' indicates that there is no difference in performance between SSMA algorithm and comparison algorithm for each example in 20 runs. The results highlight the obvious advantages of SSMA over all other competitors in two different dimensional examples.
It can be seen from Tables 14-16 that the performance of our proposed SSMA is better than that of other algorithms. In addition, in Table 17, we use the Friedman test to rank several algorithms (Ashcroft & Pereira, 2002). According to the mean rank obtained by the Friedman test, the maximum mean rank of SSMA variables is 6.95, which indicates that the overall performance of the proposed SSMA in WSN coverage optimization is the best. The rank of other test algorithms is MPA > GWO > FPA > WOA > SMA > PSO.

Conclusions and future direction
In this paper, an SSMA is proposed. It is based on the standard SMA to solve the coverage optimization problem of WSNs. For the SSMA, we have mainly made two improvements. The first was to the improved position-updating equation, mainly to improve the optimization performance of the SSMA and improve the coverage rate; the second is the improvement of the oscillation factor. The second part of the improvement was mainly improved the optimization speed in the early stage of search and the convergence speed. In the experiment section, we mainly conducted 13 groups of experiments: lowdimensional experiments and high-dimensional experiments. Low-dimensional experiments are mainly divided into two categories: small radius and large radius; There were mainly six groups in high-dimensional experiments. The experimental part is mainly compared with six stateof-the-art algorithms, namely PSO, FPA, GWO, WOA, MPA and SMA. In addition, Wilcoxon rank-sum test and Friedman test are used to determine the significant difference between the results of SSMA and other competitors. Experimental results show that these improvements to Figure 11. Node distribution optimization on C 5 and C 7. SSMA can improve search efficiency and speed up convergence. On this optimization problem, the proposed SSMA was superior to most test algorithms. The application shows that the SSMA is an efficient and reliable optimization algorithm. For future work, first, the proposed SSMA can be applied to other real-world problems, such as route path planning, transportation safety management, job shop scheduling, neural networks (Huang, 1999;Huang & Du, 2008;Zhang et al., 2006), clustering analysis (Du et al., 2007), and function approximate (Han & Huang, 2006), graph colouring problem. Second, other versions of SSMA can be extended, such as   multi-objective version, complex version, binary version, quantum coding version, etc. Finally, combining SSMA with other algorithms may be a promising aspect. Fourth, the updating formula and parameters of SSMA can be improved, and the correctness can be verified by experiments on benchmark problems.