Coverage Maximization of WSNs in 3D Space Based on Hybrid Lion Swarm Optimization

Abstract

The wireless sensor network (WSN) collects all the physical information required by the user for the target area and transmits the data to the sink or base station via multihop communication. With a limited number of nodes, it is important to reduce the overlapping area between nodes to improve coverage. However, increasing the coverage rate tends to create isolated points, which reduces the connection efficiency. Therefore, considering coverage and connection efficiency simultaneously is a challenging optimization task. In order to solve the above problems, this paper proposes a spatial configuration optimization strategy based on the hybrid lion swarm optimization (HLSO) algorithm. The proposed algorithm is experimented on a large number of benchmark graphs. The experimental results prove very competitive with and even better than different algorithms. Finally, in order to avoid isolated points in the node deployment process, the optimal deployment scheme of the nodes is found by considering both conditions of node coverage and connection efficiency at the same time.

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1. Introduction

The Internet of Things is a new type of network that interconnects things through wireless sensor networks. The nodes in the network have computing, storage, and wireless communication capabilities to collect various physical information (e.g., environmental data such as temperature, light intensity, and humidity) required by the user for the target area and transmit it to the sink or base station. If the node is located within the transmission range of the base station, then it transmits directly. On the contrary, other nodes are considered relay points for data transmission through multihop communication. It is widely used in biomedicine, crop growth, fire monitoring systems, traffic control, and indoor condition monitoring.

To meet the coverage requirements of the target area, a large-scale random deployment in the target area is generally used. However, high node deployment density causes energy waste in receiving, processing, and forwarding data, while low density tends to create blind spots and affect monitoring quality. Therefore, to avoid overlapping node coverage that causes monitoring and communication blind areas and reduces the waste of network resources transmission, thus extending the network life cycle, this paper uses the swarm intelligence algorithm to optimize sensor node deployment and improve sensor network coverage [1–4].

In recent years, there are a number of related research papers on the WSN coverage optimization problem. Liang and Lin [5] proposed a genetic algorithm based on WSN topological and node mobility requirements with rational crossover and mutation to improve the coverage problem of mobile objects. In [6], the multiobjective optimization problem of maximizing the WAN coverage and the shortest moving distance of the initial node is considered at the same time, and the firefly algorithm is used for optimization. To avoid long-distance communication between nodes causing excessive consumption of battery power of sensors and reducing the lifetime of nodes, Zhang and Lin [7] proposed an efficient crossover method based on an evolutionary algorithm to solve such complex optimization problems. Donta et al. proposed a new algorithm to solve the point-sweep coverage (NP-hard) problem. The goal of this approach is to visit each point of interest with a minimum number of mobile sensor nodes. It can improve efficiency and reduce unnecessary energy loss compared to static sensor deployment [8].

The swarm intelligence algorithm originated from particle swarm optimization (PSO) in 1995 [9], which mainly achieves the solution space search by moving inertia, individual optimal vectors, and population optimal vectors in three directions for updating. The dragonfly algorithm (DA) [10] is a simulation of the five behavioral patterns of separation, alignment, cohesion, attraction, and distraction outwards from an enemy in dragonfly ecology to search in the solution space [11, 12]. In addition, there is the sine cosine algorithm (SCA) [13, 14] that uses sine and cosine mathematical models to realize algorithmic individual exploration and exploitation to solve optimization problems. Mirjalili et al. proposed the whale optimization algorithm (WOA) in 2016, a novel optimization algorithm developed with reference to the whale’s spiral bubble net attack and whale hunting behavior [15, 16]. In 2017, the population was divided into leaders and followers with reference to the ecology of barrel animals, and the concept of Newton’s equations of motion was introduced for mathematical modeling, called the salp swarm algorithm (SSA) [17, 18]. The lion swarm optimization (LSO) [19] divided the lion population into lion king, lioness, and lion cubs, and assigned different update patterns to achieve local and full domain search performance. The specific contributions of this study are as follows:

(1) Combining the spiral bubble net attack increases the diversity of update mechanisms and adjusts the perturbation factor parameters to improve the switching of the algorithm between exploration and exploitation operations

(2) A set of 100 benchmarks collected from [20–25] was used to test the performance of the HLSO algorithm. The used benchmark set contains the fixed-dimensional benchmark functions, the high-dimensional benchmark functions, and the CEC-C06 2019 benchmark functions. The test results show that the HLSO has excellent search performance

(3) The HLSO is applied to WSN in the 3-dimensional space coverage optimization problem compared with other optimization algorithms, which can effectively improve the coverage of nodes

2. Related Works

The LSO algorithm proposed in 2018 is a novel biomimetic algorithm that simulates the ecology of a lion population [19], where the population is sorted by fitness value and then proportionally divided into adult lions and cubs. In order to jump out of the local optimal solution and exact search, the evolutionary process develops different update methods according to different identities and combines the periodicity to reset the individual update mechanism. This algorithm has been widely used in various engineering optimization problems [26–30].

Qiao et al. [31] combined the LSO algorithm with the reproduction and mutation mechanism to optimize the traditional least squares support vector machines (LS-SVM) model and applied the optimization model to predict the CO₂ emissions of each country. The equilibrium factor [32] is proposed for the algorithm under the need for exploration and development, so that the lion population individual update mechanism is more dependent on the number of populations and iterations and can be effectively used for the path planning problem. The random values of individual pattern selection were analyzed, and the concept of normally distributed random numbers was proposed to optimize the lion cub update pattern strategy [33]. The test results showed better convergence accuracy compared to the original algorithm.

3. Hybrid Lion Swarm Optimization Algorithm

The original LSO algorithm is a class of population intelligence algorithms with a careful division of labor, but there are still some problems. (i) The linearly decreasing perturbation factor of the cubs affects the performance of local exploration in the later stages of the algorithm. In contrast, the adult lion’s perturbation factor switches too quickly between global exploitation and local exploration. (ii) The structure of the lion cub update formula is very similar, being the mean of targets and individuals multiplied by a random term and therefore lacking in diversity. (iii) The fixed-period mode switching mechanism does not guarantee its effectiveness for benchmark functions with different characteristics.

In this study, the structure of the original algorithm was adjusted, and an individual retention mechanism was introduced to ensure the effectiveness of individual updates, while a combination of spiral bubble net attacks was introduced to increase the diversity of lion cub updates. The details of the HLSO algorithm are as follows.

3.1. Arithmetic Strategy

First, to balance the perturbation factor switching between global exploitation and local exploration and to improve the search accuracy in the later stage of the algorithm, the perturbation factor is modified as follows $$\begin{array}{}\text{(1)}& D_f=\text{step}\times \exp -10{\frac{t}{T}}^3,\text{(2)}& \text{step}=0.1\cdot \overline{B}-\underset{¯}{B},\end{array}$$where $t$ is the current number of iterations, $T$ is the total number of iterations, and $\overline{B},\underset{¯}{B}$ are the upper and lower limits of the variables, respectively. The red line in Figure 1 shows the original perturbation factor curve, while the blue line shows the modified curve of the perturbation factor.

[figure(s) omitted; refer to PDF]

If ±0.1 is used as the basis of judgment, it can be observed from the graph that the original curve remains in the global exploration state in the first 60%. The intermediate transition period is only 20%. The exact search in the later period only accounts for 20%, which substantially affects the accuracy of the algorithm. If the curve data are converted into a box plot as in Figure 1(b), it is found that the original perturbation factor maximum, Q3, and median line are closer to 1 better indicates the unreasonable parameter settings. And our proposed parameters correspond to global search, transition, and local search in the proportional relationship of 20%, 40%, and 40%. The accuracy of the algorithmic search can be improved.

Assuming that the number of lion individuals $N_{id}$ in the $D$ dimension is divided into adults and cubs, the adults are the better-valued individuals, and their number can be described by the following equation: $$\begin{array}{}\text{(3)}& L_{\text{adult}}=\rho N_{\text{id}},\end{array}$$where $\rho$ is the scale parameter for adult lions, usually less than 0.5.

The adult lion performs a local search for the current optimal solution, and the update formula for the $i_{\text{th}}$ adult lion for the $j$ variable at the $t+1$ iteration is modified using the following rule: $$\begin{array}{}\text{(4)}& x_{i,j}^{t+1}=g_{j}1+D_f\cdot \gamma \cdot p_{i,j}-g_j,\end{array}$$where $\gamma$ is the random number generated by the normal distribution $\mathbb{N}0,1$, $p_{i,j}$ is the individual optimal solution of the current individual, and $g_j$ is the group optimal solution.

In the original algorithm, the cubs were developed with three update mechanisms to increase the capacity and diversity of the algorithm, and the update method was determined by a random value $q$. The main purpose of the updated formulation with the optimal solution as the goal is to achieve an exact local search.

In Figure 2, the red square is the current position of the individual, the black circle is the population optimal solution, the blue dots are the possible distribution of the original update mechanism, and the yellow dots are the possible distribution of the attack using the spiral bubble net. From Figure 2(a), we can observe that the original update mechanism is mostly distributed between two points, while the spiral bubble net attack is randomly distributed with the population optimal solution at the center. In particular, if the arbitrary sum of variables between the current position and the optimal solution is 0. The original update approach results in an extreme distribution, as shown in Figure 2(b), which drastically reduces the diversity of the local search. Therefore, we introduce the spiral bubble net attack to replace the first update formula to improve the diversity of local search. The $j_{\text{th}}$ variable of the $i_{\text{th}}$ lion cub is updated in the $t+1$ iteration with the following formula. $$\begin{array}{}\text{(5)}& x_{i,j}^{t+1}=\begin{array}{cc}{p}_{i,j}-g_j\cdot \exp ℝ\cdot \cos 2\pi ℝ+g_j,\hfill & \text{if}q\le \frac{1}{3},\hfill \\ \frac{1}{2}{p}_{m,j}+p_{i,j}\cdot 1+D_f\cdot \gamma ,\hfill & \text{if}\frac{1}{3}<q\le \frac{2}{3},\hfill \\ \frac{1}{2}{\overline{g}}_j+p_{i,j}\cdot 1+D_f\cdot \gamma ,\hfill & \text{if}\frac{2}{3}<q\le 1,\hfill \end{array}\text{(6)}& {\overline{g}}_j=\overline{B}+\underset{¯}{B}-g_j,\end{array}$$where $ℝ$ is a random value from -1 to 1 and $p_{m,j}$ is the optimal solution for any individual adult lion selected for collaboration. In addition, when a new better solution is found, the individual is updated; otherwise, it returns to the original position.

[figure(s) omitted; refer to PDF]

3.2. Algorithm Process

The procedure of the HLSO algorithm is shown below.

Algorithm 1: HLSO algorithm.

1:  Set the total number of iterations T, the total number of individuals N_id, the dimension D, and the lion proportion parameter ρ. Randomly generate the initial position of the lion population.

2:  while t < T or achieve stop criterion

3:   Calculate the perturbation factor according to equation (1)

4:   Calculate the number and fitness values of adult and cub lions according to equation (3)

5:   for i = 1:Number of adult lions

6:    Update the position of the adult lion according to equation (4)

7:   end for

8:   for i = 1:Number of cubs

9:    q = rand();

10:    if q < 1/3

11:      Update the location according to the spiral bubble net attack mechanism

12:    else if 1/3 < q <2/3

13:      The cub moves around the lioness and learns to hunt with the lioness

14:    Else

15:      The cub is driven away from the lion king

16:    end if

17:   end for

19:    If the new solution is better than the current solution, move to the new position

20:   end for

21:  end while

According to the above algorithm flow, it can be observed that the complexity is related to the number of iterations, the number of populations, and the algorithmic mechanism. Since the algorithmic process must be sorted by QuickSort. Therefore, the overall computational complexity is as follows: $$\begin{array}{}\text{(7)}& O\text{HLSO}=OTO\text{QuickSort}+N\times D=OT{N}^2+N\times D,\end{array}$$where $T$ is the total number of iterations, $N$ is the number of populations, and $D$ is the problem dimension.

4. Experiment and Analysis

To verify the performance of the HLSO algorithm, 100 benchmark functions were collected from the literature [20–25], containing the fixed-dimensional benchmark functions (f₀₁~f₅₄), as shown in Table 1, the high-dimensional benchmark functions (f₅₅~f₉₀), as shown in Table 2, and the CEC-C06 2019 benchmark functions (f₉₁~f₁₀₀), as shown in Table 3.

Table 1

Information of benchmark functions with fixed dimensions.

No.	Function	$D$	Range	$f^{\ast }$
f01	$$-200\exp -0.02\sqrt{x^2+y^2}+5\exp \cos 3x+\sin 3y$$	2	±32	-195.629
f02	$$\cos x\sin y-x{y^2+1}^{-1}$$	2	±2	-2.0218
f03	$$-200\exp -0.2\sqrt{x^2+y^2}$$	2	[0,10]	-6.1295
f04	$$x^2+y^2+xy+\sin x+\cos y$$	2	±500	1
f05	$${1.5-x+xy}^2+{2.25-x+{xy}^2}^2+{2.625-x+{xy}^3}^2$$	2	±4.5	0
f06	$$\sin x\exp {1-\cos y}^2+\cos y\exp {1-\sin x}^2+{x-y}^2$$	2	±2π	-106.7645
f07	$$x^2+2y^2-0.3\cos 2\pi x-0.4\cos 4\pi y+0.7$$	2	±100	0
f08	$$x^2+2y^2-0.3\cos 3\pi x\cos 4\pi y+0.3$$	2	±100	0
f09	$$x^2+2y^2-0.3\cos 3\pi x+4\pi y+0.3$$	2	±100	0
f10	$${x+2y-7}^2+{2x+y-5}^2$$	2	±10	0
f11	$$y-5.1{4{\pi }^2}^{-1}{x}^2+5{\pi }^{-1}x-6+101-{8\pi }^{-1}\cos x+10$$	2	[-5,10],[0,15]	0.398
f12	$$100\sqrt{y-0.01x^2}+0.01x+10$$	2	[-15,3]	0
f13	$$-{10}^{-4}{\sin x\sin y\exp 100-{\pi }^{-1}\sqrt{x^2+y^2}+1}^{0.1}$$	2	±10	-2.0626
f14	$$100{x_1^2-x_2}^2+{x_1-1}^2+{x_3-1}^2+90{x_3^2-x_4}^2$$$$+10.1{x_2-1}^2+{x_4-1}^2+19.8x_2-1x_4-1$$	4	±10	0
f15	$${10}^5{x}^2+y^2-{x^2+y^2}^2+{10}^{-5}{x^2+y^2}^4$$	2	±20	-24771.09
f16	$$-\cos x\cos y\exp -{x-\pi }^2+{y-\pi }^2$$	2	±100	-1
f17	$${\sum }_{i=1}^2{{\sum }_{j=1}^2{a}_{ij}\sin {\alpha }_j+b_{ij}\cos {\alpha }_j-{\sum }_{j=1}^2{a}_{ij}\sin x_j+b_{ij}\cos x_j}^2$$	2	±π	0
f18	$${\sum }_{i=1}^5{{\sum }_{j=1}^5{a}_{ij}\sin {\alpha }_j+b_{ij}\cos {\alpha }_j-{\sum }_{j=1}^5{a}_{ij}\sin x_j+b_{ij}\cos x_j}^2$$	5	±π	0
f19	$${\sum }_{i=1}^{10}{{\sum }_{j=1}^{10}{a}_{ij}\sin {\alpha }_j+b_{ij}\cos {\alpha }_j-{\sum }_{j=1}^{10}{a}_{ij}\sin x_j+b_{ij}\cos x_j}^2$$	10	±π	0
f20	$$1/0.002+{\sum }_{j=1}^{25}{j+{x_1-a_{1j}}^6+{x_2-a_{2j}}^6}^{-1}$$	2	±65.536	0.998
f21	$$1+{x+y+1}^{2}19-14x+3x^2-14y+6xy+3y^2$$$$\times 30+{2x-3y}^{2}18-32x+12x^2+4y-36xy+27y^2$$	2	±2	3
f22	$$-{\sum }_{i=1}^4{\alpha }_i\exp -{\sum }_{j=1}^3{A}_{ij}{x_j-P_{ij}}^2$$	3	[0,1]	-3.86
f23	$$-{\sum }_{i=1}^4{\alpha }_i\exp -{\sum }_{j=1}^6{A}_{ij}{x_j-P_{ij}}^2$$	6	[0,1]	-3.32
f24	$${x^2+y-11}^2+{x+y^2-7}^2$$	2	±6	0
f25	$$-\sin x\cos y\exp 1-{\pi }^{-1}\sqrt{x^2+y^2}$$	2	±10	-19.2085
f26	$$-{x^2+y^2}^{-0.5}{\sin }^{2}x-y{\sin }^{2}x+y$$	2	[0,10]	-0.6737
f27	$${\sum }_{i=1}^{11}{a_i-x_1{b}_i^2+b_i{x}_2{b_i^2+b_i{x}_3+x_4}^{-1}}^2$$	4	±5	0.00031
f28	$${\sum }_{i=1}^5{c}_i\exp -{\pi }^{-1}{\sum }_{j=1}^2{x_j-A_{ij}}^2\cos \pi {\sum }_{j=1}^2{x_j-A_{ij}}^2$$	2	[0,10]	-4.1556
f29	$$100{y-x^3}^2+{1-x}^2$$	2	[0,10]	0
f30	$${\sin }^{2}3\pi x+{x-1}^{2}1+{\sin }^{2}3\pi y+{y-1}^{2}1+{\sin }^{2}2\pi y$$	2	±10	0
f31	$$0.26x^2+y^2-48\times {10}^{-2}\times xy$$	2	±10	0
f32	$$\sin x+y+{x-y}^2-1.5x+2.5y+1$$	2	[-3,4]	-1.9133
f33	$${\sum }_{i=1}^2\sin x_i\sin {{ix}_i^2{\pi }^{-1}}^{20}$$	2	[0,π]	-1.8013
f34	$${\sum }_{i=1}^5\sin x_i\sin {{ix}_i^2{\pi }^{-1}}^{20}$$	5	[0,π]	-4.6877
f35	$${\sum }_{i=1}^{10}\sin x_i\sin {{ix}_i^2{\pi }^{-1}}^{20}$$	10	[0,π]	-9.6602
f36	$${\sum }_{i=1}^D{{\sum }_{j=1}^D{j}^i+0.5{j^{-1}{x}_j}^i-1}^2$$	4	±4	0
f37	$${\sum }_{i=1}^{D/4}{x_{4i-3}+10x_{4i-2}}^2+5{x_{4i-1}-x_{4i}}^2+{x_{4i-2}-2x_{4i-1}}^4+10{x_{4i-3}-x_{4i}}^4$$	24	[-4,5]	0
f38	$${\sum }_{i=1}^4{x_i}^{i+1}$$	4	[0,4]	0
f39	$$0.5+{\sin }^2{x}^2-y^2-0.5{1+{10}^{-3}{x}^2+y^2}^{-2}$$	2	±100	0
f40	$$0.5+{\sin }^2{x^2+y^2}^2-0.5{1+{10}^{-3}{x}^2+y^2}^{-2}$$	2	±100	0
f41	$$0.5+{\sin }^2{x}^2-y^2-0.5{1+{10}^{-3}{x}^2+y^2}^{-2}$$	2	±100	0
f42	$$0.5+{\sin }^2\cos x^2-y^2-0.5{1+{10}^{-3}{x}^2+y^2}^{-2}$$	2	±100	0.00156685
f43	$$0.5+{\cos }^2\sin x^2-y^2-0.5{1+{10}^{-3}{x}^2+y^2}^{-2}$$	2	±100	0.292579
f44	$$-{\sum }_{i=1}^5{{\sum }_{j=1}^4{x_j-C_{ji}}^2+{\beta }_i}^{-1}$$	4	[0,10]	-10.1532
f45	$$-{\sum }_{i=1}^7{{\sum }_{j=1}^4{x_j-C_{ji}}^2+{\beta }_i}^{-1}$$	4	[0,10]	-10.4028
f46	$$-{\sum }_{i=1}^{10}{{\sum }_{j=1}^4{x_j-C_{ji}}^2+{\beta }_i}^{-1}$$	4	[0,10]	-10.5363
f47	$${\prod }_{i=1}^2{\sum }_{j=1}^5\cos j+1x_i+j$$	2	±10	-186.7309
f48	$$4x_1^2-2.1x_1^4+1/3x_1^6+x_1{x}_2-4x_2^2+4x_2^4$$	2	±5	-1.03163
f49	$$25+{\sum }_{i=1}^5{x}_i$$	5	±-5.12	0
f50	$$2x^2-1.05x^4+6^{-1}{x}^6+xy+y^2$$	2	±5	0
f51	$${\sum }_{i=1}^6{x_i-1}^2-{\sum }_{i=2}^6{x}_i{x}_{i-1}$$	6	±36	-50
f52	$${\sum }_{i=1}^{10}{x_i-1}^2-{\sum }_{i=2}^{10}{x}_i{x}_{i-1}$$	10	±100	-210
f53	$$4/3{x^2+y^2-xy}^{0.75}+z$$	3	[0,2]	0
f54	$${\sum }_{i=1}^{10}{x}_i^2+{{\sum }_{i=1}^{10}0.5{ix}_i}^2+{{\sum }_{i=1}^{10}0.5{ix}_i}^4$$	10	[-5,10]	0

Table 2

Information on benchmark functions with high dimensionality.

No.	Function	$D$	Range	$f^{\ast }$
f55	$$-20\exp -0.2\sqrt{D^{-1}{\sum }_{i=1}^D{x}_i^2}-\exp D^{-1}{\sum }_{i=1}^D\cos 2\pi x_i+20+\exp 1$$	10/30/50	±32	0
f56	$$-200\exp -0.2\sqrt{{\sum }_{i=1}^D{x}_i^2}$$		±32	-200
f57	$${\sum }_{i=1}^{D-1}{e}^{-0.2}\sqrt{x_i^2+x_{i+1}^2}+3\cos 2x_i+\sin 2x_{i+1}$$		±35	-4.5901
f58	$${\sum }_{i=1}^D{x}_i\sin x_i+0.1x_i$$		±10	0
f59	$${\sum }_{i=1}^D{x_i+10}^2+\exp -{\sum }_{i=1}^D{x}_i^2$$		[-20,0]	Exp(-100D)
f60	$${\sum }_{i=1}^{D-1}{x_i^2}^{x_{i+1}^2+1}+{x_{i+1}^2}^{x_i^2+1}$$		±4	0
f61	$${{\sum }_{i=1}^D{x}_i^2}^2$$		±100	0
f62	$${x_1-1}^2+{\sum }_{i=2}^{D}i{2x_i^2-x_{i-1}}^2$$		±10	0
f63	$$-{0.5{\sum }_{i=1}^D{x}_i^2+2}^{-1}1+\cos 12\sqrt{{\sum }_{i=1}^D{x}_i^2}$$		±5.2	-1
f64	$${\sum }_{i=1}^D{x}_i^2+25{\sum }_{i=1}^D{\sin }^2{x}_i$$		±5	0
f65	$$-\exp -0.5{\sum }_{i=1}^D{x}_i^2$$		±2	-1
f66	$$1+{\sum }_{i=1}^{D}1/4000x_i^2-{\prod }_{i=1}^D\cos x_i/\sqrt{i}$$		±600	0
f67	$$\pi /D10{\sin }^2\pi y_1+{\sum }_{i=1}^{D-1}{y_i-1}^{2}1+10{\sin }^2\pi y_{i+1}+{y_D-1}^2+{\sum }_{i=1}^{D}u{x}_i,10,100,4$$		±50	0
f68	$$0.1{\sin }^2\pi 3x_1+{\sum }_{i=1}^{D-1}{x_i-1}^{2}1+{\sin }^{2}3\pi x_{i+1}+{x_D-1}^2$$$$\cdot 1+{\sin }^{2}2\pi x_D+{\sum }_{i=1}^{D}u{x}_i,5,100,4$$		±50	0
f69	$$1+{\sum }_{i=1}^D{\sin }^2{x}_i-0.1\exp -{\sum }_{i=1}^D{x}_i^2$$		±10	0.9
f70	$${\sum }_{i=1}^D{x_i^2-i}^2$$		±500	0
f71	$${\sum }_{i=1}^D{ix}_i^4+\text{random}0,1$$		±1.28	0
f72	$$10D+{\sum }_{i=1}^D{x}_i^2-10\cos 2\pi x_i$$		±5.12	0
f73	$$x_1+2{{\sum }_{i=2}^D{x}_i^2}^{0.1}$$		±5	−LB
f74	$${\sum }_{i=1}^D{10}^2{x_{i+1}-x_i^2}^2+{1-x_i}^2$$		±30	0
f75	$$1-\cos 2\pi \sqrt{{\sum }_{i=1}^D{x}_i^2}+1/10\sqrt{{\sum }_{i=1}^D{x}_i^2}$$		±100	0
f76	$$418.9829D-{\sum }_{i=1}^D{x}_i\sin \sqrt{x_i}$$		±500	-418.9829D
f77	$${\sum }_{i=1}^D{{\sum }_{j=1}^i{x}_i}^2$$		±100	0
f78	$${\sum }_{i=1}^D{x}_i$$		±100	0
f79	$$\underset{i=1,\cdots ,D}{\max }{x}_i$$		±100	0
f80	$${\sum }_{i=1}^D{x}_i+{\prod }_{i=1}^D{x}_i$$		±10	0
f81	$${\sum }_{i=1}^D{x}_i^{10}$$		±10	0
f82	$${\sum }_{i=1}^D{\sum }_{j=1}^{5}j\sin j+1x_i+j$$		±10	-14.8365D
f83	$${\sum }_{i=1}^D{x}_i^2$$		±100	0
f84	$${\sum }_{i=1}^D{x_i+0.5}^2$$		±100	0
f85	$$0.5\times {\sum }_{i=1}^D{x}_i^4-16x_i^2+5x_i$$		±5	-39.16599D
f86	$${\sum }_{i=1}^D{ix}_i^2$$		±10	0
f87	$${\sum }_{i=1}^D{\epsilon }_i{x_i}^i,{\epsilon }_i=\mathrm{rand}$$		±5	0
f88	$${\sum }_{i=1}^D{\epsilon }_i{x_i-i^{-1}}^i,{\epsilon }_i=\mathrm{rand}$$		±5	0
f89	$${\sum }_{i=1}^D{x}_i\exp -{\sum }_{i=1}^D\sin x_i^2$$		±2π	0
f90	$$\exp -{\sum }_{i=1}^D{x_i/15}^{10}-2\exp -{\sum }_{i=1}^n{x}_i^2{\prod }_{i=1}^D{\cos }^2{x}_i$$		±2π	-1

Table 3

Benchmark function information for CEC C06 2019.

No.	Function	Dim	Range	$f^{\ast }$
f91	Storn’s Chebyshev polynomial fitting problem	9	±8192	1
f92	Inverse Hilbert matrix problem	16	±16384	1
f93	Lennard-Jones minimum energy cluster	18	±4	1
f94	Rastrigin’s function	10	±100	1
f95	Griewank’s function	10	±100	1
f96	Weierstrass function	10	±100	1
f97	Modified Schwefel’s function	10	±100	1
f98	Expanded Schaffer’s F6 function	10	±100	1
f99	Happy cat function	10	±100	1
f100	Ackley function	10	±100	1

For the benchmark functions in Tables 1 and 2, the population size is equal to 30, and the number of iterations is considered to be equal to 500 iterations. In addition, the number of evaluations of the compared algorithms is equal to $1.5\times {10}^4$. For the benchmark functions of CEC-C06 2019 in Table 3, the population size is equal to 200, and the number of iterations is considered to be equal to 1000 iterations. In addition, the number of evaluations of the compared algorithms is equal $\text{to}2\times {10}^5$. The statistical results are based on 30 independent runs. This ensures a fair comparison between all algorithms.

It is well known that the updated model of the original LSO algorithm is the ability to achieve exploration and exploitation in the solution space through the division of labor and multiple update models. To validate the performance of the HLSO algorithm, we use some optimization algorithms with similar search characteristics to compare with our proposed algorithm to verify the reliability of the HLSO algorithm. For example, the original LSO algorithm [19], the classical PSO algorithm [9], the DA [11, 12], the SSA [17, 18], the WOA [15, 16], and the SCA[13, 14], and the algorithm-related parameters are set in Table 4.

Table 4

Initial parameter settings for different algorithms.

Algorithm	Parameter settings
PSO	Inertia weight is 1, individual optimal and group optimal weight is 2
DA	Inertia weight is 0.9 linearly decreasing to 0.4; separation, alignment, and cohesion weight are -0.2 to 0.2 linearly decreasing random values; food attraction weight is 0 to 2 linearly decreasing random values; enemy distraction weight is linearly decreasing from 0.1 to -0.1.
SCA	$r_1$ is 2 linearly decreasing to 0, $r_2$ is a random value from 0 to 2π, and $r_3$ is a random value from 0 to 2.
WOA	Constant of the shape of the spiral $b=1$
SSA	N/A
LSO	$\rho =0.2$, mode switching period is 10
HLSO	$$\rho =0.2$$

The relevant experimental data were completed on a Windows 10 system with 8 GB of RAM and an Intel i7-9750H, and the evaluation software was MATLAB 2021b.

4.1. Comparison of Optimization Accuracy

To test the performance of the HLSO algorithm, we perform experiments using five algorithms with similar update characteristics and the classical PSO algorithm and list all the experimental data. The experimental results of the benchmark functions (f₀₁~f₅₄) are in Table 5. And Tables 6–8 are the experimental results of the benchmark functions (f₅₅~f₉₀) in 10, 30, and 50 dimensions, and the experimental results of the CEC-C06 2019 benchmark functions in Table 9.

Table 5

Comparison of different algorithms with fixed dimensional benchmark functions.

Alg.	HLSO	PSO	DA	SCA	WOA	SSA	LSO
$$f$$	Avg.	Avg.	Avg.	Avg.	Avg.	Avg.	Avg.
$$f_{01}$$	$$-1.96\text{E}+02$$	$$-1.96\text{E}+02$$	$$-1.96\text{E}+02$$	$$-1.96\text{E}+02$$	$$-1.96\text{E}+02$$	$$-1.96\text{E}+02$$	$$-1.96\text{E}+02$$
$$f_{02}$$	$$-2.02\text{E}+00$$	$$-2.02\text{E}+00$$	$$-2.02\text{E}+00$$	$$-2.02\text{E}+00$$	$$-2.02\text{E}+00$$	$$-2.02\text{E}+00$$	$$-2.02\text{E}+00$$
$$f_{03}$$	$$-6.13\text{E}+00$$	$$-6.13\text{E}+00$$	$$-6.13\text{E}+00$$	$$-6.12\text{E}+00$$	$$-6.13\text{E}+00$$	$$-6.13\text{E}+00$$	$$-6.13\text{E}+00$$
$$f_{04}$$	$$1.00\text{E}+00$$	$$1.01\text{E}+00$$	$$1.01\text{E}+00$$	$$1.00\text{E}+00$$	$$1.00\text{E}+00$$	$$1.00\text{E}+00$$	$$1.00\text{E}+00$$
$$f_{05}$$	$$9.24\text{E}-34$$	$$1.65\text{E}-03$$	$$1.57\text{E}-06$$	$$2.61\text{E}-04$$	$$1.78\text{E}-01$$	$$8.29\text{E}-15$$	$$1.23\text{E}-06$$
$$f_{06}$$	$$-1.07\text{E}+02$$	$$-1.07\text{E}+02$$	$$-1.07\text{E}+02$$	$$-1.07\text{E}+02$$	$$-1.07\text{E}+02$$	$$-1.07\text{E}+02$$	$$-1.07\text{E}+02$$
$$f_{07}$$	$$0.00\text{E}+00$$	$$1.77\text{E}-02$$	$$1.41\text{E}-02$$	$$0.00\text{E}+00$$	$$0.00\text{E}+00$$	$$4.18\text{E}-11$$	$$1.00\text{E}-12$$
$$f_{08}$$	$$0.00\text{E}+00$$	$$1.71\text{E}-02$$	$$7.30\text{E}-03$$	$$0.00\text{E}+00$$	$$0.00\text{E}+00$$	$$2.02\text{E}-11$$	$$4.18\text{E}-16$$
$$f_{09}$$	$$0.00\text{E}+00$$	$$6.46\text{E}-03$$	$$1.21\text{E}-04$$	$$0.00\text{E}+00$$	$$9.82\text{E}-12$$	$$9.32\text{E}-12$$	$$1.19\text{E}-15$$
$$f_{10}$$	$$0.00\text{E}+00$$	$$3.00\text{E}-03$$	$$3.13\text{E}-06$$	$$1.26\text{E}-03$$	$$1.20\text{E}-03$$	$$5.00\text{E}-14$$	$$1.28\text{E}-05$$
$$f_{11}$$	$$3.98\text{E}-01$$	$$3.99\text{E}-01$$	$$3.98\text{E}-01$$	$$4.00\text{E}-01$$	$$3.98\text{E}-01$$	$$3.98\text{E}-01$$	$$3.98\text{E}-01$$
$$f_{12}$$	$$9.85\text{E}-02$$	$$1.24\text{E}+00$$	$$5.93\text{E}-02$$	$$3.12\text{E}-01$$	$$8.63\text{E}-02$$	$$4.26\text{E}-02$$	$$1.00\text{E}-01$$
$$f_{13}$$	$$-2.06\text{E}+00$$	$$-2.06\text{E}+00$$	$$-2.06\text{E}+00$$	$$-2.06\text{E}+00$$	$$-2.06\text{E}+00$$	$$-2.06\text{E}+00$$	$$-2.06\text{E}+00$$
$$f_{14}$$	$$1.77\text{E}-01$$	$$6.91\text{E}+00$$	$$1.95\text{E}+00$$	$$1.39\text{E}+00$$	$$2.56\text{E}+00$$	$$8.60\text{E}-01$$	$$8.71\text{E}-01$$
$$f_{15}$$	$$-2.48\text{E}+04$$	$$-2.48\text{E}+04$$	$$-2.48\text{E}+04$$	$$-2.48\text{E}+04$$	$$-2.48\text{E}+04$$	$$-2.48\text{E}+04$$	$$-2.48\text{E}+04$$
$$f_{16}$$	$$-1.00\text{E}+00$$	$$-9.97\text{E}-01$$	$$-1.00\text{E}+00$$	$$-9.98\text{E}-01$$	$$-1.00\text{E}+00$$	$$-1.00\text{E}+00$$	$$-9.97\text{E}-01$$
$$f_{17}$$	$$2.35\text{E}+01$$	$$6.70\text{E}+01$$	$$2.00\text{E}-06$$	$$7.20\text{E}-01$$	$$3.00\text{E}-06$$	$$5.24\text{E}-12$$	$$2.19\text{E}+00$$
$$f_{18}$$	$$2.00\text{E}+02$$	$$9.18\text{E}+02$$	$$1.31\text{E}+03$$	$$2.69\text{E}+02$$	$$5.11\text{E}+02$$	$$4.37\text{E}+02$$	$$3.13\text{E}+02$$
$$f_{19}$$	$$4.08\text{E}+03$$	$$2.77\text{E}+04$$	$$2.14\text{E}+04$$	$$1.39\text{E}+04$$	$$1.37\text{E}+04$$	$$7.57\text{E}+03$$	$$2.36\text{E}+04$$
$$f_{20}$$	$$9.98\text{E}-01$$	$$3.66\text{E}+00$$	$$1.23\text{E}+00$$	$$1.73\text{E}+00$$	$$3.13\text{E}+00$$	$$1.30\text{E}+00$$	$$1.89\text{E}+00$$
$$f_{21}$$	$$3.00\text{E}+00$$	$$3.09\text{E}+00$$	$$3.00\text{E}+00$$	$$3.00\text{E}+00$$	$$3.00\text{E}+00$$	$$3.00\text{E}+00$$	$$3.00\text{E}+00$$
$$f_{22}$$	$$-3.86\text{E}+00$$	$$-3.86\text{E}+00$$	$$-3.86\text{E}+00$$	$$-3.85\text{E}+00$$	$$-3.86\text{E}+00$$	$$-3.86\text{E}+00$$	$$-3.86\text{E}+00$$
$$f_{23}$$	$$-3.28\text{E}+00$$	$$-3.04\text{E}+00$$	$$-3.22\text{E}+00$$	$$-2.87\text{E}+00$$	$$-3.17\text{E}+00$$	$$-3.23\text{E}+00$$	$$-3.31\text{E}+00$$
$$f_{24}$$	$$1.84\text{E}-31$$	$$2.27\text{E}-02$$	$$3.55\text{E}-05$$	$$2.33\text{E}-02$$	$$9.09\text{E}-05$$	$$1.98\text{E}-13$$	$$4.23\text{E}-04$$
$$f_{25}$$	$$-1.92\text{E}+01$$	$$-1.90\text{E}+01$$	$$-1.92\text{E}+01$$	$$-1.92\text{E}+01$$	$$-1.92\text{E}+01$$	$$-1.92\text{E}+01$$	$$-1.92\text{E}+01$$
$$f_{26}$$	$$-6.74\text{E}-01$$	$$-6.74\text{E}-01$$	$$-6.74\text{E}-01$$	$$-6.74\text{E}-01$$	$$-6.74\text{E}-01$$	$$-6.74\text{E}-01$$	$$-6.74\text{E}-01$$
$$f_{27}$$	$$5.54\text{E}-04$$	$$8.12\text{E}-03$$	$$3.73\text{E}-03$$	$$1.01\text{E}-03$$	$$6.79\text{E}-04$$	$$2.83\text{E}-03$$	$$5.23\text{E}-04$$
$$f_{28}$$	$$-4.14\text{E}+00$$	$$-3.81\text{E}+00$$	$$-4.12\text{E}+00$$	$$-3.78\text{E}+00$$	$$-3.93\text{E}+00$$	$$-4.14\text{E}+00$$	$$-4.12\text{E}+00$$
$$f_{29}$$	$$3.18\text{E}-09$$	$$3.90\text{E}-01$$	$$5.22\text{E}-02$$	$$5.05\text{E}-01$$	$$1.56\text{E}-01$$	$$1.35\text{E}-07$$	$$5.00\text{E}-03$$
$$f_{30}$$	$$1.35\text{E}-31$$	$$8.78\text{E}-03$$	$$5.00\text{E}-06$$	$$1.53\text{E}-03$$	$$8.56\text{E}-06$$	$$1.19\text{E}-13$$	$$1.26\text{E}-05$$
$$f_{31}$$	$$9.45\text{E}-206$$	$$1.05\text{E}-04$$	$$6.34\text{E}-07$$	$$8.47\text{E}-55$$	$$6.60\text{E}-182$$	$$1.59\text{E}-15$$	$$2.05\text{E}-54$$
$$f_{32}$$	$$-1.91\text{E}+00$$	$$-1.89\text{E}+00$$	$$-1.91\text{E}+00$$	$$-1.91\text{E}+00$$	$$-1.91\text{E}+00$$	$$-1.91\text{E}+00$$	$$-1.91\text{E}+00$$
$$f_{33}$$	$$-1.80\text{E}+00$$	$$-1.80\text{E}+00$$	$$-1.80\text{E}+00$$	$$-1.72\text{E}+00$$	$$-1.75\text{E}+00$$	$$-1.80\text{E}+00$$	$$-1.80\text{E}+00$$
$$f_{34}$$	$$-4.54\text{E}+00$$	$$-3.08\text{E}+00$$	$$-3.86\text{E}+00$$	$$-2.73\text{E}+00$$	$$-3.68\text{E}+00$$	$$-4.12\text{E}+00$$	$$-4.56\text{E}+00$$
$$f_{35}$$	$$-8.05\text{E}+00$$	$$-4.11\text{E}+00$$	$$-5.87\text{E}+00$$	$$-3.96\text{E}+00$$	$$-5.77\text{E}+00$$	$$-7.19\text{E}+00$$	$$-7.12\text{E}+00$$
$$f_{36}$$	$$7.49\text{E}-02$$	$$1.09\text{E}+00$$	$$7.64\text{E}-01$$	$$1.52\text{E}+00$$	$$8.19\text{E}+00$$	$$2.55\text{E}-01$$	$$4.21\text{E}+00$$
$$f_{37}$$	$$6.99\text{E}-05$$	$$2.78\text{E}+03$$	$$1.04\text{E}+02$$	$$6.85\text{E}+00$$	$$3.16\text{E}-06$$	$$3.16\text{E}+00$$	$$2.01\text{E}-10$$
$$f_{38}$$	$$5.69\text{E}-03$$	$$6.19\text{E}-01$$	$$1.50\text{E}-01$$	$$4.95\text{E}+00$$	$$3.55\text{E}+00$$	$$1.66\text{E}-02$$	$$4.47\text{E}-02$$
$$f_{39}$$	$$0.00\text{E}+00$$	$$1.89\text{E}-06$$	$$7.85\text{E}-16$$	$$0.00\text{E}+00$$	$$2.32\text{E}-04$$	$$1.16\text{E}-15$$	$$3.21\text{E}-10$$
$$f_{40}$$	$$0.00\text{E}+00$$	$$9.35\text{E}-07$$	$$3.70\text{E}-17$$	$$0.00\text{E}+00$$	$$0.00\text{E}+00$$	$$1.57\text{E}-15$$	$$0.00\text{E}+00$$
$$f_{41}$$	$$0.00\text{E}+00$$	$$2.51\text{E}-06$$	$$0.00\text{E}+00$$	$$0.00\text{E}+00$$	$$4.98\text{E}-05$$	$$1.06\text{E}-15$$	$$0.00\text{E}+00$$
$$f_{42}$$	$$1.57\text{E}-03$$	$$1.61\text{E}-03$$	$$1.63\text{E}-03$$	$$1.57\text{E}-03$$	$$1.62\text{E}-03$$	$$1.57\text{E}-03$$	$$1.73\text{E}-03$$
$$f_{43}$$	$$2.93\text{E}-01$$	$$2.93\text{E}-01$$	$$2.93\text{E}-01$$	$$2.93\text{E}-01$$	$$2.93\text{E}-01$$	$$2.93\text{E}-01$$	$$2.93\text{E}-01$$
$$f_{44}$$	$$-8.09\text{E}+00$$	$$-5.26\text{E}+00$$	$$-7.03\text{E}+00$$	$$-2.74\text{E}+00$$	$$-7.71\text{E}+00$$	$$-7.56\text{E}+00$$	$$-5.61\text{E}+00$$
$$f_{45}$$	$$-8.17\text{E}+00$$	$$-5.13\text{E}+00$$	$$-8.27\text{E}+00$$	$$-3.41\text{E}+00$$	$$-7.15\text{E}+00$$	$$-8.33\text{E}+00$$	$$-5.80\text{E}+00$$
$$f_{46}$$	$$-8.32\text{E}+00$$	$$-5.10\text{E}+00$$	$$-8.15\text{E}+00$$	$$-3.95\text{E}+00$$	$$-7.56\text{E}+00$$	$$-8.45\text{E}+00$$	$$-8.89\text{E}+00$$
$$f_{47}$$	$$-1.87\text{E}+02$$	$$-1.86\text{E}+02$$	$$-1.87\text{E}+02$$	$$-1.86\text{E}+02$$	$$-1.87\text{E}+02$$	$$-1.87\text{E}+02$$	$$-1.87\text{E}+02$$
$$f_{48}$$	$$-1.03\text{E}+00$$	$$-1.02\text{E}+00$$	$$-1.03\text{E}+00$$	$$-1.03\text{E}+00$$	$$-1.03\text{E}+00$$	$$-1.03\text{E}+00$$	$$-1.03\text{E}+00$$
$$f_{49}$$	$$-5.00\text{E}+00$$	$$-5.00\text{E}+00$$	$$-4.83\text{E}+00$$	$$-5.00\text{E}+00$$	$$-5.00\text{E}+00$$	$$-5.00\text{E}+00$$	$$-5.00\text{E}+00$$
$$f_{50}$$	$$5.15\text{E}-269$$	$$1.80\text{E}-03$$	$$1.43\text{E}-07$$	$$3.56\text{E}-70$$	$$1.58\text{E}-77$$	$$4.12\text{E}-15$$	$$8.30\text{E}-68$$
$$f_{51}$$	$$-5.00\text{E}+01$$	$$-4.88\text{E}+01$$	$$-4.98\text{E}+01$$	$$-4.11\text{E}+01$$	$$-5.00\text{E}+01$$	$$-5.00\text{E}+01$$	$$-4.04\text{E}+01$$
$$f_{52}$$	$$-2.06\text{E}+02$$	$$-2.00\text{E}+02$$	$$-4.26\text{E}+01$$	$$2.94\text{E}+01$$	$$-2.10\text{E}+02$$	$$-2.10\text{E}+02$$	$$-2.25\text{E}+01$$
$$f_{53}$$	$$0.00\text{E}+00$$	$$0.00\text{E}+00$$	$$0.00\text{E}+00$$	$$0.00\text{E}+00$$	$$0.00\text{E}+00$$	$$0.00\text{E}+00$$	$$5.51\text{E}-13$$
$$f_{54}$$	$$1.10\text{E}-17$$	$$3.32\text{E}+01$$	$$9.55\text{E}+00$$	$$1.44\text{E}-05$$	$$1.28\text{E}+01$$	$$3.21\text{E}-11$$	$$1.42\text{E}-14$$
Avg. rank	1.6204	6.1481	4.0833	4.9815	4.1574	2.6667	4.3426

Table 6

Comparison of different algorithms under 10-dimensional benchmark functions.

Alg.	HLSO	PSO	DA	SCA	WOA	SSA	LSO
$$f$$	Avg.	Avg.	Avg.	Avg.	Avg.	Avg.	Avg.
$$f_{55}$$	$$8.88\text{E}-16$$	$$4.88\text{E}+00$$	$$3.01\text{E}+00$$	$$6.25\text{E}-07$$	$$4.32\text{E}-15$$	$$9.32\text{E}-01$$	$$3.21\text{E}-10$$
$$f_{56}$$	$$-2.00\text{E}+02$$	$$-1.09\text{E}+02$$	$$-1.77\text{E}+02$$	$$-2.00\text{E}+02$$	$$-2.00\text{E}+02$$	$$-2.00\text{E}+02$$	$$-2.00\text{E}+02$$
$$f_{57}$$	$$-2.52\text{E}+01$$	$$-8.24\text{E}+00$$	$$-7.25\text{E}+00$$	$$-1.60\text{E}+01$$	$$-2.28\text{E}+01$$	$$-7.32\text{E}+00$$	$$-1.69\text{E}+01$$
$$f_{58}$$	$$1.51\text{E}-70$$	$$4.15\text{E}+00$$	$$1.67\text{E}+00$$	$$1.02\text{E}-02$$	$$5.33\text{E}-01$$	$$5.31\text{E}-01$$	$$1.79\text{E}-17$$
$$f_{59}$$	$$2.51\text{E}-16$$	$$8.31\text{E}+00$$	$$2.58\text{E}-01$$	$$5.05\text{E}+01$$	$$9.79\text{E}-02$$	$$3.84\text{E}-11$$	$$2.43\text{E}-02$$
$$f_{60}$$	$$1.91\text{E}-173$$	$$2.16\text{E}+01$$	$$2.71\text{E}-02$$	$$3.43\text{E}-13$$	$$3.08\text{E}-80$$	$$2.60\text{E}-12$$	$$2.67\text{E}-46$$
$$f_{61}$$	$$3.30\text{E}-146$$	$$6.72\text{E}+01$$	$$4.40\text{E}+02$$	$$1.32\text{E}-22$$	$$1.16\text{E}-150$$	$$9.62\text{E}-19$$	$$2.31\text{E}-19$$
$$f_{62}$$	$$6.67\text{E}-01$$	$$4.30\text{E}+02$$	$$1.58\text{E}+01$$	$$6.67\text{E}-01$$	$$6.49\text{E}-01$$	$$6.81\text{E}-01$$	$$6.67\text{E}-01$$
$$f_{63}$$	$$-1.00\text{E}+00$$	$$-2.77\text{E}-01$$	$$-7.25\text{E}-01$$	$$-9.36\text{E}-01$$	$$-9.15\text{E}-01$$	$$-7.90\text{E}-01$$	$$-9.80\text{E}-01$$
$$f_{64}$$	$$1.18\text{E}-167$$	$$8.18\text{E}+01$$	$$1.63\text{E}+01$$	$$8.73\text{E}-10$$	$$2.06\text{E}-72$$	$$1.74\text{E}+01$$	$$4.22\text{E}-41$$
$$f_{65}$$	$$-1.00\text{E}+00$$	$$-3.59\text{E}-01$$	$$-9.97\text{E}-01$$	$$-1.00\text{E}+00$$	$$-1.00\text{E}+00$$	$$-1.00\text{E}+00$$	$$-1.00\text{E}+00$$
$$f_{66}$$	$$4.03\text{E}-04$$	$$5.74\text{E}-01$$	$$6.12\text{E}-01$$	$$4.01\text{E}-02$$	$$6.33\text{E}-02$$	$$2.02\text{E}-01$$	$$1.05\text{E}+00$$
$$f_{67}$$	$$1.32\text{E}-13$$	$$8.12\text{E}-01$$	$$2.19\text{E}+00$$	$$1.01\text{E}-01$$	$$1.30\text{E}-02$$	$$9.63\text{E}-01$$	$$1.16\text{E}-01$$
$$f_{68}$$	$$1.10\text{E}-03$$	$$1.23\text{E}+00$$	$$1.46\text{E}+00$$	$$2.94\text{E}-01$$	$$2.63\text{E}-02$$	$$3.57\text{E}-03$$	$$3.21\text{E}-01$$
$$f_{69}$$	$$9.01\text{E}-01$$	$$1.89\text{E}+00$$	$$1.57\text{E}+00$$	$$1.32\text{E}+00$$	$$9.94\text{E}-01$$	$$1.00\text{E}+00$$	$$9.00\text{E}-01$$
$$f_{70}$$	$$6.77\text{E}-09$$	$$6.64\text{E}+01$$	$$9.05\text{E}+04$$	$$6.66\text{E}+01$$	$$5.49\text{E}+00$$	$$2.24\text{E}-03$$	$$1.77\text{E}+04$$
$$f_{71}$$	$$1.80\text{E}-04$$	$$1.41\text{E}+00$$	$$2.94\text{E}-02$$	$$2.84\text{E}-03$$	$$1.92\text{E}-03$$	$$1.38\text{E}-02$$	$$1.34\text{E}-04$$
$$f_{72}$$	$$0.00\text{E}+00$$	$$6.23\text{E}+01$$	$$2.95\text{E}+01$$	$$1.83\text{E}+00$$	$$2.18\text{E}+00$$	$$1.69\text{E}+01$$	$$0.00\text{E}+00$$
$$f_{73}$$	$$-5.00\text{E}+00$$	$$-2.56\text{E}+00$$	$$-3.73\text{E}+00$$	$$-4.82\text{E}+00$$	$$-5.00\text{E}+00$$	$$-4.86\text{E}+00$$	$$-4.99\text{E}+00$$
$$f_{74}$$	$$5.30\text{E}+00$$	$$2.49\text{E}+03$$	$$7.77\text{E}+02$$	$$7.34\text{E}+00$$	$$6.99\text{E}+00$$	$$1.42\text{E}+02$$	$$8.07\text{E}+00$$
$$f_{75}$$	$$2.22\text{E}-02$$	$$4.22\text{E}-01$$	$$1.06\text{E}+00$$	$$9.99\text{E}-02$$	$$1.30\text{E}-01$$	$$2.60\text{E}-01$$	$$9.73\text{E}-02$$
$$f_{76}$$	$$-2.66\text{E}+03$$	$$-2.48\text{E}+03$$	$$-2.76\text{E}+03$$	$$-2.19\text{E}+03$$	$$-3.39\text{E}+03$$	$$-2.72\text{E}+03$$	$$-1.87\text{E}+03$$
$$f_{77}$$	$$1.41\text{E}-57$$	$$1.13\text{E}+01$$	$$1.81\text{E}+02$$	$$2.54\text{E}-02$$	$$6.90\text{E}+01$$	$$2.65\text{E}-06$$	$$1.78\text{E}-05$$
$$f_{78}$$	$$1.14\text{E}-39$$	$$6.99\text{E}+00$$	$$1.04\text{E}+01$$	$$2.21\text{E}-08$$	$$2.14\text{E}-47$$	$$2.09\text{E}-01$$	$$1.25\text{E}-05$$
$$f_{79}$$	$$1.07\text{E}-33$$	$$1.75\text{E}+00$$	$$2.85\text{E}+00$$	$$4.76\text{E}-03$$	$$4.56\text{E}+00$$	$$2.25\text{E}-05$$	$$1.18\text{E}-04$$
$$f_{80}$$	$$3.49\text{E}-71$$	$$7.81\text{E}+00$$	$$1.84\text{E}+00$$	$$1.17\text{E}-09$$	$$2.08\text{E}-50$$	$$1.73\text{E}-02$$	$$7.90\text{E}-16$$
$$f_{81}$$	$$0.00\text{E}+00$$	$$6.62\text{E}+02$$	$$1.90\text{E}-04$$	$$8.95\text{E}-30$$	$$6.03\text{E}-198$$	$$1.69\text{E}-56$$	$$8.63\text{E}-142$$
$$f_{82}$$	$$-1.28\text{E}+02$$	$$-6.67\text{E}+01$$	$$-8.01\text{E}+01$$	$$-7.44\text{E}+01$$	$$-9.08\text{E}+01$$	$$-9.08\text{E}+01$$	$$-7.42\text{E}+01$$
$$f_{83}$$	$$2.98\text{E}-76$$	$$8.63\text{E}+00$$	$$1.34\text{E}+01$$	$$3.49\text{E}-13$$	$$3.45\text{E}-77$$	$$1.02\text{E}-09$$	$$6.55\text{E}-10$$
$$f_{84}$$	$$0.00\text{E}+00$$	$$9.63\text{E}+00$$	$$1.91\text{E}+01$$	$$0.00\text{E}+00$$	$$0.00\text{E}+00$$	$$1.00\text{E}+00$$	$$0.00\text{E}+00$$
$$f_{85}$$	$$-3.59\text{E}+02$$	$$-2.97\text{E}+02$$	$$-3.46\text{E}+02$$	$$-3.07\text{E}+02$$	$$-3.80\text{E}+02$$	$$-3.45\text{E}+02$$	$$-3.50\text{E}+02$$
$$f_{86}$$	$$7.19\text{E}-138$$	$$4.60\text{E}+01$$	$$6.86\text{E}-01$$	$$5.59\text{E}-13$$	$$1.67\text{E}-75$$	$$9.77\text{E}-11$$	$$4.34\text{E}-28$$
$$f_{87}$$	$$1.20\text{E}-64$$	$$8.36\text{E}+00$$	$$8.48\text{E}-02$$	$$2.19\text{E}-05$$	$$1.66\text{E}-04$$	$$2.90\text{E}-02$$	$$4.68\text{E}-38$$
$$f_{88}$$	$$9.48\text{E}-02$$	$$2.52\text{E}+00$$	$$1.02\text{E}+00$$	$$3.52\text{E}-01$$	$$3.92\text{E}-01$$	$$9.41\text{E}-01$$	$$1.21\text{E}-01$$
$$f_{89}$$	$$7.68\text{E}-04$$	$$2.79\text{E}-03$$	$$1.79\text{E}-03$$	$$2.75\text{E}-03$$	$$6.58\text{E}-04$$	$$2.19\text{E}-03$$	$$3.31\text{E}-03$$
$$f_{90}$$	$$-9.69\text{E}-01$$	$$1.55\text{E}-04$$	$$1.55\text{E}-04$$	$$1.55\text{E}-04$$	$$-1.00\text{E}+00$$	$$1.55\text{E}-04$$	$$-6.14\text{E}-01$$
Avg. rank	1.4583	6.3194	5.9028	4.0000	2.5556	4.4028	3.3611

Table 7

Comparison of different algorithms under 30-dimensional benchmark functions.

Alg.	HLSO	PSO	DA	SCA	WOA	SSA	LSO
$$f$$	Avg.	Avg.	Avg.	Avg.	Avg.	Avg.	Avg.
$$f_{55}$$	$$2.19\text{E}-15$$	$$8.70\text{E}+00$$	$$1.09\text{E}+01$$	$$1.45\text{E}+01$$	$$4.20\text{E}-15$$	$$2.70\text{E}+00$$	$$1.00\text{E}-03$$
$$f_{56}$$	$$-2.00\text{E}+02$$	$$-1.87\text{E}+01$$	$$-1.75\text{E}+01$$	$$-1.70\text{E}+02$$	$$-2.00\text{E}+02$$	$$-2.00\text{E}+02$$	$$-2.00\text{E}+02$$
$$f_{57}$$	$$-6.80\text{E}+01$$	$$4.82\text{E}+01$$	$$6.23\text{E}+01$$	$$2.53\text{E}+01$$	$$-6.40\text{E}+01$$	$$4.07\text{E}+01$$	$$8.27\text{E}-01$$
$$f_{58}$$	$$6.43\text{E}-58$$	$$3.36\text{E}+01$$	$$1.43\text{E}+01$$	$$1.41\text{E}+00$$	$$7.33\text{E}-31$$	$$3.94\text{E}+00$$	$$8.01\text{E}-12$$
$$f_{59}$$	$$2.61\text{E}-05$$	$$1.42\text{E}+02$$	$$3.11\text{E}+01$$	$$5.16\text{E}+02$$	$$8.31\text{E}+00$$	$$6.59\text{E}-09$$	$$1.29\text{E}-01$$
$$f_{60}$$	$$8.90\text{E}-155$$	$$1.92\text{E}+07$$	$$6.77\text{E}+00$$	$$8.11\text{E}-02$$	$$7.70\text{E}-76$$	$$3.53\text{E}-09$$	$$1.08\text{E}-31$$
$$f_{61}$$	$$1.13\text{E}-109$$	$$1.93\text{E}+04$$	$$6.38\text{E}+06$$	$$1.30\text{E}+03$$	$$2.49\text{E}-144$$	$$1.82\text{E}-14$$	$$1.07\text{E}-10$$
$$f_{62}$$	$$6.67\text{E}-01$$	$$7.94\text{E}+04$$	$$3.41\text{E}+03$$	$$2.79\text{E}+02$$	$$6.67\text{E}-01$$	$$4.74\text{E}+00$$	$$6.67\text{E}-01$$
$$f_{63}$$	$$-9.81\text{E}-01$$	$$-2.67\text{E}-02$$	$$-1.80\text{E}-01$$	$$-6.79\text{E}-01$$	$$-9.02\text{E}-01$$	$$-2.39\text{E}-01$$	$$-9.55\text{E}-01$$
$$f_{64}$$	$$2.32\text{E}-145$$	$$4.43\text{E}+02$$	$$9.85\text{E}+01$$	$$5.67\text{E}+00$$	$$1.27\text{E}-72$$	$$4.43\text{E}+01$$	$$1.70\text{E}-28$$
$$f_{65}$$	$$-1.00\text{E}+00$$	$$-2.10\text{E}-05$$	$$-6.48\text{E}-01$$	$$-9.97\text{E}-01$$	$$-1.00\text{E}+00$$	$$-1.00\text{E}+00$$	$$-1.00\text{E}+00$$
$$f_{66}$$	$$0.00\text{E}+00$$	$$1.04\text{E}+00$$	$$1.98\text{E}+01$$	$$1.01\text{E}+00$$	$$8.77\text{E}-03$$	$$1.71\text{E}-02$$	$$3.03\text{E}+00$$
$$f_{67}$$	$$8.86\text{E}-03$$	$$5.49\text{E}+00$$	$$5.09\text{E}+03$$	$$1.33\text{E}+05$$	$$2.31\text{E}-02$$	$$7.10\text{E}+00$$	$$4.92\text{E}-01$$
$$f_{68}$$	$$1.21\text{E}+00$$	$$2.15\text{E}+01$$	$$3.60\text{E}+03$$	$$1.43\text{E}+04$$	$$4.68\text{E}-01$$	$$2.89\text{E}+01$$	$$2.33\text{E}+00$$
$$f_{69}$$	$$9.00\text{E}-01$$	$$7.19\text{E}+00$$	$$4.74\text{E}+00$$	$$5.86\text{E}+00$$	$$1.21\text{E}+00$$	$$1.00\text{E}+00$$	$$9.24\text{E}-01$$
$$f_{70}$$	$$1.54\text{E}+03$$	$$3.67\text{E}+03$$	$$3.46\text{E}+08$$	$$1.01\text{E}+07$$	$$1.27\text{E}+03$$	$$2.36\text{E}+01$$	$$7.08\text{E}+06$$
$$f_{71}$$	$$1.51\text{E}-04$$	$$1.04\text{E}+02$$	$$7.24\text{E}-01$$	$$7.28\text{E}-02$$	$$2.00\text{E}-03$$	$$1.70\text{E}-01$$	$$1.25\text{E}-04$$
$$f_{72}$$	$$0.00\text{E}+00$$	$$3.60\text{E}+02$$	$$1.59\text{E}+02$$	$$2.98\text{E}+01$$	$$1.89\text{E}-15$$	$$5.55\text{E}+01$$	$$0.00\text{E}+00$$
$$f_{73}$$	$$-4.94\text{E}+00$$	$$-1.74\text{E}+00$$	$$-2.81\text{E}+00$$	$$-3.04\text{E}+00$$	$$-5.00\text{E}+00$$	$$-4.78\text{E}+00$$	$$-4.90\text{E}+00$$
$$f_{74}$$	$$2.65\text{E}+01$$	$$1.77\text{E}+05$$	$$4.77\text{E}+05$$	$$1.83\text{E}+04$$	$$2.78\text{E}+01$$	$$4.82\text{E}+02$$	$$2.86\text{E}+01$$
$$f_{75}$$	$$4.32\text{E}-02$$	$$1.41\text{E}+00$$	$$6.14\text{E}+00$$	$$1.09\text{E}+00$$	$$1.23\text{E}-01$$	$$1.84\text{E}+00$$	$$1.10\text{E}-01$$
$$f_{76}$$	$$-7.23\text{E}+03$$	$$-6.68\text{E}+03$$	$$-5.32\text{E}+03$$	$$-3.76\text{E}+03$$	$$-1.11\text{E}+04$$	$$-7.54\text{E}+03$$	$$-3.48\text{E}+03$$
$$f_{77}$$	$$4.36\text{E}-44$$	$$5.98\text{E}+02$$	$$1.57\text{E}+04$$	$$7.67\text{E}+03$$	$$4.63\text{E}+04$$	$$1.68\text{E}+03$$	$$4.10\text{E}-01$$
$$f_{78}$$	$$1.10\text{E}-28$$	$$5.18\text{E}+01$$	$$1.57\text{E}+02$$	$$2.58\text{E}-01$$	$$2.82\text{E}-45$$	$$1.74\text{E}+01$$	$$3.68\text{E}-03$$
$$f_{79}$$	$$1.18\text{E}-24$$	$$4.84\text{E}+00$$	$$2.97\text{E}+01$$	$$3.39\text{E}+01$$	$$4.01\text{E}+01$$	$$1.29\text{E}+01$$	$$1.48\text{E}-02$$
$$f_{80}$$	$$1.48\text{E}-58$$	$$1.72\text{E}+02$$	$$1.73\text{E}+01$$	$$1.18\text{E}-02$$	$$3.83\text{E}-47$$	$$2.26\text{E}+00$$	$$1.09\text{E}-07$$
$$f_{81}$$	$$0.00\text{E}+00$$	$$8.16\text{E}+06$$	$$6.05\text{E}+04$$	$$8.11\text{E}+06$$	$$1.76\text{E}-180$$	$$2.13\text{E}-07$$	$$2.09\text{E}-68$$
$$f_{82}$$	$$-1.89\text{E}+02$$	$$-1.02\text{E}+02$$	$$-1.43\text{E}+02$$	$$-1.25\text{E}+02$$	$$-2.15\text{E}+02$$	$$-2.12\text{E}+02$$	$$-1.22\text{E}+02$$
$$f_{83}$$	$$4.92\text{E}-56$$	$$1.36\text{E}+02$$	$$2.01\text{E}+03$$	$$1.64\text{E}+01$$	$$5.81\text{E}-72$$	$$1.31\text{E}-07$$	$$7.23\text{E}-06$$
$$f_{84}$$	$$0.00\text{E}+00$$	$$1.48\text{E}+02$$	$$2.24\text{E}+03$$	$$2.38\text{E}+01$$	$$0.00\text{E}+00$$	$$1.94\text{E}+01$$	$$0.00\text{E}+00$$
$$f_{85}$$	$$-9.97\text{E}+02$$	$$-5.64\text{E}+02$$	$$-9.45\text{E}+02$$	$$-5.91\text{E}+02$$	$$-1.09\text{E}+03$$	$$-1.00\text{E}+03$$	$$-7.46\text{E}+02$$
$$f_{86}$$	$$3.00\text{E}-112$$	$$2.06\text{E}+03$$	$$2.19\text{E}+02$$	$$1.25\text{E}+00$$	$$1.28\text{E}-72$$	$$2.76\text{E}+00$$	$$8.74\text{E}-17$$
$$f_{87}$$	$$1.13\text{E}-69$$	$$6.73\text{E}+09$$	$$7.36\text{E}+01$$	$$1.47\text{E}+01$$	$$1.80\text{E}-03$$	$$2.44\text{E}+01$$	$$1.73\text{E}-31$$
$$f_{88}$$	$$6.35\text{E}-01$$	$$2.00\text{E}+01$$	$$4.24\text{E}+00$$	$$3.06\text{E}+00$$	$$6.96\text{E}-01$$	$$5.71\text{E}+00$$	$$5.97\text{E}-01$$
$$f_{89}$$	$$1.33\text{E}-11$$	$$3.16\text{E}-11$$	$$6.74\text{E}-10$$	$$7.71\text{E}-10$$	$$4.96\text{E}-12$$	$$4.90\text{E}-11$$	$$1.25\text{E}-08$$
$$f_{90}$$	$$-9.53\text{E}-01$$	$$3.71\text{E}-12$$	$$3.71\text{E}-12$$	$$5.14\text{E}-11$$	$$-8.67\text{E}-01$$	$$5.16\text{E}-11$$	$$-4.83\text{E}-01$$
Avg. rank	1.5556	5.8750	5.9306	5.0833	2.3194	4.0833	3.1528

Table 8

Comparison of different algorithms under 50-dimensional benchmark functions.

Alg.	HLSO	PSO	DA	SCA	WOA	SSA	LSO
$$f$$	Avg.	Avg.	Avg.	Avg.	Avg.	Avg.	Avg.
$$f_{55}$$	$$3.26\text{E}-15$$	$$1.03\text{E}+01$$	$$1.24\text{E}+01$$	$$1.67\text{E}+01$$	$$4.56\text{E}-15$$	$$4.44\text{E}+00$$	$$1.69\text{E}-02$$
$$f_{56}$$	$$-2.00\text{E}+02$$	$$-3.78\text{E}+00$$	$$-2.18\text{E}+00$$	$$-4.46\text{E}+01$$	$$-2.00\text{E}+02$$	$$-1.90\text{E}+02$$	$$-2.00\text{E}+02$$
$$f_{57}$$	$$-9.38\text{E}+01$$	$$1.37\text{E}+02$$	$$1.67\text{E}+02$$	$$9.70\text{E}+01$$	$$-9.80\text{E}+01$$	$$7.68\text{E}+01$$	$$2.91\text{E}+01$$
$$f_{58}$$	$$3.01\text{E}-54$$	$$7.19\text{E}+01$$	$$2.62\text{E}+01$$	$$6.80\text{E}+00$$	$$2.37\text{E}-47$$	$$9.14\text{E}+00$$	$$1.50\text{E}-07$$
$$f_{59}$$	$$5.42\text{E}-05$$	$$4.12\text{E}+02$$	$$8.39\text{E}+01$$	$$1.01\text{E}+03$$	$$2.34\text{E}+01$$	$$2.00\text{E}-02$$	$$2.65\text{E}-01$$
$$f_{60}$$	$$5.96\text{E}-149$$	$$2.60\text{E}+11$$	$$1.99\text{E}+01$$	$$1.14\text{E}+01$$	$$2.73\text{E}-73$$	$$5.37\text{E}-03$$	$$1.49\text{E}-30$$
$$f_{61}$$	$$1.93\text{E}-102$$	$$1.56\text{E}+05$$	$$8.36\text{E}+07$$	$$2.14\text{E}+06$$	$$3.10\text{E}-134$$	$$3.80\text{E}-01$$	$$7.53\text{E}-09$$
$$f_{62}$$	$$6.67\text{E}-01$$	$$7.43\text{E}+05$$	$$4.01\text{E}+04$$	$$3.62\text{E}+04$$	$$6.67\text{E}-01$$	$$3.16\text{E}+01$$	$$6.67\text{E}-01$$
$$f_{63}$$	$$-9.83\text{E}-01$$	$$-1.39\text{E}-02$$	$$-1.01\text{E}-01$$	$$-2.60\text{E}-01$$	$$-9.09\text{E}-01$$	$$-1.11\text{E}-01$$	$$-9.45\text{E}-01$$
$$f_{64}$$	$$9.20\text{E}-141$$	$$8.19\text{E}+02$$	$$2.47\text{E}+02$$	$$2.00\text{E}+01$$	$$1.56\text{E}-73$$	$$7.49\text{E}+01$$	$$1.58\text{E}-28$$
$$f_{65}$$	$$-1.00\text{E}+00$$	$$-5.71\text{E}-10$$	$$-2.89\text{E}-01$$	$$-8.55\text{E}-01$$	$$-1.00\text{E}+00$$	$$-1.00\text{E}+00$$	$$-1.00\text{E}+00$$
$$f_{66}$$	$$0.00\text{E}+00$$	$$1.10\text{E}+00$$	$$6.67\text{E}+01$$	$$9.49\text{E}+00$$	$$1.62\text{E}-02$$	$$4.55\text{E}-01$$	$$6.47\text{E}+00$$
$$f_{67}$$	$$6.79\text{E}-02$$	$$1.09\text{E}+01$$	$$1.11\text{E}+06$$	$$1.56\text{E}+07$$	$$2.86\text{E}-02$$	$$1.31\text{E}+01$$	$$6.77\text{E}-01$$
$$f_{68}$$	$$3.76\text{E}+00$$	$$6.20\text{E}+01$$	$$1.07\text{E}+06$$	$$1.15\text{E}+07$$	$$1.06\text{E}+00$$	$$2.07\text{E}+02$$	$$4.62\text{E}+00$$
$$f_{69}$$	$$9.00\text{E}-01$$	$$1.28\text{E}+01$$	$$1.05\text{E}+01$$	$$1.19\text{E}+01$$	$$1.25\text{E}+00$$	$$1.03\text{E}+00$$	$$9.02\text{E}-01$$
$$f_{70}$$	$$1.53\text{E}+04$$	$$1.97\text{E}+04$$	$$3.78\text{E}+09$$	$$5.89\text{E}+09$$	$$1.07\text{E}+04$$	$$8.07\text{E}+03$$	$$9.77\text{E}+06$$
$$f_{71}$$	$$1.87\text{E}-04$$	$$3.94\text{E}+02$$	$$3.08\text{E}+00$$	$$3.83\text{E}+00$$	$$5.29\text{E}-03$$	$$5.22\text{E}-01$$	$$1.26\text{E}-04$$
$$f_{72}$$	$$0.00\text{E}+00$$	$$6.66\text{E}+02$$	$$3.38\text{E}+02$$	$$1.12\text{E}+02$$	$$0.00\text{E}+00$$	$$8.25\text{E}+01$$	$$0.00\text{E}+00$$
$$f_{73}$$	$$-4.80\text{E}+00$$	$$-1.47\text{E}+00$$	$$-2.47\text{E}+00$$	$$-2.44\text{E}+00$$	$$-5.00\text{E}+00$$	$$-3.32\text{E}+00$$	$$-4.79\text{E}+00$$
$$f_{74}$$	$$4.74\text{E}+01$$	$$8.93\text{E}+05$$	$$3.60\text{E}+06$$	$$7.06\text{E}+06$$	$$4.84\text{E}+01$$	$$1.56\text{E}+03$$	$$4.86\text{E}+01$$
$$f_{75}$$	$$5.42\text{E}-02$$	$$2.34\text{E}+00$$	$$9.99\text{E}+00$$	$$4.19\text{E}+00$$	$$1.27\text{E}-01$$	$$5.01\text{E}+00$$	$$1.16\text{E}-01$$
$$f_{76}$$	$$-1.16\text{E}+04$$	$$-1.08\text{E}+04$$	$$-7.30\text{E}+03$$	$$-4.86\text{E}+03$$	$$-1.81\text{E}+04$$	$$-1.23\text{E}+04$$	$$-4.43\text{E}+03$$
$$f_{77}$$	$$1.83\text{E}-41$$	$$4.61\text{E}+03$$	$$4.88\text{E}+04$$	$$4.88\text{E}+04$$	$$2.05\text{E}+05$$	$$8.45\text{E}+03$$	$$1.76\text{E}+00$$
$$f_{78}$$	$$1.85\text{E}-27$$	$$1.16\text{E}+02$$	$$3.52\text{E}+02$$	$$4.46\text{E}+00$$	$$3.20\text{E}-45$$	$$8.91\text{E}+01$$	$$1.22\text{E}-02$$
$$f_{79}$$	$$8.63\text{E}-23$$	$$8.01\text{E}+00$$	$$4.59\text{E}+01$$	$$6.90\text{E}+01$$	$$7.05\text{E}+01$$	$$2.00\text{E}+01$$	$$7.75\text{E}-02$$
$$f_{80}$$	$$7.88\text{E}-54$$	$$3.83\text{E}+10$$	$$3.41\text{E}+01$$	$$6.72\text{E}-01$$	$$1.78\text{E}-44$$	$$8.97\text{E}+00$$	$$4.07\text{E}-08$$
$$f_{81}$$	$$0.00\text{E}+00$$	$$3.99\text{E}+08$$	$$1.68\text{E}+06$$	$$4.73\text{E}+08$$	$$6.76\text{E}-184$$	$$1.01\text{E}+00$$	$$1.78\text{E}-78$$
$$f_{82}$$	$$-2.42\text{E}+02$$	$$-1.30\text{E}+02$$	$$-2.01\text{E}+02$$	$$-1.59\text{E}+02$$	$$-3.35\text{E}+02$$	$$-3.44\text{E}+02$$	$$-1.60\text{E}+02$$
$$f_{83}$$	$$5.11\text{E}-52$$	$$4.06\text{E}+02$$	$$8.06\text{E}+03$$	$$1.03\text{E}+03$$	$$2.14\text{E}-71$$	$$1.04\text{E}+00$$	$$1.99\text{E}-05$$
$$f_{84}$$	$$0.00\text{E}+00$$	$$3.96\text{E}+02$$	$$7.60\text{E}+03$$	$$6.22\text{E}+02$$	$$0.00\text{E}+00$$	$$1.25\text{E}+02$$	$$0.00\text{E}+00$$
$$f_{85}$$	$$-1.47\text{E}+03$$	$$-7.94\text{E}+02$$	$$-1.47\text{E}+03$$	$$-8.68\text{E}+02$$	$$-1.83\text{E}+03$$	$$-1.66\text{E}+03$$	$$-9.65\text{E}+02$$
$$f_{86}$$	$$3.51\text{E}-106$$	$$9.49\text{E}+03$$	$$1.35\text{E}+03$$	$$1.66\text{E}+02$$	$$7.27\text{E}-70$$	$$3.13\text{E}+01$$	$$6.83\text{E}-16$$
$$f_{87}$$	$$5.11\text{E}-60$$	$$5.63\text{E}+21$$	$$2.96\text{E}+10$$	$$5.23\text{E}+06$$	$$6.11\text{E}-04$$	$$3.34\text{E}+03$$	$$2.57\text{E}-33$$
$$f_{88}$$	$$9.84\text{E}-01$$	$$3.87\text{E}+01$$	$$9.34\text{E}+00$$	$$6.61\text{E}+00$$	$$9.41\text{E}-01$$	$$1.14\text{E}+01$$	$$9.69\text{E}-01$$
$$f_{89}$$	$$5.83\text{E}-20$$	$$1.69\text{E}-19$$	$$8.54\text{E}-16$$	$$1.15\text{E}-15$$	$$1.56\text{E}-20$$	$$2.63\text{E}-18$$	$$1.33\text{E}-14$$
$$f_{90}$$	$$-9.45\text{E}-01$$	$$8.91\text{E}-20$$	$$8.91\text{E}-20$$	$$7.87\text{E}-17$$	$$-8.33\text{E}-01$$	$$1.72\text{E}-16$$	$$-4.15\text{E}-01$$
Avg. rank	1.6528	5.6806	5.7639	5.6111	2.1806	4.0000	3.1111

Table 9

Comparison of different algorithms under the benchmark functions of CEC C06 2019.

Alg.	HLSO	PSO	DA	SCA	WOA	SSA	LSO
$$f$$	Avg.	Avg.	Avg.	Avg.	Avg.	Avg.	Avg.
$$f_{91}$$	$$3.47\text{E}+07$$	$$1.69\text{E}+10$$	$$1.76\text{E}+10$$	$$8.10\text{E}+08$$	$$4.01\text{E}+09$$	$$2.15\text{E}+08$$	$$1.54\text{E}+06$$
$$f_{92}$$	$$1.73\text{E}+01$$	$$1.20\text{E}+03$$	$$1.82\text{E}+01$$	$$1.74\text{E}+01$$	$$1.73\text{E}+01$$	$$1.73\text{E}+01$$	$$1.98\text{E}+01$$
$$f_{93}$$	$$1.27\text{E}+01$$	$$1.27\text{E}+01$$	$$1.27\text{E}+01$$	$$1.27\text{E}+01$$	$$1.27\text{E}+01$$	$$1.27\text{E}+01$$	$$1.27\text{E}+01$$
$$f_{94}$$	$$1.81\text{E}+01$$	$$6.01\text{E}+01$$	$$3.45\text{E}+02$$	$$6.21\text{E}+02$$	$$1.44\text{E}+02$$	$$1.92\text{E}+01$$	$$2.95\text{E}+02$$
$$f_{95}$$	$$1.16\text{E}+00$$	$$1.40\text{E}+00$$	$$1.69\text{E}+00$$	$$2.06\text{E}+00$$	$$1.66\text{E}+00$$	$$1.29\text{E}+00$$	$$1.66\text{E}+00$$
$$f_{96}$$	$$7.80\text{E}+00$$	$$9.46\text{E}+00$$	$$8.51\text{E}+00$$	$$9.60\text{E}+00$$	$$7.58\text{E}+00$$	$$3.09\text{E}+00$$	$$8.93\text{E}+00$$
$$f_{97}$$	$$5.50\text{E}+01$$	$$4.05\text{E}+02$$	$$3.80\text{E}+02$$	$$4.11\text{E}+02$$	$$3.49\text{E}+02$$	$$1.07\text{E}+02$$	$$5.74\text{E}+02$$
$$f_{98}$$	$$4.72\text{E}+00$$	$$4.80\text{E}+00$$	$$5.49\text{E}+00$$	$$5.28\text{E}+00$$	$$5.35\text{E}+00$$	$$4.66\text{E}+00$$	$$5.81\text{E}+00$$
$$f_{99}$$	$$2.34\text{E}+00$$	$$2.68\text{E}+00$$	$$2.32\text{E}+01$$	$$1.94\text{E}+01$$	$$4.05\text{E}+00$$	$$2.34\text{E}+00$$	$$5.98\text{E}+00$$
$$f_{100}$$	$$1.98\text{E}+01$$	$$1.93\text{E}+01$$	$$2.02\text{E}+01$$	$$2.03\text{E}+01$$	$$2.01\text{E}+01$$	$$1.93\text{E}+01$$	$$2.03\text{E}+01$$
Avg. rank	2.05	4.15	5.40	5.55	3.75	2.00	5.10

Table 5 observes that the HLSO algorithm outperforms the PSO algorithm for all experimental items and also outperforms other algorithms by about 83%, reaching an average rank of 1.6204 in this benchmark set. Table 6 observes that the HLSO algorithm outperforms the different algorithms by about 96%, with 24 functions having better performance compared to the WOA, reaching an average rank of 1.4583 in this benchmark set. It can be observed in Table 7 that the HLSO algorithm has 23 functions with better performance than the WOA, while the percentage of better performance than the other algorithms is around 94%, and the average rank of this benchmark set reaches 1.5556. Table 8 observes that the HLSO algorithm outperforms the different algorithms by about 94%, and the average rank of this benchmark function reaches 1.6528. In Table 9, it can be observed that the HLSO algorithm has the same level of search performance as SSA, outperforming other algorithms by about 90%, and the average rank of this benchmark function reaches 2.05.

Figure 3 shows the average rank of the seven algorithms in Tables 5–9, which clearly indicates that the HLSO algorithm has a better performance under the current test benchmark functions.

[figure(s) omitted; refer to PDF]

4.2. Analysis of Convergence Rate

Figure 4 illustrates the convergence curve of $f_5$, $f_{11}$, $f_{20}$, $f_{28}$, $f_{36}$,$f_{44}$, and $f_{54}$ in the fixed-dimensional benchmark functions, and Figure 5 presents the convergence curve of $f_{55}$, $f_{59}$, $f_{66}$, $f_{69}$, and $f_{77}$ in the high-dimensional (50D) benchmark functions, $f_{94}$ and $f_{97}$ in CEC-C06 2019.

[figure(s) omitted; refer to PDF]

From the experimental results, it can be observed that the convergence speed of the proposed algorithm is significantly better than other algorithms in the benchmark functions $f_{11}$, $f_{20}$, $f_{28}$, $f_{69}$, and $f_{94}$. In addition, the ability of the cubs to jump out of the local optimal solution and the local exact search is well demonstrated in the benchmark functions $f_{44}$, $f_{55}$, $f_{66}$, $f_{77}$, and $f_{97}$. In conclusion, the proposed HLSO algorithm has excellent performance compared to other optimization algorithms.

5. WSN Node Coverage Optimization

In this paper, two experimental scenarios are considered, from simple to complex.

(1) Maximizing coverage by a limited number of nodes in the region of interest under the condition that only coverage is considered

(2) Sink or base stations are set up in the space of interest, and the nodes are randomly configured while considering the connection efficiency to avoid creating isolated points leading to energy loss

5.1. WSN Node Coverage Mathematical Model

Assuming that the WSN monitoring area is digitized into a grid of $L\times W\times H$, each grid size is set to 1. Num sensors with sensing radius $R_s$ are deployed in the area, and the set of nodes can be denoted as $N=n_1,n_2,\cdots ,n_{Num}$. In this paper, the Boolean model is used as the node perception model, and the target can be successfully sensed as long as it is within the node perception range. Suppose the coordinates of a node $n_i$ in the monitored area are $n_{x,i},n_{y,i},n_{z,i}$ and the coordinates of the target point $t_j$ are $t_{x,j},t_{y,j},t_{z,j}$, then the distance between the two points is $$\begin{array}{}\text{(8)}& d{n}_i,t_j=\sqrt{{n_{x,i}-t_{x,j}}^2+{n_{y,i}-t_{y,j}}^2+{n_{z,i}-t_{z,j}}^2}.\end{array}$$

The perceptual quality of node $n_i$ to $t_j$ is denoted by$p{n}_i,t_j$. When the position of $t_j$ is within the perceptual range circle of node $n_i$, the perceptual quality is 1; otherwise, the perceptual quality of node $n_i$ to $t_j$ is 0, as shown in Equation (9). $$\begin{array}{}\text{(9)}& p{n}_i,t_j=\begin{array}{cc}1,\hfill & \text{if}d{n}_i,t_j\le R_s,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\end{array}$$

In general, the probability of sensor perception of a target is less than 1. To improve the probability of target perception, multiple sensors are needed to detect collaboratively, then the likelihood of WSN perception of a target is expressed as follows. $$\begin{array}{}\text{(10)}& pN,t_j=1-\prod _{i=1}^{\text{Num}}1-p{n}_i,t_j.\end{array}$$

The coverage ratio of wireless sensor nodes is the ratio of the number of target points covered by all sensor nodes to the total number of target points in the area, as shown in $$\begin{array}{}\text{(11)}& R_{\mathrm{cov}}=\frac{{\sum }_{j=1}^{L\times W\times H}pN,t_j}{L\times W\times H}.\end{array}$$

Isolated points cannot transmit data to the base station through multihop communication resulting in reduced connection efficiency. Therefore, the connection efficiency can be considered as the ratio of isolated points. $$\begin{array}{}\text{(12)}& R_{\text{CE}}=\frac{\text{Num}-\text{ISN}}{\text{Num}},\end{array}$$where $\text{ISN}$ denotes the number of isolated points.

5.2. Simulation Environment Setting

Let the monitoring area be a three-dimensional space of $20\text{m}\times 20\text{m}\times 20\text{m}$, the coordinates of the sink or base station are (10, 10, 10), and test the performance of different algorithms for WSN coverage optimization with the number of sensor nodes of 10, 20, 30, and 50, respectively, and their communication radius of $R_s=3m$. The relevant parameters are set as in the fixed-dimensional benchmark function experiment, containing the number of populations, the number of iterations, the number of independent tests, and the algorithm-related parameters. The average coverage rates of different algorithms are listed for comparison.

5.3. Simulation Results

5.3.1. Example 1

To be able to observe the optimization performance of different algorithms applied to the WSN coverage problem, Figures 6–12 present the results of the spatially optimized configurations of the different algorithms and plot the views in different directions for easy analysis. The red dots represent the node locations, and the yellow circles are the coverage areas.

[figure(s) omitted; refer to PDF]

Figure 6 shows that the wireless sensor nodes are evenly distributed in the views in different directions, and it can be observed from the 3D view that only the corners are not completely covered. Figures 7 and 8 can be observed that the nodes are set at the edges, wasting the coverage of the nodes, and the poor coverage of the corners and edges is clearly observed from the 3D plots. Figures 9–12 show the poor coverage of nodes at the edges and corners of the space. And redundancy of node coverage occurs inside the space. Figure 13 shows the average coverage of different algorithms for the number of nodes 10/20/30/50.

[figure(s) omitted; refer to PDF]

It can be observed that the HLSO algorithm can avoid over the aggregation of nodes in some areas and effectively improve the node coverage when the number of nodes increases gradually.

5.3.2. Example 2

Under the conditions of Example 1, the base station coordinates are set to (10, 10, 10). In order to achieve effective coverage, the coverage of isolated points is not considered in this experiment. Tables 10–13 present best, worst, mean, and Std for 30 independent tests. The $R_{\mathrm{cov}}$, $R_{\text{CE}}$, and $\text{ISN}$ in the table are the simulation results that are close to the mean value from 30 independent tests. From the results of the study, it can be observed that SCA produces the highest number of isolated points and the lowest coverage rate under the condition of multi-objective optimization. DA and WOA have the same level of optimization results as HLSO, but their stability is not enough.

Table 10

Simulation results of different algorithms considering both connectivity and coverage (N10).

Alg.	Best	Worst	Mean	Std	$R_{\mathrm{cov}}$	$R_{\text{CE}}$	$\text{ISN}$
HLSO	1.1106	1.0926	1.1005	0.0047	10.00%	100%	0
PSO	0.8791	0.6535	0.7292	0.0766	6.05%	70%	3
DA	1.1038	0.7713	1.0438	0.0840	8.48%	100%	0
SCA	0.4505	0.2296	0.3085	0.0740	2.70%	30%	7
WOA	1.1053	0.2308	0.8597	0.2560	7.70%	80%	2
LSO	1.0924	1.0826	1.0870	0.0026	6.60%	100%	0
SSA	1.0996	0.2300	1.0112	0.2163	7.06%	90%	1

Table 11

Simulation results of different algorithms considering both connectivity and coverage (N20).

Alg.	Best	Worst	Mean	Std	$R_{\mathrm{cov}}$	$R_{\text{CE}}$	$\text{ISN}$
HLSO	1.1740	1.1323	1.1518	0.0098	15.24%	100%	0
PSO	0.8135	0.4755	0.6448	0.0923	9.45%	55%	9
DA	1.1611	0.8679	1.0710	0.0824	12.60%	95%	1
SCA	0.5225	0.1820	0.2707	0.0855	5.01%	20%	16
WOA	1.1654	0.3700	1.0490	0.1745	13.73%	90%	2
LSO	1.1409	1.1198	1.1309	0.0057	9.95%	100%	0
SSA	1.1550	1.0823	1.1162	0.0163	11.63%	100%	0

Table 12

Simulation results of different algorithms considering both connectivity and coverage (N30).

Alg.	Best	Worst	Mean	Std	$R_{\mathrm{cov}}$	$R_{\text{CE}}$	$\text{ISN}$
HLSO	1.2165	1.1528	1.1913	0.0143	19.11%	100%	0
PSO	0.8509	0.5273	0.6526	0.0709	11.80%	53%	14
DA	1.2141	0.8565	1.1269	0.0819	18.32%	93%	2
SCA	0.5142	0.1708	0.2925	0.0810	6.04%	23%	23
WOA	1.2225	0.9750	1.1614	0.0568	18.99%	97%	1
LSO	1.1820	1.1485	1.1641	0.0089	14.59%	97%	1
SSA	1.1770	1.1223	1.1489	0.0144	14.74%	100%	0

Table 13

Simulation results of different algorithms considering both connectivity and coverage (N50).

Alg.	Best	Worst	Mean	Std	$R_{\mathrm{cov}}$	$R_{\text{CE}}$	$\text{ISN}$
HLSO	1.3060	1.2183	1.2554	0.0201	25.47%	100%	0
PSO	0.9083	0.6443	0.7858	0.0727	20.50%	58%	21
DA	1.3074	0.9676	1.1918	0.0690	25.17%	94%	3
SCA	0.6860	0.1976	0.4105	0.1016	8.58%	32%	34
WOA	1.3096	1.0678	1.2450	0.0467	24.42%	100%	0
LSO	1.2506	1.2079	1.2272	0.0108	21.20%	98%	1
SSA	1.2855	1.1855	1.2234	0.0206	22.31%	100%	0

6. Conclusion

In this paper, we propose a HLSO algorithm based on the original LSO algorithm structure. First, the local exploration capability is improved by adjusting the perturbation factor and the adult lion update formula. Then, a spiral bubble net attack is introduced to replace the cub first update mechanism to improve diversity.

The average ranking of the algorithm performance is calculated for different sets of evaluation benchmarks, and from Tables 5–9, it can be observed that the proposed algorithm is approximately between 1.45 and 2. A comparison with other optimization algorithms presents that the algorithm has better search and exploitation capabilities. The long bar graph in Figure 3 better shows the stable performance of the improved algorithm.

Finally, it can be applied to overcome the problems of large coverage blind areas and uneven distribution often faced by wireless sensor networks in random node deployment. It can be observed from Figure 12 that HLSO can achieve a large coverage area with different number of nodes considering only the coverage rate. If base stations are installed in the specified space, it can be observed from Tables 10–13 that HLSO can avoid the generation of isolated points and maintain the maximum coverage while considering both connection efficiency and coverage.

Acknowledgments

This work was partially supported by the Natural Science Foundation of Fujian Province under Grant 2020J01843, 2020J05170, and 2021J011017 and Minjiang University under Grant MJY192026 and 103952022087.

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