2.1. Coati optimization algorithm (COA)

This section describes the mathematical model of the original COA. The COA is a novel swarm intelligence algorithm proposed in 2022, which primarily simulates the behavior of coatis hunting iguanas. Initially, half of the population will climb trees to approach their food source, the iguana, and this behavior is defined by Eq. (1):

(1) \begin{align} Xnew_{i}\colon xnew_{i,j}=x_{i,j}+r1\cdot \left(Ig_{j}-I\cdot x_{i,j}\right),i=1,2,3,\ldots \left\lfloor \frac{N}{2}\right\rfloor \end{align}

In the above equation, x _i,j represents the position of the i-th individual in the j-th dimension of the solution space, Ig denotes the position of the best solution in the current population, N represents the population size, r ₁ is a random number between [0, 1], and I takes a random value of either 1 or 2.

Faced with the siege of coatis, the iguana will randomly drop to the ground. Eq. (2) expresses this behavior in the solution space. Meanwhile, the other half of the coatis will search for their prey, as represented by Eq. (3):

(2) \begin{align} Ig^{G}\colon Ig_{j}^{G} & = lbj+r2\cdot \left(ubj-lbj\right),\ j=1,2,3,\ldots m \\[-4pt] \nonumber \end{align}

(3) \begin{align} Xnew_{i}\colon xnew_{i,j} & = \left\{\begin{array}{l} x_{i,j}+r3\cdot \left(Ig_{j}^{G}-I\cdot x_{i,j}\right),F_{Ig}^{G}\lt F_{i}\\[3pt] x_{i,j}+r3\cdot \left(I\cdot x_{i,j}-Ig_{j}^{G}\right),else \end{array},\ i=\right.\left\lfloor \frac{N}{2}\right\rfloor +1,\ldots N \\[8pt] \nonumber \end{align}

where $Ig_{j}^{G}$ represents the random dropping position of the iguana, with lb and ub denoting the upper and lower bounds of the solution space, respectively. r ₂ and r ₃ are both random numbers between [0,1]. In Eq. (3), $F_{Ig}^{G}$ represents the fitness value of Ig and F _i denotes the current fitness value of the i-th individual. Subsequently, if the fitness value Fnew _i of the newly generated individual is superior to the current individual, its position is replaced; otherwise, the original position is retained, as expressed in Eq. (4):

(4) \begin{align} X_{i}=\left\{\begin{array}{l} Xnew_{i},Fnew_{i}\lt F_{i}\\[3pt] X_{i},else \end{array}\right. \end{align}

After locating the iguana, all the coatis will slowly surround their prey. This biological characteristic is represented by equations (5) and (6):

(5) \begin{align} lb_{\mathrm{j}}^{L} & =\frac{lbj}{t},ub_{\mathrm{j}}^{\mathrm{L}}=\frac{ubj}{t},t=1,2,3,\ldots T \\[-4pt] \nonumber \end{align}

(6) \begin{align} Xnew_{i}\colon xnew_{i,j} & = x_{i,j}+\left(1-2r4\right)\cdot \left(lb_{\mathrm{j}}^{L}+r5\cdot \left(ub_{\mathrm{j}}^{L}-ub_{\mathrm{j}}^{L}\right)\right),i=1,2,3,\ldots N \\[8pt] \nonumber \end{align}

where t denotes the current iteration number, while T represents the maximum iteration number, and r ₄ and r ₅ are both random numbers between [0, 1]. Subsequently, the new position of the current individual is also calculated using Eq. (4). The pseudocode of the basic framework of the COA is represented by Algorithm 1.

Algorithm 1 Pseudocode of COA.

2.2. Differential evolution (DE)

This section describes the mathematical model of original DE. DE is a well-known classic metaheuristic algorithm, mainly consisting of three evolutionary processes: mutation, crossover, and selection.

2.2.1. Mutation

In DE, the mutation process involves randomly selecting two individuals and using the difference of their position vectors as the step size for updating the target carrier position, as represented by Eq. (7):

(7) \begin{align} \mathrm{y}_{i,j}=x_{i,j}+F\cdot \left(x_{R1,j}-x_{R2,j}\right),i=1,2,3,\ldots, \mathrm{N} \end{align}

In the above equation, x _i represents the i-th individual in the population, x _R1 and x _R2 are two different individuals randomly selected from the population, j denotes the j-th dimension of the solution, and F is the scaling factor.

2.2.2. Crossover

The crossover process involves exchanging the j-th dimension component between the mutated individual y _i and the target carrier x _i , generating a crossover individual z _i , as represented by Eq. (8):

(8) \begin{align} \mathrm{Z}_{i}\colon z_{i,j}=\left\{\begin{array}{l} y_{i,j},r\lt CR||j_{r}=j\\[3pt] x_{i,j},else \end{array}\right. \end{align}

where CR denotes the crossover probability, j _r represents a randomly selected number between 1 and the maximum dimensions, and r is a random number between [0, 1].

2.2.3. Selection

Similar to Eq. (4), after comparing the fitness values of the target carrier and the crossover individual, one of them is randomly selected. The better individual is chosen as the position vector for the next generation, achieving the goal of evolution:

(9) \begin{align} X_{i}=\left\{\begin{array}{l} \mathrm{Z}_{i},F_{{z_{i}}}\lt F_{i}\\[3pt] X_{i},else \end{array}\right. \end{align}

where $F_{{z_{i}}}$ represents the fitness value of the crossover individual and F _i represents the fitness value of the target carrier.

2.3. Real-world constrained optimization problems

This paragraph introduces the basic definition of constrained optimization problems, which typically refer to minimization problems under several equality and inequality constraints. The real-world constrained optimization problems in this paper are all derived from CEC2020, covering various fields such as industrial chemical processes, synthesis and design, mechanical engineering, power systems, power electronics, and livestock feed optimization. The definitions of these problems are referenced from literature [Reference Kumar, Wu, Ali, Mallipeddi, Suganthan and Das34] and all meet the following function definition.

(10) \begin{align} \textrm{Minimize} \quad & \mathrm{f}\left(X\right),X=\left(x_{1},x_{2},\ldots, x_{n}\right)\\[3pt] \nonumber\textrm{Subject to:} \quad & \begin{array}{l} g_{\mathrm{i}}(X)\leq 0,i=1,\ldots, n\\[3pt] h_{j}(X)=0,j=n+1,\ldots, m \end{array} \end{align}

In the above equation, X denotes the solution vector of the problem, g represents inequality constraints, n denotes the number of g, h represents equality constraints, and m signifies the total number of all constraints.

2.4. Multi-robot online path planning problem

This article studies the multi-robot path planning problem with multiple static and dynamic obstacles; therefore, it is necessary to consider the safe distance between robots and other robots, robots and static obstacles, and robots and dynamic obstacles. The mathematical model of this problem is provided in literature [Reference Akay and Mustafa35], where each robot has its own starting point and destination. During the robots movement, there are static and dynamic obstacles of various sizes and shapes. All robots not only need to avoid these obstacles but also must not collide with each other. The robots need to reach their destinations with the minimum cost step by step.

Minimize,

(11) \begin{align} Fit=F_{1}+F_{2}+F_{3}+F_{4} \end{align}

where F ₁ represents the shortest distance, F ₂ represents avoiding static obstacles, F ₃ represents avoiding dynamic obstacles, and F ₄ represents avoiding other robots. F ₁ can be expressed as Eq. (12):

(12) \begin{align} F_{1}=\sum _{i=1}^{NR}\left(f_{i}+g_{i}\right) \end{align}

where

(13) \begin{align} f_{i}=\sqrt{\left(x_{i}^{n}-x_{i}^{c}\right)^{2}+\left(y_{i}^{n}-y_{i}^{c}\right)^{2}} \\[-4pt] \nonumber \end{align}

(14) \begin{align} g_{i}=\sqrt{\left(x_{i}^{n}-x_{i}^{g}\right)^{2}+\left(y_{i}^{n}-y_{i}^{g}\right)^{2}} \\[8pt] \nonumber \end{align}

where NR represents the number of robots and ( $x_{i}^{g}, y_{i}^{g}$ ) represents the destination coordinates of the i-th robot. F ₂ can be expressed as Eq. (15):

(15) \begin{align} F_{{_{2}}}=\left\{\begin{array}{l} \varepsilon, d_{i}^{s}\leq d_{s}\\[3pt] 0,else \end{array}\right. \end{align}

where $\varepsilon$ is a large penalty value, d _s represent the safety distance between any two objects, and $d_{i}^{s}$ is the sum of distances from all static obstacles to the i-th robot, as derived from Eq. (16):

(16) \begin{align} d_{i}^{s}=\sum _{i=1}^{NR}\sum _{j=1}^{NS}\sqrt{\left(x_{i}^{n}-x_{j}^{s}\right)^{2}+\left(y_{i}^{n}-y_{j}^{s}\right)^{2}} \end{align}

where NS is the number of static obstacles and ( $x_{j}^{s}, y_{j}^{s}$ ) is the coordinate position of the j-th static obstacle. F ₃ can be expressed as Eq. (17):

(17) \begin{align} F_{{_{3}}}=\left\{\begin{array}{l} \varepsilon, d_{i}^{D}\leq d_{s}\\[3pt] 0,else \end{array}\right. \end{align}

where $d_{i}^{D}$ is the sum of distances from all dynamic obstacles to the i-th robot, as obtained from Eq. (18):

(18) \begin{align} d_{i}^{D}=\sum _{i=1}^{NR}\sum _{j=1}^{ND}\sqrt{\left(x_{i}^{n}-x_{j}^{D}\right)^{2}+\left(y_{i}^{n}-y_{j}^{D}\right)^{2}} \end{align}

where ND is the number of dynamic obstacles, ( $x_{j}^{D}, y_{j}^{D}$ ) is the coordinate position of the j-th dynamic obstacle, and the movement of dynamic obstacles at each step is determined by equations (19) and (20):

(19) \begin{align} x_{j}^{D}=x_{j}^{D}+v_{j}^{D}\cos \left(\alpha _{j}\right) \\[-4pt] \nonumber \end{align}

(20) \begin{align} y_{j}^{D}=y_{j}^{D}+v_{j}^{D}\sin \left(\alpha _{j}\right) \\[8pt] \nonumber \end{align}

In the above equation, $v_{j}^{D}$ represents the movement speed of dynamic obstacle j and $\alpha _{j}$ represents its radial position relative to the target position. F ₄ can be expressed as Eq. (21):

(21) \begin{align} F_{{_{4}}}=\left\{\begin{array}{l} \varepsilon, d_{i}^{R}\leq d_{s}\\[3pt] 0,else \end{array}\right. \end{align}

where $d_{i}^{R}$ is the sum of distances between different robots and ( $x_{j}^{R}, y_{j}^{R}$ ) is the coordinate position of the j-th robot, as obtained from Eq. (22):

(22) \begin{align} d_{i}^{R}=\sum _{i=1}^{NR-1}\sum _{j=i+1}^{NR}\sqrt{\left(x_{i}^{n}-x_{j}^{R}\right)^{2}+\left(y_{i}^{n}-y_{j}^{R}\right)^{2}} \end{align}

3.1. Crossing

In traditional DE algorithms, the crossing probability CR is a fixed value, with literature [Reference Draa, Samira and Imene36] suggesting a range of values for CR between [0, 1]. When CR approaches zero, the changes in individuals between adjacent generations are minimal, leading to a high presence of similar individuals in the population and potentially causing the algorithm to get stuck in local optima. Conversely, when CR is close to 1, the individuals update positions fluctuate greatly, hindering efficient convergence. To address these issues, this paper proposes an adaptive dynamic adjustment method, calculated as follows:

(23) \begin{align} CR\left(\mathrm{t}\right)=CR_{\min }+\left(CR_{\max }-CR_{\min }\right)\cdot \left(1-\frac{t}{T}\right) \end{align}

where $CR_{\max }$ equals 0.95, $CR_{\min }$ equals 0.05, t represents the current iteration number, and T is the maximum number of iterations.

3.2. Variant update strategies of DE

The first introduced update strategy, rand < CR, was proposed in literature [Reference Das and Ponnuthurai37], while the other part was inspired by the GWO [Reference Kumar, Wu, Ali, Mallipeddi, Suganthan and Das34] and is represented by Eq. (24):

(24) \begin{align} Xnew_{i}\colon xnew_{i,j}=\left\{\begin{array}{l} x_{i,j}+FF\cdot \left(x_{best,j}-x_{i,j}+x_{R1,j}-x_{R2,j}\right);\ if\ rand\lt CR\\[3pt] \frac{1}{3}\left(x_{best,j}+x_{\sec ond,j}+x_{\textit{third},j}\right),else \end{array}\right. \end{align}

In the above equation, x _best , x _second , and x _third represent the current population’s top three individuals, FF is the scaling factor with a value of 0.8, while R1 and R2 are randomly selected individuals that are different from the i-th individual.

The second introduced update strategy was proposed in literature [Reference Wang, Zixing and Qingfu39] and is represented by Eq. (25):

(25) \begin{align} Xnew_{i}\colon xnew_{i,j}=x_{i,j}+L\cdot \left(x_{R1,j}-x_{i,j}\right)+FF\cdot \left(x_{R2,j}-x_{R3,j}\right) \end{align}

where L is a random number between [0, 1], and R1, R2, and R3 are also three distinct random individuals.

3.3. Algorithm implementation

In the algorithm proposed, during the exploration phase, Eqs. (1), (24), and (25) are randomly used, with their probabilities determined by Eq. (26). During the invocation process, a counter records which update strategy an individual has been used. If an individual achieves better results in this iteration, the score of the strategy used by that individual is incremented. The probabilities for the next roulette wheel selection are calculated based on the total score obtained by each strategy in this iteration:

(26) \begin{align} p_{j}\left(t+1\right)=\frac{S_{j}\left(t\right)}{\sum _{i=1}^{SN}S_{i}\left(t\right)},\,j\in \left\{1,2,\ldots, SN\right\} \end{align}

where $p_{j}(t+1)$ represents the probability of strategy j being used in the next iteration, $S_{i}(t)$ represents the score of strategy i in the current iteration, and SN represents the number of strategies, which is set to 3 in this paper. The update strategy for each individual is determined by a roulette wheel selection. The pseudocode and flow chart of SDECOA are, respectively, shown in Algorithm 1 and Figure 1.

Algorithm 2 Pseudocode of SDECOA.

Figure 1. Flow chart of SDECOA.

3.4. Time complexity analysis

The operation of SDECOA includes population initialization, fitness calculation, population position update, individual probability update, and strategy selection, so by analyzing the complexity of each process, the time complexity of SDECOA can be obtained. Assuming the population size is N, the number of iterations is T, and the dimension of the problem is D, the time complexity of population initialization is O(ND). In each iteration process, the time complexity of population position update is O(ND+ND), the time complexity of fitness value calculation and comparison is O(ND+ND), and the time complexity of individual probability update and strategy selection is O(ND). Therefore, the time complexity of SDECOA can be concluded as O(ND+T×(ND+ND+ND+ND+ND+ND)) ≈ O(TND).

4.1. Experimental configuration and design

The programming language used in this experiment is MATLAB, and the compilation software is MATLAB 2021a. The computer configuration consists of a 64-bit Windows 10 operating system, 8.0G RAM, and a processor with a base frequency of 2.10 GHz. To demonstrate the effectiveness of the improved algorithm, our algorithm was initially compared with several classic and advanced algorithms on the 20-dimensional CEC2022 benchmark functions. To further validate the performance of our algorithm, SDECOA was employed to solve 48 real-world constrained optimization problems from CEC2020, and comparisons were made with several algorithms that won in the CEC2020 competition. Finally, our algorithm was applied to the problem of multi-robot online path planning.

4.2. SDECOA for CEC2022 benchmark functions

Solving benchmark test functions is a common method for testing algorithm performance, and this section presents the experimental results of SDECOA solving this test set. The CEC2022 benchmark test functions are the latest test functions, and basic information is shown in Table 1. This test suite mainly includes three dimensions (dim = {2, 10, 20}), with function types primarily including single-peak functions (F1), multimodal functions (F2–F5), hybrid functions (F6-F8), and composite functions (F9–F12). To validate the algorithm’s effectiveness, we compare SDECOA with the original DE and COA. Furthermore, to evaluate its performance, we also test and analyze it against some well-known classical algorithms and recent advanced algorithms. The algorithms involved in testing the CEC2022 benchmark functions include simulated annealing (SA) [Reference Kirkpatrick, Gelatt and Vecchi40], PSO [Reference Kennedy and Russell7], GWO [Reference Mirjalili, Mirjalili, Lewis and wolf optimizer38], sine-cosine algorithm (SCA) [Reference Mirjalili41], whale optimization algorithm (WOA) [Reference Mirjalili and Andrew42], Harris hawk optimizer (HHO) [Reference Heidari, Mirjalili, Faris, Aljarah, Mafarja and Chen43], SSA [Reference Xue and Bo44], modified adaptive sparrow search algorithm (MASSA) [Reference Mirjalili41], self-adaptive differential sine-cosine algorithm (sdSCA) [Reference Akay and Mustafa35], hybrid algorithm of differential evolution and flower pollination (HADEFP) [45], etc. In this set of experiments, all algorithms were independently run 25 times. Except for SA, the iteration count for all algorithms was set to 1000 iterations, with a population size of 50. The other parameters for each algorithm are shown in Table 2.

Table I. Basic information of CEC2022 benchmark test functions.

Table II. Parameter settings of different algorithms.

As shown in Table 3, the means and standard deviations of different algorithms are recorded. In the experimental results across all test functions, the proposed algorithm outperforms the original COA and DE by a significant margin, particularly achieving the best results in solving F1, F6, F7, F8, and F9. Among all the compared algorithms, SDECOA proves to be the most competitive. Table 3 also presents the rankings of all algorithms based on the Friedman rank-sum test, with the data confirming that the proposed algorithm achieved the best performance in solving the CEC2022 test functions.

Table III. Experimental results of different algorithms on CEC2022 benchmark functions in 20 dimensions.

The convergence curves for different CEC2022 test functions are shown in Figure 2. From these curves, it is evident that SDECOA exhibits substantial improvements in both optimization capability and convergence speed compared to DE and COA. Compared with other algorithms, SDECOA shows the best convergence speed and results for most test functions. The boxplots for all test functions are shown in Figure 3, where it can be seen that SDECOA performs better in terms of median values and lower variance for most functions. Compared to other algorithms, SDECOA is more stable, thus demonstrating the superior robustness of the proposed algorithm.

Figure 2. Convergence curves of different algorithms.

Figure 3. Boxplots of different algorithms.

4.3. SDECOA for CEC2020 problems

To further demonstrate the effectiveness of SDECOA, this section presents the experimental results and data analysis of SDECOA applied to real-world constrained optimization problems. The real-world constrained problems used in this study are based on the CEC2020 real-world constrained optimization problems. The mathematical models for all problems in this study are derived from reference [Reference Sharma and Jabeen46]. Table 4 displays the basic information of 48 problems, where D represents the dimension of the problem, g represents the number of inequality constraints, h represents the number of equality constraints, and f^* represents the known optimal value. These problems are mainly designed in fields such as Industrial Chemical Processes, Process Synthesis and Design, Mechanical Engineering, Power Systems, Power Electronics, and Livestock Feed Ration Optimization, with dimensions ranging from 2 to 118 and varying numbers of constraints from 1 to 108.

In this set of experiments, SDECOA is compared with several advanced algorithms that won in the CEC2020 competition. These top-ranking algorithms, in descending order, include SASS [Reference Kumar, Das and Zelinka47], COLSHADE [Reference Gurrola-Ramos, Hernàndez-Aguirre and Dalmau-Cedeño48], sCMAgES [Reference Kumar, Das and Zelinka49], EnMODE [Reference Sallam, Elsayed, Chakrabortty and Ryan50], and BpMAgES [Reference Hellwig and Beyer51]. Each algorithm is independently run 25 times, with 1000 iterations, a population size of 60, and other control parameters consistent with the literature [Reference Xu and Lihong52]. A dynamic adaptive penalty function is used. Table 5 presents the optimal values, mean values, and standard deviations of these algorithms. It is worth noting that shaded cells in the table indicate that the proposed algorithm’s best result exceeds the f* for that problem. In this paper, the Friedman rank-sum test is also conducted on the average values of these 48 optimization problems, and the ranking of the proposed algorithm, obtained through calculation, is superior to the top five algorithms that won the CEC2020 competition.

The introduction in the literature [Reference Ezugwu, Adewumi and Frîncu53] determines the quality of the solutions by calculating the percentage deviation of the mean with respect to the known optimal solution (PDmean). As can be seen from Eq. (27), the smaller the average value obtained from 25 runs, the smaller the PDmean value, indicating higher solution quality. Table 6 presents the PDmean values when different algorithms solve different problems:

(27) \begin{align} \textit{PDmean}=\left\{\begin{array}{l} \frac{mean-f^{*}}{f^{*}+\varepsilon };if \quad f^{*}\geq 0\\[3pt] \frac{f^{*}-mean}{f^{*}},else \end{array}\right. \end{align}

where f* represents the best know solution, mean represents the average value of the test results, and $\varepsilon$ represents a very small constant. An analysis of Table 6 shows that the proposed algorithm generally demonstrates superior solution quality compared to these advanced algorithms. Based on the experimental results, the proposed algorithm achieved ideal results in most cases, with results for 26 problems surpassing the known optimal solution. These experimental results are sufficient to demonstrate that the proposed algorithm not only performs well but also excels in addressing real-world constrained optimization problems using SDECOA. Therefore, in comparison with these state-of-the-art algorithms, our algorithm demonstrates excellent competitiveness.

4.4. SDECOA for multi-robot path planning problem

In this section, the experimental content applies SDECOA to the multi-robot path planning problem, the mathematical model code of which can be obtained from reference [Reference Akay and Mustafa35].

4.4.1. Design of the scenario for multi-robot path planning problem

This section introduces the simulation scenario layout for the robot path planning problem. In the simulation model of this problem, all robots are circular with equal radii, and collisions between them are not allowed. Each robot has its own starting point and end point. As for static obstacles, each one has a different size and shape. Similarly, each dynamic obstacle is also circular with equal radii and has its own starting point and end point. Each dynamic obstacle moves at a constant speed without deviating during the motion. In this study, three different scenarios are created, with the numbers of robots, static obstacles, and dynamic obstacles shown in Table 7. Except for Scenario 1, which references literature [Reference Akay and Mustafa35], the other scenarios are arranged independently in this study. The layouts of these three scenarios are shown in Figures 4, 5, and 6. In these scenarios, the black paths represent the planned paths of robots, the blue paths represent the paths of dynamic obstacles, and ‘×’ indicates the respective end points of objects.

In the experiments of the three scenarios mentioned above, in addition to the proposed algorithm, the comparison algorithms include DE, PSO, SCA, COA, and sdSCA, with a population size of 30, and the parameter settings are consistent with Section 4.2. It is worth noting that in this experiment, the iteration of all algorithms stops when all robots reach their end points. Each algorithm runs independently 20 times.

Table IV. Basic information of 48 CEC2020 real-world constrained optimization problems.

4.4.2. Simulation experiment in Scenario 1

In the simulation experiment in Scenario 1, Figure 7 illustrates the travel paths of each algorithm. In solving the multi-robot path planning problem, the stronger the algorithm’s performance, the smoother the robot’s path and the shorter the average travel distance. As seen in Figure 7, the path corresponding to SDECOA is notably the smoothest. Table 8 records the average travel distance and standard deviation for each robot from the starting point to the destination and also calculates the total average travel distance for the six robots in Scenario 1. For ease of comparison, a histogram of the average travel distances for different algorithms is shown in Figure 8. From this, it is evident that SDECOA demonstrates the best performance. Table 9 lists the number of steps taken by each robot, with fewer steps indicating a smoother path.

Table V. Experimental results of different algorithms solving 48 CEC2020 real-world constrained optimization problems (shaded areas indicate results superior to the known optimal solution).

The experimental results from Scenario 1 show that, compared to DE and COA, the algorithm’s performance improved by 33.2% and 22.4%, respectively, while the average number of steps taken was reduced by 26.9% and 15.8%.

4.4.3. Simulation experiment in Scenario 2

Figure 9 shows the travel paths of each algorithm in Scenario 2. It is clearly observed in Figure 9 that, as the number of robots increases, the differences in path smoothness between the algorithms become more pronounced. Compared to the other algorithms, the path corresponding to SDECOA is the smoothest. Table 10 records the average travel distance and corresponding standard deviation for the 12 robots from the starting point to the destination. Figure 10 presents a histogram of the average travel distances for different algorithms. From the analysis of Figure 10, it can be seen that, despite the increased complexity of the problem, SDECOA still demonstrates strong performance. Table 11 records the number of steps taken by each robot.

Table VI. Percentage deviation values (PDmean) obtained through solutions using different algorithms.

Table VII. Number of robots and obstacles in different scenarios.

Figure 4. Scenario 1.

The experimental results from Scenario 2 show that, compared to DE and COA, the algorithm’s performance improved by 46.1% and 50.2%, respectively, while the average total number of steps taken was reduced by 40.0% and 48.3%.

4.4.4. Simulation experiment in Scenario 3

In Scenario 3, the number of robots was increased to 20, and the quantities of both dynamic and static obstacles were significantly raised, making the path planning problem more complex. It should be noted that in this experiment, COA led to the phenomenon of robots getting lost, demonstrating subpar performance. Therefore, the experimental results involving COA are excluded from consideration in this section. In this experiment, Figure 11 illustrates the trajectories of each algorithm in Scenario 3. From Figure 11, it is clearly observable that, even as the problem scale increased, the proposed algorithm still managed to find effective paths compared to the other algorithms. Table 12 records the average travel distance and corresponding standard deviation for 20 robots from the starting point to the destination, while Figure 12 shows the histogram of the average travel distances for different algorithms. The histogram clearly indicates that the average travel distance of SDECOA is significantly lower than that of the other algorithms. Table 13 documents the number of steps taken by each robot.

Figure 5. Scenario 2.

Figure 6. Scenario 3.

Figure 7. Paths of different algorithms in Scenario 1.

Table VIII. Experimental results of the travel distance for each robot in Scenario 1.

Figure 8. Average travel distance of each robot in Scenario 1.

From the experimental results in Scenario 3, it can be concluded that, compared to DE, the proposed algorithm improved performance by 45.3%, with the total average movement steps reduced by 37.0%.

Through these experimental data, it can be seen that regardless of the number of robots, the proposed algorithm always achieves the best performance. Moreover, as the number of robots increases, the values of other algorithms become very large, while SDECOA can still manage reasonably. The path diagrams in the three scenarios clearly show that the proposed algorithm’s results in a shorter travel distance and smoother paths for the robot, particularly demonstrating significant advantages when the robot operates in more complex environments. Therefore, it can be proven that our algorithm outperforms these algorithms in terms of performance and competitiveness.