Abstract
To address the limitations of the standard equilibrium optimizer (EO) in terms of insufficient optimization capability, multiple strategies are proposed to enhance its performance. These include a reverse equilibrium state pool, a non-uniform equilibrium state selection strategy, and an equilibrium state mutation strategy. The reverse equilibrium state pool is introduced to encourage candidate solutions with poorer positions to search in a wider search space, under such considerations the global search ability of the improved EO can be enhanced. The non-uniform equilibrium state selection strategy is proposed to select equilibrium state. Under the proposed selection strategy, the candidate solutions with better positions are more likely to be chosen as the equilibrium state, allowing for sufficient exploration of positions near the current optimal point. The equilibrium state mutation strategy leads to cross mutation between candidate solutions and equilibrium state, increasing the likelihood of the group exploring the global optimal solution. To verify and further analyze the performance and superiority of the improved EO, i.e., reverse equilibrium states EO (R
O), 29 benchmark functions are adopted. It is verified theoretically from the experimental results that the R
O is with a significant improvement in performance by comparison between the standard EO and certain frequently-used heuristic optimization algorithms. Finally, the R
O is successfully applied in path planning for surface marine vehicles under the situations of both dynamic and static obstacles.
Keywords: Equilibrium optimizer, Multiple strategies, Reverse equilibrium, Path planning, Surface marine vehicles
Subject terms: Electrical and electronic engineering, Information technology
Introduction
Nowadays, application-oriented optimization problems have attracted numerous researchers to study and seek for solutions, as highlighted in recent review papers1–5. In the real word, being directed against specific engineering problems, various heuristic optimization algorithms have already demonstrated their excellent performances in dealing with optimization problems, such as differential evolution6, genetic algorithm7,8, ant colony optimization9, simulated annealing algorithm10, equilibrium optimizer (EO)11, particle swarm optimization (PSO)12 and so on.
The EO is a heuristic optimization algorithm introduced by the authors in11. Owing to the outstanding optimization performance, EO is with strong application potential and has been successfully applied in various practical applications to cope with the optimization issues, see for instance13–21. In14, by sufficiently considering slime mould algorithm, an EO-based job shop scheduling solution has been proposed, which can solve bigger-scale job shop scheduling problem with faster rate of convergence. An EO-based k-nearest neighbor classifier has been designed in15 to maintenance internet security and further reduce the losses caused by cyber-attacks. By introducing EO into slime mould algorithm, the algorithm efficiency has been improved by means of optimizing the search of slime mould algorithm in17, and the modified algorithm has well solved nine engineering design problems by seeking out feasible solutions under all engineering constraints. Moreover, the power flow optical calculation for power systems in18 and the capacitated vehicle routing problems in20 have been also addressed by employing EO, respectively.
Although EO has been widely applied and has shown effectiveness in engineering practices, significant research efforts are still focused on improving its accuracy. And so far, the performance of the EO has been improved from different perspectives. For example, in order to avoid duplicated solutions and conserve valuable evaluation opportunities, an improved EO algorithm based on dual population of non revisit mechanism has been proposed in22. For the sake of balancing the abilities of exploration and exploitation, a hybrid algorithm EO-grey wolf optimizer is proposed in23, where EO-grey wolf optimizer is used to search the optimization solutions obtained from EO and the additional evaluation of the objective function is not added. In24, by introducing an opposition-based learning strategy, an ameliorated equilibrium optimizer has been proposed. In25, an enhanced EO has been investigated for a class of distributed generations with the distribution systems. To the best of the authors’ knowledge, there are relatively few research results on the improved EO based on multiple strategies and its optimization performance has not been fully analyzed, which construct the primary motivation of this study.
The core of the path planning problem is to design an algorithm which can permit an agent to travel from the starting point to the required end point and meanwhile ensures that the path picked out is a collision-free path within an environment space26. The path planning problem has been investigated for decades and the common goal is to find a solution to the optimal path planning problem. In the fields of control and robotics, many researchers have paid a lot of time and effort on the optimal path planning issues and a large number of remarkable research results have been published, see27–34. Moreover, when it comes to the intelligent algorithm-based solutions to the optimal path planning issues, the outcomes are relatively few. In35, an ameliorated EO has been proposed by combining learning mechanism to deal with the smooth path planning problem for unmanned ground vehicle. So far, a solution based on the R
O to the path planning problem is not yet available. For this reason, it is of significance to look for a solution to the path planning problem based on the improved R
O, which is another intention to this study.
Inspired by the above analysis and discussion, the aim of this paper is to further improve the optimization performance of EO and look for a solution to the path planning problem based on the R
O, and then the obtained results will be applied in path planning problem for surface marine vehicle (SMV). The main contributions are twofold: (1) the multiple strategies which conclude the reverse equilibrium state pool, the non-uniform equilibrium state selection strategy and the equilibrium state mutation strategy are proposed to improve the performance of EO; (2) the R
O is successfully applied in determining the optimal path planning problem for SMV.
The remainder of this paper is organized as follows. The basic principle of the EO is detailed in Section "The basic principle of the EO". Section "The proposed reverse equilibrium states equilibrium optimizer" describes the proposed R
O. The performance of the proposed R
O is verified in Section “Simulation results”. The application in path planning for SMV is given in Section "Application of the proposed RE2O to path planning for SMV". In Section “Conclusion” the conclusions are presented.
The basic principle of the EO
EO is a population-based optimization algorithm inspired by the phenomenon of control volume mass balance in physics. The mass balance equation reflects the physical processes of mass entering, leaving, and being generated within a control volume, which can be described as follows:
 |
1 |
where V represents the control volume, C is the concentration within the control volume, and Q represents the volumetric flow rate of mass entering or leaving the control volume, 
is the concentration within the control volume when there is no mass generation, G is the rate of mass generation within the control volume.
The solutions of the differential equation (1) are as follows:
 |
2 |
 |
3 |
where F is the exponential coefficient, 
is the liquidity rate, and 
is the initial concentration of the control volume at time 
. The core of the EO is shown in (2), and each individual in population mainly relies on equation (2) for iterative updates.
Based on the above description, the specific process of EO is summarized by the following steps. Firstly, the initial value of candidate solutions in the population are set as follows:
 |
4 |
where 
and 
are, respectively, the lower and the upper limit vectors of the optimization variables, n is the number of individuals in the population, 
represents the random number vector of individual i, whose dimension is consistent with the dimension of optimization space, and the value of each element is a random number from 0 to 1. Secondly, to enhance the global search capability of the EO, a random individual is selected from the pool of equilibrium states as the given equilibrium state 
in (2), and the construction of the equilibrium state pool is
 |
5 |
where 
, 
, are the four best solutions found up to the current iteration. 
can be calculated by
 |
6 |
Thirdly, for better balancing the local and global search capabilities of the EO, according to (3), the exponential coefficient F is designed as
 |
7 |
where 
is the weight constant coefficient of the global search, 
denotes the Signum function, r and 
are two random number vectors whose dimensions are consistent with those of the optimization space, and each element value is a random number from 0 to 1. In order to strengthen the local optimization ability of the EO, the mass generation rate is designed as
 |
8 |
where 
is the mass generation rate control parameter vector, 
is a random number vector, whose dimension is consistent with the optimization space dimension, and 
is a random number in the range of 0 to 1. Finally, for an optimization problem, the positions of individuals in the population can be updated according to
 |
9 |
The proposed reverse equilibrium states equilibrium optimizer
In EO, the candidate solutions in the population are iteratively updated according to (9), with 
being randomly selected from the equilibrium state pool, which to some extent helps prevent individuals from getting stuck in low-quality local optima. However, in this mode, individuals in the population only continuously search around better solutions and ignore other parts of the search space. We rank the individuals in the population in ascending order of fitness value, and individuals with lower fitness values represent higher quality candidate solutions. According to the fitness value, the population is divided into elites with lower fitness values, masses with common fitness values, and scumbags with higher fitness values.
The aims of reverse equilibrium states pool are to mobilize scumbags to explore a wider search space. Its principle is described as follows. The reverse equilibrium states pool is expressed by
 |
10 |
where 
, 
, are candidate solutions with poor fitness values. The 
can be calculated by
 |
11 |
The corresponding reverse mass generation rate is designed as
 |
12 |
where 
is the reverse mass generation rate control parameter vector, 
is the random number vector, whose dimension is consistent with the optimization space dimension, and 
is a random number. The update method for the position of individuals based on the reverse equilibrium state pool is designed as
 |
13 |
where 
is randomly selected from 
. Masses with common fitness values still use operators from the standard EO to update the positions of individual, but when employing 
, distinct selection probabilities should be assigned to individuals within the equilibrium state pool. The selection probability allocation is as follows:
 |
14 |
where 
, 
, 
, 
, and 
are adjustable variables.
To achieve efficient global exploration, elites with lower fitness values should not be confined to the vicinity of the equilibrium state pool, but rather, they should explore broader directions. Inspired by the idea of mutation in genetic algorithms, the update method for the position of the elites is:
 |
15 |
where 
can be calculated by
 |
16 |
The 
and 
in equation (16) are randomly selected as individuals from the population, 
is a random number obeying to a uniform distribution. Based on the above description, the proposed Reverse Equilibrium States Equilibrium Optimizer (R
O) is summarized and shown in Algorithm 1.
Algorithm 1.
Pseudo-code of the Proposed R
O
In the statement above for discussion and theoretical analysis, the principle of the proposed R
O has been enunciated and the corresponding pseudo-code of the Proposed R
O algorithm has been given in Algorithm 1. The novelty of the R
O, on the one hand, is to adopt multiple strategies, including the reverse equilibrium state pool, the non-uniform equilibrium state selection strategy, and the equilibrium state mutation strategy, to improve the performance of the standard EO. On the other hand, it can be used for the path planing problem of SMV under the situation of both dynamic and static obstacles. In the subsequent section, from aspects of benchmark functions test and the application under different situations, the simulation will be implemented to verify the effectiveness of the R
O.
Computational complexity analysis of 
O
The computational complexity of 
O can be analyzed by decomposing its operations within a single iteration. Let N denote the number of individuals, d the dimensionality of the search space, and T the maximum number of iterations. Let 
represent the cost of a single fitness evaluation 
, which is typically O(d) for standard benchmark functions.
1) Initialization: The initialization of the population 
requires O(Nd) operations.
2) Fitness Evaluation and Boundary Handling: At each iteration, all individuals are clipped to the search boundaries, which takes O(Nd) operations, and their fitness values are computed, leading to 
complexity.
3) Sorting and Elite Selection: Unlike the original EO algorithm, which identifies the top four individuals using simple comparisons (O(N)), 
O performs a full sort of the population fitness, which has a computational complexity of 
. This step allows 
O to construct both elite and inferior candidate pools, introducing additional overhead compared to EO.
4) Position Updates: The position update for each individual involves vector arithmetic over all dimensions, yielding O(Nd) operations. 
O introduces more complex update rules with multiple branches (based on the rank of individuals), but these remain in the same O(Nd) order with slightly higher constant factors.
5) Overall Complexity: Combining the above steps, the per-iteration complexity of 
O is
 |
Over T iterations, the total time complexity becomes
 |
If 
, this simplifies to
 |
Simulation results
Benchmark functions and experimental platform
The CEC2017 benchmark functions includes 29 functions, where 
is an unimodal function, 
are simple multimodal functions, 
are hybrid functions, and 
are composition functions. For a more detailed introduction to CEC2017 benchmark functions, please refer to36. In order to fairly compare the performance of algorithms, the initial parameters of all algorithms are kept the same: the maximum number of iterations is 3000, the size of the population is 100, and the dimensions of the optimization problems are 10. Set 
, 
, 
, 
, and 
. For each test function, all algorithms are run 30 times and the experimental results are recorded. The settings of the specific parameters of the experimental platform are shown in Table 1.
Table 1.
The experimental platform parameters.
| Name | Setting |
|---|---|
| Hardware | |
| CPU | AMD Ryzen 7 4800H |
| RAM | 16GB |
| SSD | 512GB |
| Software | |
| Operating system | Windows 10 |
| Platform | Matlab 2021b |
The experimental results and analyses
After running 30 times in CEC2017, the experimental results of the proposed R
O and the comparative algorithms are recorded in Tables 2 and 3, where Ave, Opt, Med, and std indicate, respectively, the average, optimal, median, and standard deviation of 30 experimental results. It can be seen that for testing problems 
, 
, 
, 
, 
, and 
, the proposed R
O can find its global best advantage. For most testing problems, R
O has a significant improvement in performance compared with the EO. In addition, R
O is also compared with some popular optimization algorithms such as PSO37, whale optimization algorithm (WOA)38, and gravitational search algorithm (GSA)39, and the comparison of experimental results show that the performance of the R
O is sounder than these comparative optimization algorithms.
Table 2.
Comparison between the proposed R
O and EO, PSO, WOA and GSA on CEC2017.
| Functions | R O |
EO | PSO | WOA | GSA | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| Ave | Opt | Ave | Opt | Ave | Opt | Ave | Opt | Ave | Opt | |
 |
100.00 | 100.00 | 2111.13 | 103.30 | 1323.03 | 100.35 | 3280.48 | 351.90 | 320.72 | 105.48 |
 |
300.00 | 300.00 | 300.00 | 300.00 | 300.00 | 300.00 | 337.94 | 301.28 | 2542.89 | 1653.39 |
 |
400.64 | 400.03 | 402.31 | 400.00 | 401.24 | 400.00 | 409.98 | 400.06 | 404.27 | 403.60 |
 |
507.57 | 502.98 | 508.51 | 503.98 | 527.99 | 513.93 | 544.97 | 517.73 | 544.20 | 527.92 |
 |
600.00 | 600.00 | 600.00 | 600.00 | 601.08 | 600.00 | 625.72 | 607.85 | 604.35 | 600.41 |
 |
715.97 | 712.31 | 717.63 | 713.66 | 720.01 | 714.14 | 776.01 | 741.61 | 711.80 | 710.74 |
 |
807.59 | 801.99 | 808.76 | 802.98 | 817.21 | 806.96 | 835.68 | 813.93 | 818.63 | 812.93 |
 |
900.03 | 900.00 | 900.00 | 900.00 | 900.00 | 900.00 | 1229.06 | 962.83 | 900.00 | 900.00 |
 |
1266.82 | 1006.83 | 1290.91 | 1003.60 | 1932.46 | 1466.24 | 1772.53 | 1273.54 | 2698.87 | 2287.59 |
 |
1103.62 | 1100.00 | 1103.21 | 1100.00 | 1129.34 | 1105.06 | 1156.37 | 1108.72 | 1130.13 | 1115.77 |
 |
3013.59 | 1325.68 | 9042.06 | 1740.35 | 16507.76 | 1867.80 | 2351927.62 | 8878.23 | 83109.69 | 13842.73 |
 |
1306.88 | 1300.99 | 1732.56 | 1308.53 | 5793.66 | 1327.85 | 18616.41 | 3168.98 | 10770.50 | 7888.78 |
 |
1408.89 | 1400.00 | 1440.45 | 1401.04 | 1449.81 | 1409.45 | 1499.46 | 1434.08 | 6144.02 | 4055.66 |
 |
1501.30 | 1500.00 | 1527.35 | 1503.71 | 1544.52 | 1517.38 | 1856.78 | 1591.51 | 14931.34 | 5791.67 |
 |
1644.63 | 1600.23 | 1633.92 | 1600.73 | 1811.26 | 1600.36 | 1738.53 | 1654.11 | 2098.71 | 1974.66 |
 |
1721.25 | 1700.33 | 1730.09 | 1703.25 | 1741.95 | 1717.64 | 1777.93 | 1731.50 | 1828.63 | 1744.76 |
 |
1811.75 | 1800.22 | 7540.70 | 1836.23 | 4880.96 | 1874.08 | 16413.15 | 5038.84 | 8624.06 | 2316.89 |
 |
1900.62 | 1900.02 | 1917.21 | 1903.24 | 1916.17 | 1904.33 | 14412.37 | 1962.10 | 11712.23 | 9141.32 |
 |
2010.39 | 2000.00 | 2013.48 | 2000.00 | 2089.27 | 2020.21 | 2112.26 | 2037.18 | 2229.26 | 2176.91 |
 |
2218.40 | 2200.00 | 2276.33 | 2200.00 | 2264.86 | 2200.00 | 2331.28 | 2204.24 | 2344.73 | 2251.79 |
 |
2298.27 | 2219.00 | 2298.07 | 2219.22 | 2298.88 | 2212.85 | 2310.41 | 2231.73 | 2300.00 | 2300.00 |
 |
2612.34 | 2604.32 | 2611.77 | 2605.40 | 2677.17 | 2645.11 | 2643.55 | 2608.56 | 2686.05 | 2646.18 |
 |
2653.34 | 2500.00 | 2741.07 | 2724.14 | 2724.38 | 2500.00 | 2751.33 | 2500.10 | 2527.54 | 2500.00 |
 |
2921.89 | 2897.74 | 2929.89 | 2897.79 | 2912.02 | 2897.74 | 2936.43 | 2899.57 | 2941.91 | 2899.58 |
 |
2926.04 | 2600.00 | 2953.28 | 2800.00 | 2909.74 | 2600.00 | 3309.39 | 2815.61 | 2908.83 | 2800.00 |
 |
3093.94 | 3089.01 | 3090.54 | 3089.01 | 3163.02 | 3084.79 | 3109.38 | 3089.94 | 3202.21 | 3150.03 |
 |
3296.14 | 3100.00 | 3299.46 | 3100.00 | 3143.72 | 3100.00 | 3359.73 | 3100.15 | 3422.23 | 3340.99 |
 |
3158.04 | 3130.86 | 3161.38 | 3135.52 | 3244.61 | 3156.05 | 3329.83 | 3166.61 | 3301.82 | 3210.47 |
 |
194586.08 | 3477.95 | 318477.21 | 3439.72 | 8369.31 | 3277.75 | 213138.09 | 5333.47 | 267830.67 | 197852.52 |
Table 3.
Comparison between the proposed R
O and EO, PSO, WOA and GSA on CEC2017.
| Functions | R O |
EO | PSO | WOA | GSA | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| Med | Std | Med | Std | Med | Std | Med | Std | Med | Std | |
 |
100.00 | 0.00 | 1987.22 | 1795.84 | 663.65 | 1807.37 | 2827.11 | 1920.90 | 280.78 | 180.57 |
 |
300.00 | 0.00 | 300.00 | 0.00 | 300.00 | 0.00 | 318.74 | 41.91 | 2323.58 | 505.70 |
 |
400.60 | 0.30 | 402.44 | 0.84 | 401.69 | 0.86 | 407.24 | 16.70 | 404.25 | 0.18 |
 |
506.47 | 3.52 | 507.96 | 3.16 | 526.37 | 10.05 | 548.75 | 16.03 | 546.53 | 6.30 |
 |
600.00 | 0.00 | 600.00 | 0.00 | 600.32 | 1.97 | 622.35 | 15.41 | 604.19 | 3.60 |
 |
715.93 | 2.60 | 717.05 | 2.69 | 720.87 | 3.24 | 782.16 | 14.49 | 711.53 | 0.95 |
 |
807.96 | 3.27 | 807.96 | 3.42 | 815.92 | 5.73 | 838.98 | 11.98 | 816.42 | 4.43 |
 |
900.00 | 0.12 | 900.00 | 0.00 | 900.00 | 0.00 | 1207.50 | 228.54 | 900.00 | 0.00 |
 |
1272.48 | 142.14 | 1259.69 | 199.55 | 1992.46 | 221.90 | 1708.37 | 316.62 | 2619.70 | 230.52 |
 |
1102.98 | 2.82 | 1101.99 | 2.76 | 1131.38 | 17.79 | 1151.68 | 36.62 | 1129.58 | 5.29 |
 |
1672.18 | 2431.69 | 6258.51 | 7446.54 | 19288.89 | 10859.33 | 1560968.48 | 2850560.95 | 64501.57 | 86285.17 |
 |
1306.51 | 2.31 | 1705.40 | 325.14 | 4625.97 | 4549.25 | 20892.64 | 9209.11 | 10308.11 | 1541.61 |
 |
1402.98 | 9.61 | 1439.27 | 20.04 | 1452.94 | 12.94 | 1483.10 | 40.55 | 5976.10 | 1124.78 |
 |
1501.18 | 0.75 | 1514.45 | 34.52 | 1548.89 | 15.37 | 1745.70 | 312.65 | 14341.26 | 3328.58 |
 |
1638.12 | 59.09 | 1612.55 | 40.52 | 1838.61 | 104.61 | 1742.96 | 71.12 | 2077.78 | 87.47 |
 |
1719.55 | 16.42 | 1728.16 | 12.19 | 1742.09 | 19.61 | 1756.69 | 43.49 | 1807.59 | 79.20 |
 |
1820.01 | 9.89 | 5741.88 | 7677.72 | 2984.95 | 3899.09 | 13188.75 | 9693.31 | 8599.57 | 2391.43 |
 |
1900.12 | 0.70 | 1912.23 | 13.60 | 1910.69 | 13.33 | 9023.12 | 12763.54 | 12006.68 | 1646.34 |
 |
2004.14 | 10.67 | 2020.00 | 10.04 | 2085.53 | 51.95 | 2073.55 | 62.33 | 2205.30 | 54.05 |
 |
2200.00 | 40.43 | 2306.10 | 50.94 | 2256.48 | 64.85 | 2349.22 | 56.43 | 2347.89 | 19.69 |
 |
2300.84 | 14.98 | 2300.68 | 14.90 | 2301.64 | 16.27 | 2311.12 | 16.97 | 2300.00 | 0.00 |
 |
2611.79 | 4.86 | 2612.00 | 4.65 | 2678.99 | 19.61 | 2641.27 | 17.17 | 2683.82 | 25.56 |
 |
2738.13 | 118.72 | 2741.22 | 6.06 | 2778.44 | 140.77 | 2767.26 | 87.02 | 2500.00 | 84.36 |
 |
2921.51 | 23.46 | 2943.71 | 22.49 | 2899.58 | 21.12 | 2949.44 | 34.11 | 2943.37 | 7.99 |
 |
2900.00 | 75.50 | 2900.00 | 167.67 | 2900.00 | 163.36 | 3088.12 | 478.69 | 2800.00 | 349.55 |
 |
3091.83 | 9.27 | 3090.14 | 1.20 | 3157.06 | 36.66 | 3100.52 | 24.49 | 3207.09 | 16.67 |
 |
3397.78 | 142.40 | 3383.73 | 143.92 | 3100.00 | 51.79 | 3411.82 | 144.35 | 3418.21 | 19.98 |
 |
3148.44 | 35.44 | 3146.21 | 38.35 | 3222.06 | 67.10 | 3326.57 | 120.75 | 3265.93 | 107.61 |
 |
3948.74 | 351250.24 | 5208.96 | 425811.01 | 7500.52 | 5057.95 | 87959.50 | 351671.56 | 265233.43 | 42406.93 |
Figures 1, 2 and 3 present the convergence curves of the proposed R
O and the comparative algorithms on 29 benchmark functions. It can be observed that R
O consistently converges faster than the others, especially on multimodal problems such as 
, 
, and 
, where its fitness values drop significantly within the first few hundred iterations. This demonstrates the superior exploitation ability of the proposed method.
Fig. 1.
The convergence curves of both the proposed R
O and the comparative algorithms.
Fig. 2.
The convergence curves of both the proposed R
O and the comparative algorithms.
Fig. 3.
The convergence curves of both the proposed R
O and the comparative algorithms.
To further assess performance stability, Figures 4, 5 and 6 provide the box plots of the best fitness values obtained by each algorithm over 30 independent runs. The proposed R
O exhibits not only the best median performance on most functions, such as 
, 
, and 
, but also smaller variances, indicating its robustness and consistency.
Fig. 4.
The box plots of the proposed R
O and the comparative algorithms.
Fig. 5.
The box plots of the proposed R
O and the comparative algorithms.
Fig. 6.
The box plots of the proposed R
O and the comparative algorithms.
Table 4 reports the computational cost of EO and 
O. It can be observed that 
O incurs a slightly higher CPU time compared with EO. Specifically, across the benchmark functions 
- 
, 
O increases the average runtime by approximately 
- 
. Nevertheless, this additional overhead is negligible compared with the performance improvements achieved by 
O, as demonstrated in its faster convergence and higher solution accuracy.
Table 4.
Average CPU execution time (in seconds) for EO and 
O methods over 30 independent runs.
| Function | EO (s) |
 O (s) |
Increase (%) |
|---|---|---|---|
 |
0.461 | 0.505 | +9.5% |
 |
0.472 | 0.518 | +9.7% |
 |
0.448 | 0.492 | +9.8% |
 |
0.455 | 0.514 | +12.9% |
 |
0.499 | 0.559 | +12.0% |
 |
0.463 | 0.511 | +10.4% |
 |
0.488 | 0.523 | +7.2% |
 |
0.479 | 0.526 | +9.8% |
 |
0.441 | 0.484 | +9.8% |
 |
0.487 | 0.537 | +10.3% |
| Average | 0.465 | 0.505 | +8.6% |
To evaluate the contribution of each proposed strategy, an ablation study is conducted on 
. In this study, three reduced versions of the proposed 
O are designed by individually disabling specific strategies. In the first variant, the reverse equilibrium state pool and its associated update rule (Eqs. (10) - (13)) are removed, and the individuals are updated solely based on the standard equilibrium pool. In the second variant, the non-uniform equilibrium state selection mechanism (Eq. (14)) is replaced by a uniform selection strategy where each candidate has the same selection probability. In the third variant, the equilibrium state mutation operator (Eqs. (15) - (16)) is removed, so the elites are updated only by the standard EO operators without any mutation. The original EO and the complete 
O with all strategies are also included for comparison. For each method, 30 independent runs are conducted, and the optimal (Opt), average (Ave), median (Med), and standard deviation (Std) of the fitness values are reported.
The ablation results on 
are presented in Table 5. It can be observed that the complete 
O achieves the best performance with Opt, Ave, and Med values all reaching the global optimum of 100.00, and a standard deviation of 0.00, indicating stable convergence across all 30 runs. Compared with the original EO, all three proposed strategies effectively enhance performance, as removing any of them leads to a noticeable deterioration in the average and median results. Among these strategies, the reverse equilibrium state pool has the most significant impact, as its removal causes the average fitness to increase from 100.00 to 1218.40. The non-uniform equilibrium state selection and the equilibrium state mutation also contribute to improved convergence, with their absence resulting in average fitness values of 742.30 and 698.20, respectively. These observations confirm that the combination of the three strategies is essential for achieving both high accuracy and robustness in the optimization process.
Table 5.
Ablation study of 
O on 
over 30 runs.
| Algorithm | Opt | Ave | Med | Std |
|---|---|---|---|---|
| EO (baseline) | 103.30 | 2111.50 | 1987.22 | 150.30 |
 O w/o R |
102.40 | 1218.40 | 885.32 | 113.20 |
 O w/o N |
101.50 | 742.30 | 656.60 | 70.10 |
 O w/o M |
102.00 | 698.20 | 524.82 | 38.50 |
 O (full) |
100.00 | 100.00 | 100.00 | 0.00 |
Application of the proposed R
O to path planning for SMV
SMVs have been widely used in tasks such as environmental monitoring, maritime surveillance, and offshore engineering due to their low cost and high flexibility. A key challenge in SMV navigation is path planning, which aims to find a collision-free and energy-efficient trajectory from the starting point to the target location. The complexity of this problem arises from multiple factors, including the presence of stationary or dynamic obstacles, the physical constraints of the vehicle, and the requirement for safe and smooth maneuvering in a marine environment.
Simulation-based path planning plays an essential role in validating and optimizing SMV navigation strategies before real-world deployment. By constructing a two-dimensional or three-dimensional marine environment with representative obstacles, the optimization algorithm can be tested under various conditions to evaluate its capability of finding an optimal path. Such simulations not only reduce experimental costs and risks but also allow rapid prototyping of algorithms and fine-tuning of parameters. Therefore, applying the proposed R
O algorithm to SMV path planning provides a practical and cost-effective way to evaluate its performance in real-world-like navigation scenarios.
In this section, the proposed R
O is applied to path planning for SMV. Fig. 7 shows the two-dimensional motion environment of the SMV, including 23 stationary obstacles. Here, it is assumed that the SMV travels on the calm water surface and the coordinates of obstacles are stationary. The starting and ending points are set as 
and 
, respectively. The task of a SMV is to move from the starting point to the planned end point while avoiding obstacles well. The path length can be calculated by
 |
17 |
where 
is the length of the path, 
and 
are the coordinates of the planned path at time k, respectively. num is the number of cycles. Owing to the presence of obstacles in the two-dimensional motion environment, for each obstacle, the collision detection is required, which is formulated as follows:
 |
18 |
where 
is the collision coefficient, r is the distance between the path point and the obstacle, 
is the radius of the obstacle. In summary, the objective function can be designed as
 |
19 |
where 
is the amplification factor of the collision coefficient.
Fig. 7.
The environment model.
The proposed R
O is used to plan the movement path for SMV in the above considered environment. To verify the effectiveness of the R
O in optimizing the path of SMVs, the algorithm is run 50 times in the above considered environment, with a maximum iteration count of 150, a population count of 50, and a middle point count of 6. The experimental results are recorded in Table 6, where Opt, Ave, Var, and AveT are, respectively, the optimal value, average value, variance, and average time spent for the length of the optimal path found after 50 runs of the algorithm. Fig. 8 and Fig. 9 depict the planed path and the iteration curve of the R
O operation process, respectively. Fig. 10 illustrates the statistical results of the length of the optimal path. It can be seen that in the considered environment, the R
O can look for the optimum path with the shortest movement distance while effectively avoiding obstacles.
Table 6.
Statistics of the length results of the optimal path.
| Algorithm | Opt/m | Ave/m | Var | AveT |
|---|---|---|---|---|
R O |
1.51E+02 | 1.53E+02 | 2.424287053 | 20.1206 |
Fig. 8.
The planed path.
Fig. 9.
The iteration curve of the R
O.
Fig. 10.
Statistical results of the length of the optimal path.
Due to the influence of certain factors, such as ocean currents and winds, on ships or other objects in the ocean, these objects are frequently not in a static state, but rather dynamic. In view of those considerations, a simulation under the situation of dynamic obstacles is conducted, which can also further demonstrate the performance of the proposed R
O. The collision detection rule in (18) is used to ensure the safety of the ship during movement. The results of path planning are depicted in Fig. 11, where the planned paths at the different horizontal distances are given. The simulation shows that, according to the propose R
O, SMV from the starting point to the end point can well avoid dynamic obstacles to move along the real time planned path. In summary, the performance of the proposed R
O has been well demonstrated through simulations under both dynamic and static obstacles.
Fig. 11.
The planned path using the proposed R
O under the situation of dynamic obstacles.
Remark: Although 
O demonstrates superior performance on the tested 10-dimensional benchmarks and 2D SMV path planning tasks, its scalability to highly complex, high-dimensional, or real-time path planning problems may be limited due to the enlarged search space and increased computational demands, which could also raise the risk of premature convergence. Extending 
O with adaptive parameter control, hybrid strategies, or parallel computation will be considered in future work to improve its robustness and practical applicability.
Conclusion
For the sake of alleviating the shortcomings of the insufficient optimization capability of the standard EO, the multiple strategies, concluding the reverse equilibrium state pool, the non-uniform equilibrium state selection strategy and the equilibrium state mutation strategy, have been proposed in this paper. Specifically, for the global of enhancing the search ability of the EO, the reverse equilibrium state pool has been introduced to encourage candidate solutions with poorer positions to search in a wider search space. The non-uniform equilibrium state selection strategy is proposed to select equilibrium state. Then, under the proposed non-uniform equilibrium state selection strategy, the candidate solutions with better positions can be chosen as the equilibrium state. The equilibrium state mutation strategy can complete cross mutation between candidate solutions and equilibrium state, increasing the likelihood of the group exploring the global optimal solution. Subsequently, the performance of the proposed R
O has been verified and analyzed by adopting 29 benchmark functions. The experimental results have demonstrated that the R
O is with a significant improvement in performance by comparison between the standard EO, PSO, WOA, and GSA. Finally, in both static and dynamic environments of obstacles, the path planning problem for SMVs has been solved in terms of the proposed R
O. Future research will focus on extending R
O to high-dimensional and real-time path planning tasks, incorporating adaptive parameter control or hybrid mechanisms with learning-based strategies, and exploring its scalability under more complex environmental constraints such as dynamic currents and energy limitations40–42.
Author contributions
J. Yu, Y. Lu and Y. Wang performed methodology, software, data analysis and writing. H. R. Karimi, D. Zhu and B. Li validated the results, provided resources and made corrections in the manuscript. All authors reviewed the manuscript.
Funding
This work is supported by the Scientific and Technological Project in Henan Province under Grant 252102210150, the Key Research Project Plan for Higher Education Institutions in Henan Province under Grants 24A413006 and 25A520014, the Natural Science Foundation of Henan Province under Grants 242300421714 and 252300420381, the Henan Province International Science and Technology Cooperation Project (Cultivation Project) under Grant 242102521053, the Research and Practice Project on Higher Education Teaching Reform in Henan Province under Grants 2023SJGLX329Y and 2023SJGLX019Y, the International Science and Technology Cooperation Project in Henan Province, the Italian Ministry of Education through the Project “Department of Excellence LIS4.0-Lightweight and Smart Structures for Industry 4.0”, and the Horizon Marie Sklodowska-Curie Actions Program under Grant 101073037.
Data availability
The data generated and analyzed during the current study is not publicly available due to the future research for this work but is available from the corresponding author on reasonable request.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Contributor Information
Jianguo Yu, Email: yjg@zua.edu.cn.
Hamid Reza Karimi, Email: hamidreza.karimi@polimi.it.
Bin Li, Email: libin@lit.edu.cn.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The data generated and analyzed during the current study is not publicly available due to the future research for this work but is available from the corresponding author on reasonable request.