Velocity clamping-assisted adaptive salp swarm algorithm: balance analysis and case studies

1.   Introduction

Global optimization problems exist widely in scientific research and engineering applications. In the past, optimization problems were solved using methods such as Newton's method, Lagrange multiplier method, quasi-Newton, and other accurate calculation methods. Although the effectiveness of these approaches has long been proven, due to the increasing of data complexity and dimensionality, all information in the feature space is unknown in various fields such as machine learning and engineering design, so the optimal solutions to all optimization problems cannot be obtained using exact mathematical models. Finding optimal solution has become a challenge in the field of optimization. To solve this problem, researchers have created metaheuristic algorithms. It is a stochastic optimization technique that does not require any prior knowledge and can obtain approximate optimal solutions through established steps 1. Swarm intelligence-based algorithms are an essential branch of nature inspired algorithms and have become a research hot topic in the metaheuristic community, which include but are not limited to: particle swarm algorithm (PSO) 2, marine predators algorithm (MPA) 3, ant colony optimization (ACO) 4, earthworm optimization algorithm (EWA) 5, artificial bee colony (ABC) algorithm [6, 7, moth flame optimizer (MFO) 8, firefly algorithm (FA) 910, whale optimization algorithm (WOA) 11, cuckoo search (CS) algorithm 12, grey wolf optimization (GWO) algorithm 13, krill herd (KH) algorithm 14, harris hawks optimizer (HHO) 15, monarch butterfly optimization (MBO) 16, grasshopper optimization algorithm (GOA) 17, elephant herding optimization (EHO) 18, slime mould algorithm (SMA) 19, hunger games search (HGS) 20, Runge Kutta method (RUN) 21, colony predation algorithm (CPA) 22, weighted mean of vectors (INFO) 23.

In this work, we focus on a relatively novel and well-established population intelligence-based approach, called salp swarm optimizer (SSA) 24, which was developed by Mirjalili et al. in 2017. Compared with other well-known metaheuristics, SSA has several merits, including few control parameters, straightforward framework, and distinctive search pattern. Numerous studies conducted on SSA have shown that the optimizer demonstrates competitive performance on function optimization and real-world engineering optimization problems. Consequently, SSA is extensively adopted to tackle various global optimization tasks. In 25, Wang et al. developed a dynamic version of SSA by introducing a boosted orthogonal opposition-based learning component and ranking-based learning tactic to the basic SSA. The boosted orthogonal opposition-based learning component helps the leader to search for unknown promising regions to improve the convergence speed. The dynamic learning mechanism enhances the dynamic nature of the follower's position update pattern, thus helping the algorithm avoid getting trapped in local optima. A series of experiments, including numerical function optimization, engineering design problems, and parameter estimation of PV model, validate that this SSA variant outperforms the basic SSA, well-known SSA variants, and frontier metaheuristic techniques. In addition, the developed SSA-based algorithm is employed to solve the mobile robot path planning problem and its performance is tested on five environmental maps. Simulation results show that the proposed path planning approach has competitive performance. In 26, Ewees et al. proposed an enhanced SSA algorithm based on the operators of FA, called SSAFA. In SSAFA, the operators of FA are used to improve the exploitation ability of the basic SSA. The performance of this SSA variant was verified on the unrelated parallel machine scheduling problem. In 27, Tu et al. developed a quantum-behaved SSA algorithm. The proposed method introduces quantum mechanics theories and trajectory analysis in the position update mechanism to enhance the local search ability of followers. The algorithm is applied to solve the localization problem in anisotropic wireless sensor networks. Experimental results show that this method outperforms its competitors in terms of accuracy and robustness against network anisotropy. In 28, Tubishat et al. suggested a novel SSA variant that includes two modifications. The first refinement includes the use of opposition-based learning mechanism during the population initialization period to increase the population diversity and to speed up the convergence of the algorithm. The second enhancement includes the introduction of a new local search mechanism to boost the local search ability. The performance of the developed method was validated using the feature selection problem. The experimental results indicate that this approach achieves better performance in terms of fitness values, accuracy, convergence curves, and feature reduction on most of the employed 18 datasets from UCI repository compared with the other four baseline approach.

Although SSA has demonstrated more competitive performance than some well-known metaheuristic approaches in solving numerical and engineering optimization problems, it still has some drawbacks, including unbalanced exploitation and exploration properties, poor convergence performance, and prone to fall into local optimal. To mitigate the above-mentioned limitations, many improved SSA-based algorithms have been developed. In 29, a novel SSA variant incorporating three perturbation mechanism was developed. Gaussian mutation was utilized to enhance neighborhood-informed ability. Cauchy mutation was employed to improve the global search capability. Levy-flight mechanism was introduced to boost randomness. The performance of the method was investigated using 23 classical benchmark functions. The experimental results show that the three mutation schemes can effectively augment the exploration and exploitation abilities. In 30, a hybrid improved whale optimization salp swarm algorithm was introduced. First, the linear relationships in WOA are replaced with exponential relationships to improve its search ability. Second, the improved WOA algorithm is fused with the basic SSA, and the combined algorithm is more flexible and has a faster convergence rate and higher convergence accuracy. The performance of the algorithm is verified using 23 classical benchmark functions and compared with 8 swarm intelligent algorithms. The comparison results demonstrate that the algorithm has good performance. In addition, the method was used to tune the adaptive PID controller. From the simulation results, the enhancement of the algorithm is demonstrated by different performance metrics. In 31, an improved SSA injected with random opposition-based learning, multiple leadership, and simulated annealing mechanisms was developed. The random opposition-based learning strategy facilitates the expansion of the population search. The multi-leader mechanism can improve the exploration ability. The simulated annealing component enhances the local search ability and is beneficial to improve the solution accuracy. The performance of the algorithm was tested on CEC2015 benchmark function. The algorithm was also used to train feedforward neural networks. The experimental results show that the developed approach has excellent performance on both numerical optimization problems and training feedforward neural network problems. In 32, the SSA algorithm was enhanced by adding three simple but effective strategies. Firstly, the control parameter in SSA responsible for equilibrating exploration and exploitation is chaotically altered to catch a better tradeoff between global search and local search. Secondly, a mutualistic strategy is used to augment the neighborhood interaction between leaders. Finally, a stochastic technique is adopted to enrich the diversity of the population to expand the search range of the followers. To verify the effectiveness of this algorithm, numerical optimization problems and practical engineering design cases are applied. Experimental results demonstrate that the algorithm outperforms its competitors. In 33, an improved SSA algorithm with Gaussian Barebone mechanism and Stochastic Fractal Search strategy, called GBSFSSA, was introduced. In GBSFSSA, Gaussian Barebone mechanism and Stochastic Fractal Search strategy effectively enhance the balance between exploration and exploitation of the basic SSA algorithm. Experimental results on CEC2017 functions show that GBSFSSA has competitive performance. Moreover, this approach is applied to solve image segmentation problems and the effectiveness of the proposed methodology is tested on COVID-19 CT images. The experimental results show that GBSFSSA is a trustworthy image segmentation tool. In 34, differential evolution and chaotic structure were embedded in the basic SSA to improve its convergence performance. Chaotic initialization can enrich the population diversity of the initial population and thus speed up the convergence rate. The differential evolution mechanism is introduced in the search process to make a more solid balance between exploration and exploitation. The improved SSA algorithm is tested using the CEC2014 benchmark functions and five classical engineering design problems, and the results show that the advocated method outperforms the basic SSA and some popular swarm intelligent algorithms. In addition, this SSA variant was used to solve the feature selection problem and obtained desirable results. In 35, the location update mechanism of the basic SSA was revised and an elite gray wolf domination strategy was inserted in the final stage of population location update. The modified location update pattern enhances the local search ability of the algorithm, and the elite domination strategy improves the convergence speed of the basic SSA. A series of experiments including numerical optimization problems, engineering design cases, and feature selection tasks verified the superior performance of the proposed algorithm. In 36, a novel annealing salp swarm algorithm named SASSA was proposed. In SASSA, the simulated annealing strategy is introduced in the follower position update phase, random walk in simulated annealing is implemented using levy flight, and the number of leaders is increased. With the help of these three mechanisms, the SSA algorithm achieves a better and solid balance between global search and local search. Experimental results on 30 CEC2017 test functions show that the SASSA algorithm outperforms the standard SSA and 23 peer algorithms. Additionally, the SASSA method is used to solve five engineering design problems and the fertilizer effect function problem, and the experimental results indicate that SASSA has strong competitiveness. In 37, a levy flight-based population position update mechanism was proposed to replace the location update pattern of the basic SSA. The new location update pattern improves the population diversity. Then, the search technique of HHO was introduced for improving the global search capability of the algorithm. The results of comparison experiments on CEC2014 test function set validate the superiority of the proposed method. This algorithm is also used to segment breast cancer microscopic images and the experimental results demonstrate its superior performance.

The existing SSA variants mainly focus on achieving a delicate balance between cohesion and alignment patterns. Despite the development of many effective SSA variants, a research gap still exists in this field. While the existing SSA variants improve the overall performance of the basic SSA, unbalanced exploitation and exploration operators still exist. This is mainly because the existing SSA variants mainly emphasize on improving the exploration or development capability of the algorithm in expectation of achieving a balance between the two, but there is no algorithm that can smoothly switch between development and exploration. Therefore, in this study, a novel SSA variant, called VC-SSA, is reported to intensify the overall performance of the standard SSA. The main contributions of this paper are summarized as follows:

1) A novel version of SSA named VC-SSA is proposed, which embeds three effective but simple mechanisms, namely, velocity clamping mechanism, reduction factor tactic, and adaptive weight strategy. The ablation study in subsection 4.5 verifies the role of the three components in VC-SSA.

2) To verify the effectiveness of the proposed VC-SSA algorithm, comparison experiments with a number of state-of-the-art swarm intelligent algorithms and SSA variants are carried out using 23 classical benchmark functions and 30 complex optimization problems from IEEE CEC 2017. The results are statistically analyzed by the Friedman test and the Wilcox sign-rank test.

3) To further prove the superiority of VC-SSA, it is employed to five engineering problems and the mobile robot path planning task. The experimental results demonstrate that VC-SSA can achieve satisfactory solutions for the above-mentioned cases.

4) The remainder of this study is arranged as follows: The fundamental principle of the basic SSA is explained in Section 2. Three proposed modified mechanisms are described and the developed VC-SSA algorithm is depicted in Section 3. Section 4 investigates the validity of VC-SSA on twenty-three well-known benchmark problems. Section 5 focus on a VC-SSA-based mobile robot path planning approach. Finally, the conclusions and future directions of our paper are summarized in Section 6. The detailed flow of the work is illustrated in Figure 1.

2.   Salp swarm algorithm

SSA is a swarm-intelligence based algorithm with a minimalist style developed by Mirjalili et al. ^[24] in 2017, which is motivated by the unique swarming behavior of the salps found in oceans. In SSA, the search agents performing optimization are divided into leading salps and followers according to their position. The individual at the top of the salp group is considered as the leader, and those behind it are classified as followers. The location updating formula for the leading salp is as follows:

where x_j¹ represent the location of the leading salp, F_j is the parameter that depicts the jth dimension of the food source, ub_j denotes the upper limit of the search space and lb_j shows the lower limit of the search landscape, c₂, and c₃ are random numbers between [0, 1], c₁ is an essential parameter in SSA, which is responsible for maintaining the balance between global search and local exploitation during the search process, and its value is updated by the following mathematical model.

where t and T represent the current and the maximum iterations, respectively.

In follower position update phase, the mathematical model is formulated as follows:

where x_jⁱ indicates the jth dimension in the position of the ith search agent.

3.   Structure of the proposed VC-SSA

The SSA is a population-intelligence based global optimizer with straightforward framework and relatively outstanding performance. However, the strong minimalist style of SSA destabilizes its inherent tradeoff between convergence and diversity and slows down the convergence rate. Therefore, a modified SSA called VC-SSA is proposed for further augmenting the overall performance of the standard SSA in this section. In addition, the NFL theorem 38 demonstrates that the average performance of any optimization technique on all optimization cases is equal, which is the main motivation for us to conduct this study. It is important to state that VC-SSA improves the performance of SSA by introducing new operators while keeping its original framework intact. Consecutive subsections will elaborate on each of the components in detail.

3.1.   Velocity clamping strategy

For swarm-intelligence based techniques, the exploration-exploitation trade-off is a core imperative that directly affects the convergence rate and the solution precision. In the standard SSA, the parameter c₁ acting on the leader position update phase is responsible for this aspect. Its value decreases nonlinearly over the course of iterations with the hope of helping the salp chain to develop the solution landscape during the early iterations and to mine the rough position of the global optimal after the lapse of iterations. From Eq (1), the term assisted by c₁, i.e., the vector ((ub_j-lb_j)c₂+lb_j), can essentially be understood as the leader's velocity. Based on this analysis, Eq (1) can be reformulated as

where V_j means the jth dimension in the velocity of the leading salp.

From Eq (4), during the search process, the leader generates a step size depending on the velocity V and moves around the food source at that step. Nevertheless, according to Eq (5), the step size stemming from the velocity is completely random since the only parameter involved, c₂, is an arbitrary number between [0, 1]. This would lead to the velocity V escaping the constrain of c₁, thus defeating the expectation that a smaller step size should be used during the early search process, while the step size should reduce as the search progresses. To alleviate this problem, this paper proposes a velocity clamping strategy. After the leader generates the velocity, it is adjusted according to the following rule and the position is updated in accordance with the modified velocity.

where V_jmax and V_jmin are the jth dimension in the maximum and minimum legal velocity of the leader, respectively. To initialize them, Eqs (7) and (8) are employed.

where δ is a constant number called clamping factor.

Overall, with the help of the velocity clamping strategy, the role of the parameter c₁ is enhanced, so that the leader can adequately search the solution space using lager step sizes in the early search phase, while moving with smaller step sizes to improve the explored area in the later search stage.

3.2.   Reduction factor strategy

The performance of the basic SSA is improved with the addition of the velocity clamping strategy, but it also brings a pitfall. If the velocity in each iteration is always equal to V_max or V_min, the search agent may be committed to searching on the boundaries of the search region, or it may continue to hunt on the boundaries of the already located current global optimal area, but unable to refine this promising zone. There are many alternatives to address this challenge. In the current study, we have tackled it by inserting a reduction coefficient into the leader position update equation. The mathematical expression for this new procedure is presented below.

where H(t) is the reduction factor, and this attenuation is expressed using Eq (10).

Here, the value of s is defined by Eq (11)

where c₁ is the conversion parameter of SSA.

From Eq (11), s decreases nonlinearly during the iterations, and consequently, H increases dynamically. As a result, the initial H is favorable for global exploration, while after the lapse of iterations, it is preferable for local exploitation. Consequently, the above notion can help the algorithm to obtain a subtle tradeoff between the diversification and intensification operators.

3.3.   Adaptive inertia weight mechanism

In the basic SSA, the follower position update pattern adopts a minimalist style, which makes the algorithm easy to accomplish, but also poses some limitations. From Eq (3), the followers update their states only pursuant to the neighboring followers, and this procedure does not have any adaptivity, and the interaction between followers is relatively inflexible. Once a neighboring individual is trapped in the sub-optimal region, all followers behind it will inevitably be lured to that locally optimal region, eventually leading to premature convergence of the optimizer.

To tackle this limitation, we focus on modifying the position update formula of followers and propose an adaptive position update pattern to replace Eq (3). The new mathematical expression is:

where ω is an inertia weight.

The concept of inertia weights was first proposed for controlling the velocity of particles in PSO and obtained satisfactory results. Inspired by this idea, researchers have introduced this operator to other nature-inspired metaheuristic techniques. For example, Jena et al. 39 designed a sigmoid-adaptive inertia weight and introduced it into the social group optimization (SGO) 40 to improve its overall performance. Sun et al. 41 employed the nonlinear inertia weight factor to reinforce the equilibrium between exploration and exploitation properties of the atom search optimization (ASO) algorithm 42. Ma et al. 43 amended the position update pattern of MFO by utilizing the inertia weights. Li et al. 44 adopted the dynamic inertia weight to amend the position equation of sine cosine algorithm (SCA) 45. In this work, we design a new adaptively inertia weight coefficient and the mathematical formulation is formulated as below:

where ω_max and ω_min represent the maximum and minimum weight values, respectively, λ and μ are constant number.

Figure 2 draws the evolution of the proposed inertia weight over 500 iterations. The figure indicates that the value of ω decreases nonlinearly during the search process, which is advantageous for the algorithm to explore in the early phase of evolution and switch smoothly to exploit after the iterations have elapsed. Accordingly, the follower can adaptively update the state to balance diversification and intensification.

3.4.   VC-SSA algorithm

Based on the above analysis, our modifications to the SSA algorithm are mainly based on the three components introduced. For a more systematic understanding of the proposed VC-SSA, its pseudo-code is given in Algorithm 1, and Figure 3 displays the flowchart of the proposed VC-SSA.

To analyze the calculation complexity of VC-SSA, the steps to calculate the complexity are as follows:

1) The computational complexity of population initialization is O(N), where N is the population scale.

2) The computational complexity of assessing the fitness values of the initial population is O(N).

3) The computational complexity of sorting the initial population according to fitness values and finding the food source position is O(N²).

4) In the iterative search, the position of each salp is changed and their fitness values are computed. The computational complexity of this part is O(2N).

5) In the final stage of the iteration, the population is sorted and the food source is updated. The computational complexity of this operation if O(N²).

Therefore, the total time complexity of the proposed VC-SSA is as follows:

The generalized time complexity of the VC-SSA on the problem with a d-dimensional search space is given by

According to the above analysis, the computational complexity of VC-SSA does not increase compared to the basic SSA. Analyzed from another perspective, the VC-SSA algorithm focuses on adjusting the position update mechanism of SSA and does not include any additional search phase, so its time complexity is the same as that of the basic SSA.

4.   Simulation and comparisons

To strictly investigate the effectiveness of the developed VC-SSA, a series of experiments are performed to tackle various benchmark functions. Any involved algorithm is coded on MATLAB R2016b software under Windows 10 operating system with Intel(R) Xeon(R) W-2102 CPU @ 2.90 GHz with 8 GB RAM.

4.1.   Benchmark test functions

For comparison purposes, a recognized benchmark set including 11 unimodal and 12 multi-modal functions is selected. Details of the covered classical baseline functions are reported in Table 1.

In Table 1, these 23 classical benchmarks have different characteristics, and by testing on them, the global search and partial exploration abilities of the optimizer can be comprehensively investigated. The unimodal problems possess only one global optimal solution and are appropriate for testing the approach's capability to exploit in a narrow region. On the contrary, the multimodal cases have multiple local minima and are appropriate for examining the global exploration ability of the optimizer 46.

4.2.   Compared against SSA and SSA variants

To evaluate the effectiveness of the developed VC-SSA approach, we conducted tests on 23 classical benchmark functions with 100 dimensions, the details are presented in Table 1. The VC-SSA approach was compared with 11 SSA variants, including the basic SSA 24, ISSA 28, GSSA 29, IWOSSA 30, ASSA 47, RDSSA 48, CSSA 49, ESSA 50, LSSA 51, OBSSA 52, ASSO 53. The general parameters of all the mentioned approaches identical, i.e., the population size and the maximum evaluations were 30 and 500, respectively. The specific parameters of the involved approaches were extracted from the respective source work. The details are reported in Table 2. In the proposed VC-SSA algorithm, δ = 0.003, μ = 0.04, ω_max = 0.9, ω_min = 0.2, λ = 3, where the value of δ is set using test and trail. Each algorithm was executed 30 trials and the average minimum (Mean) and standard deviation (Std) were employed as metrics to evaluate the effectiveness of the algorithms. To strengthen the belief in the VC-SSA algorithm, Friedman's rank test was employed. In addition, to observed the statistical differences in outcomes of VC-SSA and its competitors, the Wilcoxon signed rank test with significance level 5% has been considered. The comparison results of VC-SSA with its competitors and the average ranking of each method are reported in Table 3. The p-value from Wilcoxon signed rank test are provided in Table 4. The symbols "+/-/=" in Table 4 indicate that the VC-SSA performs significantly superior, inferior, or similar to its peers.

From Table 3, VC-SSA finds theoretical optimal solutions on all benchmarks except f₇ and f₁₃. Compared with SSA, IWOSSA, CSSA, ISSA, ASSA, and LSSA, VC-SSA provides superior values on all test cases. Concerning GSSA, VC-SSA finds superior and similar scores on 21 and two cases, respectively. VC-SSA outperforms ESSA and OBSSA on 17 test functions, and for f₅, f₁₂~f₁₄, f₁₇, and f₂₁, they achieve similar results. With respect to ASSO and RDSSA, VC-SSA shows better and similar performance on 18 and 5 test cases, respectively. Moreover, the average ranking presented at Table 3 indicates that VC-SSA gets the highest ranking while the basic SSA has the lowest ranking, which demonstrates that all SSA variants intensify the performance of the standard SSA, with VC-SSA enhancing it the most. As shown in Table 4, there is almost no p-value lager than 0.05, which proves that the differences between VC-SSA and the comparison methods are statistically significant.

In addition, Figure 4 illustrates the average ranking of the involved approaches on the selected 23 tested functions in the form of a radar plot. The radar chart shows that the VC-SSA algorithm is ahead of the comparison algorithms on all benchmark problems.

4.3.   Scalability test

In this subsection, the effect of dimensionality variation on the performance of the algorithm is discussed through scalability test. For further experiments, VC-SSA is employed to tackle the 23 benchmark problems with 10000 dimensions in Table 1. The parameter settings are identical to those in Subsection 4.2. The VC-SSA approach is performed 30 times on per test case, and the optimal (Best), worst (Worst), average minimum (Mean), and standard deviation (Std) of the objective function values are reported in Table 5. Moreover, the success rate indicator was introduced to estimate the performance of the VC-SSA algorithm in dealing with high-dimensional problems. For this purpose, the following flag is designed to distinguish whether a solution is successful or not.

where S_A indicates the result achieved by the optimizer on the benchmark problem, and S_T represents the theoretical optimal solution of the corresponding case. If the solution provided by the proposed approach on the test case satisfies the above equation, it means that this solution is successful. Note that the success rate listed in Table 5 is the proportion of the successful times out of 30 experiments.

From Table 5, the VC-SSA algorithm obtains global optimal solutions on 21 benchmarks; for f₇ and f₁₃, although VC-SSA fails to converge to the theoretical optimum, the solutions obtained are still satisfactory. From another perspective, comparing the results in Table 5 with those reported in Table 2, we can see that the suggested approach exhibits similar performance on large-scale problems as on 100-dimensional functions, which represents that the impact caused by dimensionality variation on the VC-SSA algorithm is negligible. Also, in terms of SR%, VC-SSA can achieve 100% success rate on 22 tested functions; for f₇, the VC-SSA algorithm has 80% success rate. According to the above analysis, the proposed optimizer has excellent scalability and is available as as an auxiliary tool for tackling large-scale optimization cases.

4.4.   Comparison against other meta-heuristic algorithms

To further validate the effectiveness of the anticipated VC-SSA algorithm, 11 cutting-edge metaheuristic techniques, including TSA 54, SOGWO 55, MPA 3, HGS 20, EO 46, ArOA 57, AOA 58, IGWO 59, WEMFO 60, DMMFO 43, and OGWO 61, were utilized as the comparison approaches. The general parameters of the approaches involved are consistent with those mentioned in Subsection 4.2, and each method was performed 30 times to mitigate the effect of random values. The specific parameter settings of the competitors were adopted from the respective original work, as detailed in Table 6. The detailed results with 100 dimensions, including the average minimum (Mean), standard deviation (Std), and the average ranking are listed in Table 7. The Wilcoxon signed-rank test with significance level 5% are presented in Table 8 to check the significant variances between the VC-SSA and its competitors on the benchmark functions.

From Table 7, the performance of VC-SSA is better than TSA and DMMFO on all test functions. Compared to IGWO and OGWO, VC-SSA provides superior and similar outcomes on 22 and one problems, respectively. Regarding EO, AOA and WEMFO, the suggested method presents better and similar performance on 19 and four benchmarks, respectively. Compared with MPA, VC-SSA gets the better results on 18 cases; for f₅, f₁₂, f₁₄, f₁₇ and f₂₁, they successfully find the global optimal solution. Concerning HGS, VC-SSA provides better results on 17 functions; for the other seven problems, two optimizers converge to the theoretical optimum. VC-SSA outperforms ArOA on 16 cases; for the other seven benchmarks, two algorithms find the theoretical optimum. Moreover, according to the average ranking results of the 12 methods, VC-SSA obtained the first ranking, TSA ranked last, and the rest of the approaches ranked second to eleventh. The above analysis reveals that the overall performance of the proposed VC-SSA algorithm outperformed its peers.

From Table 8, the reported p-values are less than 0.05 except for six pairwise comparisons (ArOA versus VC-SSA on f₂, ArOA versus VC-SSA on f₇, ArOA versus VC-SSA on f₂₁, HGS versus VC-SSA on f₆, EO versus VC-SSA on f₁₄, and AOA versus VC-SSA on f₁₄). This implies that VC-SSA outperforms its peers and that the difference between VC-SSA and the comparison algorithms is statistically significant.

Figure 5 draws a radar plot to highlight the rank variation of VC-SSA and other state-of-the-art metaheuristic techniques over 23 benchmark problems. From the radar diagram, the developed approach achieves better rankings on all cases compared to the competitors, which represents that the suggested VC-SSA outperforms all the competitors.

4.5.   Component analysis

The developed VC-SSA algorithm enhances the overall performance of the standard SSA by introducing three novel components, namely, velocity clamping strategy, reduction factor tactic, and adaptive inertia weight mechanism. In this subsection, we perform several suites of comparison experiments to check the validity of the modifications. For this purpose, 23 classical numerical functions from Table 1 were used, and the dimensionality was set to 100. The parameter settings were identical to those in Subsection 4.2. The naming rules for the SSA-based approaches equipped with a single improvement strategy are: the enhanced SSA using only the velocity clamping strategy is called VSSA; the improved SSA adopting only the reduction factor is noted as RSSA; and the boosted SSA applying only the adaptive inertia weight mechanism is known as ISSA. Each method is executed 30 times independently for each benchmark problem, and the corresponding average minimum (Mean) and standard deviation (Std) as well as average rankings are selected as performance indicators and recorded in Table 9.

From Table 9, ISSA, RSSA and VSSA outperform the standard SSA on all benchmark functions, which proves that the three developed components are effective. Moreover, according to the pairwise comparison between the single-strategy algorithm and the VC-SSA, VC-SSA obtains better outcomes than VSSA on all benchmark functions. With respect to ISSA, VC-SSA shows better performance on 17 benchmark functions, and for the other 6 problems, both algorithms get theoretically optimal solutions. VC-SSA and RSSA obtain similar results on most benchmark functions, while for f₇, VC-SSA gets better results than RSSA, which indicates that the reduction factor can improve the performance of the basic SSA, but the exploitation capability is still insufficient. It is necessary to use velocity clamping and adaptive inertia weighting strategies to further improve the ability to refine the already explored region. Finally, the average ranking derived from the Friedman's rank test indicates that the VC-SSA algorithm ranks first, followed by RSSA, ISSA, and VSSA, which further validates that the proposed strategies are efficient and can maximize the performance improvement when the three introduced components collaborate to assist SSA.

4.6.   Convergence analysis

The proposed VC-SSA algorithm aims to expedite the convergence speed and convergence precision of the standard SSA, thus providing a well-performance optimization tool for global optimization tasks. In this subsection, the convergence performance of the VC-SSA approach will be investigated. In Figure 6, the convergence curves of VC-SSA and SSA-based algorithms on 15 representative functions in Table 1 are plotted. In Figure 7, the convergence graphs of VC-SSA and 11 frontier methods on 15 typical functions in Table 1 are drawn. In Figure 8, the convergence curves of VC-SSA and SSA using different optimization mechanisms are plotted. In all experiments, the dimensionality of the function is set to 100, the algorithm terminates after 500 iterations, and the specific parameters of all algorithms are set as before.

From Figure 6, the VC-SSA algorithm progressively converges during the early evaluation process and becomes faster over the course of iterations. The main reason is that a well-established swarm intelligence algorithm needs to allocate a large number of steps in the initial stage to explore the search domain and find the rough location of the global optimum, and in the later phase, it needs to shift from global search to local exploitation and precisely search the already explored region precisely to improve the solution accuracy. Therefore, as shown in Figure 6, the comparison algorithms fall into search stagnation after several iterations, while VC-SSA continues to converge, and the overall convergence rate and convergence precision are better than the involved peer approaches. From Figure 7, the suggested algorithm converges sluggishly during the early search phase. After the lapse of few iterations, the converge speed becomes faster. Compared with other frontier algorithms, VC-SSA shows a greater advantage in terms of convergence rate and solution accuracy, which is due to that the developed method can maintain a proper equilibrium between exploration and exploitation. According to Figure 8, the convergence performance of all three SSA algorithm using different optimization mechanisms outperforms the basic SSA, which proves that the proposed optimization mechanisms are effective. In addition, the VC-SSA algorithm outperforms the SSA using a single strategy in terms of convergence speed and convergence accuracy, which indicates that the three strategies can maximize the performance of SSA when they help SSA collaboratively. According to the convergence graph, RSSA facilitates to accelerate the convergence speed of the algorithm in the later iteration, VSSA contributes to improve the search ability of the algorithm in the early iteration stage, and ISSA solidifies the balance between exploitation and exploration of the algorithm. Overall, the convergence performance of VC-SSA outperforms the other SSA variants and the compared cutting-edge methods.

4.7.   Experiment on CEC2017 benchmark functions

In this section, the effectiveness of VC-SSA is verified using the CEC 2017 test function set. The performance of VC-SSA is compared with the state-of-the-art swarm intelligent algorithms, including SSA, MFO 8, MVO 72, SCA 45, GWO 13, HGS 20, WOA 11, and HHO 15. In this experiment, the population size is set to 30 and the maximum number of evaluations is set to D × 10,000, where D denotes the dimension of the problem. The parameter settings of the compared algorithms are the same as those recommended in the respective original literature. The parameters of the VC-SSA algorithm are the same as those used in Subsection 4.2. All algorithms are run 30 times independently on each function, and the mean (Mean) and standard deviation (Std) for all fitness are recorded. In addition, the results of the Friedman's rank test are also provided. To investigate the effect of the problem dimensions on the algorithms, the algorithms are tested using test functions with different dimensions. The statistical results are shown in Table 10 when D is set to 30, and the statistical results are reported in Table 11 when D is set to 50.

As shown in Table 10, VC-SSA outperforms the SCA, WOA, and HHO algorithms for all test functions. Compared with SSA, MFO, and GWO algorithms, VC-SSA achieves better and inferior results on 28 and one cases, respectively. VC-SSA beats MVO on 27 benchmarks and is inferior to MVO on two cases. With respect to HGS, VC-SSA achieves superiority on 23 problems and is worse than HGS on six functions. According to the average ranking, the proposed VC-SSA received the top rank, followed by MVO, HGS, GWO, SSA, MFO, HHO, WOA, and SCA. From Table 11, VC-SSA outperforms SSA on 23 test functions and underperforms SSA on the remaining six problems. Compared to MFO and HHO, VC-SSA wins on 28 test cases and loses to the competitor on only one function. With respect to MVO, VC-SSA shows better performance on 26 benchmark problems and slightly worse performance on three functions. The proposed VC-SSA beats SCA and WOA on all test functions. Compared to GWO, VC-SSA achieves superiority on 21 test functions and shows inferiority on eight problems. VC-SSA beat HGS on 12 test problems and inferior to HGS on 17 benchmarks.

5.   Application to engineering disciplines

In this section, five restricted practical engineering problems are chosen to testify the applicability of VC-SSA on real applications. These five problems namely, car side impact design (CSID) 62, pressure vessel design (PVD) 57, cantilever beam design (CBD) 63, speed reducer design (SRD), and three-bar truss design (TBTD) 57, which have been extensively studied to evaluate the ability of the metaheuristic approach to address real-life problems. The applied practical design cases have different constrains, and it is indispensable to use adequate methods to deal with them to successfully obtain feasible solutions. In this experiment, we use the penalty function approach 64 to cope with the optimization constrains. The parameter settings are as consistent with Subsection 4.2.

5.1.   CSID problem

The CSID problem was originally introduced by Gu et al. 65. For this case, the vehicle finite element model illustrated in Figure 9 is adopted to investigate the crashworthiness of the vehicle. The object of the problem is to optimize the gross weight of the vehicle by identifying eleven hybrid variables. All the decision variables are continuous except for the eighth and ninth variables. This challenging problem is restricted by ten unequal constrains.

The mathematical model of the CSID problem can be described as follows:

Objective function:

Subject to:

where 0.5 ≤ x₁-x₇ ≤ 1.5, x₈, x₉∈(0.192, 0.345) and -30 ≤ x₁₀, x₁₁ ≤ 30.

The VC-SSA is utilized to address this challenging problem, and compared with eight well performed algorithms, namely, GA 66, PSO 2, FA 9, ABC 6, GWO 13, WCA 67, SAFA 68, and SSA 24. The optimal results achieved by the approaches for the problem are reported in Table 12.

From Table 12, VC-SSA obtains the best value among the methods. This indicates that the total weight of the vehicle can be 22.8412 when the eleven parameters are set as 0.5017, 1.1172, 0.5000, 1.2959, 0.5001, 1.5000, 0.5035, 0.3450, 0.1920, -18.6991, and 0.0379. Therefore, VC-SSA can be used as an effective tool for solving the optimal vehicle crash design problem.

5.2.   PVD problem

The second engineering design problem that we study in this work is PVD. The goal of this case is to minimize the overall cost, which is strictly correlated with material, forming and welding, as illustrated in Figure 10. In the PVD project, there are four decision variables to be authorized, including the thickness of the shell (T_s), the thickness of the head (T_h), the inner radius (R), and the cylinder section length (L). The mathematical expression can be formulated as shown below:

Consider:

Objective function:

Subject to:

where 0 ≤ x₁ ≤99, 0 ≤x₂ ≤99, 10 ≤x₃ ≤200, 10 ≤x₄ ≤200.

This task has been addressed by the advocated VC-SSA approach, and compared with nine classical methods, namely, GSA 69, ABC 6, BA ^[70], MFO 8, HHO 15, GWO 13, WOA 11, AOS 71, and SSA 24. The best-obtained solutions from all involved optimizers are reported in Table 13. From Table 13, VC-SSA can get design a pressure vessel with the optimal total cost, i.e., 5886.9631. In addition, MBA and ASO can provide competitive results, which are 5889.3216 and 5888.4579, respectively.

5.3.   CBD problem

The third real-world engineering problem that we investigate is CBD. In this problem, the objective function is to minimize the weight of the cantilever beam. As illustrated in Figure 11, this issue has five structure parameters. There is one more vertical displacement constraint that requires be considered. The CBD problem can be modelled as follows:

Objective function:

Subject to:

where 0.01 ≤ x₁, x₂, x₃, x₄, x₅ ≤ 100.

This problem has been tackled by MFO 8, MVO 72, SOS 73, CS 12, ALO 74, MMA 75, and SSA 24. The optimal results obtained by VC-SSA and other algorithms on this issue are presented in Table 14. From Table 14, VC-SSA outperforms all competitors. Therefore, the developed algorithm can provide strong aid for the CBD problem.

5.4.   SRD problem

In this subsection, the SRD issue is considered, whose main objective is to devise a speed reducer with the minimum weight. Seven structure variables x = (b, m, z, l₁, l₂, d₁, d₂) are concluded in this problem, where b is the surface width, m is the module of gear teeth, z represents the quantity of teeth on pinion, l₁ and l₂ indicate the lengths of the first and second shafts between bearings, respectively, and d₁ and d₂ denote the diameters of the first and second shafts, respectively. For solving this problem, four design constrains needed to be considered. The configuration of this problem is illustrated in Figure 12, and the mathematical expression can be expressed as follows:

Consider:

Objective function:

Subject to:

where 2.6 ≤ x₁ ≤ 3.6, 0.7 ≤ x₁ ≤ 0.8, 17 ≤ x₃ ≤ 28, 7.3 ≤ x₄ ≤ 8.3, 7.3 ≤ x₅ ≤ 8.3, 3.9 ≤ x₆ ≤ 3.9, 5.0 ≤ x₇ ≤ 5.5.

This problem has been addressed by several nature-inspired metaheuristics including ABC 6, CS 15, SCA 45, MBFPA 76, WWO 77, and PVS 78. Table 15 tabulates the generated optimal solution by VC-SSA and its peer algorithms. From Table 15, VC-SSA can obtain the best weight, i.e., 2841.600123. The other compared algorithms achieve similar results.

5.5.   TBTD problem

The last constrained engineering design problem, named TSTD, has two decision variables with the target function of minimize the weight. In this problem, there are three constraints that need to be considered. The schematic of this problem is plotted in Figure 13. The corresponding expression of this project is as follows:

Consider:

Objective function:

Subjective to:

where 0 ≤ x₁, x₂ ≤1, L = 100cm, P = 2km/cm², σ = 2km/cm².

The algorithms HHO 15, MVO 72, GOA 17, MFO 8, SSA 24, ALO 74, CS 12 and WWO 77 are respectively employed to solve the TBTD problem. The outcomes gained by these algorithms and VC-SSA are shown in Table 16. With respect to the comparison results, VC-SSA can offer the optimal results, i.e., 263. 89584337713. The outstanding outputs proves that the proposed approach can effectively solve the TBTD problem.

6.   Robot path planning

Mobile robot path planning (MRPP) is a key problem in the field of robotics, where the main goal is to generate a collision-free path from the starting position to the destination for an autonomous mobile robot (AMR). Since the inception of mobile robots, researchers have been conducting corresponding research with the hope of designing efficient MRPP methods. Topologically, the MRPP problem is closely related to the problem of finding the shortest path between two points in the plane 79. In recent years, nature-inspired swarm intelligence metaheuristic techniques have become increasingly popular in MRPP tasks due to their strong stochasticity, robust search capability, and ease of implementation 80. In this section, we propose a VC-SSA-based MRPP approach for AMRs to help robots move along the shortest collision-avoiding path from one position to another.

6.1.   Robot path planning problem description

The MRPP problem can be treated as a constrained optimization task, where the trajectory between two locations is the objective to be optimized, and the threatening region is the constraint to be considered. Nature-inspired swarm intelligence metaheuristic approaches are effective tools for solving optimization problems by optimizing the objective function to acquire the feasible solution to the issue. The results obtained using the metaheuristic technique for MRPP tasks are influenced by two factors, namely, the performance of the metaheuristic method and the effectiveness of the goal function. According to the previous experimental results, the VC-SSA algorithm has outstanding optimization performance; therefore, the main factor that affects the effectiveness of the VC-SSA-based MRPP algorithm is the objective function. We need to establish an efficient objective function according to the characteristics of the MRPP problem, and the VC-SSA algorithm completes the path planning task by evaluating and optimizing the established objective function. Based on the above analysis, an objective function is designed considering the trail length and obstacle-avoiding for evaluating the quality of the paths produced by the VC-SSA-based MRPP approach. The proposed objective function is as follows:

where$\varpi$is the scaling factor used to motivate conflict-free paths and prevent routes containing threatening regions, and L denotes the route length. To calculate L, the following formula is applied.

where (x_i, y_i) represents the coordinates of the ith interpolation point.

In Eq (28), η is a marker variable that is used to evaluate the security condition of the current route. To calculate it, the following mathematical formula is utilized.

where nobs is the number of barriers, m represents the quantity of interpolant points in the path, d_{j, k} denotes the distance between the interpolant spot j to barrier k, and robs_k indicates the radius of the kth threatening region. It is clear from Eq (30) that if the current route includes no obstacles, then the calculated η-value is smaller, and vice versa.

6.2.   Experiment and results

In this subsection, we implement the VC-SSA-based MRPP algorithm, aiming to plan a globally optimal collision-avoiding route with the shortest path length. To validate the feasibility and superiority of the developed approach, five maps provided by 81 are adopted. These environmental maps have different characteristics, such as different number and size of obstacles and different distances between the starting and target points, as detailed in Table 17. After study the feasibility of VC-SSA in MRPP problems, the routes planned by this approach were compared with those generated by other metaheuristics, namely SSA, GWO, PSO, FA and ABC. For a fair comparison, the general parameter settings of the algorithms are identical, while the specific parameter setting of each method are taken from the source paper. The path lengths generated by the six methods in different environments are presented in Table 18, and the respective trajectories are presented in Figures 14–18.

From Table 18, the lengths of the paths planned by VC-SSA for all five terrains are the shortest compared to the comparison algorithms. This proves that the proposed VC-SSA-based MRPP method is an effective path planning tool that can be applied in both simple and complex environmental setups. Next, we analyzed the simulation results in detail based on the specific trajectories planned for each type of environment by the six algorithms.

Optimal paths planned by all approaches for the first environmental setup is exhibited in Figure 14(a–f). From the figure, all the algorithms can plan a collision-free path for this map. In terms of route length, SSA takes the longest route from the initial point to destination, while VC-SSA follows the shortest obstacle-avoiding route. Regarding the generated trajectories, FA and SSA design similar paths, while PSO, GWO, ABC and VC-SSA create another type of straightforward paths. Overall, VC-SSA avoids the local optimum and designs the most reasonable obstacle-free path for the first environmental setup.

Optimal paths designed by all algorithms for the second environmental setup is shown in Figure 15(a–f). The comparison results reveal that VC-SSA outperforms the compared approaches for this environment setting with the shortest path length of 14.3120, followed by GWO, FA, PSO, ABC and SSA. With respect to the trajectory, FA, ABC, GWO, SSA and VC-SSA produce similar routes, while PSO provides a different style of route. Based on the above analysis, all algorithms generate a path without collisions, while VC-SSA circumvents the local optima and thus outperforms its peers.

A comparison of the best trajectories produced by all approaches for the third environment map is displayed in Figure 16(a–f). The figure depicts that all approaches can generate an unobstructed route. FA, GWO and SSA fall into the local optimum, which leads to the generated path having redundancy. Nevertheless, PSO and VC-SSA provide satisfactory routes for this map. Since VC-SSA plans a more sensible trajectory, it provides a shorter collision-avoiding path than PSO.

The best performance of all algorithms for the fourth environment map with 30 obstacles of different scales are displayed in Figure 17(a–f). From the figure, PSO, ABC, GWO and SSA try to take the same trajectory during the maneuver from the starting point to the terminal point to produce a collision avoidance optimal path, while FA and VC-SSA take another more subtle trajectory. Compared with the trajectories provided by PSO, ABC, GWO and SSA, the curved routes generated by FA and VCSSA are smoother. Whereas FA falls into a sub-optimal solution, which affects the quality of the generated paths, VC-SSA obtains the optimal path with the length of 15.8750.

The simulation results of optimal routes produced by all methods for the fifth landscape are shown in Figure 18. From the figure, PSO, FA, ABC and SSA attempt to take the same trajectory during the maneuver from the initial position to goal to produce an obstacle-free shortest path, while GWO and VC-SSA attend to take a more promising path. However, GWO is trapped in a local optimum, while VC-SSA avoids the local optimal solution and exhibits better performance than the competitors with the path length of 21.5506.

7.   Conclusions and scope for future work

In this work, a novel SSA variant, called the VC-SSA approach, has been proposed with the hope of enhancing the solution quality and convergence rate of the standard salp swarm optimizer. We first introduce the velocity clamping strategy to boost the exploitation ability and the solution precious. Reduction factor mechanism is designed to expedite the convergence of the basic SSA. Finally, a novel position update equation is introduced by injecting an inertia weight to establish a counterbalance between exploration and exploitation, thus effectively expanding the landscape of global exploration during the early search, and refining the rough region of the global optima with the hope of advancing the solution accuracy after the lapse of iterations. According to the experimental results implemented in this paper, each of the introduced modifications can improve the performance of the basic SSA. More importantly, the components are novel, easy to accomplish, and do not destroy the minimalist style of the standard SSA. To evaluate the effectiveness and applicability of VC-SSA, we tested it on four classes of problems including 23 widely used benchmark functions, 30 latest benchmark problems from CEC 2017, five real-life engineering design problems, and a mobile robot path planning problem, and thoroughly compared it with the basic SSA, the well-performance SSA variants, and the frontier metaheuristic techniques. Based on the comparison results and non-parametric test results, our novel approach is able to provide superior, or at least competitive, performance on most test cases compared to its peer algorithms.

However, this study still has some limitations. For example, the value of the parameter δ was determined through a large number of experiments, and although the algorithm shows superior performance for δ = 0.003, this way of taking the value needs to be improved. In future research, the optimal value of δ will be trained using reinforcement learning. In addition, this paper proposes a VC-SSA-based mobile robot path planning approach in autonomous mobile robots, and although the proposed method is able to plan a reasonable optimal path for the robot in different environmental settings, it cannot be applied to multi-robot systems. In future work, we will focus on developing this path planning approach to multi-robot systems, aiming to plan shortest collision-free paths for multiple robots acting in concert. Finally, we expect researchers to use our proposed VC-SSA algorithm to tackle optimization problems in different disciplines.

Acknowledgments

This work was supported by the National Natural Science Foundation of China [Grant Numbers 61461053, 61461054, and 61072079], Yunnan Provincial Education Department Scientific Research Fund Project [2022Y008].

Conflict of interest

All authors declare no conflicts of interest in this paper.