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 "excerpt": "The salp swarm algorithm (SSA) is a bio-inspired metaheuristic for optimizing single- and multi-objective continuous functions, proposed by Seyedali Mirjalili and colleagues in Advances in Engineering Software in 2017.",
 "snippet": "The salp swarm algorithm (SSA) is a bio-inspired metaheuristic for optimizing single- and multi-objective continuous functions, proposed by Seyedali Mirjalili and colleagues in Advances in Engineering Software in 2017.",
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 "markdown": "# Salp swarm algorithm\n\nThe salp swarm algorithm (SSA) is a bio-inspired metaheuristic (general search method for finding good-enough solutions to hard problems) that optimizes single- and multi-objective continuous functions by simulating the chain-like swarming of salps in the ocean, producing the best solution found for a given objective function. It belongs to the swarm-intelligence family alongside ant colony optimization, particle swarm optimization, and the Grey Wolf Optimizer, and was proposed for engineering design problems where function evaluations can be expensive.\n\n| Key fact | Detail |\n|---|---|\n| Original proposal | Seyedali Mirjalili, Amir H. Gandomi, Seyedeh Zahra Mirjalili, Shahrzad Saremi, Hossam Faris, and Seyed Mohammad Mirjalili, Advances in Engineering Software, 2017 <sup>[1](https://doi.org/10.1016/j.advengsoft.2017.07.002)</sup> |\n| Core mechanism | A leader salp explores toward a food source; followers average their position with the salp ahead <sup>[2](https://www.ejpam.com/index.php/ejpam/article/view/7030)</sup> |\n| Control parameter | A single coefficient, \\( c_{1} = 2e^{-(4t/T)^{2}} \\), decays over iterations to shift from exploration to exploitation <sup>[3](https://vigir.missouri.edu/~gdesouza/Research/Conference_CDs/IEEE_WCCI_2020/CEC/Papers/E-24466.pdf)</sup> |\n| Parameter count | Fewer control parameters than PSO, which requires inertia weight plus cognitive and social coefficients <sup>[2](https://www.ejpam.com/index.php/ejpam/article/view/7030)</sup> |\n| Complexity | \\( O(T \\cdot (n \\cdot d + n \\cdot C_{f})) \\) per run for a fitness-dependent variant, versus \\( O(T \\cdot (n \\cdot (d + C_{f}))) \\) for standard PSO <sup>[3](https://vigir.missouri.edu/~gdesouza/Research/Conference_CDs/IEEE_WCCI_2020/CEC/Papers/E-24466.pdf)</sup> |\n| Main critique | The original formulation is not shift-invariant and was outperformed by a simple Differential Evolution on 27 of 30 CEC 2017 functions <sup>[4](https://pure.tue.nl/ws/files/191758766/1_s2.0_S0957417421013750_main.pdf)</sup> |\n| Variants | Binary, multi-objective, chaotic, quantum-inspired, adaptive, and many hybrid forms <sup>[2](https://www.ejpam.com/index.php/ejpam/article/view/7030)</sup> |\n\n## How it works\n\nA salp population is split into a leader, the front individual, and followers. The leader's position in each dimension is drawn around a food source \\( F \\), the current best solution:\n\n\\[ x_{1,j} = \\begin{cases} F_{j} + c_{1}((ub_{j} - lb_{j})c_{2} + lb_{j}) & c_{3} \\geq 0.5 \\\\ F_{j} - c_{1}((ub_{j} - lb_{j})c_{2} + lb_{j}) & c_{3} < 0.5 \\end{cases} \\]\n\nwhere \\( ub_{j} \\) and \\( lb_{j} \\) are the bounds, and \\( c_{2} \\), \\( c_{3} \\) are uniform random numbers in \\( [0,1] \\); \\( c_{3} \\) decides whether the step points toward \\( -\\infty \\) or \\( +\\infty \\).<sup>[2](https://www.ejpam.com/index.php/ejpam/article/view/7030)</sup> The exploration-exploitation coefficient decays as\n\n\\[ c_{1} = 2e^{-(4t/T)^{2}} \\]\n\nwith \\( t \\) the current iteration and \\( T \\) the maximum number of iterations, so early iterations explore widely and later ones exploit around the food source.<sup>[3](https://vigir.missouri.edu/~gdesouza/Research/Conference_CDs/IEEE_WCCI_2020/CEC/Papers/E-24466.pdf)</sup> [Followers](https://www.edgepedia.org/followers) are updated by Newtonian motion, implemented as averaging with the salp ahead:\n\n\\[ x_{i}^{j,t+1} = \\tfrac{1}{2}\\left(x_{i-1}^{j,t+1} + x_{i}^{j,t}\\right) \\]\n\nThe division of labor is that the leader explores for promising regions while followers exploit the area around the leader.<sup>[4](https://pure.tue.nl/ws/files/191758766/1_s2.0_S0957417421013750_main.pdf)</sup> A critical review by Mauro Castelli and colleagues documents two defects in this formulation: the printed equation thresholds \\( c_{3} \\) at 0, which since \\( c_{3} \\in [0,1] \\) makes the second case unreachable, while the source code uses 0.5; and the follower update's acceleration formula \\( a = (x_{\\mathrm{final}} - x_{0})/t^{2} \\) is not the correct average acceleration \\( (v(t+\\Delta t) - v(t))/\\Delta t \\).<sup>[4](https://pure.tue.nl/ws/files/191758766/1_s2.0_S0957417421013750_main.pdf)</sup> A comprehensive review also prints the decay schedule without the factor 4 in the exponent, \\( k_{1} = 2e^{-t/T_{\\max}} \\), an inconsistency across the literature.<sup>[2](https://www.ejpam.com/index.php/ejpam/article/view/7030)</sup>\n\n## How it is done\n\nThe pseudocode printed with the original algorithm runs as follows <sup>[5](https://inria.hal.science/hal-02197771v1/file/473854_1_En_16_Chapter.pdf)</sup>:\n\n1. Initialize the salp population \\( X_{i} \\), \\( i = 1, \\dots, n \\), uniformly within the bounds \\( lb \\) and \\( ub \\).\n2. While the stopping condition is not met, calculate the fitness of each search agent.\n3. Set the food source \\( F \\) to the best search agent.\n4. Update \\( c_{1} \\) from the iteration schedule.\n5. Update the leader (agent 1) with the leader equation; update every other agent as a follower.\n6. Amend positions that violate the bounds.\n7. Return \\( F \\) when the stopping condition is satisfied.\n\nInitialization costs \\( O(N \\times D) \\), and fitness evaluation costs \\( O(N \\times C_{f}) \\), where \\( C_{f} \\) is the cost of one objective-function evaluation; the worst-case movement phase costs \\( O(\\mathrm{MaxIt} \\times (N \\times D)) \\).<sup>[6](https://www.nature.com/articles/s41598-025-09345-9)</sup>\n\n## Origin\n\nThe SSA and its multi-objective version MSSA were proposed by Seyedali Mirjalili and colleagues in *Advances in Engineering Software* in 2017.<sup>[1](https://doi.org/10.1016/j.advengsoft.2017.07.002)</sup> The paper cites the No-Free-Lunch theorem, which states that no algorithm solves all optimization problems, as the motivation for proposing new metaheuristics.<sup>[1](https://doi.org/10.1016/j.advengsoft.2017.07.002)</sup> It built on earlier swarm methods, including the Grey Wolf Optimizer proposed by Seyedali Mirjalili, Seyed Mohammad Mirjalili, and Andrew Lewis in 2014 <sup>[7](https://doi.org/10.1016/j.advengsoft.2013.12.007)</sup> and the Whale Optimization Algorithm proposed by Seyedali Mirjalili and Andrew Lewis in 2016 <sup>[8](https://doi.org/10.1016/j.advengsoft.2016.01.008)</sup>, alongside ant colony optimization, particle swarm optimization, and other swarm-intelligence techniques.<sup>[1](https://doi.org/10.1016/j.advengsoft.2017.07.002)</sup> The algorithm changed names several times in the literature, appearing as Salp Swarm Algorithm, Salp Swarm Optimizer, Salp Swarm Optimization, and Salp Optimization Algorithm.<sup>[4](https://pure.tue.nl/ws/files/191758766/1_s2.0_S0957417421013750_main.pdf)</sup>\n\n## Variants\n\nA 2025 review catalogs binary versions, hybrid models, multi-objective extensions, and parameterless adaptations.<sup>[2](https://www.ejpam.com/index.php/ejpam/article/view/7030)</sup> Named variants include the Binary SSA (BSSA), Improved SSA with Gaussian perturbation, polynomial mutation, and Laplace crossover, Multi-objective SSA, Chaotic SSA using chaotic maps, Quantum-inspired SSA, and Adaptive SSA.<sup>[2](https://www.ejpam.com/index.php/ejpam/article/view/7030)</sup> The BSSA of Hossam Faris and colleagues applies eight transfer functions plus a crossover operator on the leader, and significantly outperformed five wrapper methods on about 90% of 22 UCI datasets.<sup>[9](https://www.sciencedirect.com/science/article/abs/pii/S0950705118302132)</sup> Mutation-scheme versions apply Gaussian, Cauchy, and Levy-flight mutations, which respectively add neighborhood-informed moves, large global-search steps, and increased randomness.<sup>[10](https://doi.org/10.1007/s00366-020-01252-z)</sup> SMSSA updates the worst salp with the simplex method each iteration.<sup>[5](https://inria.hal.science/hal-02197771v1/file/473854_1_En_16_Chapter.pdf)</sup> FDSSA weights \\( c_{1} \\) by leader fitness as \\( c_{1} = 4w_{i}\\exp(-(4t/T)^{2}) \\), adds a spiral follower trajectory, and applies DE/best/2 mutation to followers.<sup>[3](https://vigir.missouri.edu/~gdesouza/Research/Conference_CDs/IEEE_WCCI_2020/CEC/Papers/E-24466.pdf)</sup> CL-SSA hybridizes the Competitive Swarm Optimizer, splitting solutions into winners and losers, with winners updated by the SSA leader equation.<sup>[11](https://www.mdpi.com/2227-7390/11/6/1362)</sup>\n\n## Applications\n\nThe original paper applied SSA and MSSA to airfoil design and marine propeller design, where each function evaluation can take up to five minutes.<sup>[1](https://doi.org/10.1016/j.advengsoft.2017.07.002)</sup> Review-level coverage lists feature selection, neural network training, task scheduling, power system control, renewable energy management, and image segmentation and enhancement among applications.<sup>[2](https://www.ejpam.com/index.php/ejpam/article/view/7030)</sup> In machine learning, SLSSA (2024) targets multi-layer perceptron training <sup>[12](https://www.nature.com/articles/s41598-024-77440-4)</sup>, EKSSA (2025) tunes SVM hyperparameters for seed classification with higher accuracy than baselines <sup>[13](https://pmc.ncbi.nlm.nih.gov/articles/PMC12467668/)</sup>, and m-SSA variants (2024) optimize an MLP classifier for defective software module classification.<sup>[14](https://link.springer.com/article/10.1007/s00521-024-10131-3)</sup> The surrogate-assisted MSA-MPSSA reached 100% classification accuracy on the high-dimensional Leukemia1 dataset with more than 5000 features.<sup>[15](https://proceedings.mlr.press/v222/yu24a/yu24a.pdf)</sup>\n\n## Limitations and alternatives\n\nDocumented weaknesses include premature convergence to local optima, slow convergence rates for specific functions, and difficulty balancing exploration and exploitation <sup>[6](https://www.nature.com/articles/s41598-025-09345-9)</sup>, plus an exploitation issue that slows convergence <sup>[2](https://www.ejpam.com/index.php/ejpam/article/view/7030)</sup> and insufficient convergence precision and missing crossover-like exploration.<sup>[11](https://www.mdpi.com/2227-7390/11/6/1362)</sup> [Performance](https://www.edgepedia.org/performance) degrades on highly multimodal problems, and fixed control parameters favor exploitation while constraining exploration over time.<sup>[16](https://encyclopedia.pub/entry/54371)</sup> The most consequential critique is the loss of shift invariance: the lower-bound term in the leader update biases the search proportionally to the distance between the lower bound and the origin, and on functions shifted by a large constant, even random search outperformed SSO.<sup>[4](https://pure.tue.nl/ws/files/191758766/1_s2.0_S0957417421013750_main.pdf)</sup> The same authors propose the Amended Salp Swarm Optimizer (ASSO), which removes the additive \\( lb_{j} \\) term from the leader update while retaining \\( c_{1} \\), addressing the shift sensitivity of the original formulation, and it outperformed the original on benchmarks.<sup>[4](https://pure.tue.nl/ws/files/191758766/1_s2.0_S0957417421013750_main.pdf)</sup> On the CEC 2017 suite, all SSO-based versions were outperformed by a simple Differential Evolution on almost all functions (27 of 30), and CMA-ES beat them on more than half; the critical review's authors conclude the community \"can safely abandon SSO\".<sup>[4](https://pure.tue.nl/ws/files/191758766/1_s2.0_S0957417421013750_main.pdf)</sup> By contrast, FDSSA achieved the best solutions on functions F1, F3, and F5 and ranked best on the [Friedman test](https://www.edgepedia.org/friedman-test) except against PSO, with 30 agents, 1000 iterations, and 30 runs.<sup>[3](https://vigir.missouri.edu/~gdesouza/Research/Conference_CDs/IEEE_WCCI_2020/CEC/Papers/E-24466.pdf)</sup> CL-SSA reached an overall effectiveness of 82.75% across CEC 2017 (dimensions 50 and 100), CEC 2008 large-scale (dimensions 200, 500, 1000), and seven constrained CEC 2020 design problems, outperforming SSA, CSO, and other algorithms on Friedman and Wilcoxon rank-sum tests.<sup>[11](https://www.mdpi.com/2227-7390/11/6/1362)</sup> Across comparative testing, no single algorithm consistently outperformed the others in all test cases, so algorithm choice depends on problem characteristics.<sup>[2](https://www.ejpam.com/index.php/ejpam/article/view/7030)</sup> Head-to-head benchmark comparisons against the GWO and WOA are available; for example, one study selected and thoroughly described the Grey Wolf Optimizer (GWO), the Whale Optimization Algorithm (WOA), and the Salp Swarm Algorithm (SSA), and performed a comparative study against widely known test functions and antenna synthesis problems. Post-2023 remedies include ESSA with multi-search strategies and memory <sup>[6](https://www.nature.com/articles/s41598-025-09345-9)</sup>, SLSSA's multiple food sources and generalized oppositional learning to counter the single-elite strategy <sup>[12](https://www.nature.com/articles/s41598-024-77440-4)</sup>, EKSSA's adaptive \\( c_{1} \\), Gaussian walk, and dynamic mirror learning <sup>[13](https://pmc.ncbi.nlm.nih.gov/articles/PMC12467668/)</sup>, and m-SSA's dynamic manipulation of \\( c_{1} \\).<sup>[14](https://link.springer.com/article/10.1007/s00521-024-10131-3)</sup>\n\n## References\n\n1. [Seyedali Mirjalili and colleagues (2017). Salp Swarm Algorithm: A bio-inspired optimizer for engineering design problems. Advances in Engineering Software.](https://doi.org/10.1016/j.advengsoft.2017.07.002)\n2. [Salp Swarm Optimization: A Comprehensive Review of Recent Advances, Variants, Applications, and Future Research Directions (Nasar & El-Shorbagy, EJ-PAM)](https://www.ejpam.com/index.php/ejpam/article/view/7030)\n3. [A fitness dependent salp swarm algorithm (FDSSA, IEEE WCCI 2020 CEC)](https://vigir.missouri.edu/~gdesouza/Research/Conference_CDs/IEEE_WCCI_2020/CEC/Papers/E-24466.pdf)\n4. [Salp Swarm Optimization: A critical review (Castelli/Manzoni/Mariot/Nobile/Tangherloni, Expert Systems with Applications)](https://pure.tue.nl/ws/files/191758766/1_s2.0_S0957417421013750_main.pdf)\n5. [A Simplex Method-Based Salp Swarm Algorithm for Numerical and Engineering Optimization (Wang, Zhou, Jiang, Liu, IIP 2018)](https://inria.hal.science/hal-02197771v1/file/473854_1_En_16_Chapter.pdf)\n6. [Evolutionary salp swarm algorithm (ESSA) with multi-search strategies and advanced memory mechanism for solving global optimization and complex engineering problems (Scientific Reports, 2025)](https://www.nature.com/articles/s41598-025-09345-9)\n7. [Seyedali Mirjalili and colleagues (2014). Grey Wolf Optimizer. Advances in Engineering Software.](https://doi.org/10.1016/j.advengsoft.2013.12.007)\n8. [Seyedali Mirjalili, Andrew Lewis (2016). The Whale Optimization Algorithm. Advances in Engineering Software.](https://doi.org/10.1016/j.advengsoft.2016.01.008)\n9. [An efficient binary Salp Swarm Algorithm with crossover scheme for feature selection problems (Knowledge-Based Systems, 2018)](https://www.sciencedirect.com/science/article/abs/pii/S0950705118302132)\n10. [Bhaskar Nautiyal and colleagues (2021). Improved Salp Swarm Algorithm with mutation schemes for solving global optimization and engineering problems. Engineering With Computers.](https://doi.org/10.1007/s00366-020-01252-z)\n11. [Large-Scale Competitive Learning-Based Salp Swarm for Global Optimization and Solving Constrained Mechanical and Engineering Design Problems (CL-SSA, Mathematics, 2023)](https://www.mdpi.com/2227-7390/11/6/1362)\n12. [Self-learning salp swarm algorithm (SLSSA) for global optimization and its application in multi-layer perceptron model training (Scientific Reports, 2024)](https://www.nature.com/articles/s41598-024-77440-4)\n13. [An Enhanced Knowledge Salp Swarm Algorithm (EKSSA) for numerical optimization and seed classification](https://pmc.ncbi.nlm.nih.gov/articles/PMC12467668/)\n14. [Optimizing beyond boundaries: empowering the salp swarm algorithm for global optimization and defective software module classification (m-SSA, Neural Computing and Applications, 2024)](https://link.springer.com/article/10.1007/s00521-024-10131-3)\n15. [A Multi-Surrogate Assisted Salp Swarm Feature Selection Algorithm with Multi-Population Adaptive Generation Strategy for Classification (MSA-MPSSA, PMLR, 2024)](https://proceedings.mlr.press/v222/yu24a/yu24a.pdf)\n16. [Salp Swarm Algorithm | Encyclopedia MDPI](https://encyclopedia.pub/entry/54371)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Algorithms and computational methods › Optimization and dynamic programming › Swarm intelligence optimizers*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n",
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