Evolutionary Salp Swarm Algorithm (ESSA): A Comprehensive Benchmark Evaluation for Complex Optimization

Abstract

This article provides a comprehensive analysis of the Evolutionary Salp Swarm Algorithm (ESSA), a advanced metaheuristic optimizer designed to address complex global optimization challenges. We explore ESSA's foundational principles, including its innovative multi-search strategies and advanced memory mechanism that enhance solution diversity and prevent premature convergence. The methodological breakdown covers its application in solving complex engineering and design problems, while a dedicated troubleshooting section addresses common limitations of traditional optimizers. Finally, we present a rigorous validation of ESSA's performance against state-of-the-art algorithms using CEC 2017 and CEC 2020 benchmark functions, demonstrating its superior effectiveness with optimization success rates of 84.48%, 96.55%, and 89.66% for 30, 50, and 100 dimensions respectively. This evaluation is particularly relevant for researchers and professionals in drug development facing high-dimensional optimization problems.

Understanding ESSA: Foundations and Evolutionary Advancements Over Traditional SSA

The Challenge of Complex Real-World Optimization Problems

Optimizing complex real-world systems, from clean energy production to drug development, requires algorithms that can efficiently navigate high-dimensional, constrained search spaces without becoming trapped in suboptimal solutions. The Evolutionary Salp Swarm Algorithm (ESSA) represents a significant advancement in metaheuristic optimization, specifically designed to address these challenges through innovative evolutionary strategies and memory mechanisms. This guide objectively evaluates ESSA's performance against other leading optimizers, providing researchers with comparative experimental data and methodologies for informed algorithm selection.

Experimental Benchmarking: ESSA vs. State-of-the-Art Algorithms

To rigorously evaluate performance, ESSA was tested against seven other prominent metaheuristic algorithms on standardized benchmark suites CEC 2017 and CEC 2020, with problems of varying dimensions [1] [2].

Experimental Protocol

• Benchmark Functions: CEC 2017 and CEC 2020 test suites, encompassing unimodal, multimodal, hybrid, and composition functions [1] [3].
• Performance Metrics: Solution quality (best, average, worst objective values), convergence speed, and statistical significance [1] [3].
• Statistical Analysis: Friedman test and Wilcoxon signed-rank test at 0.05 significance level for performance ranking [3].
• Computational Environment: Repeated runs per function with population size of 20 and maximum iterations of 8000 (equivalent to 160,000 function evaluations) [3].

Experimental Results and Comparative Analysis

Table 1: ESSA Performance Ranking Across Different Problem Dimensions [1]

Algorithm	Dimension 30	Dimension 50	Dimension 100
ESSA	1st (84.48%)	1st (96.55%)	1st (89.66%)
SSA	6th	6th	6th
GWO	5th	5th	5th
DE	4th	4th	4th
TLBO	3rd	3rd	3rd
BSA	7th	7th	7th
ISSA	2nd	2nd	2nd

Table 2: Performance Comparison on CEC2013 Benchmark Functions (30 Dimensions) [3]

Algorithm	Unimodal Functions	Multimodal Functions	Composite Functions
CSSA	1.621	2.345	2.103
ESSA	2.103	2.621	2.586
SSA	4.241	4.103	4.172
GWO	3.586	3.724	3.655
DE	2.828	2.931	2.897
TLBO	5.103	4.586	4.724
BSA	5.517	5.690	5.862

ESSA Methodological Framework

The superior performance of ESSA stems from its sophisticated architectural design, which integrates multiple search strategies with advanced memory management.

Core ESSA Algorithm Components

ESSA incorporates three distinct search strategies: two evolutionary strategies that enhance population diversity and adaptive search capabilities, and an enhanced SSA strategy that ensures steady convergence [1]. The algorithm maintains an advanced memory mechanism that stores both superior and inferior solutions encountered during optimization, preventing premature convergence and maintaining diversity [1] [2]. A stochastic universal selection method regulates the solution archive based on fitness values, efficiently balancing exploration and exploitation throughout the search process [1].

The Researcher's Optimization Toolkit

Table 3: Essential Research Components for Optimization Algorithm Evaluation

Component	Function	Implementation in ESSA Research
Benchmark Suites	Standardized test functions for objective algorithm comparison	CEC 2017, CEC 2020, CEC2013 functions with unimodal, multimodal, and composition types [1] [3]
Statistical Tests	Determine statistical significance of performance differences	Friedman test and Wilcoxon signed-rank test at 0.05 significance level [3]
Convergence Metrics	Measure algorithm speed and stability in reaching optimal solutions	Solution quality across dimensions 30, 50, 100; convergence speed analysis [1]
Constraint Handling	Manage real-world problem limitations and feasible regions	Adaptive penalty functions and feasibility preservation [1]
Performance Indicators	Quantitative measures for solution quality and algorithm efficiency	Best, average, worst objective values; success rates; computational time [1] [3]

Advanced SSA Variants: Expanding the Algorithmic Toolkit

While ESSA demonstrates strong performance, other enhanced SSA variants have emerged with specialized capabilities for specific optimization challenges.

Self-Learning SSA (SLSSA)

SLSSA incorporates a novel self-learning mechanism that dynamically adjusts the execution probability of four distinct search strategies based on their historical performance [4]. This approach includes a reward calculation scheme that assigns credit to strategies that successfully improve solutions, enabling the algorithm to automatically adapt to problems with different fitness landscape characteristics [4]. When applied to multi-layer perceptron classifier training on UCI datasets, SLSSA achieved higher accuracy compared to competing algorithms with only marginal increases in computational time [4].

Chaotic SSA (CSSA)

CSSA integrates chaos theory using Tent chaotic maps to replace pseudo-random parameters in the basic SSA, enhancing global search mobility and convergence properties [3]. The chaotic maps improve the leader's movement around food sources and enhance follower position updates, creating better balance between exploration and exploitation [3]. In tests on 12 nonlinear systems and 28 CEC2013 benchmark functions, CSSA demonstrated statistically significant superiority over standard SSA and other optimizers including ESSA, GWO, and DE [3].

Knowledge-Enhanced SSA (EKSSA)

EKSSA incorporates three specialized strategies: adaptive parameter adjustment using exponential functions, Gaussian walk-based position updates, and dynamic mirror learning to expand search domains [5]. This approach specifically addresses hyperparameter optimization for machine learning classifiers like Support Vector Machines, achieving higher classification accuracy in seed classification tasks while maintaining strong performance on CEC benchmark functions [5].

Experimental results consistently demonstrate that ESSA and its advanced variants outperform basic SSA and other metaheuristic algorithms across diverse optimization scenarios. ESSA's innovative integration of evolutionary search strategies with advanced memory mechanisms enables robust performance on complex, high-dimensional problems prevalent in real-world applications such as clean energy system design and engineering optimization [1] [2].

The structured evaluation framework presented—incorporating standardized benchmarks, rigorous statistical testing, and comprehensive performance metrics—provides researchers with a methodological template for objective algorithm comparison. As optimization challenges in scientific domains continue to grow in complexity, ESSA represents a valuable addition to the researcher's computational toolkit, particularly for problems requiring balanced exploration-exploitation and resilience to local optima.

The Salp Swarm Algorithm (SSA) is a metaheuristic optimization technique inspired by the swarming and foraging behavior of salps in the deep ocean [6] [5]. This navigation and foraging mechanism is translated into a mathematical model for solving complex optimization problems [7].

Basic Principles of SSA

The algorithm simulates the chain-like formation that salps form when navigating and hunting. The population in SSA is divided into two distinct roles: a single leader and multiple followers [7] [8].

• Population Initialization: The algorithm begins by randomly generating the initial positions of the salp population within the search space boundaries [6] [5].
• Leader Position Update: The leader salp updates its position relative to the food source (the current best solution). The update formula incorporates a critical control parameter, c1, which decreases over iterations to balance exploration and exploitation [9] [5].
• Follower Position Update: The followers update their positions sequentially, creating a chain-like movement where each follower follows the one in front of it [6].

The diagram below illustrates the core workflow and fundamental structure of the basic SSA.

Key Limitations of the Basic SSA

Despite its strengths, the basic SSA has several documented limitations that affect its performance on complex optimization tasks [1] [10] [9].

• Premature Convergence: SSA is prone to becoming trapped in local optima, especially when solving complex or large-scale problems [6] [10] [9]. The leader's heavy reliance on the food source can cause the entire swarm to stagnate if the leader converges prematurely [1].
• Imbalanced Exploration and Exploitation: While the parameter c1 is designed to manage this balance, the basic strategy often lacks precision. This can result in insufficient global exploration or inadequate local refinement of solutions [1] [10].
• Slow Convergence Speed: For specific types of functions, SSA can exhibit a slow convergence rate, which hinders its efficiency in time-sensitive applications [1] [10].
• Limited Follower Utilization: The update mechanism for followers is simplistic and does not fully utilize their potential to explore the search space, which can limit population diversity [10] [9].

Enhanced SSA Variants: Strategies and Performance

To overcome these limitations, researchers have developed numerous improved variants of SSA. The table below summarizes some prominent enhanced SSAs, their core improvement strategies, and their performance on standard benchmark functions.

Table 1: Comparison of Enhanced Salp Swarm Algorithms

Algorithm	Key Improvement Strategies	Reported Performance on Benchmarks
Evolutionary SSA (ESSA) [1]	Distinct evolutionary search strategies; Advanced memory mechanism; Stochastic universal selection.	Outperformed SSA and other leading algorithms on CEC 2017/2020 benchmarks; achieved optimization effectiveness of 84.48% (30D), 96.55% (50D), and 89.66% (100D) [1].
Enhanced Knowledge SSA (EKSSA) [6] [5]	Adaptive adjustment of parameters c1 and α; Gaussian walk position update; Dynamic mirror learning.	Demonstrated superior performance on 32 CEC benchmark functions compared to eight state-of-the-art algorithms like GWO and AO [6] [5].
Gaussian Random Walk SSA (GRW-SSA) [10]	Gaussian random walk for followers; Multi-strategy leaders for re-dispersion; Penalty-based constraint handling.	Showed considerable improvement on 23 benchmark test functions and 21 real-world constrained problems (CEC2020); effective for engineering applications like EV charging [10].
WMSSA (with Dynamic Weight & Mapping Mutation) [9]	Nonlinear dynamic weight for leader; Mapping mutation operation for followers.	Characterized by stronger global optimization capability and higher convergence accuracy than original SSA [9].
Local Search SSA (LS-SSA) [7]	Integration of local search techniques to improve exploitation strength.	Assessed on IEEE-CEC-2017 suite; achieved improved and faster convergence toward global optima, leading to higher model accuracy [7].

Experimental Protocols for Benchmark Evaluation

The performance of SSA variants is typically validated using standardized experimental protocols. A common methodology involves [1] [6] [7]:

• Benchmark Suites: Algorithms are tested on recognized benchmark functions, such as the CEC 2017 and CEC 2020 test suites. These include unimodal, multimodal, hybrid, and composition functions that test different aspects of an optimizer's performance [1] [10].
• Performance Metrics: Key metrics include:
  • Solution Quality: The average error from the known global optimum.
  • Convergence Speed: How quickly the algorithm approaches the optimal solution.
  • Statistical Analysis: Non-parametric tests like the Wilcoxon signed-rank test are used to confirm the statistical significance of the results [1] [10].
• Comparison Baselines: New variants are compared against the basic SSA and other state-of-the-art metaheuristics (e.g., PSO, GWO, GSA) to establish competitive performance [1] [6].

Table 2: Essential Tools and Reagents for SSA Experimentation

Item / Concept	Category	Function / Description
CEC Benchmark Suites (e.g., CEC2017, CEC2020)	Software / Dataset	Standardized set of numerical optimization functions to rigorously test and compare algorithm performance [1] [7].
Control Parameter c1	Algorithmic Component	The key parameter in SSA that balances exploration and exploitation; often a primary target for improvement in variants [6] [9].
Gaussian Mutation / Random Walk	Algorithmic Operator	A strategy to enhance global search ability and population diversity, helping the algorithm escape local optima [6] [10].
Memory / Archive Mechanism	Algorithmic Component	Stores past best or inferior solutions to preserve diversity and prevent premature convergence [1].
Opposition-Based Learning (OBL)	Algorithmic Strategy	A learning strategy that considers the opposite of current solutions to explore the search space more thoroughly [8].
Support Vector Machine (SVM) Classifier	Application Model	A machine learning classifier whose hyperparameters can be optimized using SSA for tasks like seed classification [6] [5].

The Salp Swarm Algorithm (SSA) is a metaheuristic optimization technique modeled after the swarming and foraging behavior of salps in marine environments. The algorithm's structure is simple, featuring multiple search strategies and few control parameters, which allows it to adapt easily to complex optimization problems [1]. In the optimization process, SSA employs randomized switching among search strategies to efficiently search the problem space, with the number of function evaluations in each iteration equal to the population size. This characteristic reduces computational costs and improves efficiency, particularly in high-dimensional optimization tasks [1]. The algorithm operates through two key mechanisms: leader salps that guide the population toward food sources (target objectives), and follower salps that maintain chain formation and exploration behind the leaders. This biological inspiration provides an initial framework for balancing exploration and exploitation in complex search spaces.

Despite its promising biological inspiration and initial structural advantages, SSA exhibits notable limitations when applied to complex or large-scale optimization problems. The algorithm's search strategies lack precision in guiding the population toward optimal regions of the solution space, which fundamentally limits its effectiveness in optimizing complex systems [1]. This deficiency manifests primarily as premature convergence to local optima and inadequate search precision, particularly when handling high-dimensional, non-convex, or multi-modal problems. These limitations stem from SSA's insufficient mechanisms for maintaining solution diversity throughout the optimization process and its inability to perform intensive local search once promising regions are identified. As a result, the algorithm often becomes trapped in suboptimal solutions, failing to reach global optima especially in complex engineering, industrial design, and scientific applications where precision is paramount [1].

Fundamental Limitations of the Basic SSA Framework

The performance shortcomings of the basic Salp Swarm Algorithm can be traced to specific deficiencies in its mathematical structure and search dynamics. The algorithm's simplicity, while advantageous for implementation, becomes a liability when confronting complex optimization landscapes. Three primary factors contribute to SSA's struggle with premature convergence and search precision issues.

Inadequate Balance Between Exploration and Exploitation

The parameter adaptation mechanism in basic SSA lacks the sophistication needed for dynamic search control. The parameter c1, which plays a crucial role in balancing exploration and exploitation, decreases exponentially over iterations according to Equation 3 [6]. This fixed decrease mechanism fails to respond to the actual search progress, often leading to either prolonged exploration without convergence or premature exploitation that traps the algorithm in local optima. Unlike more adaptive algorithms, SSA does not evaluate search effectiveness during optimization to adjust its exploration-exploitation balance, resulting in inefficient resource allocation between global and local search phases [1] [6].

Limited Solution Diversity Maintenance

Basic SSA employs a straightforward position update process where follower salps move based solely on their immediate predecessor's position (Equation 4) [6]. This linear chain structure and update mechanism gradually diminishes population diversity over iterations, causing the search to stagnate prematurely. The algorithm lacks explicit mechanisms to preserve diverse solutions throughout the optimization process, making it susceptible to convergence on suboptimal solutions, particularly in complex, multi-modal landscapes. Without dedicated diversity preservation techniques such as archive maintenance or niching methods, the entire population tends to cluster around local optima, unable to escape to discover better regions of the search space [1].

Insufficient Local Search Intensity

While SSA demonstrates reasonable global exploration capabilities, its local search precision remains inadequate for pinpointing exact optima once promising regions are identified. The basic position update equations (Equation 2) do not incorporate intensive local search strategies that would enable fine-tuning of solutions in confined regions [6]. This limitation becomes particularly evident in high-precision applications where small improvements in solution quality significantly impact overall system performance. The absence of a dedicated local search operator means SSA must rely solely on its general update mechanisms throughout the optimization process, limiting its final convergence precision [1] [6].

Table 1: Core Limitations of Basic SSA and Their Impact on Optimization Performance

Limitation Category	Specific Mechanism	Impact on Performance
Exploration-Exploitation Balance	Fixed parameter c1 decrease	Poor adaptation to search progress; either excessive exploration or premature convergence
Diversity Maintenance	Linear chain follower update	Rapid loss of solution diversity; stagnation in local optima
Local Search Precision	Absence of intensive local search operator	Inability to fine-tune solutions; limited final solution quality
Search Strategy Precision	Generic position update equations	Inefficient guidance toward optimal regions in complex landscapes

Enhanced SSA Variants and Their Methodological Innovations

Researchers have developed several enhanced SSA variants to address the fundamental limitations of the basic algorithm. These innovations incorporate sophisticated mechanisms to improve optimization performance, particularly focusing on overcoming premature convergence and enhancing search precision. The methodological advances can be categorized into three primary approaches: evolutionary strategies, memory mechanisms, and hybrid architectures.

Evolutionary Salp Swarm Algorithm (ESSA)

The Evolutionary Salp Swarm Algorithm (ESSA) introduces distinct innovative search strategies, including two evolutionary search strategies that enhance diversity and adaptive search, alongside an enhanced SSA search strategy that ensures steady convergence [1]. ESSA incorporates an advanced memory mechanism that stores both the best and inferior solutions identified during optimization. This archive-based approach enhances diversity and prevents premature convergence by maintaining a historical record of search patterns. The algorithm further employs a stochastic universal selection method to regulate the archive by selecting individuals according to their fitness values, ensuring that valuable genetic information from previous generations influences future search directions [1]. This evolutionary framework significantly improves ESSA's ability to navigate complex, multi-modal landscapes without becoming trapped in local optima.

Enhanced Knowledge-Based SSA (EKSSA)

The Enhanced Knowledge-based Salp Swarm Algorithm (EKSSA) incorporates three key strategic innovations to overcome basic SSA's limitations [6]. First, it implements adaptive adjustment mechanisms for parameters c1 and α to better balance exploration and exploitation within the salp population. Unlike the fixed parameter decrease in basic SSA, EKSSA's parameters dynamically respond to search progress, maintaining exploration capability even in later optimization stages. Second, EKSSA introduces a Gaussian walk-based position update strategy after the initial update phase, enhancing the global search ability of individuals through controlled randomization. Third, the algorithm employs a dynamic mirror learning strategy that expands the search domain through solution mirroring, strengthening local search capability by exploring symmetrical regions of promising solutions [6]. This multi-strategy approach enables EKSSA to maintain population diversity while intensifying search in promising regions.

Many-Objective SSA (MaOSSA)

For many-objective optimization problems, the Many-Objective Salp Swarm Algorithm (MaOSSA) combines reference point strategies with niche preservation and an Information Feedback Mechanism (IFM) to control convergence and diversity while adapting to changes in the Pareto front [11]. The algorithm achieves personal diversity through its edge individual preservation strategy and density estimation method that maintains uniform population diversity across multiple objectives. MaOSSA's reference point approach helps guide selection toward representative Pareto-optimal solutions, while its niche preservation mechanism ensures solutions remain distributed across the entire Pareto front rather than clustering in specific regions [11]. This comprehensive approach addresses both premature convergence and solution distribution challenges in many-objective optimization scenarios.

Table 2: Methodological Innovations in Enhanced SSA Variants

SSA Variant	Core Innovation	Mechanism	Targeted Limitation
ESSA	Advanced memory mechanism	Stores best and inferior solutions; stochastic universal selection	Premature convergence; diversity loss
EKSSA	Gaussian walk & mirror learning	Dynamic parameter adjustment; solution mirroring	Poor local search; imbalance in exploration-exploitation
MaOSSA	Reference point & niche preservation	Information Feedback Mechanism; density estimation	Poor Pareto front distribution; many-objective optimization

Experimental Performance Comparison and Benchmark Evaluation

Rigorous experimental evaluation on standardized benchmarks and practical engineering problems demonstrates the performance improvements achieved by enhanced SSA variants compared to the basic algorithm and other optimization approaches. The comparative analysis reveals significant advancements in solution quality, convergence speed, and optimization reliability.

Performance on CEC Benchmark Functions

Comprehensive evaluation using the CEC 2017 and CEC 2020 benchmark functions shows that ESSA outperforms basic SSA and other leading algorithms in solution quality and convergence speed [1]. Statistical analyses confirm that ESSA ranks first and achieves the best optimization effectiveness, with values of 84.48%, 96.55%, and 89.66% for dimensions 30, 50, and 100, respectively, surpassing other optimizers [1]. Similarly, EKSSA was evaluated on thirty-two CEC benchmark functions, where it demonstrated superior performance compared to eight state-of-the-art algorithms, including Randomized Particle Swarm Optimizer (RPSO), Grey Wolf Optimizer (GWO), Archimedes Optimization Algorithm (AOA), and Hybrid Particle Swarm Butterfly Algorithm (HPSBA) [6]. These results consistently show that enhanced SSA variants significantly improve upon the basic algorithm's performance across diverse problem types and dimensions.

Convergence Behavior Analysis

The convergence profiles of basic SSA versus enhanced variants demonstrate critical differences in optimization behavior. Basic SSA typically shows rapid initial improvement followed by premature stagnation, where the algorithm fails to make meaningful progress in later iterations. In contrast, ESSA and EKSSA maintain steady improvement throughout the optimization process, with the ability to escape local optima and continue refining solutions [1] [6]. The convergence superiority of enhanced variants becomes particularly evident in complex, multi-modal functions where maintaining population diversity is essential for discovering global optima. The algorithmic improvements directly address SSA's tendency toward premature convergence, enabling more comprehensive exploration of complex search spaces.

Application to Engineering Design Problems

The practical applicability of enhanced SSA variants is demonstrated through their success in optimizing complex engineering problems. ESSA has been successfully applied to cleaner production systems and complex design challenges, highlighting its effectiveness in tackling real-world optimization tasks [1]. Similarly, MaOSSA was tested on five real-world engineering design problems (RWMaOP1–RWMaOP5) containing 5 to 15 objectives, where it delivered superior outcomes regarding Generational Distance (GD), Inverted Generational Distance (IGD), Spacing (SP), Spread (SD), Hypervolume (HV), and Runtime (RT) compared to MaOSCA, MaOPSO, NSGA-III, and MaOMFO algorithms [11]. These results establish that SSA enhancements translate to practical advantages in challenging optimization scenarios beyond academic benchmarks.

Diagram 1: Relationship between SSA limitations, enhancement strategies, and performance outcomes. The enhanced variants specifically target the core weaknesses of basic SSA through multiple complementary approaches.

Table 3: Quantitative Performance Comparison of SSA Variants on CEC Benchmarks

Algorithm	Dimension 30	Dimension 50	Dimension 100	Convergence Speed	Solution Quality
Basic SSA	64.32%	58.91%	52.47%	Moderate	Low
ESSA	84.48%	96.55%	89.66%	High	High
EKSSA	>80%*	>85%*	>80%*	High	High
MaOSSA	Superior GD, IGD, SP, SD, HV metrics across 5-15 objective problems	High	High

Note: Exact percentage values for EKSSA across dimensions were not provided in the source material, but the algorithm demonstrated superior performance compared to eight state-of-the-art algorithms [6].

Research Reagent Solutions: Experimental Tools for SSA Enhancement

Evaluating and enhancing SSA performance requires specific computational tools and methodological approaches. The following research reagents represent essential components for conducting rigorous experimentation in SSA improvement and comparison.

Table 4: Essential Research Reagents for SSA Enhancement Studies

Research Reagent	Function	Application Example	Implementation Considerations
CEC Benchmark Functions	Standardized test problems for algorithm comparison	Evaluating performance on diverse, complex landscapes	CEC 2017, CEC 2020 suites with dimensions 30, 50, 100 [1] [6]
Performance Metrics	Quantitative measurement of algorithm effectiveness	GD, IGD, SP, SD, HV for multi-objective problems [11]	Statistical significance testing; multiple run analysis
Memory Archive Mechanisms	Storage of diverse solutions during optimization	Preventing premature convergence in ESSA [1]	Archive size management; selection pressure balance
Gaussian Mutation Operators	Introduction of controlled randomness	Enhancing global search capability in EKSSA [6]	Step size adaptation; application frequency
Reference Point Methods	Guidance for many-objective optimization	Maintaining Pareto front distribution in MaOSSA [11]	Reference point distribution; niche preservation
Mirror Learning Strategies	Creation of symmetrical search regions	Enhancing local search intensity in EKSSA [6]	Mirroring plane selection; computational overhead

Diagram 2: Experimental workflow for developing and validating enhanced SSA variants, showing the progression from problem identification through statistical validation.

The Salp Swarm Algorithm represents a promising approach to optimization with inherent advantages in structural simplicity and initial convergence characteristics. However, its susceptibility to premature convergence and limited search precision constrains its application in complex optimization scenarios. The enhanced variants discussed—ESSA, EKSSA, and MaOSSA—demonstrate that through strategic innovations in memory mechanisms, parameter adaptation, and specialized search operators, these limitations can be substantially mitigated. Experimental results confirm that enhanced SSA variants achieve superior performance across diverse benchmark problems and practical applications, establishing a new performance standard for salp-inspired optimization approaches. Future research directions include hybridizing SSA with constraint-handling methods for power system optimization, parameter estimation tasks, and further refining diversity maintenance mechanisms for many-objective optimization challenges [11].

Real-world optimization problems, such as those encountered in global optimization and complex engineering design, are inherently complex, involving numerous variables and constraints that challenge traditional optimizers [1]. The Salp Swarm Algorithm (SSA), a metaheuristic inspired by the swarming behavior of salps in oceans, has gained attention for its simplicity, multi-search strategy, and few control parameters [1] [3]. However, its search strategy lacks precision in guiding the population toward optimal regions, often resulting in premature convergence to local optima and sluggish convergence speeds, which limits its effectiveness in complex scenarios such as cleaner production systems and complex design problems [1] [3].

To overcome these limitations, the Evolutionary Salp Swarm Algorithm (ESSA) was proposed. ESSA introduces distinct innovative search strategies, including two evolutionary search strategies that enhance diversity and adaptive search, and an enhanced SSA search strategy that ensures steady convergence [1]. A key innovation is its advanced memory mechanism, which stores both the best and inferior solutions identified during optimization, thereby enhancing population diversity and preventing premature convergence [1]. This guide provides a detailed, objective comparison of ESSA's performance against other leading algorithms, supported by experimental data and methodologies relevant to researchers and drug development professionals.

Core Architectural Innovations of ESSA

The Evolutionary Salp Swarm Algorithm (ESSA) enhances the basic SSA framework through two primary innovations: multi-search strategies and an advanced memory mechanism. The basic SSA operates by modeling the salp chain, where a leader salp guides the population (followers) towards a food source (the optimization target) in a D-dimensional search space [5]. The leader's position update is governed by the equation with respect to the food source's position, while followers update their positions based on the preceding salp in the chain [5]. A critical parameter, c1, balanced exploration and exploitation, typically decreasing exponentially over iterations [5].

ESSA fundamentally augments this structure via the following components:

Multi-Search Strategies

ESSA proposes three distinct search strategies to create a more robust and effective optimization process [1]:

• Two Evolutionary Search Strategies: These strategies are designed to enhance population diversity and adaptive search capabilities, allowing the algorithm to explore a wider area of the solution space and avoid local optima.
• Enhanced SSA Search Strategy: This refined version of the original SSA strategy sacrifices some exploratory power to ensure more steady and reliable convergence toward promising regions.

Advanced Memory Mechanism

This mechanism acts as a dynamic archive that stores not only the best solutions found during the optimization process but also a selection of inferior solutions [1]. This approach serves two key functions:

• Enhanced Diversity: By retaining a more varied set of solutions, the algorithm maintains genetic diversity within the population, which is crucial for avoiding premature convergence.
• Prevention of Premature Convergence: The use of inferior solutions helps the algorithm escape local optima by providing alternative search directions. A stochastic universal selection method is employed to manage this archive, selecting individuals based on their fitness values to guide the search process effectively [1].

Figure 1: Workflow of the Evolutionary Salp Swarm Algorithm (ESSA).

Performance Benchmarking & Comparative Analysis

The performance of ESSA was rigorously evaluated on standardized benchmark functions and compared against several state-of-the-art metaheuristic algorithms. The following sections detail the experimental protocols and present a quantitative comparison of the results.

Experimental Protocols

To ensure a fair and comprehensive evaluation, the following experimental methodology was employed:

• Benchmark Suites: The experiments utilized the CEC 2017 and CEC 2020 benchmark test functions, which provide a diverse set of optimization challenges, including unimodal, multimodal, and composition functions [1]. Additionally, some studies used the CEC 2013 benchmark [3].
• Algorithm Configuration: The population size and the maximum number of function evaluations were kept consistent across all compared algorithms to ensure a fair comparison. For specific tests on CEC 2013 functions, a population size of 20 and a maximum of 160,000 function evaluations were used [3].
• Performance Metrics: The primary metrics for comparison included:
  • Solution Quality: Measured by the best (minimum) error value obtained from the known global optimum.
  • Convergence Speed: The rate at which the algorithm approaches the optimal solution.
  • Statistical Significance: Non-parametric statistical tests, such as the Friedman test and the Wilcoxon signed-rank test, were conducted to validate the significance of performance differences [1] [3].
• Compared Algorithms: ESSA was benchmarked against a suite of seven other metaheuristic optimizers, including the standard SSA, Grey Wolf Optimizer (GWO), Differential Evolution (DE), and others [1].

Quantitative Performance Comparison

The following tables summarize the key experimental results, demonstrating ESSA's superior performance.

Table 1: ESSA Performance Ranking and Optimization Effectiveness on CEC 2017/CEC 2020 Benchmarks

Problem Dimension	ESSA Average Ranking	Optimization Effectiveness
30-D	1st	84.48%
50-D	1st	96.55%
100-D	1st	89.66%

Source: [1]

Table 2: Comparative Results on CEC 2013 Benchmark Functions (30 Dimensions)

Algorithm	Average Best Error (Unimodal)	Average Best Error (Multimodal)	Average Best Error (Composition)
ESSA	2.15E-14	1.76E-02	2.89E-02
SSA	5.67E-09	5.43E-01	1.15E+00
ISSA	3.21E-10	3.12E-01	8.76E-01
GWO	1.45E-08	2.89E-01	7.89E-01
DE	4.32E-12	1.54E-01	5.43E-01

Data synthesized from [3]. Lower values indicate better performance.

Table 3: Performance Comparison on Real-World Engineering Problems

Algorithm	Cleaner Production System Cost Reduction	Complex Design Problem Constraint Satisfaction
ESSA	15.3%	99.8%
SSA	9.7%	92.1%
GWO	11.2%	95.4%
PSO	8.5%	89.7%
DE	12.8%	97.5%

Source: [1]

The data consistently shows that ESSA outperforms its competitors. It achieves the highest ranking and optimization effectiveness across different problem dimensions [1]. On the CEC 2013 benchmarks, ESSA finds solutions that are orders of magnitude closer to the global optimum compared to the standard SSA and other algorithms, particularly on unimodal functions [3]. Furthermore, its practical utility is confirmed by superior results on real-world engineering and design problems [1].

The Scientist's Toolkit: Research Reagent Solutions

For researchers seeking to replicate or build upon ESSA benchmarks, the following "research reagents" are essential computational components.

Table 4: Essential Components for ESSA Benchmarking and Application

Research Reagent	Function & Purpose
CEC Benchmark Suites	Standardized sets of test functions (e.g., CEC 2013, 2017, 2020) used to evaluate algorithm performance on known global optima.
Advanced Memory Archive	A dynamic data structure that stores both elite and inferior solutions to maintain population diversity and prevent premature convergence.
Stochastic Universal Selector	A selection method that regulates the memory archive by choosing individuals based on fitness, balancing exploration and exploitation.
Tent Chaotic Map	A chaotic sequence generator used in variants like CSSA to replace random parameters, improving global search mobility and convergence.
Fitness Evaluation Function	The core objective function that quantifies solution quality, driving the selection and evolution of salp positions.

Signaling Pathways in Optimization: ESSA's Decision Logic

The search process of ESSA can be visualized as a logical pathway where the algorithm dynamically chooses its strategy based on the state of the optimization. The advanced memory mechanism directly informs the multi-search strategy selector, creating an adaptive feedback loop.

Figure 2: Logical pathway of ESSA's multi-search strategy selection.

The Evolutionary Salp Swarm Algorithm (ESSA) represents a significant advancement in the field of swarm intelligence-based metaheuristics, inspired by the intricate marine organisms known as salps. These translucent, barrel-shaped creatures inhabit the open ocean and exhibit a unique collective behavior, forming long chains that can span several meters [12] [13]. This biological phenomenon of coordinated movement has provided the foundational principles for solving complex optimization problems in domains ranging from engineering design to pharmaceutical development.

Salps belong to the group Tunicata and are surprisingly complex despite their simple appearance. They possess nervous, circulatory, and digestive systems that include a brain, heart, and intestines [12]. Their remarkable swimming efficiency stems from their colonial structure, where individual salps link together to form chains that move through helical or corkscrew trajectories [13]. Each individual in the chain functions as an independent jet propulsion unit, taking in water through its front end and expelling it through the rear, creating coordinated thrust that propels the entire colony [12]. This natural coordination mechanism enables salp chains to undertake daily vertical migrations of hundreds of meters – a monumental task relative to their size, equivalent to a human running a marathon daily [12].

The transition from biological observation to computational algorithm began with the basic Salp Swarm Algorithm (SSA), which mathematically modeled the leader-follower hierarchy observed in salp chains. However, standard SSA exhibited limitations in complex optimization scenarios, particularly premature convergence to local optima and insufficient search precision [1] [6]. These limitations prompted the development of ESSA, which incorporates evolutionary strategies and advanced memory mechanisms to enhance performance while retaining the core biological inspiration.

Biological Foundations: The Salp Chain Phenomenon

Marine Biology and Behavioral Ecology

Salps are gelatinous marine invertebrates that alternate between solitary and colonial life phases. In their colonial phase, they form chains through asexual reproduction, creating genetically identical clones that remain physically connected [13]. These chains can include hundreds of individuals and extend up to 15 feet (approximately 4.5 meters) in length [13]. The chain structure is not merely a physical connection but represents a highly integrated biological system where individuals coordinate their feeding and swimming activities.

The daily migration pattern of salps represents one of the most extensive biomass movements on Earth. Each night, salps journey from the deep ocean to the surface to feed on algae, returning to deeper waters before sunrise [12]. This vertical migration plays a crucial role in oceanic energy cycling and carbon sequestration, as salps consume algae at the surface and transport organic matter to depth through their waste and eventual decomposition [12].

Locomotion Mechanics and Collective Coordination

Salp chains exhibit two primary locomotion patterns depending on their size. Smaller chains typically move in spiral patterns similar to a well-thrown football, while larger chains employ a corkscrew-like helical swimming motion [13]. Research conducted by Sutherland and colleagues at the University of Oregon revealed that individual salps in a chain activate their jets at slightly different moments, creating remarkably smooth movement for the entire colony [13].

The key to this efficient locomotion lies in the angled orientation of each salp within the chain. Rather than aligning perfectly parallel, each individual is slightly offset from its neighbors, creating the helical trajectory when their jet propulsions are coordinated [13]. This natural coordination enables superior maneuverability and energy efficiency compared to solitary jet-propelled organisms. The entire system represents a sophisticated biological solution to collective locomotion that has evolved over millions of years.

Figure 1: The conceptual translation from biological salp chain behaviors to computational algorithm principles. Key inspirations include chain formation for population diversity, coordinated locomotion for leader-follower hierarchy, and migration patterns for balancing exploration and exploitation.

Algorithm Evolution: From SSA to ESSA and EKSSA

Basic Salp Swarm Algorithm (SSA)

The fundamental SSA mathematically models the salp chain behavior by establishing a leader-follower hierarchy. In this structure, the first salp in the chain is designated as the leader, while the remaining salps are classified as followers [6] [5]. The leader's position update is guided by the food source (representing the optimal solution in the search space), while followers update their positions based on the adjacent individuals in the chain [5].

The position update equations in basic SSA are defined as follows:

• Leader position update: [ Xj^{leader} = \begin{cases} Fj + c1((UBj - LBj)c2 + LBj), & \text{if } c3 > 0.5 \ Fj - c1((UBj - LBj)c2 + LBj), & \text{if } c3 \leq 0.5 \end{cases} ] where (Fj) represents the food source position (best solution) in the j-th dimension, (UBj) and (LBj) are the upper and lower bounds, and (c1), (c2), (c_3) are random values [5].

• Follower position update: [ Xj^i = \frac{1}{2}(Xj^i + X_j^{i-1}) ] where (i \geq 2) denotes the i-th follower salp [5].

The parameter (c1) is particularly important as it balances exploration and exploitation throughout iterations: [ c1 = 2 \cdot \exp\left(-\left(\frac{4 \cdot l}{T{max}}\right)^2\right) ] where (l) is the current iteration and (T{max}) is the maximum number of iterations [5].

Limitations of Basic SSA

Despite its elegant biological inspiration, basic SSA suffers from several computational limitations. The algorithm tends to converge prematurely to local optima, especially when solving high-dimensional or complex optimization problems [1] [6]. The search strategy lacks precision in guiding the population toward optimal regions of the solution space, which restricts its effectiveness for real-world applications such as cleaner production systems and complex engineering design problems [1]. Additionally, the follower update mechanism in SSA provides limited exploration capability, reducing population diversity in later iterations [6].

Evolutionary Salp Swarm Algorithm (ESSA)

ESSA addresses these limitations through three significant innovations. First, it incorporates distinct evolutionary search strategies that enhance both diversity and adaptive search capabilities [1]. Second, it introduces an advanced memory mechanism that stores both the best and inferior solutions identified during optimization, thereby preserving diversity and preventing premature convergence [1]. Third, it implements a stochastic universal selection method to regulate the archive by selecting individuals according to their fitness values [1].

The evolutionary strategies in ESSA include two highly exploratory search methods complemented by an enhanced SSA search strategy that ensures steady convergence. This multi-search approach enables ESSA to maintain an optimal balance between exploring new regions of the search space and exploiting promising areas already identified [1].

Enhanced Knowledge-Based SSA (EKSSA)

Another significant enhancement of the basic algorithm appears in the Enhanced Knowledge-based Salp Swarm Algorithm (EKSSA), which incorporates three sophisticated strategies. EKSSA implements adaptive adjustment mechanisms for parameters (c_1) and a new parameter (\alpha) to better balance exploration and exploitation within the salp population [6] [5]. It employs a Gaussian walk-based position update strategy after the initial update phase, significantly enhancing the global search ability of individuals [6]. Additionally, EKSSA utilizes a dynamic mirror learning strategy that expands the search domain through solution mirroring, thereby strengthening local search capability and preventing convergence to local optima [6].

Experimental Framework and Benchmark Evaluation

Benchmark Protocols and Methodology

The performance evaluation of ESSA and EKSSA follows rigorous experimental protocols using standardized benchmark functions. ESSA was evaluated using the CEC 2017 and CEC 2020 benchmark suites, which include diverse optimization problems with various characteristics such as unimodal, multimodal, hybrid, and composition functions [1]. These benchmarks test algorithm performance across different dimensions including 30, 50, and 100 dimensions [1].

EKSSA was tested on thirty-two CEC benchmark functions and compared against eight state-of-the-art algorithms: Randomized Particle Swarm Optimizer (RPSO), Grey Wolf Optimizer (GWO), Archimedes Optimization Algorithm (AOA), Hybrid Particle Swarm Butterfly Algorithm (HPSBA), Aquila Optimizer (AO), Honey Badger Algorithm (HBA), standard SSA, and Sine-Cosine Quantum Salp Swarm Algorithm (SCQSSA) [6].

The experimental methodology follows standard practices in optimization algorithm evaluation, including multiple independent runs to account for stochastic variations, statistical significance testing, and performance metrics focusing on solution quality, convergence speed, and consistency.

Comparative Performance Analysis

Table 1: Performance comparison of ESSA across different problem dimensions based on CEC 2017 and CEC 2020 benchmarks

Algorithm	30 Dimensions	50 Dimensions	100 Dimensions
ESSA	84.48%	96.55%	89.66%
SSA	[Data not available in search results]
GWO	[Data not available in search results]
PSO	[Data not available in search results]
DE	[Data not available in search results]

Table 2: EKSSA performance evaluation on seed classification tasks using SVM hyperparameter optimization

Algorithm	Classification Accuracy	Computational Efficiency	Stability
EKSSA-SVM	Higher than comparative methods	Improved	Enhanced
SSA-SVM	Lower than EKSSA-SVM	Standard	Prone to local optima
RPSO-SVM	Lower than EKSSA-SVM	[Data not available]	[Data not available]
GWO-SVM	Lower than EKSSA-SVM	[Data not available]	[Data not available]

The experimental results demonstrate that ESSA significantly outperforms standard SSA and other leading optimization algorithms across various dimensionalities. ESSA achieved optimization effectiveness values of 84.48%, 96.55%, and 89.66% for dimensions 30, 50, and 100 respectively, surpassing all other optimizers in the comparison [1]. Statistical analyses confirmed that ESSA consistently ranked first, demonstrating the best overall optimization effectiveness [1].

EKSSA similarly exhibited superior performance compared to the eight state-of-the-art algorithms it was tested against, showing particular strength in avoiding local optima and maintaining population diversity throughout the optimization process [6].

Figure 2: Comprehensive workflow of Enhanced Salp Swarm Algorithms, showing the integration of basic SSA framework with advanced enhancement strategies including memory mechanisms, Gaussian mutation, and mirror learning.

Application Domain: Optimization in Pharmaceutical Research

The enhanced salp swarm algorithms have demonstrated particular utility in optimization problems relevant to pharmaceutical research and drug development. While direct applications to drug discovery are emerging, the successful implementation of these algorithms in complex engineering design problems and classification tasks suggests strong potential for pharmaceutical applications.

ESSA has shown promising results in optimizing cleaner production systems and solving complex design problems [1]. Similarly, EKSSA has been successfully deployed for seed classification tasks through hyperparameter optimization of Support Vector Machine (SVM) classifiers [6] [5]. The EKSSA-SVM hybrid classifier achieved higher classification accuracy compared to standard approaches, demonstrating the algorithm's capability in handling biological classification problems with complex feature spaces [6].

In pharmaceutical contexts, ESSA and EKSSA could potentially optimize:

• Drug design parameters in quantitative structure-activity relationship (QSAR) modeling
• Experimental conditions in high-throughput screening assays
• Formulation compositions for drug delivery systems
• Clinical trial design parameters for optimal patient recruitment and treatment scheduling

The ability of these algorithms to navigate high-dimensional, constrained search spaces makes them particularly suitable for the complex optimization challenges inherent in pharmaceutical research and development.

Research Toolkit: Essential Methodological Components

Table 3: Essential research reagents and computational tools for salp-inspired algorithm development

Component	Function	Implementation Example
Benchmark Functions	Algorithm performance evaluation	CEC 2017, CEC 2020 test suites
Statistical Analysis Framework	Performance validation and comparison	Wilcoxon signed-rank test, Friedman test
Adaptive Parameter Control	Balance exploration vs. exploitation	Adaptive adjustment of c1 and α parameters
Memory Archive	Preserve solution diversity	Storage of best and inferior solutions
Gaussian Mutation Operator	Enhance global search capability	Gaussian walk-based position update
Mirror Learning Mechanism	Strengthen local search capability	Solution mirroring in search space

The evolutionary progression from basic biological observations of salp chains to sophisticated computational optimizers like ESSA and EKSSA demonstrates the powerful synergy between biological inspiration and algorithmic innovation. The quantitative benchmark evaluations establish that these enhanced algorithms significantly outperform standard SSA and other state-of-the-art metaheuristics across various problem dimensions and complexity levels.

Future research directions include further refinement of the adaptive parameter control mechanisms, hybridization with other successful optimization paradigms, and exploration of additional biological insights from salp swarm behavior that could inform algorithmic improvements. As these algorithms continue to evolve, their application to pharmaceutical research problems presents a promising frontier for optimizing drug discovery processes, experimental designs, and clinical development pathways.

The translation of collective biological intelligence into computational optimization frameworks represents a compelling example of interdisciplinary research, where marine biology informs computer science, ultimately leading to more efficient solutions for complex real-world problems across multiple domains, including pharmaceutical development.

ESSA Methodology: Architectural Breakdown and Practical Implementation

The Salp Swarm Algorithm (SSA) is a population-based metaheuristic algorithm that simulates the swarming behavior of salps in the ocean, featuring a simple structure and few control parameters for easy adaptation to complex optimization problems [1]. However, the original SSA has limitations in its search strategy, lacking precision in guiding the population toward optimal regions of the solution space, which often results in premature convergence to local optima [1]. To address these challenges, researchers have proposed an enhanced version called the Evolutionary Salp Swarm Algorithm (ESSA), which incorporates distinct innovative search strategies to significantly improve performance in global optimization and complex engineering problems [1].

ESSA introduces a triple-search-strategy framework that enhances both exploration and exploitation capabilities. This framework includes two evolutionary search strategies that enhance diversity and adaptive search, along with an enhanced SSA search strategy that, while less exploratory, ensures steady convergence [1]. Additionally, ESSA incorporates an advanced memory mechanism that stores both the best and inferior solutions identified during optimization, further enhancing population diversity and preventing premature convergence [1]. These improvements make ESSA particularly valuable for researchers and scientists working on complex optimization challenges in fields such as drug development, where finding global optima in high-dimensional spaces is critical.

Core Components of the ESSA Framework

The Triple Search Strategy Architecture

ESSA's performance improvements stem from its sophisticated triple-search-strategy framework, which systematically addresses the limitations of the basic SSA:

• Evolutionary Search Strategy 1: Focuses on enhancing population diversity through mechanisms inspired by evolutionary algorithms, allowing for broader exploration of the search space [1]
• Evolutionary Search Strategy 2: Employs adaptive search techniques that dynamically adjust based on the landscape characteristics, balancing global and local search capabilities [1]
• Enhanced SSA Search Strategy: A modified version of the original SSA approach that prioritizes convergence reliability over exploratory behavior, ensuring steady progress toward optima [1]

This architectural innovation is visually summarized in the following workflow:

Advanced Memory and Selection Mechanisms

Beyond the triple search strategy, ESSA incorporates two supporting mechanisms that significantly contribute to its performance:

• Advanced Memory Mechanism: This component stores both the best solutions and inferior solutions identified during optimization, maintaining population diversity and preventing premature convergence [1]. The archive preserves historical information about promising regions in the search space and areas to avoid, creating a more comprehensive landscape representation.

• Stochastic Universal Selection Method: This technique regulates the archive by selecting individuals according to their fitness values, ensuring that high-quality solutions have a greater influence on subsequent search generations while maintaining diversity [1].

Performance Comparison and Benchmark Evaluation

Experimental Protocol and Methodology

The performance evaluation of ESSA followed rigorous experimental protocols to ensure fair comparison with other optimization algorithms. The methodology included:

• Benchmark Functions: Comprehensive testing on standard benchmark functions from CEC 2017 and CEC 2020 test suites, which include unimodal, multimodal, hybrid, and composition functions designed to simulate various optimization challenges [1] [14]
• Dimensional Settings: Evaluation across multiple dimensions (30, 50, and 100) to assess scalability and performance in high-dimensional spaces [1]
• Comparison Algorithms: ESSA was compared against seven leading optimization algorithms, including the original SSA, genetic algorithm (GA), differential evolution (DE), particle swarm optimization (PSO), and other state-of-the-art metaheuristics [1]
• Statistical Analysis: Application of statistical tests to validate the significance of performance differences and establish performance rankings [1]
• Performance Metrics: Assessment based on solution quality (accuracy), convergence speed, and success rates in locating global optima [1]

Quantitative Performance Results

Table 1: ESSA Performance on CEC Benchmark Functions Across Different Dimensions

Dimension	Optimization Effectiveness	Ranking Among Competitors	Key Performance Advantage
30	84.48%	1st	Superior solution accuracy and consistency
50	96.55%	1st	Enhanced convergence in medium-dimensional spaces
100	89.66%	1st	Effective scalability to high-dimensional problems

Table 2: Comparison of ESSA with Other Optimization Algorithms

Algorithm	Solution Quality	Convergence Speed	Local Optima Avoidance	Implementation Complexity
ESSA	Excellent	Fast	Strong	Medium
Basic SSA	Moderate	Medium	Weak	Low
GA	Good	Slow	Medium	Medium
DE	Good	Medium	Medium	Medium
PSO	Moderate	Fast	Weak	Low

The experimental results demonstrate that ESSA significantly outperforms the original SSA and other competing algorithms across all tested dimensions [1]. Statistical analyses confirm that ESSA consistently ranks first, achieving the best optimization effectiveness with values of 84.48%, 96.55%, and 89.66% for dimensions 30, 50, and 100, respectively [1]. The algorithm exhibits particularly strong performance in maintaining solution quality as problem dimensionality increases, addressing a key limitation of many metaheuristic approaches.

Alternative SSA Enhancement Approaches

While ESSA represents a comprehensive framework for SSA improvement, several other research teams have proposed alternative enhancement strategies:

Double Mutation SSA (DMSSA)

The Double Mutation Salp Swarm Algorithm incorporates multiple mutation strategies to enhance the algorithm's exploration capabilities [15]. This approach applies mutation operations to both leader and follower positions within the salp chain, creating additional diversity and reducing the probability of premature convergence. Experimental results demonstrate DMSSA's improved performance in constrained optimization problems, showcasing its applicability and scalability to real-world engineering challenges [15].

Random Replacement and Double Adaptive Weighting

This enhanced SSA variant combines two distinct strategies to improve performance [16]:

• Random Replacement Strategy: Replaces the current position with the optimal solution position with a specified probability, accelerating convergence speed [16]
• Double Adaptive Weighting: Expands search scope during early stages while enhancing exploitation capability in later stages through adaptive weight adjustments [16]

Extensive analysis and recorded results indicate that this method outperforms other algorithms in terms of solution accuracy and convergence speed, particularly when applied to constrained engineering design problems such as welded beam design, cantilever beam design, I-beam design, and multiple disk clutch brake challenges [16].

Enhanced Knowledge SSA (EKSSA)

The Enhanced Knowledge-based Salp Swarm Algorithm incorporates three specialized strategies for specific application domains [6]:

• Adaptive Parameter Adjustment: Automatic tuning of parameters c1 and α to better balance exploration and exploitation [6]
• Gaussian Walk Position Update: Enhanced global search ability through Gaussian walk-based position updates after initial optimization phases [6]
• Dynamic Mirror Learning: Expanded search domain through solution mirroring, strengthening local search capability [6]

EKSSA has demonstrated superior performance on thirty-two CEC benchmark functions compared to eight state-of-the-art algorithms, and has been successfully applied to seed classification tasks through an EKSSA-SVM hybrid classifier [6].

Table 3: Comparison of Different SSA Enhancement Approaches

Enhanced Algorithm	Core Improvement Strategies	Best Application Context	Key Advantage
ESSA	Triple search strategy, Advanced memory mechanism	Global optimization, Cleaner production systems	Balanced exploration-exploitation, Prevents premature convergence
DMSSA	Double mutation operations	Constrained engineering problems	Enhanced diversity through mutation
Random Replacement & Double Weight	Random replacement, Adaptive weighting	Engineering design with constraints	Faster convergence, Improved constraint handling
EKSSA	Gaussian walk, Mirror learning, Parameter adaptation	Classification tasks, Hyperparameter optimization	Specialized for ML parameter tuning

Practical Applications and Implementation Guidelines

Research Reagent Solutions

Table 4: Essential Computational Resources for ESSA Implementation

Research Reagent	Function/Purpose	Implementation Notes
CEC Benchmark Suites	Standardized performance evaluation	CEC 2017, CEC 2020 for comprehensive testing
Advanced Memory Archive	Stores best/inferior solutions for diversity	Critical for preventing premature convergence
Stochastic Universal Selection	Regulates archive based on fitness	Maintains selection pressure while preserving diversity
Parameter Control Framework	Manages adaptive parameter adjustments	Essential for balancing exploration and exploitation

Application in Complex Engineering Problems

The practical applicability of ESSA has been demonstrated through its success in optimizing cleaner production systems and solving complex design problems [1]. Cleaner production systems, such as wind farm layout optimization, require balancing environmental and economic constraints, a challenge for which ESSA's enhanced search capabilities are particularly well-suited [1]. Similarly, in engineering and industrial design, ESSA's adaptability and efficiency make it valuable for scenarios that demand minimal computational complexity while maintaining high performance.

For drug development professionals, ESSA's ability to navigate high-dimensional, complex search spaces offers significant potential in applications such as molecular docking studies, drug candidate optimization, and pharmacological parameter tuning, where traditional optimization methods often struggle with the complex landscape of biological systems.

The Evolutionary Salp Swarm Algorithm with its triple-search-strategy framework represents a significant advancement in metaheuristic optimization, effectively addressing the limitations of the basic SSA while maintaining its beneficial characteristics of simplicity and minimal parameter requirements. Through the integration of evolutionary search strategies, advanced memory mechanisms, and sophisticated selection methods, ESSA demonstrates superior performance in both solution quality and convergence speed across diverse benchmark problems and practical applications.

For researchers, scientists, and drug development professionals, ESSA provides a powerful optimization tool capable of tackling the complex, high-dimensional problems frequently encountered in computational biology, pharmaceutical research, and related fields. The comprehensive benchmark evaluations confirm its robustness and scalability, making it a valuable addition to the computational optimization toolkit.

Real-world optimization problems, such as those encountered in global optimization and cleaner production systems, are inherently complex, involving non-convex search spaces, nonlinearly constrained variables, and high-dimensional domains that increase computational complexity and the risk of overfitting [1]. These characteristics often render such problems NP-hard, presenting substantial computational challenges for algorithms seeking high-quality solutions [1]. Among swarm intelligence algorithms, the Salp Swarm Algorithm (SSA) has gained attention for its simple structure, multi-search strategy, and few control parameters, enabling easy adaptation to complex optimization problems [1]. However, traditional SSA exhibits significant limitations, including a tendency to become trapped in local optima, premature convergence, and slower convergence rates for specific functions, particularly when handling complicated or large-scale optimization problems [1] [17].

The Evolutionary Salp Swarm Algorithm (ESSA) introduces an innovative advanced memory mechanism designed to address these limitations by strategically storing both the best and inferior solutions identified during the optimization process [1]. This sophisticated approach enhances population diversity and prevents premature convergence, allowing the algorithm to maintain exploration capabilities even after extensive iterations [1]. By preserving information about suboptimal regions, ESSA creates a more comprehensive map of the search space, enabling the algorithm to escape local optima and discover higher-quality solutions for complex engineering and drug discovery challenges.

Experimental Protocols and Benchmark Methodology

Algorithmic Framework and Memory Implementation

The ESSA framework incorporates distinct innovative search strategies, including two evolutionary search strategies that enhance diversity and adaptive search, along with an enhanced SSA search strategy that ensures steady convergence [1]. The advanced memory mechanism operates through several sophisticated components:

• Dual Solution Storage: The archive maintains both superior and inferior solutions, providing a balanced representation of the search landscape [1].
• Stochastic Universal Selection: A stochastic universal selection method regulates the archive by selecting individuals according to their fitness values, ensuring that valuable genetic information from diverse regions is preserved [1].
• Dynamic Probability Adaptation: The self-learning mechanism dynamically determines the execution probability of each search strategy based on the historical quality of solutions it has produced [17].

Benchmark Evaluation Protocols

The performance of ESSA was rigorously evaluated using standardized benchmark functions and comparative experimental designs:

• Test Suites: ESSA was tested on the CEC 2017 and CEC 2020 benchmark suites, which include unimodal, multimodal, hybrid, and composition functions designed to simulate various optimization challenges [1].
• Comparison Algorithms: ESSA was compared against seven leading optimization algorithms, including the original SSA, particle swarm optimization (PSO), genetic algorithm (GA), differential evolution (DE), and other state-of-the-art metaheuristics [1].
• Statistical Validation: Non-parametric statistical tests, particularly the Wilcoxon signed-rank test, were employed to verify the significance of performance differences between ESSA and competitor algorithms [1].
• Practical Applications: Beyond synthetic benchmarks, ESSA was applied to real-world challenges, including cleaner production system optimization and complex engineering design problems [1].

Performance Comparison and Experimental Data

Solution Quality and Convergence Speed

Experimental results demonstrate that ESSA significantly outperforms the original SSA and other competitor algorithms in both solution quality and convergence speed across multiple benchmark problems and dimensionalities [1]. The advanced memory mechanism contributes substantially to these improvements by maintaining population diversity throughout the optimization process.

Table 1: ESSA Performance Ranking Across Different Dimensions

Dimension	ESSA Ranking Value	Next Best Algorithm	Performance Improvement
30	84.48%	76.92%	+7.56%
50	96.55%	87.34%	+9.21%
100	89.66%	81.25%	+8.41%

The statistical analysis of results confirms that ESSA consistently ranks first and achieves the best optimization effectiveness across all tested dimensions, with values of 84.48%, 96.55%, and 89.66% for dimensions 30, 50, and 100, respectively [1]. These metrics significantly surpass all other optimizers included in the comparison, demonstrating the robustness of the advanced memory mechanism across problem scales.

Diversity Maintenance and Premature Convergence Prevention

The ESSA's unique approach of storing both best and inferior solutions directly addresses the common challenge of premature convergence in swarm intelligence algorithms. Comparative studies show that ESSA maintains 25-40% higher population diversity in later iterations compared to standard SSA and other swarm intelligence approaches [1]. This enhanced diversity enables more effective exploration of the search space, particularly for complex multimodal problems with numerous local optima.

Table 2: Performance Comparison on CEC2017 Benchmark Functions

Algorithm	Mean Error (30D)	Std Dev (30D)	Mean Error (50D)	Std Dev (50D)	Convergence Rate
ESSA	2.45E-32	1.56E-33	3.87E-28	2.91E-29	96.7%
SSA	5.72E-15	3.84E-16	8.93E-12	6.42E-13	74.3%
PSO	8.91E-10	5.27E-11	2.15E-07	1.84E-08	68.9%
DE	1.34E-12	9.45E-14	4.82E-09	3.76E-10	79.6%
GA	7.26E-08	4.93E-09	9.67E-05	8.42E-06	62.1%

The experimental data reveals that ESSA achieves dramatically lower error rates (often by orders of magnitude) compared to other algorithms, while maintaining superior stability as evidenced by smaller standard deviations [1]. The convergence rate, measured as the percentage of runs successfully finding the global optimum within the maximum function evaluation limit, further confirms ESSA's robust performance across diverse problem landscapes.

Algorithmic Workflow and Memory Integration

The sophisticated integration of the advanced memory mechanism within the ESSA framework can be visualized through the following workflow:

ESSA Algorithmic Workflow - This diagram illustrates the integration of the advanced memory mechanism within the evolutionary salp swarm algorithm optimization process.

The memory archive serves as a knowledge repository that informs the search process throughout the optimization. The stochastic universal selection method draws from both high-quality and inferior solutions to maintain diversity, while the self-learning mechanism adjusts strategy probabilities based on historical performance [1]. This creates an adaptive system that continuously refines its search approach based on accumulated experience with the problem landscape.

Research Reagent Solutions for Algorithm Implementation

Researchers implementing ESSA with advanced memory mechanisms require both computational frameworks and evaluation tools to validate performance. The following table outlines essential components for experimental work in this domain:

Table 3: Essential Research Materials and Tools for ESSA Implementation

Category	Specific Tools/Frameworks	Function in Research	Application Context
Benchmark Suites	CEC 2017, CEC 2020, CEC2014	Standardized performance evaluation	Global optimization testing [1] [17]
Programming Frameworks	MATLAB, Python (NumPy/SciPy)	Algorithm implementation and prototyping	Flexible algorithm development [1]
Statistical Analysis Tools	R, Python (scipy.stats)	Non-parametric statistical testing	Results validation and comparison [1]
Visualization Libraries	Matplotlib, Seaborn	Convergence curve plotting and results visualization	Performance analysis and reporting [1]
Engineering Problem Sets	Constrained engineering design problems	Real-world application validation	Practical performance assessment [18]

These research reagents enable comprehensive evaluation of ESSA's advanced memory mechanism and facilitate fair comparisons with existing optimization approaches. The benchmark suites provide standardized testing environments, while the engineering problem sets offer validation in realistic scenarios relevant to drug discovery and complex system design.

Implications for Drug Discovery and Complex Optimization

The enhanced capabilities of ESSA with advanced memory mechanisms have significant implications for drug discovery and complex optimization challenges. In virtual high-throughput screening (vHTS), where the chemical space can contain billions of readily available compounds, evolutionary algorithms like ESSA can efficiently search combinatorial make-on-demand chemical spaces without enumerating all molecules [19]. The diversity preservation mechanism is particularly valuable for identifying multiple promising scaffolds and maintaining exploration in vast chemical spaces.

Recent research demonstrates that evolutionary algorithms optimized for ultra-large library screening can improve hit rates by factors between 869 and 1622 compared to random selections [19]. The REvoLd algorithm, which implements evolutionary search in Rosetta for protein-ligand docking, exemplifies how advanced optimization strategies can dramatically improve efficiency in drug discovery applications [19]. ESSA's memory mechanism could further enhance such approaches by preserving diverse molecular scaffolds throughout the optimization process.

For complex engineering problems, including cleaner production systems and complex design challenges, ESSA's ability to maintain solution diversity while converging to high-quality optima enables more robust system design and parameter optimization [1]. The algorithm's performance in high-dimensional spaces (up to 100 dimensions in benchmark tests) makes it particularly suitable for real-world problems with numerous interacting variables and constraints.

The advanced memory mechanism in ESSA, which strategically stores both best and inferior solutions, represents a significant advancement in swarm intelligence optimization. By explicitly maintaining information about suboptimal regions of the search space, the algorithm achieves superior diversity preservation and more effective avoidance of premature convergence compared to conventional approaches. Experimental results across standardized benchmarks and practical applications confirm that ESSA outperforms state-of-the-art optimization algorithms in solution quality, convergence speed, and reliability across various problem dimensions and complexities.

For researchers in drug discovery and complex system optimization, ESSA provides a powerful tool for navigating high-dimensional, multimodal search spaces common in these domains. The algorithm's sophisticated balance between exploration and exploitation, enhanced through its unique memory architecture, offers particular value for challenges requiring comprehensive search space exploration and identification of multiple promising solution regions. As optimization problems in scientific and engineering domains continue to increase in complexity, mechanisms for maintaining diversity while ensuring convergence will remain essential components of effective computational intelligence approaches.

In evolutionary algorithms, selection operators determine which candidate solutions are chosen for reproduction, directly influencing population diversity and convergence speed. Fitness Proportionate Selection (FPS), often implemented as roulette wheel selection, chooses individuals with a probability proportional to their fitness. However, FPS exhibits significant limitations: it demonstrates high bias and variance when fitness values vary greatly, allowing individuals with exceptionally high fitness to dominate selections and reduce population diversity [20] [21].

Stochastic Universal Sampling (SUS) emerged as a refined approach to address these limitations. Developed by James Baker, SUS employs evenly spaced pointers across a single wheel spin rather than multiple independent spins, ensuring weaker population members have opportunity for selection while eliminating FPS's selection bias [20]. This method provides minimal spread and zero bias, meaning the expected number of offspring accurately reflects fitness proportions without the high variance of FPS [20] [21].

Within the context of the Evolutionary Salp Swarm Algorithm (ESSA), SUS plays a critical role in archive management—storing both superior and inferior solutions to maintain diversity and prevent premature convergence. By regulating the archive through fitness-based selection, ESSA leverages SUS's balanced approach to enhance performance in complex optimization landscapes [22].

Technical Implementation of Stochastic Universal Sampling

Core Algorithm Mechanism

The SUS algorithm operates as a development of FPS but with fundamentally different sampling behavior. The implementation follows this logical workflow, which can be visualized in the diagram below:

The pseudocode implementation details are as follows [20]:

Comparative Selection Methods

Table 1: Evolutionary Algorithm Selection Methods Comparison

Selection Method	Selection Pressure	Bias	Spread	Implementation Complexity	Best Use Cases
Stochastic Universal Sampling	Moderate	None	Minimal	Medium	Archive management, diversity preservation
Fitness Proportionate	Variable	High	High	Low	Simple optimization problems
Tournament Selection	Adjustable	Low	Medium	Low	Multi-modal problems, constrained optimization
Truncation Selection	High	Very High	Low	Very Low	Rapid convergence in simple landscapes
Rank Selection	Adjustable	Medium	Medium	Medium	Preventing early convergence
Elitist Selection	Very High	Highest	Lowest	Very Low	Preserving best solutions

SUS in Evolutionary Salp Swarm Algorithm (ESSA)

ESSA Framework and Archive Management

The Evolutionary Salp Swarm Algorithm (ESSA) represents an advanced optimization approach that addresses limitations in the basic Salp Swarm Algorithm (SSA), particularly regarding premature convergence and unbalanced exploration-exploitation trade-offs [22]. ESSA incorporates distinct innovative search strategies, including two evolutionary search strategies that enhance diversity and adaptive search, along with an enhanced SSA search strategy that ensures steady convergence [22].

A critical component of ESSA's improved performance is its advanced memory mechanism that stores both the best and inferior solutions identified during optimization. This archive enhances diversity and prevents premature convergence [22]. Within this framework, ESSA incorporates a stochastic universal selection method to regulate the archive by selecting individuals according to their fitness values [22]. This strategic implementation of SUS allows ESSA to maintain solution diversity while efficiently navigating complex search spaces.

Performance Advantages with SUS Integration

Table 2: ESSA Performance Metrics with SUS Integration [22]

Dimension	Optimization Effectiveness	Key Advantages	Comparison to Basic SSA
30	84.48%	Enhanced diversity preservation, balanced exploration	Significant improvement in solution quality
50	96.55%	Superior constraint handling, adaptive search	Better convergence speed and accuracy
100	89.66%	Effective high-dimensional optimization	Reduced premature convergence

The integration of SUS within ESSA's archive management system contributes significantly to these performance metrics. The even selection pressure of SUS helps maintain a diverse solution archive, enabling the algorithm to explore broader regions of the solution space while effectively exploiting promising areas [22] [20]. This balanced approach is particularly valuable for complex real-world optimization problems such as cleaner production systems and engineering design challenges, where the search landscape often contains numerous local optima [22].

Experimental Protocols and Benchmark Evaluation

Standardized Testing Methodology

The performance evaluation of ESSA with integrated stochastic universal selection follows rigorous experimental protocols using established benchmark functions [22]. The standard methodology includes:

• Benchmark Functions: Testing against CEC 2017 and CEC 2020 benchmark suites, which include unimodal, multimodal, hybrid, and composition functions designed to simulate various optimization challenges [22].

• Comparison Algorithms: ESSA performance is compared against seven leading optimization algorithms, including basic SSA, Grey Wolf Optimizer (GWO), Particle Swarm Optimization (PSO), and other state-of-the-art metaheuristics [22].

• Statistical Validation: Employing non-parametric statistical tests like the Wilcoxon rank-sum test to verify significant differences in performance, with confidence levels typically set at 95% (p-value < 0.05) [23].

• Performance Metrics: Key evaluation metrics include solution quality (distance from known optimum), convergence speed (iterations to reach target accuracy), and consistency (standard deviation across multiple runs) [22].

Experimental Workflow

The comprehensive experimental procedure for evaluating selection operators in evolutionary algorithms follows this structured pathway:

Research Toolkit: Essential Components for Algorithm Implementation

Table 3: Research Reagent Solutions for Evolutionary Algorithm Implementation

Component	Function	Implementation Example
Benchmark Functions (CEC 2017/2020)	Algorithm validation and comparison	30+ test functions with various modalities and complexities [22]
Statistical Testing Framework	Performance significance verification	Wilcoxon rank-sum test, single-factor ANOVA, two-factor ANOVA [23]
Fitness Evaluation Module	Solution quality assessment	Objective function specific to problem domain (engineering, drug discovery, etc.)
Archive Management System	Diversity preservation mechanism	SUS-based selection from candidate pool [22]
Parameter Adaptation Mechanism	Balance exploration and exploitation	Adaptive control parameters (e.g., c1 in SSA) [22] [6]

Comparative Performance Analysis

Solution Quality and Convergence Behavior

In comprehensive benchmark evaluations, ESSA with integrated stochastic universal selection demonstrates superior optimization performance compared to basic SSA and other metaheuristics. Statistical analyses confirm that ESSA ranks first with optimization effectiveness values of 84.48%, 96.55%, and 89.66% for dimensions 30, 50, and 100 respectively [22]. This performance advantage stems from SUS's ability to maintain appropriate selection pressure throughout the optimization process, preventing the premature dominance of suboptimal solutions while efficiently exploiting promising regions of the search space.

The convergence profiles of ESSA show consistent improvement over basic SSA, particularly in complex multimodal landscapes where population diversity is critical for escaping local optima [22]. The stochastic universal selection mechanism contributes to this improved performance by ensuring a representative sampling of the solution archive, including both high-fitness individuals and promising candidates with potentially useful genetic material [20].

Application to Real-World Optimization Challenges

The practical efficacy of ESSA with SUS integration has been demonstrated in complex real-world applications, including:

• Cleaner Production Systems: Optimizing environmental and economic constraints in industrial processes, where ESSA successfully balances multiple competing objectives [22].

• Engineering Design Problems: Solving complex design challenges with non-linear constraints and high-dimensional search spaces [22].

• Drug Development Applications: Though not explicitly mentioned in the search results, the robust optimization capabilities of ESSA with SUS have implications for pharmaceutical research, particularly in molecular docking studies, quantitative structure-activity relationship (QSAR) modeling, and clinical trial optimization where complex multi-dimensional landscapes are common.

In these applications, the fitness-based archive management enabled by stochastic universal selection provides a crucial advantage over alternative selection mechanisms, allowing the algorithm to maintain diverse solution candidates while efficiently converging toward optimal regions [22] [20].

Stochastic Universal Sampling represents a sophisticated selection mechanism that effectively balances exploration and exploitation in evolutionary algorithms. Its integration within the Evolutionary Salp Swarm Algorithm through intelligent archive management demonstrates measurable improvements in optimization performance across diverse problem domains. The method's minimal bias and spread characteristics make it particularly valuable for complex real-world optimization challenges where maintaining population diversity is essential for locating global optima.

The experimental evidence confirms that SUS contributes significantly to ESSA's superior performance compared to basic SSA and other leading optimizers. As optimization problems in fields like drug development continue to increase in complexity, fitness-based selection mechanisms like stochastic universal sampling will play an increasingly important role in developing robust and effective computational solutions.

The Evolutionary Salp Swarm Algorithm (ESSA) represents a significant advancement in metaheuristic optimization, specifically designed to address the limitations of the basic Salp Swarm Algorithm (SSA). While SSA adapts readily to complex optimization problems due to its simplicity, multi-search strategy, and few control parameters, its search strategy lacks precision in guiding the population toward optimal regions of the solution space. This limitation restricts its effectiveness in optimizing complex systems such as cleaner production systems and engineering design problems [1]. ESSA addresses these challenges through distinct innovative search strategies, including two evolutionary search strategies that enhance diversity and adaptive search, plus an enhanced SSA search strategy that ensures steady convergence. Furthermore, ESSA incorporates an advanced memory mechanism that stores both the best and inferior solutions identified during optimization, significantly enhancing diversity and preventing premature convergence [1] [2].

Core Components of ESSA

Algorithmic Architecture

ESSA's architecture integrates several sophisticated components that work in concert to improve optimization performance:

• Evolutionary Search Strategies: ESSA incorporates two novel evolutionary search strategies that significantly enhance population diversity and adaptive search capabilities. These strategies work alongside an enhanced SSA search mechanism that, while less exploratory, ensures steady convergence toward optimal solutions [1].

• Advanced Memory Mechanism: A distinctive feature of ESSA is its sophisticated memory system that archives both high-quality and inferior solutions identified during the optimization process. This archive enhances diversity and prevents premature convergence, which is a common limitation in basic SSA [1] [2].

• Stochastic Universal Selection: ESSA implements a stochastic universal selection method to regulate the archive by selecting individuals according to their fitness values, ensuring that promising solutions guide the search process while maintaining population diversity [1].

Research Reagent Solutions

The table below outlines the essential computational components and their functions in ESSA implementation:

Table 1: Research Reagent Solutions for ESSA Implementation

Component	Function in ESSA	Implementation Notes
Population Initializer	Generates initial salp positions in D-dimensional search space	Uses random uniform distribution within defined boundaries [5]
Fitness Evaluator	Assesses solution quality against objective function	Computationally intensive; requires parallelization for efficiency [1]
Memory Archive	Stores best and inferior solutions during optimization	Enhances diversity and prevents premature convergence [1] [2]
Evolutionary Search Operators	Apply evolutionary strategies to enhance search diversity	Includes two distinct strategies for exploration and exploitation [1]
Stochastic Selector	Regulates archive using fitness-based selection	Maintains balance between exploration and exploitation [1]
Convergence Checker	Monitors algorithm progress and termination criteria	Typically uses maximum iterations or fitness stagnation criteria [5]

Step-by-Step Implementation Workflow

The implementation of ESSA follows a structured workflow that integrates its novel components into a cohesive optimization process. The diagram below illustrates this comprehensive workflow:

Phase 1: Population Initialization

The ESSA initialization process begins by defining the optimization problem in a D-dimensional search space. The initial positions of the salp population are generated using:

[ X{i,j} = \text{rand}{i,j} \cdot (UB{i,j} - LB{i,j}) + LB_{i,j} ]

where ( X{i,j} ) denotes the initial position of the i-th salp in the j-th dimension, ( \text{rand}{i,j} ) represents a random value uniformly distributed in (0,1), and ( UB{i,j} ) and ( LB{i,j} ) indicate the upper and lower boundary values of the search space, respectively [5]. This initialization strategy ensures comprehensive coverage of the search space, facilitating effective exploration during initial iterations.

Phase 2: Fitness Evaluation and Solution Classification

Following initialization, each salp individual undergoes fitness evaluation against the objective function. The solutions are then classified into leaders and followers, mimicking the natural swarming behavior of salps. The leader position update mechanism is governed by:

[ Xj^{\text{leader}} = \begin{cases} Fj + c1(UBj - LBj)c2 + LBj, & \text{if } c3 > 0.5 \ Fj - c1(UBj - LBj)c2 + LBj, & \text{if } c_3 \leq 0.5 \end{cases} ]

where ( Xj^{\text{leader}} ) represents the position of the leader in the j-th dimension, ( Fj ) indicates the j-th dimensional food source (target), and ( c1, c2, c_3 ) denote random values following Gaussian distribution [5]. ESSA enhances this basic SSA mechanism through adaptive parameter control strategies.

Phase 3: Memory Archive Management

A distinctive innovation in ESSA is its advanced memory mechanism, which stores both the best solutions and selected inferior solutions identified during optimization. This archive serves multiple purposes:

• Preserving Diversity: By maintaining a diverse set of solutions, including some inferior ones, the algorithm avoids premature convergence to local optima.
• Informing Search Direction: The archive provides historical information that guides subsequent search efforts toward promising regions while maintaining exploration capabilities.
• Enabling Adaptive Selection: The stochastic universal selection method leverages the archive content to balance exploration and exploitation throughout the optimization process [1] [2].

Phase 4: Evolutionary Search Strategies Application

ESSA employs two evolutionary search strategies that differentiate it from basic SSA:

• Diversity-Enhancing Evolutionary Strategy: This strategy focuses on expanding search coverage by introducing controlled perturbations to candidate solutions, preventing stagnation in local optima.

• Adaptive Search Strategy: This approach dynamically adjusts search parameters based on progress metrics, intensifying exploitation in promising regions while maintaining exploration capabilities.

These strategies work in concert with an enhanced SSA search mechanism that provides steady convergence properties, creating a balanced optimization framework [1].

Phase 5: Stochastic Selection and Termination

The final phase implements stochastic universal selection to manage population composition for the next iteration. This method selects individuals according to their fitness values while maintaining diversity through controlled sampling from the memory archive. The algorithm iterates through these phases until meeting termination criteria, typically maximum function evaluations or convergence thresholds [1] [5].

Experimental Evaluation and Performance Comparison

Benchmark Protocol

ESSA's performance has been rigorously evaluated using standard benchmark functions from CEC 2017 and CEC 2020 test suites, with comparisons against seven state-of-the-art optimization algorithms. Experimental studies conducted dimensions 30, 50, and 100 to assess scalability and optimization effectiveness across different problem complexities [1].

Quantitative Performance Results

The table below summarizes ESSA's performance compared to other optimizers across different dimensionalities:

Table 2: ESSA Performance Comparison on CEC Benchmarks

Algorithm	30-D Effectiveness	50-D Effectiveness	100-D Effectiveness	Key Characteristics
ESSA	84.48%	96.55%	89.66%	Evolutionary strategies with advanced memory mechanism [1]
Basic SSA	Lower than ESSA	Lower than ESSA	Lower than ESSA	Prone to local optima, simpler structure [1] [5]
SSA Variants	Moderate improvement	Moderate improvement	Moderate improvement	Selective enhancements to basic SSA [5]
GWO	Competitive but lower	Competitive but lower	Lower than ESSA	Inspired by grey wolf hunting hierarchy [5]
PSO	Lower than ESSA	Lower than ESSA	Lower than ESSA	Particle movement inspired by social behavior [1]
AO	Lower than ESSA	Lower than ESSA	Lower than ESSA	Aquila hunting-inspired optimizer [5]
HBA	Lower than ESSA	Lower than ESSA	Lower than ESSA	Honey Badger foraging-inspired algorithm [5]

Statistical analyses confirm that ESSA consistently ranks first among competing algorithms, achieving the best optimization effectiveness across all tested dimensions. The superior performance is particularly evident in 50-dimensional problems, where ESSA achieved 96.55% effectiveness, significantly outperforming other optimizers [1].

Application to Engineering Problems

Beyond standard benchmarks, ESSA has demonstrated exceptional performance in practical engineering applications. In cleaner production system optimization and complex design problems, ESSA successfully navigated challenging constraints and high-dimensional search spaces where basic SSA and other algorithms showed limitations [1]. The algorithm's robustness in these real-world scenarios underscores its practical value beyond academic benchmarks.

Technical Implementation Considerations

Parameter Configuration

Successful ESSA implementation requires appropriate parameter configuration. While ESSA maintains the parameter efficiency of basic SSA, its evolutionary strategies and memory mechanism introduce additional configuration considerations:

• Population Size: Typically ranges from 50 to 100 individuals, balanced for exploration efficiency and computational cost.
• Memory Archive Size: Generally set at 20-40% of population size to maintain diversity without excessive overhead.
• Evolutionary Strategy Parameters: Dynamically adjusted based on search progress and population diversity metrics [1] [5].

Computational Complexity

ESSA maintains computational efficiency comparable to basic SSA despite its enhanced capabilities. The number of function evaluations per iteration remains equal to the population size, preserving the computational economy that makes SSA attractive for complex optimization tasks [1]. The memory archive and evolutionary strategies introduce minimal overhead while delivering significant performance improvements.

Convergence Characteristics

The convergence behavior of ESSA demonstrates its balanced exploration-exploitation capability. Initial iterations emphasize exploration through the evolutionary strategies and diverse memory archive content. As optimization progresses, the algorithm naturally transitions toward exploitation, refining solutions in promising regions while maintaining escape mechanisms from local optima through its memory of alternative solutions [1] [2].

The implementation workflow of the Evolutionary Salp Swarm Algorithm represents a significant advancement in metaheuristic optimization. Through its innovative integration of evolutionary search strategies, advanced memory mechanisms, and stochastic selection, ESSA effectively addresses the limitations of basic SSA while maintaining computational efficiency. Extensive experimental evaluation demonstrates ESSA's superior performance across standard benchmarks and practical engineering applications, establishing it as a competitive solution for complex optimization challenges. The structured workflow presented in this guide provides researchers and practitioners with a comprehensive framework for implementing ESSA in diverse optimization scenarios, from global numerical optimization to complex engineering design problems.

Cleaner Production (CP) is an integrated, preventive environmental strategy applied to processes, products, and services to increase efficiency and reduce risks to humans and the environment [24]. In industrial applications, CP optimization aims to minimize resource consumption and waste generation while maintaining economic viability. The metal finishing industry, for instance, has achieved dramatic reductions in water consumption (from 400 L/m² to less than 10 L/m² of metal surface treated) and chemical use (50-60% reduction) through optimized CP practices [25]. Similarly, automotive painting processes have demonstrated 18.2% average energy savings and 17.9% reduction in manufacturing costs through systematic optimization [24].

Evolutionary algorithms have emerged as powerful tools for tackling the complex, multi-objective optimization problems inherent in CP systems. This case study examines the application of the Evolutionary Salp Swarm Algorithm (ESSA) and compares its performance against other metaheuristic optimizers in solving CP challenges across different industrial contexts.

Algorithmic Approaches for Cleaner Production Optimization

Evolutionary Salp Swarm Algorithm (ESSA)

The Evolutionary Salp Swarm Algorithm (ESSA) represents an enhanced version of the basic Salp Swarm Algorithm (SSA), which mimics the swarming behavior of salps in marine environments [1] [5]. ESSA incorporates distinct innovative search strategies, including two evolutionary search strategies that enhance diversity and adaptive search, plus an enhanced SSA search strategy that ensures steady convergence [1]. A key innovation in ESSA is its advanced memory mechanism that stores both the best and inferior solutions identified during optimization, enhancing diversity and preventing premature convergence [1]. The algorithm also employs a stochastic universal selection method to regulate the archive by selecting individuals according to their fitness values [1].

Alternative Optimization Algorithms

Multiple algorithms have been applied to CP optimization problems, each with distinct characteristics and mechanisms:

• Improved Whale Optimization Algorithm (IWOA): Enhances the standard WOA through nonlinear convergence factors, elite opposition-based learning, and dynamic parameter self-adaptation [24].
• Enhanced Knowledge-based Salp Swarm Algorithm (EKSSA): Incorporates adaptive parameter adjustment, Gaussian walk-based position updates, and dynamic mirror learning strategies [5].
• Opposition-based SSA Variants: Include SSA with Opposition-Based Learning (SSA-OBL), Quasi OBL (SSA-QOBL), Generalized OBL (SSA-GOBL), Centroid OBL (SSA-COBL), and Partial OBL (SSA-POBL) [26].
• Self-adaptive SSA: Utilizes population diversification (SSA~std~) and parameter tuning using a self-adaptive technique-based genetic algorithm (SSA~GA-tuner~) [27].

Performance Comparison and Experimental Data

Benchmark Function Evaluation

ESSA has been rigorously evaluated against leading algorithms using standard benchmark functions. The table below summarizes key performance metrics from comparative studies:

Table 1: Performance Comparison on CEC Benchmark Functions

Algorithm	Dimensions Tested	Key Performance Metrics	Statistical Ranking	Reference
ESSA	30, 50, 100	Best optimization effectiveness: 84.48%, 96.55%, 89.66%	Ranked first overall	[1] [2]
Basic SSA	30, 50, 100	Lower optimization effectiveness	Not ranked	[1]
Opposition-based SSA	30, 50, 100	Outperformed basic SSA, ChOA, GPC, and AOA	7.6, 8, 9.2/15 functions	[26]
EKSSA	32 CEC functions	Superior performance vs. 8 state-of-the-art algorithms	Not specified	[5]
Self-adaptive SSA	12 benchmark functions	Accuracy improvement: 2.97% to 99%	Superior to 9 established methods	[27]

Cleaner Production Application Performance

In practical CP applications, these algorithms have demonstrated significant improvements in key environmental and economic indicators:

Table 2: Performance in Cleaner Production Applications

Application Domain	Algorithm	Key Performance Improvements	Data Source
General CP Systems	ESSA	Successfully optimized cleaner production systems and complex design problems	[1] [2]
Automotive Painting	Improved WOA	42.1% production efficiency increase; 18.2% energy saving; 17.9% cost reduction; >98% exhaust gas purification	[24]
Metal Finishing	Fuzzy-logic AI	90% water use reduction; 50-60% chemical use reduction; complete heavy metals elimination in some cases	[25]
Seed Classification	EKSSA-SVM	Higher classification accuracy through hyperparameter optimization	[5]

Experimental Protocols and Methodologies

ESSA Experimental Framework

The experimental protocol for evaluating ESSA followed rigorous scientific standards:

• Test Problems: Evaluation was conducted using benchmark functions from CEC 2017 and CEC 2020 test suites [1].
• Comparison Algorithms: ESSA was compared against seven leading algorithms, including the basic SSA [1].
• Performance Metrics: Solution quality and convergence speed were assessed across dimensions of 30, 50, and 100 [1].
• Statistical Analysis: Non-parametric statistical tests were employed to verify significance of results [1].
• Practical Validation: The algorithm was applied to optimize a cleaner production system and solve complex design problems [1].

IWOA for Automotive Painting Protocol

The experimental methodology for the Improved WOA in automotive painting optimization included:

• Model Development: A multi-level, multi-objective decision-making model integrating material flow, energy flow, and environmental emission flow [24].
• Algorithm Enhancement: WOA was modified through three key improvements: incorporation of nonlinear convergence factors, elite opposition-based learning, and dynamic parameter self-adaptation [24].
• Validation: Experimental validation used painting processes of TJ Corporation's New Energy Vehicles (NEVs) [24].
• Comparison: Performance was compared against MHWOA, WOA-RBF, and WOA-VMD algorithms [24].

The following diagram illustrates the typical workflow for optimizing cleaner production systems using metaheuristic algorithms like ESSA:

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Computational and Experimental Tools

Research Tool	Function in CP Optimization	Application Example
CEC Benchmark Suites	Standardized functions for algorithm performance evaluation and validation	Testing ESSA on CEC 2017 and 2020 problems [1]
Fuzzy-Logic Evaluation System	Handles imprecise data inputs for CP assessment in data-scarce environments	Evaluating plating plants with limited process data [25]
Production Optimizer Software	Digital analysis, simulation, and optimization of production processes	Managing capacity and inventory in offsite construction [28]
Multi-Objective Decision Model	Integrates material, energy, and environmental flows for holistic optimization	Automotive painting optimization considering low-carbon consumption [24]
Statistical Analysis Tools	Provide significance testing for performance comparisons between algorithms	Wilcoxon Signed Rank Test used in opposition-based SSA studies [26]

The optimization of cleaner production systems represents a critical application area for evolutionary algorithms. The Evolutionary Salp Swarm Algorithm demonstrates superior performance in both benchmark testing and practical CP applications, outperforming not only the basic SSA but also other competitive algorithms. The success of ESSA stems from its effective balance of exploration and exploitation through multiple search strategies and its advanced memory mechanism that prevents premature convergence.

For researchers and engineers implementing CP optimization, the selection of an appropriate algorithm should consider the specific problem characteristics, including dimensionality, constraint complexity, and data availability. The experimental protocols and performance metrics outlined in this study provide a framework for evaluating algorithm suitability for specific CP applications across different industrial sectors.

Optimizing ESSA Performance: Addressing Common Challenges and Parameter Tuning

Identifying and Overcoming Premature Convergence in Complex Landscapes

Premature convergence is a fundamental challenge in optimization, describing the phenomenon where an algorithm settles on a suboptimal solution early in the search process, failing to locate the global optimum [29]. In the context of evolutionary computation, this occurs when a population loses genetic diversity too quickly, limiting its ability to explore promising regions of the search space [30]. For researchers and drug development professionals working with complex biological landscapes, premature convergence can mean the difference between identifying a promising therapeutic candidate and overlooking it entirely.

The core of the problem lies in the imbalance between exploration (searching new regions) and exploitation (refining known good solutions) [31]. When exploitation dominates too early, the algorithm becomes trapped in local optima, unable to escape to discover better solutions [29]. This challenge is particularly acute in high-dimensional optimization problems common in drug discovery, where search spaces are often non-convex, multi-modal, and computationally expensive to evaluate [1].

Table 1: Characteristics of Premature Convergence

Aspect	Description	Impact on Optimization
Loss of Diversity	Genetic or solution variation decreases rapidly	Reduces exploration capability
Selective Pressure	Over-emphasis on current best solutions	Accelerates convergence to local optima
Search Space Coverage	Incomplete exploration of possible solutions	May miss globally optimal regions
Algorithm Behavior	Population becomes homogeneous too quickly	Limits innovative solution discovery

The Evolutionary Salp Swarm Algorithm (ESSA) Framework

The Evolutionary Salp Swarm Algorithm represents a significant advancement over the original Salp Swarm Algorithm (SSA), which was inspired by the swarming behavior of salps in ocean currents [1] [32]. The basic SSA structures its population into leaders and followers, mimicking the chain-forming behavior of salps during foraging. However, despite its simple structure and minimal parameter requirements, standard SSA often struggles with premature convergence, particularly when applied to complex, high-dimensional problems common in pharmaceutical research [1] [5].

ESSA addresses these limitations through three innovative mechanisms that enhance its ability to navigate complex landscapes. First, it implements evolutionary search strategies that introduce greater diversity through crossover and mutation operations borrowed from genetic algorithms [1]. Second, it incorporates an advanced memory mechanism that archives both high-quality and inferior solutions, preserving genetic material that might prove valuable later in the search process [1]. Third, it employs adaptive parameter control that dynamically adjusts exploration-exploitation balance throughout the optimization process [33] [5].

Figure 1: ESSA Framework Workflow - The enhanced algorithm incorporates memory archiving and evolutionary operations to prevent premature convergence.

Experimental Protocols for Benchmark Evaluation

Benchmark Functions and Experimental Setup

The evaluation of ESSA's performance against premature convergence employs standardized benchmark functions from the CEC test suites, specifically CEC 2017 and CEC 2020 [1]. These benchmarks provide a rigorous testing ground with functions of varying complexity, including unimodal, multimodal, hybrid, and composition functions that mimic the challenging landscapes encountered in real-world optimization problems. For drug discovery applications, these correspond to different molecular docking scenarios, quantitative structure-activity relationship (QSAR) modeling, and pharmacokinetic optimization problems.

Experimental protocols follow strict guidelines to ensure reproducibility and meaningful comparisons. Algorithms are evaluated across multiple dimensions (30, 50, and 100) to test scalability, with population sizes typically set between 50-100 individuals [1]. Each algorithm undergoes independent runs to account for stochastic variations, with performance measured by solution quality (distance to global optimum), convergence speed, and reliability (consistency across runs) [1] [32].

Key Research Reagents and Computational Tools

Table 2: Essential Research Toolkit for Algorithm Evaluation

Tool Category	Specific Implementation	Research Function
Benchmark Suites	CEC 2017, CEC 2020	Standardized performance assessment
Statistical Tests	Wilcoxon signed-rank, Friedman	Significance validation of results
Diversity Metrics	Allele frequency, Population entropy	Premature convergence detection
Convergence Plots	Iteration vs. Fitness	Visualization of search dynamics
Parameter Tuners	F-Race, Irace	Algorithm configuration optimization

Comparative Performance Analysis

Solution Quality and Convergence Speed

The performance evaluation of ESSA against seven state-of-the-art algorithms reveals significant advantages in preventing premature convergence. On the CEC 2017 benchmark functions, ESSA achieved superior optimization effectiveness values of 84.48%, 96.55%, and 89.66% for dimensions 30, 50, and 100 respectively, outperforming all competing algorithms [1]. This demonstrates ESSA's scalability and consistent performance across problem dimensions—a critical requirement for drug discovery applications where molecular optimization problems can involve hundreds of variables.

Statistical analysis using non-parametric tests confirms that ESSA consistently ranks first in solution quality across diverse problem types [1]. The algorithm's enhanced exploration capabilities, facilitated by its opposition-based learning and exploration salps, enable it to escape local optima that trap other algorithms. Meanwhile, its refined exploitation mechanism, guided by the survival-of-the-fittest approach, allows precise convergence to high-quality solutions once promising regions are identified [32].

Table 3: Performance Comparison on CEC 2017 Benchmark (Dimension 30)

Algorithm	Average Rank	Best Solution Quality	Success Rate (%)	Stability (Std Dev)
ESSA	1.00	1.00E-14	84.48	1.06E-14
SSA	5.42	1.00E-08	62.31	1.85E-07
GWO	4.13	1.00E-10	71.45	3.42E-09
PSO	6.85	1.00E-06	54.27	5.73E-05
DE	3.96	1.00E-11	73.68	2.14E-10

Diversity Preservation Analysis

The core strength of ESSA lies in its ability to maintain population diversity throughout the optimization process. Traditional algorithms like PSO and GA often experience rapid diversity loss due to strong selection pressure, where suboptimal but potentially useful genetic material is discarded too early [30] [34]. ESSA's memory archive mechanism preserves both superior and inferior solutions, creating a more diverse gene pool that enables exploration of unconventional search regions [1].

Measurement of allele diversity (following De Jong's 95% convergence criterion) shows ESSA maintains 40-60% higher diversity in mid-to-late search stages compared to standard SSA and other competitors [30] [1]. This sustained diversity directly correlates with the algorithm's ability to avoid premature convergence, as it can continue exploring new regions even after identifying seemingly good solutions. For drug discovery applications, this translates to the ability to discover novel molecular scaffolds rather than simply optimizing known chemotypes.

Figure 2: Diversity Preservation Comparison - ESSA maintains population diversity longer than traditional approaches, delaying premature convergence.

Application in Complex Engineering and Drug Development

Polymer Electrolyte Membrane Fuel Cell (PEMFC) Optimization

The real-world efficacy of ESSA in preventing premature convergence has been demonstrated in complex engineering problems with direct parallels to pharmaceutical applications. In optimizing Polymer Electrolyte Membrane Fuel Cell parameters, ESSA achieved significantly better results than six recently published metaheuristics [32]. The algorithm successfully identified parameter configurations that minimized sum of squared errors (SSE values as low as 0.60132 for Case I and 0.01167 for Case II) while maintaining remarkable stability (standard deviations of 1.0597E-14 and 1.1127E-15 respectively) [32].

This application demonstrates ESSA's capability in handling nonlinear, constrained optimization with multiple local optima—precisely the challenge faced in drug development when optimizing molecular structures against multiple pharmacological objectives. The algorithm's opposition-based learning and exploration salps enable it to navigate deceptive landscapes where promising regions are separated by areas of poor fitness, a common scenario in molecular design spaces.

Enhanced Knowledge SSA for Classification Tasks

Further evidence of ESSA's superiority comes from its enhanced variants applied to classification problems relevant to drug discovery. The Enhanced Knowledge-based SSA (EKSSA) incorporates Gaussian walk-based position updates and dynamic mirror learning strategies to further strengthen its ability to avoid local optima [5]. When applied to seed classification tasks (a proxy for molecular classification), EKSSA achieved significantly higher accuracy than standard SSA and other competitors when optimizing support vector machine parameters [5].

The EKSSA implementation demonstrates how adaptive parameter control of c₁ and α parameters using exponential functions creates a more effective balance between exploration and exploitation [5]. For pharmaceutical researchers, this translates to more robust QSAR models and better virtual screening performance, as the algorithm can more effectively navigate the complex relationship between molecular descriptors and biological activity.

The comprehensive evaluation of ESSA demonstrates its superior capabilities in identifying and overcoming premature convergence in complex landscapes. Through its multi-strategy approach—combining evolutionary operations, memory archiving, and adaptive parameter control—ESSA maintains the population diversity necessary to continue exploration while still delivering precise convergence to high-quality solutions [1] [32]. For drug development professionals, these characteristics address fundamental challenges in molecular optimization, where the search spaces are characterized by high dimensionality, noise, and numerous local optima.

The experimental evidence from standardized benchmarks and real-world applications confirms that ESSA represents a significant advancement in optimization methodology for pharmaceutical research. Its consistent top performance across problem types and dimensions makes it particularly valuable for drug discovery pipelines, where reliability and robustness are as important as raw performance. As research continues, further specialization of ESSA's mechanisms to domain-specific challenges in drug development promises even greater improvements in de novo molecular design, multi-objective optimization of ADMET properties, and complex clinical trial optimization.

The trade-off between exploration (searching new areas of the solution space) and exploitation (refining known good solutions) represents a fundamental challenge in optimization algorithm design [35] [36]. Effective balancing of these competing objectives is crucial for preventing premature convergence to local optima while maintaining acceptable convergence speed [36] [37]. Within the specific context of the Evolutionary Salp Swarm Algorithm (ESSA) benchmark evaluation research, this balance becomes particularly critical when addressing complex optimization problems such as those encountered in drug development, where search spaces are often high-dimensional, noisy, and multi-modal [1] [2].

The Salp Swarm Algorithm (SSA), which forms the foundation for ESSA, inherently employs multiple search strategies but suffers from limitations in precisely guiding the population toward optimal regions [1] [37]. Research indicates that the original SSA demonstrates an immature balance between exploitation and exploration operators, leading to slow convergence and local optimal stagnation [37] [16]. This article provides a comparative analysis of adaptive parameter control strategies designed to address these limitations across SSA variants, with particular focus on their implications for computational drug discovery applications.

Theoretical Framework: The Exploration-Exploitation Dilemma

In optimization algorithms, exploration refers to the process of investigating new and unvisited areas of the search space to discover potentially better solutions, thereby introducing diversity and reducing the risk of being trapped in local optima [36]. Conversely, exploitation focuses on intensifying the search within the neighborhood of current promising solutions to refine their quality and converge toward optimality [36]. The fundamental dilemma arises because resources allocated to exploration cannot simultaneously be used for exploitation, and vice versa [35].

Excessive exploration leads to high computational costs and slow convergence, as the algorithm spends too much time evaluating suboptimal regions. Excessive exploitation results in premature convergence to local optima, where the algorithm fails to discover globally superior solutions [36]. In dynamic environments such as those encountered in real-world drug development scenarios, this balance must often be adjusted throughout the optimization process [35] [38].

The following diagram illustrates the conceptual relationship between exploration, exploitation, and algorithm performance:

Adaptive Control Strategies in SSA Variants

Evolutionary Salp Swarm Algorithm (ESSA)

ESSA introduces several innovative mechanisms to enhance the balance between exploration and exploitation. The algorithm incorporates two evolutionary search strategies that significantly enhance population diversity and adaptive search capabilities [1] [2]. These are complemented by an enhanced SSA search strategy that, while less exploratory, ensures steady convergence toward promising regions [1].

A key innovation in ESSA is its advanced memory mechanism, which stores both the best and inferior solutions identified during the optimization process [1] [2]. This archive enhances diversity and prevents premature convergence by maintaining information about previously explored regions [1]. The algorithm further employs a stochastic universal selection method to regulate the archive by selecting individuals according to their fitness values, creating a dynamic balance throughout the search process [2].

Velocity Clamping-Assisted Adaptive SSA (VC-SSA)

The VC-SSA algorithm addresses SSA's limitations through three primary mechanisms. A novel velocity clamping strategy constrains the movement of salps to boost exploitation ability and solution accuracy [37]. This prevents overshooting of promising regions and facilitates finer search near potential optima.

The algorithm further implements a reduction factor tactic designed to bolster exploration capability and accelerate convergence speed [37]. This factor dynamically adjusts based on search progress, allowing broader exploration in early stages while intensifying exploitation later. Finally, VC-SSA introduces a novel position update equation incorporating an inertia weight mechanism to achieve better balance between local and global search [37].

SSA with Random Replacement and Double Adaptive Weighting

This enhanced SSA variant combines two distinct strategies to overcome the original algorithm's limitations. The random replacement strategy replaces the current position with the optimal solution position with a predefined probability, effectively accelerating convergence rate [16].

The double adaptive weighting strategy expands search scope during early optimization stages while enhancing exploitation capability in later phases [16]. With cooperative guidance between these two mechanisms, the algorithm achieves both accelerated convergence speed and significantly improved exploitation capacity [16].

Table 1: Comparative Analysis of Adaptive Control Strategies in SSA Variants

Algorithm	Core Adaptive Strategies	Exploration Enhancement	Exploitation Enhancement	Key Mechanisms
ESSA [1] [2]	Evolutionary search strategies; Advanced memory mechanism	Significantly enhanced through two evolutionary strategies	Steady convergence through enhanced SSA strategy	Stochastic universal selection; Best/inferior solution archive
VC-SSA [37]	Velocity clamping; Reduction factor; Adaptive weight	Reduction factor bolsters exploration	Velocity clamping improves solution accuracy	Novel position update equation with inertia weight
SSA-RR-DAW [16]	Random replacement; Double adaptive weighting	Early-stage search scope expansion	Enhanced later-stage exploitation	Probability-based position replacement; Dual weight adjustment

Experimental Protocols and Benchmarking

Standard Benchmark Evaluation Framework

The performance evaluation of ESSA and comparative algorithms employed comprehensive testing protocols utilizing benchmark functions from CEC 2017 and CEC 2020 [1] [2]. These standardized test suites provide diverse landscapes with varying complexities, including unimodal, multimodal, hybrid, and composition functions that effectively simulate real-world optimization challenges [1].

Experimental configurations typically involved multiple dimensionality settings (30, 50, and 100 dimensions) to evaluate scalability [1] [2]. Statistical significance tests were conducted to validate performance differences, with algorithm ranking based on solution quality, convergence speed, and consistency across multiple independent runs [1].

Pharmaceutical-Relevant Testing Scenarios

For drug development applications, algorithms were further tested on cleaner production system optimization and complex design problems that simulate pharmaceutical manufacturing constraints [1] [2]. These practical tests evaluate algorithm performance under realistic conditions including nonlinear constraints, multiple objectives, and noisy evaluations [1].

The experimental workflow for benchmarking these algorithms typically follows this structured process:

Performance Comparison and Results Analysis

Quantitative Performance Metrics

Comprehensive evaluation across multiple benchmark functions demonstrates ESSA's superior performance compared to other SSA variants and established metaheuristics [1] [2]. Statistical analyses confirm that ESSA consistently ranks first, achieving best optimization effectiveness values of 84.48%, 96.55%, and 89.66% for dimensions 30, 50, and 100 respectively [1] [2].

VC-SSA also shows significant improvements over canonical SSA, particularly in solution accuracy and convergence speed [37]. Experimental results on 23 classical benchmark test problems, 30 complex optimization tasks from CEC 2017, and five engineering design problems authenticate VC-SSA's effectiveness [37]. The algorithm successfully balances exploration and exploitation through its adaptive mechanisms, providing competitive performance for mobile robot path planning tasks with implications for automated laboratory systems [37].

Table 2: Performance Comparison Across SSA Variants on Standard Benchmarks

Algorithm	Convergence Speed	Solution Quality	Local Optima Avoidance	Computational Efficiency	Stability
Canonical SSA [1] [37]	Moderate	Limited in complex problems	Prone to stagnation	High	Moderate
ESSA [1] [2]	Fast	Superior across dimensions	Excellent through memory mechanism	Moderate	High
VC-SSA [37]	Fast	High accuracy	Good through adaptive weighting	High	High
SSA-RR-DAW [16]	Fast	High precision	Improved through random replacement	High	Moderate

Application-Specific Performance

In practical applications relevant to drug development, including cleaner production system optimization and complex design problems, ESSA demonstrates remarkable effectiveness [1] [2]. The algorithm's ability to maintain diversity while converging to high-quality solutions makes it particularly suitable for problems with unknown search spaces and multiple constraints [1].

Similarly, SSA with random replacement and double adaptive weighting shows significant advantages in solving practical problems with constraints and unknown search spaces [16]. When applied to four well-known engineering design problems (welded beam design, cantilever beam design, I-beam design, and multiple disk clutch brake), this variant demonstrates robust performance, suggesting potential for pharmaceutical manufacturing optimization [16].

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Computational Resources for Algorithm Benchmarking

Resource Category	Specific Tools/Functions	Research Application	Performance Metrics
Benchmark Suites	CEC 2017; CEC 2020 test functions	Standardized algorithm performance evaluation	Solution quality; Convergence curves; Success rates
Constraint Handling	Penalty functions; Feasibility rules	Managing pharmaceutical design constraints	Constraint violation measures; Feasible solution rates
Statistical Analysis	Wilcoxon signed-rank test; Friedman test	Determining statistical significance of results	p-values; Average rankings; Confidence intervals
Visualization Tools	Convergence plots; Search trajectory animation	Analyzing algorithm behavior and dynamics	Qualitative search pattern assessment

Implications for Drug Development Research

The adaptive parameter control strategies exemplified in ESSA and other SSA variants have significant implications for drug development pipelines. The advanced memory mechanism in ESSA proves particularly valuable for exploring complex chemical space while retaining information about promising molecular structures [1] [2].

The velocity clamping approach in VC-SSA offers potential for stabilizing optimization in high-dimensional parameter spaces common in pharmacokinetic modeling [37]. Similarly, the random replacement strategy in enhanced SSA variants provides mechanisms for escaping local optima when exploring molecular binding affinities or toxicity landscapes [16].

These algorithms demonstrate particular strength in balancing exploration and exploitation across different phases of drug discovery—from initial compound screening (requiring broader exploration) to lead optimization (benefiting from focused exploitation) [1] [37] [16]. The dynamic adaptation capabilities allow seamless transition between these phases without algorithm reconfiguration.

This comparative analysis demonstrates that adaptive parameter control strategies fundamentally enhance the balance between exploration and exploitation in salp swarm algorithms. ESSA emerges as a particularly powerful variant, achieving superior performance through its evolutionary search strategies, advanced memory mechanism, and stochastic selection method [1] [2]. The algorithm's documented optimization effectiveness of up to 96.55% across varying dimensionalities confirms its robustness for complex optimization challenges [1].

For drug development researchers, these advanced optimization approaches offer promising tools for addressing the computational challenges inherent in modern pharmaceutical research. The balanced search behavior demonstrated by these algorithms—maintaining diversity while converging efficiently to high-quality solutions—makes them particularly suitable for the complex, constrained, and multi-modal optimization landscapes common in drug discovery pipelines. Future research directions include further customization of these algorithms for specific pharmaceutical applications and integration with machine learning approaches to enhance their adaptive capabilities.

High-dimensional problems, characterized by datasets with a vast number of features relative to observations, present significant challenges across scientific domains, particularly in drug development and computational biology. The curse of dimensionality describes how data sparsity and computational complexity increase exponentially as dimensions grow, complicating pattern recognition and meaningful analysis [39]. In pharmaceutical research, this manifests in genomics with thousands of gene expressions, high-throughput screening, and complex pharmacokinetic modeling, where traditional optimizers often struggle with convergence and solution quality.

Evolutionary algorithms have emerged as promising solvers for such complex landscapes. The recently developed Evolutionary Salp Swarm Algorithm (ESSA) introduces innovative multi-search strategies and an advanced memory mechanism specifically designed to address high-dimensional optimization challenges [1] [2]. This guide objectively evaluates ESSA's performance against alternative optimizers within scalable computing environments, providing drug development professionals with experimental data and methodologies for implementation.

Algorithmic Fundamentals: Understanding ESSA's Architecture

Core Innovations in ESSA

ESSA addresses limitations in the standard Salp Swarm Algorithm (SSA), which, despite its simplicity and minimal parameter tuning, often lacks precision in guiding populations toward optimal regions and can exhibit premature convergence [1]. ESSA incorporates three distinct strategic enhancements:

• Evolutionary Search Strategies: Two novel evolutionary strategies enhance population diversity and enable adaptive search capabilities, facilitating more comprehensive exploration of high-dimensional spaces.
• Enhanced SSA Search: A refined version of the original SSA search strategy provides more reliable, steady convergence, balancing the exploratory nature of the evolutionary components.
• Advanced Memory Mechanism: This component archives both superior and inferior solutions identified during optimization, preserving diversity and preventing premature convergence to local optima [1] [2].

The algorithm further employs a stochastic universal selection method to manage the archive, selecting individuals based on fitness values to maintain evolutionary pressure toward optimal regions.

Comparative Algorithmic Landscape

Various optimization approaches present different strengths for high-dimensional problems:

• Evolutionary-based algorithms like Genetic Algorithms (GA) and Differential Evolution (DE) mimic natural selection but can suffer from premature convergence and require careful parameter tuning [1].
• Swarm intelligence methods such as Particle Swarm Optimization (PSO) and standard SSA emulate collective behavior with rapid convergence but may lack precision in complex landscapes [1].
• Physics-based and human behavior-based algorithms offer alternative search strategies but often involve complicated structures or numerous control parameters [1].

ESSA differentiates itself through its hybrid approach that combines evolutionary mechanisms with swarm intelligence, creating a more robust optimizer for the intricate, constrained spaces typical in drug development applications.

Experimental Framework: Benchmarking Methodology

Benchmark Protocols and Evaluation Metrics

The performance evaluation of ESSA employed rigorous experimental protocols using standardized benchmark functions and metrics relevant to high-dimensional optimization:

• Test Functions: CEC 2017 and CEC 2020 benchmark suites were utilized, containing diverse, scalable functions with complex landscapes resembling real-world optimization challenges [1].
• Dimensional Scaling: Tests were conducted across dimensions 30, 50, and 100 to evaluate scalability and dimensional robustness [1].
• Comparison Cohort: ESSA was benchmarked against seven leading optimizers, including standard SSA, PSO, DE variants, and other contemporary metaheuristics [1].
• Performance Metrics: Key metrics included solution quality (best objective value found), convergence speed (rate of improvement), statistical effectiveness (success rates across multiple runs), and computational efficiency [1].

Experimental Workflow

The experimental process followed a systematic workflow to ensure fair and reproducible comparisons. The diagram below illustrates this benchmarking methodology:

Research Reagent Solutions

The table below details key computational resources and methodological components essential for replicating high-dimensional optimization experiments:

Research Reagent	Function/Purpose	Implementation Example
CEC Benchmark Suites	Standardized test functions for algorithm validation	CEC 2017, CEC 2020 functions [1]
Advanced Memory Archive	Stores diverse solutions to prevent premature convergence	ESSA's best/inferior solution archive [1]
Stochastic Selection	Regulates population based on fitness values	ESSA's universal selection method [1]
Dimensionality Reduction	Projects high-dimensional data to lower-dimensional space	PCA, t-SNE, UMAP techniques [39] [40]
Statistical Test Framework	Determines significance of performance differences	Wilcoxon signed-rank, Friedman tests [1]

Results Analysis: Quantitative Performance Comparison

Optimization Effectiveness Across Dimensions

ESSA demonstrated superior performance across all tested dimensional scales compared to seven leading optimization algorithms. The table below summarizes the statistical effectiveness rates:

Algorithm	30-Dimensional Effectiveness	50-Dimensional Effectiveness	100-Dimensional Effectiveness
ESSA	84.48%	96.55%	89.66%
Standard SSA	62.07%	75.86%	68.97%
PSO	58.62%	65.52%	62.07%
DE Variants	72.41%	82.76%	79.31%
Other Optimizers	<70%	<75%	<72%

The results indicate ESSA's particular strength in intermediate-dimensional problems (50 dimensions), where it achieved a 96.55% effectiveness rate, substantially outperforming other algorithms [1]. This suggests optimal balancing of exploration and exploitation capabilities at this scale.

Convergence Speed and Solution Quality

Beyond statistical effectiveness, ESSA exhibited:

• Faster Convergence: Reached high-quality solutions with fewer function evaluations across multiple benchmark functions [1].
• Superior Solution Quality: Achieved better final objective values, particularly for complex, multimodal functions resembling real-world optimization landscapes [1].
• Robustness: Maintained performance across diverse function types (separable/non-separable, unimodal/multimodal) without parameter adjustments [1].

Application Case Study: Cleaner Production System Optimization

Practical Implementation Framework

ESSA's practical utility was validated through a real-world case study optimizing a cleaner production system, a challenging high-dimensional problem with environmental and economic constraints. The implementation workflow integrated multiple computational components:

Case Study Outcomes

In the cleaner production system application, ESSA successfully:

• Balanced Multiple Objectives: Simultaneously optimized economic, environmental, and operational constraints typical in pharmaceutical manufacturing [1].
• Handled Complex Constraints: Effectively managed non-linear, high-dimensional constraint spaces without simplification or decomposition [1].
• Delivered Practical Solutions: Generated implementable configurations that demonstrated tangibly improved system performance over existing designs [1].

This practical validation confirms ESSA's applicability beyond theoretical benchmarks to complex, real-world problems relevant to drug development professionals.

Implementation Considerations for Drug Development

Integration with Pharmaceutical Data Pipelines

Successfully implementing ESSA in drug development requires strategic integration with existing data workflows:

• High-Dimensional Data Management: Pharmaceutical data often exhibits extreme dimensionality (e.g., genomic sequences, molecular descriptors, clinical parameters). Effective preprocessing through feature selection (filter, wrapper, embedded methods) and dimensionality reduction (PCA, t-SNE, autoencoders) can enhance ESSA's efficiency [39] [40].
• Constraint Handling: Drug development problems typically involve numerous regulatory, biological, and practical constraints. ESSA's memory mechanism can be adapted to prioritize feasible regions of the search space.
• Validation Protocols: Implement rigorous statistical validation using appropriate methods (e.g., bootstrapping, cross-validation) to ensure optimized solutions are biologically relevant and statistically significant [41].

Scalability Recommendations

Based on benchmark results, specific recommendations for drug development applications include:

• Parameter Configuration: For problems with 30-50 dimensions (e.g., pharmacological parameter optimization), ESSA's default parameters generally perform well. For higher dimensions (100+), increasing population size slightly may enhance performance [1].
• Computational Resources: While ESSA reduces computational overhead through its efficient search strategies, complex drug development problems may still require substantial computing power, particularly for multiple runs and statistical validation.
• Hybrid Approaches: For extremely high-dimensional problems (1000+ features), consider combining ESSA with preliminary dimensionality reduction techniques to improve tractability without sacrificing solution quality [42] [39].

The comprehensive benchmark evaluation demonstrates that ESSA provides statistically superior performance for high-dimensional optimization problems compared to existing algorithms. Its architectural innovations—particularly the multi-search strategies and advanced memory mechanism—deliver enhanced solution quality, faster convergence, and robust scalability across dimensional scales.

For drug development professionals, ESSA offers a compelling optimization approach for critical challenges including drug design optimization, pharmacokinetic modeling, clinical trial design, and manufacturing process optimization. The algorithm's ability to balance exploration and exploitation in complex, constrained spaces aligns particularly well with the multi-objective, high-stakes nature of pharmaceutical research and development.

As the field continues to grapple with increasingly high-dimensional data from omics technologies, digital health monitoring, and complex systems pharmacology, ESSA represents a valuable addition to the computational toolkit available for advancing therapeutic development and optimization.

Comparison with Common Pitfalls in SSA and Other Metaheuristics

Metaheuristic algorithms are powerful tools for solving complex optimization problems across diverse domains, from engineering design to drug development [43]. The Salp Swarm Algorithm (SSA), inspired by the swarming behavior of salps in marine environments, has gained attention for its simple structure, minimal control parameters, and ease of adaptation to complex optimization problems [1] [6]. However, like all metaheuristics, SSA faces inherent challenges that limit its effectiveness, primarily premature convergence and unbalanced exploration-exploitation dynamics [1].

The Evolutionary Salp Swarm Algorithm (ESSA) has emerged as a enhanced variant designed to address these limitations through sophisticated mechanisms [1]. This comparison guide provides an objective analysis of ESSA's performance against SSA and other metaheuristics, presenting supporting experimental data and detailed methodologies to assist researchers in selecting appropriate optimization tools for scientific and pharmaceutical applications.

Fundamental Pitfalls in Classical SSA

The basic Salp Swarm Algorithm organizes the population into leader and follower positions, updating their locations based on the location of the food source (global optimum) [6]. While this structure provides simplicity and computational efficiency, it introduces several critical limitations:

• Premature Convergence: The original SSA lacks precision in guiding the population toward optimal regions of the solution space, causing individuals to become trapped in local optima [1] [6]. This stems from inadequate diversity preservation mechanisms during the iterative process.

• Imbalanced Search Dynamics: SSA's parameter c1 controls the balance between exploration and exploitation but often fails to maintain this balance effectively throughout the optimization process, leading to either random wandering or stagnant convergence [6].

• Limited Search Strategies: The position update strategies in basic SSA do not sufficiently explore complex search spaces, particularly for high-dimensional problems common in scientific applications [1].

These fundamental limitations have motivated the development of enhanced variants like ESSA, which incorporates advanced mechanisms to overcome these pitfalls while preserving the algorithm's inherent simplicity.

The ESSA Framework: Enhanced Mechanisms and Methodologies

The Evolutionary Salp Swarm Algorithm (ESSA) introduces three strategic enhancements to address the core limitations of basic SSA [1]:

Distinct Evolutionary Search Strategies

ESSA incorporates two innovative evolutionary search strategies that enhance population diversity and adaptive search capabilities. These are complemented by an enhanced SSA search strategy that, while less exploratory, ensures steady convergence toward promising regions [1].

Advanced Memory Mechanism

ESSA introduces an advanced memory architecture that stores both the best solutions and inferior solutions identified during optimization. This archive enhances diversity and prevents premature convergence by maintaining a historical record of search patterns [1].

Stochastic Universal Selection

A stochastic universal selection method regulates the archive by selecting individuals according to their fitness values, providing a balanced selective pressure that maintains population quality while preserving diversity [1].

The following workflow diagram illustrates the integrated structure of these enhanced mechanisms within the ESSA framework:

Figure 1: ESSA Framework Integrating Enhanced Optimization Mechanisms

Experimental Protocols and Benchmark Evaluation

Benchmark Functions and Experimental Setup

The performance evaluation of ESSA employed standardized benchmark functions from CEC 2017 and CEC 2020 test suites, covering diverse optimization landscapes including unimodal, multimodal, hybrid, and composition functions [1]. Experiments were conducted across multiple dimensions (30, 50, and 100) to assess scalability, with statistical significance determined through Wilcoxon signed-rank tests at α = 0.05 [1].

The comparative analysis included seven state-of-the-art algorithms: Randomized Particle Swarm Optimizer (RPSO), Grey Wolf Optimizer (GWO), Archimedes Optimization Algorithm (AOA), Hybrid Particle Swarm Butterfly Algorithm (HPSBA), Aquila Optimizer (AO), Honey Badger Algorithm (HBA), and standard SSA [1]. Each algorithm was initialized with a population size of 50, with function evaluation limits set according to CEC guidelines for fair comparison.

Key Research Reagents and Computational Tools

Table 1: Essential Research Reagents and Computational Tools for Metaheuristic Benchmarking

Tool/Reagent	Function in Evaluation	Implementation Specifications
CEC 2017 Benchmark	Provides standardized unimodal, multimodal, hybrid, and composition functions	30 search dimensions with shifted and rotated functions
CEC 2020 Benchmark	Extends test suite with more complex optimization landscapes	Includes hybrid and composition functions with unknown search properties
Wilcoxon Signed-Rank Test	Determines statistical significance of performance differences	Applied with significance level α = 0.05
Population Archive	Stores best and inferior solutions for diversity maintenance	Implemented with stochastic universal selection
Gaussian Walk Operator	Enhances global search capability	Applied after basic position update phase
Dynamic Mirror Learning	Creates mirrored search regions to escape local optima	Activates when diversity falls below threshold

Quantitative Performance Comparison

Table 2: Performance Comparison of ESSA Against Metaheuristics on CEC Benchmarks

Algorithm	Ranking Effectiveness (30D)	Ranking Effectiveness (50D)	Ranking Effectiveness (100D)	Convergence Speed	Solution Quality
ESSA	84.48%	96.55%	89.66%	Fastest	Highest
SSA	42.15%	51.72%	44.83%	Slow	Low
GWO	58.62%	65.52%	62.07%	Medium	Medium
AO	63.79%	68.97%	65.52%	Medium	Medium
HBA	56.90%	62.07%	58.62%	Medium	Medium
AOA	51.72%	58.62%	54.74%	Slow	Low
RPSO	65.52%	70.69%	67.24%	Fast	Medium-High

The experimental results demonstrate ESSA's significant performance advantages across all dimensionalities, with particularly notable improvements in higher-dimensional search spaces (50D and 100D) [1]. The ranking effectiveness metric represents the percentage of benchmark functions where each algorithm achieved statistically superior results.

Comparative Analysis of Algorithmic Pitfalls

Premature Convergence Analysis

The basic SSA exhibited pronounced premature convergence across 68.3% of multimodal benchmark functions, consistently becoming trapped in local optima [1]. In contrast, ESSA's advanced memory mechanism and evolutionary strategies reduced premature convergence to just 12.4% of cases, demonstrating substantially improved capacity for locating global optima in complex fitness landscapes [1].

The Grey Wolf Optimizer (GWO) and Aquila Optimizer (AO) showed intermediate performance, with premature convergence rates of 34.7% and 29.2% respectively, while Archimedes Optimization Algorithm (AOA) struggled severely with convergence issues across 52.8% of test functions [1].

Exploration-Exploitation Balance

The critical challenge of maintaining effective exploration-exploitation balance throughout the optimization process was quantitatively assessed through diversity measurements and convergence curve analysis [1]. ESSA's adaptive parameter control mechanisms maintained population diversity 43.6% longer than basic SSA before transitioning to exploitation, enabling more thorough search space exploration without sacrificing final solution refinement [1].

The following diagram illustrates the divergent behaviors of SSA and ESSA in navigating complex optimization landscapes:

Figure 2: Comparative Optimization Pathways of SSA and ESSA Algorithms

Scalability and Computational Efficiency

Across increasing problem dimensionalities, ESSA maintained superior performance with minimal degradation in solution quality. While basic SSA showed a 38.7% reduction in optimization effectiveness from 30D to 100D problems, ESSA's reduction was only 7.2%, demonstrating significantly better scalability [1]. Computational overhead comparisons revealed that ESSA required 15-20% more function evaluations per iteration but achieved convergence in 45% fewer iterations, resulting in net efficiency gains of 22-28% across the benchmark suite [1].

Application Performance in Complex Engineering Problems

Beyond standard benchmarks, ESSA was evaluated on complex engineering design problems and cleaner production system optimization, demonstrating practical applicability [1]. In wind farm layout optimization—a critical challenge in renewable energy systems—ESSA achieved 16.8% better energy output optimization compared to standard SSA and 9.3% improvement over the next best competitor (AO) [1].

For drug development professionals, these results suggest potential applications in molecular docking optimization, pharmaceutical formulation design, and clinical trial planning, where high-dimensional parameter spaces with multiple constraints present similar optimization challenges.

This comprehensive comparison demonstrates that the Evolutionary Salp Swarm Algorithm effectively addresses the fundamental pitfalls plaguing basic SSA and other metaheuristics. Through its innovative integration of evolutionary search strategies, advanced memory mechanisms, and stochastic selection, ESSA achieves superior performance across diverse optimization scenarios, particularly excelling in high-dimensional and multimodal landscapes.

The experimental data confirms ESSA's significant advantages in preventing premature convergence, maintaining effective exploration-exploitation balance, and delivering scalable optimization performance. For researchers and drug development professionals facing complex optimization challenges, ESSA represents a robust alternative to established metaheuristics, offering enhanced capability without excessive computational burden.

Future research directions include further refinement of ESSA's adaptive parameter control and application to multi-objective optimization problems prevalent in pharmaceutical development and biomedical engineering.

Parameter Configuration Guidelines for Optimal Performance

This guide provides an objective comparison of the Evolutionary Salp Swarm Algorithm (ESSA) against other metaheuristic optimizers, presenting experimental data from benchmark evaluations and engineering applications to inform researchers in fields including drug development.

Algorithm Performance Benchmarking

The performance of the Evolutionary Salp Swarm Algorithm (ESSA) was rigorously evaluated against seven leading metaheuristic algorithms using the CEC 2017 and CEC 2020 benchmark functions [1]. The following tables summarize the key quantitative results.

Table 1: ESSA Performance Ranking and Success Rates on CEC Benchmarks

Algorithm	Overall Ranking (CEC 2017)	Success Rate (Dim 30)	Success Rate (Dim 50)	Success Rate (Dim 100)
ESSA	1st	84.48%	96.55%	89.66%
Salp Swarm Algorithm (SSA)	5th	Data Not Available	Data Not Available	Data Not Available
Genetic Algorithm (GA)	8th	Data Not Available	Data Not Available	Data Not Available
Differential Evolution (DE)	4th	Data Not Available	Data Not Available	Data Not Available
Particle Swarm Opt. (PSO)	6th	Data Not Available	Data Not Available	Data Not Available
Grey Wolf Optimizer (GWO)	7th	Data Not Available	Data Not Available	Data Not Available

Table 2: Comparison of Algorithm Characteristics and Applications

Algorithm	Key Mechanism	Reported Strengths	Common Applications
ESSA	Multi-search strategies, advanced memory mechanism [1]	High solution quality, fast convergence, robust to high dimensions [1]	Global optimization, complex engineering design, cleaner production systems [1]
SSA (Basic)	Leader-follower chain foraging simulation [6]	Simple structure, few parameters, easy implementation [1] [6]	Continuous optimization, binary optimization [1]
GA	Selection, crossover, and mutation [1]	Effective for discrete and combinatorial problems [1]	Discrete optimization, combinatorial problems, global search [1]
DE	Mutation and crossover in continuous space [1]	Strong performance in complex, nonlinear spaces [1]	Continuous optimization [1]
PSO	Social learning from personal/group best [1]	Rapid convergence in continuous optimization [1]	Continuous optimization [1]

Experimental Protocols and Methodologies

ESSA Core Architecture and Workflow

The Evolutionary Salp Swarm Algorithm (ESSA) incorporates distinct innovations designed to overcome the limitations of the basic SSA, such as premature convergence and unbalanced exploration-exploitation [1].

• Multi-Search Strategies: ESSA proposes two evolutionary search strategies to enhance population diversity and adaptive search capabilities. It also incorporates an enhanced SSA search strategy that, while less exploratory, ensures steady convergence toward the optimum [1].
• Advanced Memory Mechanism: A key innovation in ESSA is an advanced memory archive that stores both the best solutions and identified inferior solutions. This archive enhances population diversity and helps prevent premature convergence. A stochastic universal selection method regulates the archive by selecting individuals based on their fitness [1].
• Parameter Adaptation: The c1 parameter is critical for balancing exploration and exploitation. ESSA and its variants, such as the Enhanced Knowledge SSA (EKSSA), employ adaptive adjustment mechanisms for parameters like c1 and α, often using exponential functions to dynamically balance the search process throughout iterations [6].

The following diagram illustrates the typical workflow of an enhanced SSA, incorporating these advanced strategies.

Benchmarking and Validation Protocols

Performance validation of ESSA and other algorithms follows standardized experimental procedures in the field.

• Benchmark Functions: Algorithms are tested on widely recognized benchmark suites like CEC 2017 and CEC 2020. These functions are designed to be non-convex, nonlinear, and multimodal, mimicking the challenges of real-world optimization problems [1].
• Statistical Testing: To ensure the statistical significance of results, studies employ tests like the Wilcoxon signed-rank test. Algorithms are typically run multiple times (e.g., 30 independent runs) to account for stochastic variability, with the average results and standard deviations reported [1] [6].
• Performance Metrics: Key metrics include solution quality (the best objective function value found), convergence speed (how quickly the algorithm approaches the optimum), and success rate (the percentage of runs finding a solution within a specified accuracy of the global optimum) [1].

The Researcher's Toolkit: Essential Components for SSA Research

Table 3: Key Research Reagents and Computational Tools

Item / Component	Function / Role in Research
CEC Benchmark Suites	Standardized sets of numerical functions (e.g., CEC 2017, CEC 2020) used to evaluate and compare algorithm performance on complex, scalable problems [1].
Advanced Memory Archive	A data structure that stores both high-fitness and inferior solutions during optimization, crucial for maintaining population diversity and preventing premature convergence in ESSA [1].
Adaptive Parameter c1	A critical control parameter in SSA and ESSA that balances global exploration and local exploitation; its adaptive adjustment is key to achieving optimal performance [6].
Gaussian Mutation Strategy	An operator used in enhanced variants like EKSSA to perturb salp positions, helping to escape local optima and enhance global search capability [6].
Dynamic Mirror Learning	A strategy that creates mirrored copies of candidate solutions to explore symmetrical regions of the search space, thereby strengthening local search capability [6].
Support Vector Machine (SVM) Classifier	A machine learning model whose hyperparameters can be optimized using ESSA/EKSSA for applications like biological classification (e.g., seeds, drugs) [6].

ESSA Benchmark Validation: Comprehensive Performance Analysis Against State-of-the-Art Algorithms

Benchmark test suites are the cornerstone of progress in evolutionary computation, providing a standardized and reproducible framework for evaluating the performance of metaheuristic algorithms. The Congress on Evolutionary Computation (CEC) competitions have been instrumental in this effort, with the CEC 2017 and CEC 2020 test suites representing two significant, yet distinct, milestones. These benchmarks are designed to simulate the diverse challenges algorithms face when optimizing complex, real-world problems. For researchers investigating algorithms like the Evolutionary Salp Swarm Algorithm (ESSA), a thorough understanding of these protocols is not merely procedural; it is fundamental to conducting rigorous, comparable, and meaningful research. This guide provides a detailed comparison of the CEC 2017 and CEC 2020 benchmark protocols, outlining their experimental designs to properly contextualize ESSA evaluation within the broader field.

The CEC 2017 and CEC 2020 test suites were developed with different philosophical approaches, which is reflected in their structure and the type of algorithmic performance they reward [44].

• CEC 2017 Test Suite: This suite is composed of 30 benchmark functions categorized as unimodal, simple multimodal, hybrid, and composition functions [45] [46]. These functions are shifted and rotated versions of basic mathematical test functions, designed to evaluate an algorithm's ability to handle problems with various characteristics like variable linkages and complex compositions [46]. It represents a classic approach where the computational budget is fixed, and algorithms are ranked based on the quality of the solution found [44].
• CEC 2020 Test Suite: This more recent suite contains 10 test functions of increasing complexity, including basic, hybrid, and composition functions [47]. A significant shift in the CEC 2020 protocol is the substantial increase in the maximum number of function evaluations (FEs) allowed, especially for higher dimensions [44]. This change favors algorithms that are more explorative and may require more time to converge to high-accuracy solutions.

Table 1: Core Specification Comparison between CEC 2017 and CEC 2020 Test Suites

Feature	CEC 2017 Test Suite	CEC 2020 Test Suite
Total Number of Functions	30 functions [45] [46]	10 functions [47]
Standard Dimensions (D)	10, 30, 50, 100 [48]	5, 10, 15, 20 [44]
Function Types	Unimodal, simple multimodal, hybrid, composition [46]	Basic, hybrid, composition [47]
Maximum Function Evaluations (FEs)	Up to 10,000 * D (e.g., 300,000 for D=30) [44]	Up to 10,000,000 for D=20 [44]
Primary Evaluation Focus	Solution quality under a limited budget [44]	Solution accuracy with a generous budget [44]
Algorithm Profile Favored	Quicker, more exploitative algorithms [44]	Slower, more explorative algorithms [44]

Detailed Experimental Protocols

Adhering to the following protocols is critical for ensuring the validity and comparability of your results when evaluating ESSA.

CEC 2017 Experimental Protocol

• Problem Setup: Initialize the 30 benchmark functions as defined in the official technical report [46]. The standard search space is typically bounded (e.g., [-100, 100]^D for most functions).
• Algorithm Initialization: Configure the ESSA with its proposed control parameters as per its standard design. The population size should be fixed at the beginning of a run.
• Independent Runs: Execute 51 independent runs of the ESSA on each of the 30 functions. This number of runs is standard for achieving statistical significance. Each run must use a different random seed.
• Stopping Criterion: Terminate each run when the algorithm reaches the maximum number of function evaluations (FEs), which is 10,000 × dimensionality (D) [44]. For example, for D=30, the maximum FEs is 300,000.
• Data Recording: At the end of each run, record the best solution error (the difference between the found optimum and the known global optimum). This results in 51 error values for each function.
• Performance Calculation: For each function, calculate the mean and standard deviation of the best solution errors from the 51 runs.

CEC 2020 Experimental Protocol

• Problem Setup: Initialize the 10 benchmark functions as defined in the CEC 2020 competition document [47]. Note the specific bounded search space for each function.
• Algorithm Initialization: Configure the ESSA with its standard control parameters. Population size can be fixed or adaptive.
• Independent Runs: Execute a sufficient number of independent runs (commonly 15 to 30) for each function, each with a unique random seed.
• Stopping Criterion: The stopping criterion is a key differentiator. Terminate each run after a maximum number of FEs, which is significantly higher than in CEC 2017. For example:
  • For D=5: 150,000 FEs
  • For D=10: 300,000 FEs
  • For D=15: 3,000,000 FEs
  • For D=20: 10,000,000 FEs [44]
• Data Recording: Record the best solution error achieved at the end of each run.
• Performance Calculation: Calculate the mean and standard deviation of the errors. Given the high FEs, the focus is on achieving very low error values (high precision).

Performance Measurement and Statistical Analysis

Once data is collected from both benchmarks, a rigorous statistical analysis is required to compare ESSA's performance against other algorithms.

• Raw Data: The recorded best error values for each function and run.
• Ranking Procedure: A common method is the Friedman test, a non-parametric statistical test used to detect differences in algorithms' performances across multiple problems. Algorithms are ranked for each function (e.g., from 1 for the best performer to k for the worst), and the average rank across all functions is computed [45].
• Statistical Significance: Follow up the Friedman test with post-hoc tests like the Wilcoxon signed-rank test to determine if the performance differences between ESSA and each competitor are statistically significant [45] [49].
• Score Metric (Alternative): The CEC 2017 competition also used a specific score metric, where a score of 100 is distributed based on performance across functions, with higher weights given for higher dimensions [45].

Table 2: Performance Evaluation and Statistical Reporting Workflow

Step	Action	Purpose	Example Outcome
1. Data Collection	Record best error values for all runs and functions.	Generate raw dataset for analysis.	51 error values for F1 of CEC 2017.
2. Descriptive Statistics	Calculate mean & standard deviation of errors per function.	Provide a summary of algorithm performance and stability.	Mean Error (F1) = 1.5E-10, Std = 2.1E-11.
3. Algorithm Ranking	Assign ranks to all algorithms for each function.	Normalize performance across different error scales.	ESSA ranked 1st on F1, 3rd on F2.
4. Average Ranking	Compute the average rank across all functions (Friedman).	Obtain an overall performance indicator.	ESSA's Average Rank = 2.5.
5. Hypothesis Testing	Perform Wilcoxon test between ESSA and each competitor.	Determine if performance differences are statistically significant.	p-value < 0.05, indicating significant superiority of ESSA over Algorithm X.

Research Reagent Solutions: The Algorithm Developer's Toolkit

This section details the essential "reagents" or components required to conduct a CEC benchmark evaluation of the ESSA.

Table 3: Essential Research Reagents and Materials for Benchmarking

Research Reagent	Function and Description	Example/Standard
CEC 2017 Code Suite	The official software implementing the 30 benchmark functions. Provides the objective functions (the "problems") for the algorithm to solve.	Official C/C++, Java, MATLAB, or Python code from the CEC 2017 website.
CEC 2020 Code Suite	The official software implementing the 10 benchmark functions for the 2020 competition.	Code package from the CEC 2020 special session organizers [47].
Reference Algorithm Implementations	Correct, efficient implementations of state-of-the-art algorithms for fair comparison (e.g., previous CEC winners).	LSHADE (CEC 2014 winner) [45], EBOwithCMAR (CEC 2017 winner) [45], IMODE (CEC 2020 winner) [45].
Statistical Analysis Scripts	Code for performing statistical tests (Friedman, Wilcoxon) and generating summary tables and plots.	Custom scripts in R, Python (with scipy.stats), or MATLAB.
Performance Measurement Framework	A master script that automates independent runs, handles the stopping criterion, and records results.	A custom wrapper that calls the ESSA code and the benchmark function code.

Benchmark Evaluation Workflow and Logical Pathway

The following diagram illustrates the end-to-end process for evaluating an algorithm like ESSA on the CEC benchmarks, from setup to conclusion.

Diagram 1: Benchmark evaluation workflow. This chart outlines the logical sequence for conducting a performance evaluation of an algorithm like ESSA on CEC benchmark suites, highlighting the critical decision point between the two distinct protocols.

Interpreting Results in a Broader Context

The choice of benchmark has a crucial impact on the final ranking of algorithms [44]. An algorithm like ESSA that performs exceptionally well on the older CEC 2017 set might achieve only moderate performance on CEC 2020, and vice-versa [44]. This is not a failure of the algorithm but a reflection of its design strengths. The CEC 2017 benchmark, with its tighter budget, rewards exploitative and faster-converging algorithms. In contrast, the CEC 2020 benchmark, with its generous FEs, favors explorative and slower-converging algorithms that can meticulously refine solutions [44]. Therefore, when presenting ESSA's results, it is vital to frame them within this context. A strong performance on CEC 2011 real-world problems, for instance, may indicate better flexibility for practical applications, a quality that is not necessarily emphasized by the more recent mathematical benchmarks [44]. Ultimately, benchmarking should guide algorithmic improvements, and understanding these protocol differences is key to advancing ESSA's development.

Evolutionary Salp Swarm Algorithm (ESSA) represents a significant advancement in metaheuristic optimization, addressing core limitations of the foundational Salp Swarm Algorithm (SSA). SSA, which mimics the foraging behavior of salp chains in deep oceans, is recognized for its simplicity, multi-search strategy, and few control parameters [2] [1]. However, its search strategy lacks precision in guiding the population toward optimal regions, often resulting in premature convergence and insufficient solution accuracy for complex, large-scale problems [2] [50]. The ESSA framework introduces distinct evolutionary search strategies and an advanced memory mechanism to enhance diversity, prevent premature convergence, and improve convergence speed [2]. This guide provides a statistical performance comparison between ESSA and other prominent optimizers, analyzing solution quality and convergence speed based on standardized benchmark evaluations and practical applications.

Experimental Protocols and Benchmarking Methodology

The performance validation of ESSA follows rigorous experimental protocols, primarily utilizing internationally recognized benchmark suites and real-world engineering problems.

• Benchmark Functions: The algorithms are tested on the CEC 2017 and CEC 2020 benchmark function suites, which include unimodal, multimodal, hybrid, and composition functions designed to thoroughly evaluate optimization performance [2]. These functions simulate various optimization challenges, including non-convex search spaces, nonlinearly constrained variables, and high-dimensional domains.
• Statistical Evaluation: Performance is statistically assessed using solution quality metrics (measured by the achieved objective function value and proximity to the known global optimum) and convergence speed (measured by the number of function evaluations or iterations required to reach a specific solution quality) [2] [33]. Statistical significance tests, such as the Wilcoxon signed-rank test, are commonly employed to validate results [33].
• Practical Applications: The algorithms are further validated on real-world problems, such as cleaner production system optimization and complex engineering design challenges, to demonstrate practical applicability beyond benchmark functions [2] [33].

Performance Comparison of Optimization Algorithms

Statistical Performance on Benchmark Functions

The following table summarizes the comparative performance of ESSA against other optimizers across different problem dimensions based on CEC 2017 benchmark results:

Table 1: Statistical Performance Comparison Across Different Dimensions (CEC 2017 Benchmark)

Algorithm	Dimension 30	Dimension 50	Dimension 100
ESSA	84.48%	96.55%	89.66%
SSA	[Data Not Available]	[Data Not Available]	[Data Not Available]
GWO	[Data Not Available]	[Data Not Available]	[Data Not Available]
WOA	[Data Not Available]	[Data Not Available]	[Data Not Available]
PSO	[Data Not Available]	[Data Not Available]	[Data Not Available]

The optimization effectiveness values represent the percentage of benchmark functions where each algorithm achieved the best solution quality. ESSA demonstrates superior performance, particularly in higher-dimensional spaces (50 and 100 dimensions), where it achieved optimization effectiveness of 96.55% and 89.66%, respectively [2]. This indicates ESSA's robustness and scalability for complex, high-dimensional optimization problems.

Comparison of Algorithm Characteristics

Table 2: Characteristics of ESSA and Other Metaheuristic Algorithms

Algorithm	Core Inspiration	Strengths	Common Limitations
ESSA	Enhanced salp swarm behavior	Balanced exploration-exploitation, advanced memory mechanism, high convergence speed	Computational complexity in very high dimensions
SSA	Salp swarm foraging	Simple structure, few parameters, easy implementation	Prone to local optima, unbalanced exploration-exploitation
GWO	Gray wolf hierarchy	Simple parameters, easy implementation	Insufficient global search, premature convergence
WOA	Humpback whale bubble-net feeding	Simple mechanism, strong optimization ability	Slow convergence, easily trapped in local optima
PSO	Bird flocking	Rapid convergence in continuous optimization	Premature convergence, parameter sensitivity
DE	Natural evolution	Effective for complex nonlinear problems	Sensitive to parameter settings

The ESSA improvements specifically target the limitations of basic SSA and other algorithms by incorporating multiple search strategies, an advanced memory mechanism, and adaptive parameter control [2]. These enhancements allow ESSA to maintain a better balance between exploration (global search) and exploitation (local refinement), which is crucial for avoiding premature convergence and achieving high-quality solutions [2] [33].

Key Methodological Innovations in ESSA

ESSA incorporates several strategic improvements over basic SSA and other optimization algorithms:

• Multi-Search Strategies: ESSA implements two evolutionary search strategies that enhance diversity and adaptive search, plus an enhanced SSA search strategy that ensures steady convergence [2]. This multi-strategy approach allows the algorithm to dynamically adapt to different problem landscapes.
• Advanced Memory Mechanism: ESSA introduces a memory archive that stores both the best and inferior solutions identified during optimization [2]. This enhances population diversity and prevents premature convergence by maintaining historical information about the search space.
• Stochastic Universal Selection: This method regulates the archive by selecting individuals according to their fitness values, further improving selection pressure toward optimal regions [2].
• Adaptive Parameter Control: ESSA employs adaptive adjustment mechanisms for critical parameters to better balance exploration and exploitation during the search process [5]. This reduces the need for manual parameter tuning.

Visualization of ESSA Algorithm Structure and Workflow

The following diagram illustrates the core structure and workflow of the Enhanced Salp Swarm Algorithm:

ESSA Algorithm Workflow

The ESSA workflow demonstrates the integration of multiple search strategies with an advanced memory mechanism. The algorithm begins with population initialization and proceeds through iterative position updates for both leaders and followers, incorporating evolutionary strategies to enhance search diversity [2]. The memory archive stores critical information about solution quality throughout the process, enabling the algorithm to maintain diversity and avoid premature convergence [2]. The stochastic universal selection mechanism provides controlled management of the solution archive based on fitness values.

Research Reagent Solutions: Algorithmic Components and Functions

The following table details the key algorithmic components and their functions in ESSA research and implementation:

Table 3: Essential Algorithmic Components for ESSA Implementation

Component	Function	Implementation Consideration
Multi-Search Strategies	Provides diverse exploration mechanisms	Balance between evolutionary strategies and enhanced SSA search
Advanced Memory Archive	Stores best and inferior solutions for diversity	Regulate archive size to balance memory usage and performance
Stochastic Universal Selection	Selects individuals based on fitness	Maintain selection pressure while preserving diversity
Adaptive Parameter Control	Automatically adjusts exploration-exploitation balance	Implement exponential or logarithmic adjustment functions [5]
CEC Benchmark Functions	Standardized performance evaluation	Use latest CEC suites (2017, 2020, 2022) for comprehensive testing
Statistical Testing Framework	Validates significance of results	Implement Wilcoxon signed-rank test or Friedman test

These components represent the essential "research reagents" for conducting rigorous ESSA performance analysis. The multi-search strategies and memory archive are particularly critical for achieving the documented performance improvements over basic SSA and other metaheuristic algorithms [2].

The statistical performance analysis confirms that ESSA demonstrates superior solution quality and convergence speed compared to basic SSA and other established metaheuristic algorithms. The incorporation of multiple search strategies, an advanced memory mechanism, and adaptive parameter control enables ESSA to effectively balance exploration and exploitation throughout the optimization process. The algorithm's performance advantage is particularly evident in high-dimensional problems and complex engineering applications, where it achieves significantly better optimization effectiveness than competing approaches. These characteristics position ESSA as a highly competitive optimizer for challenging real-world problems in fields including engineering design, clean production systems, and complex numerical optimization.

The pursuit of robust and efficient optimization algorithms is a cornerstone of computational science, directly impacting fields as diverse as drug discovery, logistics, and geophysical modeling. Swarm Intelligence (SI) and Evolutionary Algorithms (EAs) have emerged as powerful tools for solving complex, non-linear optimization problems where traditional methods falter. Within this landscape, the Salp Swarm Algorithm (SSA) and its enhanced versions have garnered significant interest. This guide provides a rigorous, head-to-head comparison of an Enhanced Sparrow Search Algorithm (ESSA) against other prominent metaheuristics, including the standard SSA, Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Whale Optimization Algorithm (WOA), and Differential Evolution (DE). The analysis is framed within a broader thesis on evolutionary algorithm benchmark evaluation, providing researchers and drug development professionals with objective experimental data and performance insights to inform their selection of computational tools.

The Enhanced Sparrow Search Algorithm (ESSA)

The ESSA represents a significant refinement of the standard Sparrow Search Algorithm. The enhancements are strategically designed to overcome common limitations in metaheuristics, such as premature convergence and imbalance between exploration and exploitation. Key improvements integrated into ESSA include [51]:

• Strengthened Random Jump: The producer's position update incorporates a reinforced random jump mechanism, which bolsters the algorithm's global search capability and reduces the probability of becoming trapped in local optima.
• Vintage Experience Learning: Each scrounger (follower) in the population continuously learns from the historical best positions discovered by the producers, facilitating a more efficient convergence.
• Individual Difference Threat Perception: For the best-positioned sparrow, when it perceives a threat, the algorithmic update incorporates the difference between the best and worst individuals. This strategy accelerates the search process and refines solution accuracy.
• Elite Reverse Search: This strategy increases population diversity by considering the opposite of elite solutions, thereby helping the algorithm explore neglected regions of the search space.

Competing Algorithm Workflows

The algorithms chosen for comparison are well-established in the literature, each with a unique search mechanism.

• Particle Swarm Optimization (PSO): Simulates the social behavior of bird flocking. Particles navigate the search space by adjusting their trajectories based on their own personal best position and the global best position found by the swarm [52] [53].
• Genetic Algorithm (GA): An evolutionary algorithm inspired by natural selection. It relies on genetic operators—selection, crossover, and mutation—to evolve a population of candidate solutions over generations [54].
• Whale Optimization Algorithm (WOA): Mimics the bubble-net hunting behavior of humpback whales. Its search process is characterized by three phases: encircling prey, spiral bubble-net attacking behavior (exploitation), and searching for prey (exploration) [54] [55].
• Differential Evolution (DE): A population-based stochastic optimizer that generates new candidates by combining existing ones according to a specific formula. It is known for its simple structure and effective mutation strategy [53].

The following diagram illustrates the core logical relationships and high-level search workflows of these competing algorithms.

Experimental Performance Data

Benchmark Function Testing

The performance of the ESSA algorithm was rigorously validated on a suite of 10 fundamental benchmark functions. In these controlled tests, ESSA demonstrated superior capability by finding the optimal value for 7 out of the 10 test functions. When compared against 12 other algorithms, ESSA's average performance consistently ranked first, establishing its robustness and high solution accuracy [51].

Performance in Applied Scenarios

UAV Path Planning: The ESSA was applied in a hybrid model, the PESSA (Parallel PSO and ESSA), and tested in four distinct scenarios (2D and 3D environments). The results, which can be interpreted as a measure of path optimality (with lower values being better), are summarized below [51]:

Table: UAV Path Planning Optimization Results

Environment	Scenario	Average Optimization Result (PESSA)
2D	Case 1	0.0165
2D	Case 2	0.0521
3D	Case 1	0.6635
3D	Case 2	0.5349

The study concluded that the PESSA algorithm acquired "more feasible and effective" routes than the other compared algorithms [51].

Postman Delivery Routing Problem: A head-to-head comparison between PSO and DE was conducted for a real-world Vehicle Routing Problem (VRP). The objective was to minimize the total travel distance for delivery vehicles. The study found that while both PSO and DE "clearly outperformed" the current manual routing practices, the performance of DE was "notably superior" to that of PSO in this specific application [53]. This highlights the importance of problem context in algorithm selection.

Geophysical Inversion: A comprehensive review of PSO applications in geophysics noted that several studies have demonstrated PSO can outperform GA in terms of "accuracy and convergence" for various geophysical inverse problems, including electromagnetic and seismic data interpretation [52].

Detailed Experimental Protocols

To ensure the reproducibility of the results cited in this guide, this section outlines the general experimental methodologies common to rigorous algorithm benchmarking.

• Problem Formulation: The optimization problem is defined, including the objective function and any constraints. For path planning, this involves defining the environment (2D/3D space with obstacles). For VRP, it involves customer locations and vehicle parameters [51] [53].
• Algorithm Initialization: Common parameters are set, such as population size and maximum number of iterations, to ensure a fair comparison. For stochastic algorithms, multiple independent runs are performed to account for random variation [51].
• Fitness Evaluation: In each iteration, every candidate solution in the population is evaluated using the objective function. For UAV path planning, this typically involves calculating the path length and penalizing collisions [51].
• Solution Update: Each algorithm applies its unique operators (e.g., PSO's velocity update, GA's crossover, WOA's spiral update) to generate a new population of solutions [51] [54] [53].
• Termination and Analysis: The process repeats until a stopping criterion is met (e.g., maximum iterations). The final best solution, convergence curves, and statistical performance (mean, standard deviation) across multiple runs are recorded and compared [51].

The workflow for such a comparative experiment is detailed below.

The Scientist's Toolkit: Key Research Reagents

In the context of computational optimization, "research reagents" refer to the essential software tools and datasets required to conduct and evaluate algorithm performance.

Table: Essential Tools for Algorithm Benchmarking

Tool / Resource	Function in Research	Examples / Notes
Benchmark Test Suites	Provides standardized functions to evaluate algorithm performance, exploration/exploitation balance, and convergence speed.	CEC2014, CEC2017, 23 classic benchmark functions [55].
Real-World Datasets	Validates algorithm performance on applied, complex problems with practical constraints.	UAV flight environments [51], postal delivery routes [53], geophysical survey data [52].
Simulation Frameworks	Software platforms used to model the problem environment and execute the algorithmic optimization process.	Custom simulations in MATLAB, Python, or C++ are common for path planning and VRP [51] [53].
Performance Metrics	Quantifiable measures used to compare algorithms objectively and statistically.	Best/Average Fitness, Standard Deviation, Convergence Curves, Statistical Significance Tests (e.g., Wilcoxon) [51] [55].

This head-to-head comparison reveals that the Enhanced Sparrow Search Algorithm (ESSA) presents a highly competitive optimizer, demonstrating superior average performance on benchmark functions and effective application in complex scenarios like UAV path planning. However, no single algorithm is universally best. The superior performance of DE over PSO in vehicle routing and the proven robustness of PSO in geophysical inversion underscore that the optimal choice is inherently problem-dependent. For researchers in drug development and other scientific fields, this guide emphasizes that selecting an optimization algorithm should be a deliberate decision, informed by the problem's nature, the landscape's complexity, and empirical evidence from rigorous benchmarking.

The pursuit of robust optimization algorithms remains a cornerstone of computational science, particularly for high-dimensional problems prevalent in fields such as drug discovery and complex systems engineering. The Evolutionary Salp Swarm Algorithm (ESSA) represents a significant advancement in swarm intelligence, specifically engineered to address the challenges of scalability and convergence in complex search spaces. This guide provides an objective comparison of ESSA's performance against other metaheuristic optimizers across 30, 50, and 100-dimensional problems, presenting empirical data from rigorous benchmark evaluations. Understanding dimensional scalability is crucial for researchers and development professionals who rely on optimization tools for real-world applications where problem dimensionality directly impacts solution quality and computational efficiency. The following analysis documents how ESSA addresses the common performance degradation observed in many algorithms as dimensionality increases, establishing a new benchmark for high-dimensional optimization.

Evolutionary Salp Swarm Algorithm (ESSA)

The Evolutionary Salp Swarm Algorithm (ESSA) enhances the basic Salp Swarm Algorithm (SSA) through several innovative mechanisms that collectively improve its dimensional scalability. The fundamental SSA operates by simulating the swarming behavior of salps in marine environments, utilizing a leader-follower chain structure for navigating the search space [5]. While SSA benefits from simple structure and minimal parameter tuning, it suffers from premature convergence and inadequate balance between exploration and exploitation in high-dimensional spaces [1] [5].

ESSA addresses these limitations through three primary evolutionary strategies:

• Distinct evolutionary search strategies: ESSA incorporates two novel evolutionary search strategies that enhance population diversity and adaptive search capabilities, along with an enhanced SSA search strategy that ensures steady convergence despite being less exploratory [1].
• Advanced memory mechanism: This mechanism archives both the best and inferior solutions identified during optimization, preserving diversity and preventing premature convergence [1].
• Stochastic universal selection: This method regulates the archive by selecting individuals according to their fitness values, maintaining selective pressure toward promising regions of the search space [1].

Comparative Algorithms

For comprehensive performance evaluation, ESSA was compared against seven established optimization algorithms, including the basic SSA and other state-of-the-art metaheuristics [1]. These algorithms represent diverse approaches to optimization, including swarm intelligence, evolutionary computation, and physics-inspired methods.

Benchmarking and Evaluation Protocol

The experimental methodology employed rigorous benchmarking standards to ensure statistically significant comparisons:

• Benchmark functions: Performance was evaluated using the CEC 2017 and CEC 2020 test suites, which provide standardized, challenging optimization landscapes with known global optima [1].
• Dimensionality testing: Experiments were conducted across 30, 50, and 100-dimensional problem spaces to assess scalability [1].
• Statistical validation: Comprehensive statistical analyses, including the Wilcoxon rank-sum test, were employed to validate performance differences and establish significance [1].
• Performance metrics: Solution quality (measured as proximity to known optima) and convergence speed were the primary evaluation criteria [1].

The following workflow diagram illustrates the experimental methodology for benchmarking ESSA's dimensional scalability:

Performance Comparison Across Dimensions

Quantitative Results Analysis

Experimental results demonstrate ESSA's superior performance across all dimensional problems compared to seven leading optimization algorithms. The following table summarizes ESSA's optimization effectiveness values, which represent the algorithm's capability to find high-quality solutions across diverse benchmark functions:

Table 1: ESSA Optimization Effectiveness Across Different Dimensions

Dimension	Optimization Effectiveness	Rank Among Comparative Algorithms
30D	84.48%	1st
50D	96.55%	1st
100D	89.66%	1st

Data sourced from [1] reveals that ESSA consistently outperformed all competing algorithms across all dimensional configurations, achieving the highest ranking in every category. The notable peak performance of 96.55% at 50 dimensions suggests an optimal balance between exploration and exploitation capabilities at this dimensionality. While slightly lower at 100 dimensions, ESSA maintained substantial performance superiority over alternative approaches, demonstrating exceptional scalability.

Comparative Performance Data

The following table presents a detailed comparison of ESSA against other optimization algorithms across the dimensional spectrum:

Table 2: Comprehensive Performance Comparison Across Dimensions

Algorithm Category	Representative Algorithm	30D Performance	50D Performance	100D Performance	Key Limitations
Enhanced SSA Variants	ESSA (Proposed)	84.48% (Best)	96.55% (Best)	89.66% (Best)	-
Basic SSA	Standard SSA	Significantly lower	Significantly lower	Significantly lower	Premature convergence, slow convergence rates
Swarm Intelligence	PSO, GWO, AO	Lower	Lower	Lower	Premature convergence, poor exploration-exploitation balance
Evolutionary Algorithms	DE, GA	Lower	Lower	Lower	Parameter sensitivity, computational complexity
Physics-inspired	AOA	Lower	Lower	Lower	Inefficient in high-dimensional spaces

The performance values for non-ESSA algorithms are represented qualitatively as "Lower" based on statistical ranking data reported in [1], which established ESSA's superior ranking position but did not provide exact effectiveness percentages for all compared algorithms. ESSA's performance advantages stem from its integrated strategies that specifically address high-dimensional optimization challenges, including its adaptive search strategies and memory mechanism that collectively maintain diversity while driving convergence [1].

ESSA's Technical Architecture for Enhanced Scalability

Core Technical Innovations

ESSA's superior dimensional scalability originates from several architectural innovations that collectively address the limitations of basic SSA and other metaheuristics:

• Adaptive parameter control: ESSA implements adaptive adjustment mechanisms for critical parameters c1 and α, enabling dynamic balance between exploration and exploitation throughout the optimization process [5]. This adaptability proves particularly valuable in high-dimensional spaces where fixed parameters often lead to suboptimal performance.

• Gaussian walk position update: After the initial position update phase, ESSA incorporates a Gaussian walk-based strategy that enhances global search capability, helping the algorithm escape local optima in complex fitness landscapes [5].

• Dynamic mirror learning: This strategy expands the search domain through solution mirroring, strengthening local search capability while maintaining diversity [5].

• Multi-search strategy integration: Unlike basic SSA that employs randomized switching among search strategies, ESSA proposes distinct evolutionary search strategies specifically designed to enhance diversity and adaptive search capabilities [1].

The following diagram illustrates ESSA's technical architecture and how its components interact to enable dimensional scalability:

Enhanced Knowledge SSA (EKSSA) Variant

Further supporting ESSA's architectural advantages, recent research has developed an Enhanced Knowledge-based SSA (EKSSA), which incorporates similar strategies for improved dimensional performance. EKSSA demonstrates superior results on thirty-two CEC benchmark functions compared to eight state-of-the-art algorithms, including Randomized PSO, GWO, Archimedes Optimization Algorithm, and Hybrid Particle Swarm Butterfly Algorithm [5]. This independent validation reinforces that the architectural principles embodied in ESSA effectively address dimensional scalability challenges.

Experimental Protocols and Research Toolkit

Benchmarking Methodology

To ensure reproducible and scientifically valid performance comparisons, the cited research employed rigorous experimental protocols:

• Benchmark suites: The CEC 2017 and CEC 2020 test suites were utilized, providing diverse function types including unimodal, multimodal, hybrid, and composition functions [1]. This diversity ensures comprehensive assessment of algorithm capabilities across various problem characteristics.

• Experimental setup: All experiments maintained consistent population sizes and maximum function evaluations across compared algorithms to ensure fair comparison. Statistical significance testing via the Wilcoxon rank-sum test with a 0.05 significance level confirmed the reliability of performance differences [1].

• Performance metrics: Primary evaluation criteria included solution accuracy (deviation from known optimum), convergence speed (function evaluations to reach target accuracy), and success rate (consistency in finding acceptable solutions) [1].

Research Reagent Solutions

Table 3: Essential Research Tools for Algorithm Benchmarking

Research Tool	Function in Evaluation	Application Context
CEC 2017/2020 Test Suites	Standardized benchmark functions	Algorithm performance evaluation
Wilcoxon Rank-Sum Test	Statistical significance validation	Performance comparison verification
Principal Moments of Inertia (PMI)	Molecular 3D shape characterization	Drug discovery applications
Plane of Best Fit (PBF)	Molecular structure dimensionality assessment	Drug design optimization
CORINA	3D molecular structure generation	Conformation-dependent descriptor calculation

These research tools enable comprehensive evaluation of optimization algorithms across diverse application contexts. The CEC benchmark suites provide validated testing environments, while statistical tests ensure result reliability [1]. The molecular descriptors (PMI and PBF) illustrate how optimization algorithms like ESSA can be applied in drug discovery contexts to evaluate molecular three-dimensionality, which correlates with desirable drug-like properties [56].

Implications for Research and Applications

Practical Applications

ESSA's robust performance across dimensional scales has significant implications for real-world optimization problems:

• Cleaner production systems: ESSA has demonstrated success in optimizing cleaner production systems, balancing environmental and economic constraints in complex engineering scenarios [1].

• Drug discovery and design: The algorithm's scalability makes it suitable for pharmaceutical applications, including drug classification and target identification tasks that involve high-dimensional feature spaces [57]. Enhanced three-dimensionality in drug-like molecules, which can be optimized using algorithms like ESSA, correlates with improved clinical success probability [56].

• Complex engineering design: ESSA effectively solves complex design problems with numerous variables and constraints, outperforming traditional approaches in solution quality and convergence speed [1].

Future Research Directions

While ESSA represents a significant advancement in dimensional scalability, several research directions merit further investigation:

• Ultra-high-dimensional problems: Extending evaluation beyond 100 dimensions to assess performance on modern data science problems with thousands of variables.

• Hybrid approaches: Combining ESSA's strengths with other optimization paradigms to create specialized solvers for domain-specific challenges.

• Theoretical foundations: Developing deeper theoretical understanding of why ESSA's strategies so effectively address dimensional scalability challenges.

• Automated parameter adaptation: Enhancing the adaptive capabilities to further reduce manual parameter tuning requirements.

This comparison guide has objectively evaluated the dimensional scalability of the Evolutionary Salp Swarm Algorithm across 30, 50, and 100-dimensional problems. Experimental evidence consistently demonstrates ESSA's superior performance compared to other metaheuristic optimizers, with optimization effectiveness values of 84.48%, 96.55%, and 89.66% across the dimensional spectrum. These results confirm ESSA's robust scalability and its ability to maintain solution quality as dimensionality increases. The algorithm's architectural innovations—including adaptive parameter control, Gaussian walk position updates, dynamic mirror learning, and advanced memory mechanisms—collectively address the fundamental challenges of high-dimensional optimization. For researchers and drug development professionals working with complex optimization problems, ESSA represents a state-of-the-art solution that reliably delivers superior performance across diverse dimensional challenges.

The pursuit of robust optimization techniques represents a cornerstone of scientific and engineering progress, particularly when addressing complex, real-world problems characterized by high dimensionality, non-linear constraints, and multi-modal search spaces. Within this domain, the Salp Swarm Algorithm (SSA), a metaheuristic inspired by the swarming behavior of salps in marine environments, has garnered attention for its simple structure, multi-search strategy, and minimal control parameters [1] [5]. However, its practical application is often hindered by a propensity for premature convergence and an imprecise search strategy that struggles to guide populations toward globally optimal regions [1] [33].

To overcome these limitations, an enhanced variant known as the Evolutionary Salp Swarm Algorithm (ESSA) has been developed. This article provides a comprehensive comparison of ESSA's performance against other leading optimizers, drawing on empirical evidence from standardized benchmark functions and real-world applications in engineering design and biomedical research. The analysis is framed within a broader thesis on evolutionary algorithm benchmark evaluation, objectively assessing ESSA's potential to solve complex optimization challenges in drug development and other critical fields.

Methodological Foundations of ESSA

The Evolutionary Salp Swarm Algorithm incorporates several strategic enhancements that distinguish it from its predecessor and other metaheuristics. Its architecture is designed to systematically balance global exploration and local exploitation, a critical factor for achieving high-quality solutions.

Core Algorithmic Innovations

ESSA integrates three distinct innovative search strategies to augment its performance [1]:

• Evolutionary Search Strategies: Two novel strategies enhance population diversity and enable adaptive search, allowing the algorithm to escape local optima.
• Enhanced SSA Search Strategy: A refined version of the original SSA search, which sacrifices some exploratory power to ensure steady and reliable convergence.
• Advanced Memory Mechanism: This mechanism archives both the best and inferior solutions identified during the optimization process. By storing and leveraging this information, ESSA enhances diversity and actively prevents premature convergence [1] [2].
• Stochastic Universal Selection: A selection method is employed to regulate the archive, choosing individuals based on their fitness values to efficiently guide the population's evolution [1].

Another documented enhancement, referred to as a "lifetime scheme," iteratively tunes ESSA's dominant parameters using ESSA itself. This self-adaptation refines the convergence performance throughout the evolutionary process, ensuring that search agents update their positions optimally [33].

Workflow and Process

The following diagram illustrates the integrated workflow of ESSA, showcasing how its multi-search strategies and memory mechanism interact.

Benchmark Evaluation and Comparative Performance

A rigorous evaluation of ESSA's capabilities was conducted using the widely recognized CEC 2017 and CEC 2020 benchmark functions, which test algorithms on a diverse set of optimization problems, including unimodal, multimodal, hybrid, and composite functions [1].

Experimental Protocol for Benchmark Testing

The performance validation of ESSA followed a standardized experimental protocol to ensure fairness and reproducibility [1]:

• Algorithm Configuration: ESSA and seven other state-of-the-art metaheuristic algorithms were configured with their respective optimal parameter settings.
• Test Environment: All algorithms were evaluated on the same set of benchmark functions from CEC 2017 and CEC 2020.
• Performance Metrics: The primary metrics for comparison were solution quality (the best objective function value found) and convergence speed (the rate at which the algorithm approaches the optimum).
• Dimensionality Testing: Experiments were conducted across different search space dimensions (30, 50, and 100) to assess scalability.
• Statistical Analysis: The results were subjected to statistical tests to confirm the significance of performance differences. The effectiveness of each algorithm was calculated as the percentage of benchmark functions on which it achieved the best performance.

Quantitative Performance Results

The following table summarizes the key performance metrics of ESSA compared to other leading optimizers, as reported in the literature [1].

Table 1: Benchmark Performance Comparison (CEC 2017 & CEC 2020)

Algorithm	Abbreviation	Key Principle	Ranking (Statistical Test)	Optimization Effectiveness (Dim 30)	Optimization Effectiveness (Dim 50)	Optimization Effectiveness (Dim 100)
Evolutionary Salp Swarm Algorithm	ESSA	Evolutionary multi-search & memory mechanism	1st	84.48%	96.55%	89.66%
Salp Swarm Algorithm	SSA	Swarming behavior of salps	Not 1st	Lower than ESSA	Lower than ESSA	Lower than ESSA
Genetic Algorithm	GA	Natural selection & genetics	Not 1st	Lower than ESSA	Lower than ESSA	Lower than ESSA
Differential Evolution	DE	Vector-based mutation & crossover	Not 1st	Lower than ESSA	Lower than ESSA	Lower than ESSA
Particle Swarm Optimization	PSO	Social behavior of birds/flocks	Not 1st	Lower than ESSA	Lower than ESSA	Lower than ESSA
Grey Wolf Optimizer	GWO	Hierarchical hunting of grey wolves	Not 1st	Lower than ESSA	Lower than ESSA	Lower than ESSA

The data demonstrates that ESSA consistently secured a first-place ranking in statistical analyses, achieving the best optimization effectiveness across all tested dimensions, with a peak of 96.55% for 50-dimensional problems [1]. This indicates a superior ability to navigate complex, high-dimensional search spaces compared to other established algorithms.

Engineering Design Application: Cleaner Production Systems

The true measure of an optimization algorithm's value lies in its ability to solve complex, constrained real-world problems. ESSA's performance has been validated through application to challenging engineering design cases.

Experimental Protocol for Engineering Problems

When applied to engineering problems like cleaner production system optimization and other complex design challenges, the validation protocol involves [1] [33]:

• Problem Formulation: The engineering system is mathematically modeled, defining objective functions (e.g., maximize efficiency, minimize cost or waste) and all operational constraints.
• Constraint Handling: ESSA's search strategy is integrated with a method for handling engineering constraints, such as penalty functions or feasibility-based rules.
• Solution Evaluation: The algorithm's performance is measured by its ability to find a feasible, optimal design that satisfies all constraints and improves upon existing solutions regarding key performance indicators.

Results and Industrial Relevance

In optimizing a cleaner production system, ESSA demonstrated exceptional practicality [1]. The algorithm's enhanced exploration and exploitation capabilities allowed it to efficiently determine system parameters that balance economic and environmental objectives. The advanced memory mechanism was particularly instrumental in avoiding suboptimal configurations, leading to solutions that contribute to more sustainable and economically viable production processes. Furthermore, ESSA has proven effective in solving other complex engineering design problems, highlighting its robustness and general applicability [1] [33].

Biomedical and Pharmaceutical Application Potentials

The biomedical and pharmaceutical sectors present a frontier for advanced optimization algorithms, with opportunities ranging from drug development to clinical trial design.

Optimizing Biomedical Classifiers

In one biomedical application, an enhanced SSA variant termed the Enhanced Knowledge-based Salp Swarm Algorithm (EKSSA) was used to optimize a Support Vector Machine (SVM) classifier for seed classification tasks, a proxy for broader hyperparameter optimization challenges in biomedical data analysis [5].

Table 2: Research Reagent Solutions for Computational Experiments

Research Reagent / Resource	Function in Experiment	Source / Implementation
CEC 2017 & CEC 2020 Benchmarks	Standardized test functions for evaluating algorithm performance	IEEE Computational Intelligence Society
Support Vector Machine (SVM)	A machine learning classifier whose hyperparameters are optimized	Scikit-learn, LIBSVM, etc.
Seed Classification Datasets	Real-world biomedical data for validating optimization performance	UCI Machine Learning Repository
Electronic Health Records (EHR)	Real-world data source for building external control arms	Hospital Information Systems
Historical Clinical Trial Data	Pooled data from past trials for constructing synthetic controls	Project Data Sphere, Medidata Enterprise Data Store

The experimental protocol for this application involved [5]:

• Hybrid Model Formation: Creating an EKSSA-SVM hybrid model where EKSSA optimizes the critical hyperparameters of the SVM (e.g., regularization parameter C, kernel coefficients).
• Performance Evaluation: Training and testing the hybrid classifier on seed classification datasets and comparing its accuracy against benchmarks. The EKSSA-SVM hybrid classifier achieved higher classification accuracy than standard models, demonstrating the potential of SSA variants to enhance the performance of diagnostic and predictive tools in biology and medicine [5].

Revolutionizing Clinical Trial Design with Real-World Data

A transformative application of optimization in drug development lies in the construction of External Control Arms (ECAs). Traditional randomized clinical trials (RCTs) are the gold standard but are often costly, time-consuming, and ethically challenging for diseases with high unmet need [58] [59]. ECAs use existing data from outside the trial—such as historical clinical trial data or Real-World Data (RWD) from electronic health records and patient registries—to create a synthetic control group for comparison with the group receiving the investigational drug [59].

The process of constructing and utilizing an ECA, a prime candidate for optimization, is outlined below.

The integration of ECAs can significantly reduce trial duration and cost, allow a greater proportion of patients to receive the investigational therapy, and facilitate research in rare diseases [58] [59]. Algorithms like ESSA could be critically applied to optimize the ECA construction process, such as by solving the complex, high-dimensional matching problem to create the most balanced cohorts possible from large, disparate datasets, thereby minimizing bias and improving the robustness of evidence for regulatory review.

The empirical evidence and comparative analysis presented in this guide firmly establish the Evolutionary Salp Swarm Algorithm (ESSA) as a superior optimizer in both theoretical benchmarks and practical applications. Its innovative multi-search strategy and advanced memory mechanism enable it to overcome the limitations of the basic SSA and other metaheuristics, achieving top-tier performance in solution quality and convergence speed across various dimensions.

As the biomedical and pharmaceutical industries increasingly embrace alternative evidence generation strategies, such as Real-World Evidence and External Control Arms, the need for robust, scalable, and efficient optimization tools will grow. ESSA's proven capabilities in complex engineering design and its potential for optimizing clinical trial workflows and biomedical classifiers position it as a powerful computational tool for the next horizon of drug development and biomedical innovation.

Conclusion

The Evolutionary Salp Swarm Algorithm represents a significant advancement in metaheuristic optimization, effectively addressing the fundamental limitations of traditional SSA through its innovative multi-search strategies and advanced memory mechanism. The comprehensive benchmark evaluations demonstrate ESSA's superior performance in solution quality, convergence speed, and dimensional scalability compared to leading algorithms. With optimization success rates reaching 96.55% for 50-dimensional problems, ESSA shows particular promise for complex biomedical applications including drug discovery, clinical trial optimization, and molecular design where high-dimensional parameter spaces are common. Future research directions should focus on adapting ESSA for multi-objective drug design problems, integrating domain-specific constraints for clinical applications, and exploring hybrid models that combine ESSA with machine learning for predictive modeling in pharmaceutical development. The algorithm's proven effectiveness in cleaner production systems and complex engineering design provides a strong foundation for its application in optimizing biomedical manufacturing processes and therapeutic development pipelines.

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