An enhanced connected banking system optimizer with multiple strategies for numerical optimization problems

Abstract

Connected Banking System Optimizer (CBSO) is a recently proposed meta-heuristic inspired by inter-bank financial transactions. Owing to its parameter-free nature, it has shown competitive performance on engineering constrained optimization problems. Nevertheless, the CBSO algorithm still suffers from limited inter-population information exchange and an insufficiently smooth transition between exploitation and exploration, which often leads to premature convergence due to inadequate coverage of the search space. To address these shortcomings, this paper presents an enhanced variant called ECBSO that integrates a feedback selection strategy, a regenerative population strategy, and a distribution estimation strategy. Comprehensive experiments were conducted on the CEC-2017 benchmark suite to evaluate ECBSO, encompassing parameter sensitivity analysis, ablation studies, and comparisons with various advanced variants. Statistical validation was performed using the Wilcoxon rank-sum test, Friedman test, and Nemenyi post-hoc test to confirm ECBSO’s superiority over competing algorithms. The experimental results demonstrate that ECBSO possesses high optimization efficacy and robustness, achieving average Friedman ranks of 2.103 (10D), 1.586 (30D), 1.828 (50D), and 2.103 (100D). Finally, ECBSO was applied to ten real-world engineering constrained optimization problems. The outcomes show that it not only solves practical problems effectively but also maintains remarkable stability, establishing ECBSO as an outstanding meta-heuristic variant.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-38261-9.

Introduction

Artificial intelligence is advancing rapidly¹. Optimization techniques have emerged as one of its most prominent exemplars. The essence of optimization lies in locating the extremum of an objective function subject to prescribed constraints, a pursuit whose imprint spans computational theory, mathematical sciences, engineering design, and economic analysis^2,3. Every optimization model can be decomposed into three core components: the objective function to be optimized, the constraints that delimit the feasible region, and the decision variables available for unconstrained manipulation⁴. Classical algorithms—linear programming, quadratic programming, and dynamic programming—deliver rigorous mathematical guarantees and efficient optimality certificates for problems that are deterministic, convex, and structurally well-behaved⁵. Their efficacy, however, rests on stringent assumptions: the objective must be convex, the constraints linear or analytically tractable, and the variables continuous⁶. When the landscape becomes multimodal, non-convex, non-differentiable, or high-dimensional, these methods are prone to local minima, and the quality of the final solution is highly sensitive to the initial point⁷. In contrast, metaheuristic algorithms adopt a black-box perspective that eschews gradient information and remains agnostic to analytical descriptions⁸. By emulating natural evolution, swarm cooperation, physical phenomena, or mathematical heuristics, they employ stochastic sampling and adaptive operators to alternate between exploration and exploitation across rugged landscapes, thereby escaping local basins and converging toward global optima. The continual evolution of metaheuristic algorithms has enabled both their archetypes and refinements to rapidly diffuse across diverse application domains, including unmanned aerial vehicle path- and mission-coordination optimization^9,10, high-precision lesion segmentation in medical imaging^11,12, real-time power dispatch in microgrids^13,14, risk-return balancing in financial portfolios^15,16, machine learning hyperparameter optimization^17–20,optimal reactive power dispatch^21,22, cost compression in logistics networks^23,24, instantaneous fault diagnosis in Internet-of-Things infrastructures^25,26, and maneuvering decision making in unmanned aerial vehicles^27,28. These cross-disciplinary success cases collectively corroborate the universal utility of this class of algorithms in complex, dynamic, and high-dimensional environments.

Based on algorithmic lineage and design rationale, mainstream metaheuristics can be taxonomized into four paradigms: evolutionary drivers, swarm synergies, physical metaphors, and mathematical formulations²⁹. Evolution-based algorithms (EbA) constitute a class of metaheuristics grounded in Darwinian principles of natural selection and survival of the fittest. By emulating genetic inheritance, variation, and selection, these methods evolve a population across successive generations within the solution space, progressively converging toward the global optimum. Representative paradigms include Genetic Algorithm (GA)³⁰, Differential Evolution (DE)³¹, Genetic Programming (GP)³², and Alpha Evolution (AE) [26]. Swarm-based Algorithms (SbA) constitute a family of metaheuristics rooted in swarm-intelligence theory. They emulate the localized interactions and cooperative behaviors of simple natural agents such as birds, fish, ants, bees, wolves, and other social organisms. Representative and recently proposed SbA include Particle Swarm Optimization (PSO)³³, Harris Hawks Optimization (HHO)³⁴, Tuna Swarm Optimization (TSO)³⁵, Superb Fairy-wren Optimization Algorithm (SFOA)³⁶, Circulatory System Based Optimization (CSBO)³⁷, Snow Geese Algorithm (SGA)³⁸, Wild Geese Algorithm (WGA)³⁹, Birds of Prey-Based Optimization (BPBO)⁴⁰, and Crayfish Optimization Algorithm (COA)⁴¹. Physics-based algorithms (PbA) are a class of metaheuristics that abstract physical laws or natural phenomena into search strategies, emulating processes such as thermodynamic equilibration, mechanical motion, electromagnetic field evolution, quantum transitions, and optical refraction. Examples of PbA are Simulated Annealing (SA)⁴², Polar Lights Optimizer (PLO)⁴³, and Light Spectrum Optimizer (LSO)⁴⁴, Kirchhoff’s Law Algorithm (KLA)⁴⁵, together with Fata Morgana Algorithm (FMA)⁴⁶. Mathematics-based algorithms (MbA) are a class of metaheuristics whose search mechanisms are distilled directly from mathematical theory, numerical methods, and abstract mathematical structures. Leveraging tools such as analytical geometry, convex analysis, game theory, matrix computation, stochastic processes, and algebraic topology, these algorithms construct search operators that guide the optimization trajectory. Representative MbA include Sine Cosine Algorithm (SCA)⁴⁷, Weighted Mean Of Vectors (WMOV)⁴⁸, Quasi Random Fractal Search (QRFS)⁴⁹, Arithmetic Optimization Algorithm (AOA)⁵⁰, and Sinh Cosh Optimizer (SCO)⁵¹.

Although metaheuristics demonstrate notable strengths on complex optimization tasks, their inherent propensity for premature convergence remains unresolved. The No-Free-Lunch theorem affirms that no single algorithm can uniformly dominate all others across every problem domain⁵². To devise more universally applicable solvers, researchers continuously augment the original algorithms. Cui et al. proposed a pressure selection mechanism and a covariance matrix learning strategy to strengthen the snow ablation algorithm and combined it with an extreme learning machine for flood hazard prediction⁵³. Huang et al. improved the zebra optimization algorithm by borrowing the hierarchy of the grey wolf optimizer and solved the three-dimensional path planning problem⁵⁴. Diao et al. employed adaptive parameter sequences to smooth the transition between exploitation and exploration, introducing a restart strategy to prevent premature convergence⁵⁵. Abualigah et al. introduced a reverse learning strategy and a poop search strategy to avoid the walrus Optimizer from falling into premature convergence, and verified its effectiveness on a clustering test set⁵⁶. Li et al. utilized a roulette-based fitness distance balancing strategy to optimize the hard frost phase of the RIME algorithm, thereby achieving equilibrium between exploitation and exploration⁵⁷. Gyan et al. proposed an improved SCA with a hybrid gazelle optimization algorithm and further enhanced its exploration capability with Brownian motion and Levy flight strategy⁵⁸. Binanda et al. proposed a chaotic crayfish optimization algorithm to achieve a balance between exploitation and exploration by integrating chaotic mapping⁵⁹. Djordje et al. proposed a hybrid self-adaptive Red Fox Optimization Algorithm enhanced with quasi-reflexive learning and applied it to tune classifier hyperparameters⁶⁰. Shi et al. enhanced the early global search capability of the RIME algorithm using an average guidance strategy and proposed a novel stagnation perturbation method⁶¹. Dakic et al. introduced an improved PSO variant that integrates adaptive parameter control and quasi-reflexive-learning-based initialization, and employed it to optimize the hyperparameters of extreme gradient boosting and K-nearest neighbor algorithms⁶². Ana et al. employed quasi-reflexive learning to preserve population diversity within the Crayfish Optimization Algorithm and hybridized it with Convolutional Neural Networks to address the waste-classification task⁶³. Beyond refining existing algorithms, researchers continue to devise other novel metaheuristics. Motivated by the advancement of Teaching–Learning-Based Optimization (TLBO)⁶⁴, a family of human-behavior-inspired methods—collectively termed human-based algorithms (HbA)—has emerged. In addition to TLBO, other HbA examples include Football Team Training Algorithm (FTTA)⁶⁵, Enterprise Development Optimization (EDO)⁶⁶, Catch Fish Optimization Algorithm (CFOA)⁶⁷, Lungs performance-based optimization (LPO)⁶⁸, and Human Evolutionary Optimization Algorithm (HEOA)⁶⁹. The classification of meta-heuristic algorithms is shown in Fig. 1.

Fig. 1.

The classification of meta-heuristic algorithms.

The Connected Banking System Optimizer (CBSO), introduced by Nemati in 2024, is a meta-heuristic algorithm inspired by the transaction dynamics among heterogeneous banks⁷⁰. The banking system is modeled as a network of banks that establish bilateral routes to transmit and settle transactions. Nemati abstracts these inter-bank connection patterns into a three-phase search strategy. In Phase 1, a sender bank constructs an initial route to the receiver bank based on the best route discovered so far. Phase 2 bifurcates the transaction flow: one half continues along the current best route, while the other half is routed through a randomly selected alternative path. Phase 3 refines the best route by leveraging two additional random routes, using information gathered from previously examined routes to select a path that minimizes transaction time between sender and receiver banks. To account for real-world disruptions, CBSO incorporates a 20% probability of bank failure per phase, triggering immediate route reselection to avoid dysfunctional nodes. In the absence of failure, the algorithm considers the possibility of cyberattacks, encrypting transaction information to ensure confidentiality. This mechanism dynamically modifies routes, ensuring robust and secure optimization across the search landscape. Nevertheless, CBSO still exhibits shortcomings in inter-population information exchange and the seamless transition between exploitation and exploration. Moreover, its limited exploration of the solution space renders it prone to premature convergence. To remedy these deficiencies, this study introduces the Enhanced Connected Banking System Optimizer (ECBSO). ECBSO incorporates three complementary enhancements: a feedback selection strategy for balancing exploitation and exploration, a regenerative population strategy for enriching diversity, and a distribution-estimation strategy for enhancing information exchange across the population.

To systematically evaluate the optimization capability of ECBSO, this work conducted comprehensive experiments on both the CEC-2017 benchmark and a suite of real-world engineering constrained optimization problems. Experimental results demonstrate that the proposed ECBSO consistently delivers high-quality solutions across diverse problem landscapes. The principal contributions of this study are summarized as follows:

1. To address the limitations of the CBSO algorithm, this paper proposes the Enhanced Connected Banking System Optimizer (ECBSO), which integrates three techniques: a feedback-selection strategy, a regenerative population strategy, and a distribution estimation strategy.

2. A range of improved algorithms was selected for comparison. The CEC-2017 benchmark suite across dimensions 10, 30, 50, and 100 was used as the testing functions. Multiple statistical tests were employed to analyze the experimental results of ECBSO and the comparison algorithms.

3. Engineering constrained optimization problems are employed to evaluate the practical performance of ECBSO in solving real-world optimization tasks.

The remainder of this paper is organized as follows: Section "An overview of the connected banking system optimizer" reviews the basic model of the CBSO algorithm. Section "A framework of proposed enhanced banking interconnection system optimizer" provides the details of the proposed ECBSO algorithm. Additionally, pseudocode, flowcharts, and complexity analysis are presented in this section. Section "Results and analysis on benchmark functions" conducts a comprehensive evaluation based on test functions. Section "Results and analysis on engineering constraint optimization problem" examines the capability of the CBSO algorithm in engineering constrained optimization problems. Finally, discussions and conclusions are drawn in Sect. 6, with an outlook on future research directions.

An overview of the connected banking system optimizer

The Connected Banking System Optimizer (CBSO) is a novel metaheuristic algorithm proposed by Mehrdad Nemati et al. in 2024. The CBSO mimics the various transaction methods employed by banks to achieve exploration and exploitation during the search process. The algorithm primarily consists of three search phases and a supplementary phase.

Firstly, CBSO initializes the entire population within the search space, where the initial positions are randomly determined within the bounds of decision variables, as expressed in Eq. (1).

1

In Eq. (1), denotes the j-th dimensional position of the i-th agent. and represent the lower and upper boundaries of the search domain, respectively, limiting the minimum and maximum positions any agent may occupy. is a uniformly distributed random number, is the population size, and is the problem dimensionality.

The search process of CBSO is divided into three phases: the first 20% is devoted to global exploration (Strategy 1), the last 60% to local exploitation (Strategy 3), and the intermediate 20% acts as a transition phase (Strategy 3) that simultaneously performs global and local searches.

During the exploration phase, the CBSO algorithm conducts the search using Eq. (2).

2

In Eq. (2), is drawn from a standard normal distribution with mean 0 and standard deviation 1, and denotes the best agent found so far. denotes the current iteration.

During the transition phase, the CBSO algorithm applies two complementary search strategies to refresh the population. Half of the agents are assigned to exploration, as described by Eq. (3), while the remaining half perform exploitation, governed by Eq. (4). Importantly, membership in either subpopulation is determined randomly for every generation.

3

4

In Eq. (3), is a D-dimensional vector drawn from a Lévy distribution. and are two distinct agents chosen at random from the current population. is an adaptive parameter whose value is updated by Eq. (5), with representing the maximum iteration number.

5

In the exploitation phase, the CBSO algorithm refines the search around promising regions by means of Eq. (6).

6

In addition to the three search phases, the CBSO algorithm incorporates a supplementary enhancement phase, as defined by Eq. (7), to further strengthen the search.

7

In Eq. (7), A is a constant set to 0.2 according to the source literature, representing the probability of a banking-system collapse. Algorithm 1 displays the CBSO algorithm’s pseudocode.

Algorithm 1.

Connected banking system optimizer (CBSO).

A framework of proposed enhanced banking interconnection system optimizer

This section presents the motivation and implementation details of the three proposed strategies: the feedback selection strategy, the regenerative population strategy, and the distribution estimation strategy. At the end of the section, pseudocode, a flowchart, and a complexity analysis are also provided.

Feedback selection strategy

Meta-heuristics typically begin by coarsely locating several promising solutions across the problem space and then refine the search within their vicinities. This requires broad coverage early on and intensive exploitation later. The CBSO algorithm, however, partitions the search process solely by iteration count. Such a rigid schedule cannot adapt to complex or changing landscapes, because the fixed breakpoints prevent on-the-fly adjustment of the exploration–exploitation balance. Drawing on control theory, this work therefore proposes a feedback selection strategy (FSS) to regulate this balance dynamically. This strategy differs from existing reinforcement-learning-based selection frameworks. Although RL frameworks can dynamically select operators according to instantaneous reward values, they are computationally expensive at each iteration and hence unsuitable for real-time applications; moreover, their performance is sensitive to the choice of learning rate. The adaptive DE framework is composed of multiple manually designed search operators, and its adaptability across different problems is inherently weak because the human-crafted operators are usually problem-specific and difficult to generalize. The FSS borrows the feedback concept from control theory to switch among search operators on the fly. Feedback can be positive or negative: positive feedback amplifies the current output, whereas negative feedback suppresses it. In FSS, this work regards an operator as beneficial if it improves a solution and detrimental otherwise. Taking Eq. (2) as an example, when an agent updated by this equation achieves better fitness, the trial is deemed positive feedback; Eq. (2) will be kept for the same agent in the next iteration. Conversely, if the fitness worsens, the outcome is interpreted as negative feedback, and one of the remaining operators is selected at random for that agent in the next iteration. FSS balances exploration and exploitation by assigning distinct search operators to individual agents. Consequently, CBSO abandons the fixed, iteration-based schedule and instead chooses operators according to the current state of each agent, enabling the algorithm to adapt more effectively to diverse problem landscapes. Algorithm 2 displays the FSS’s pseudocode.

Algorithm 2.

Feedback selection strategy (FSS).

Regenerative population strategy

In the CBSO algorithm, every agent undergoes the supplementary update at each iteration. When termination is based on a fixed maximum iteration count, this practice is advantageous because it gives each agent an extra update per generation. However, it becomes unfair—and detrimental—when the stopping criterion is the maximum number of function evaluations. Moreover, the strategy attempts to escape local optima by perturbing every agent, yet not all agents require such perturbation at every step. This indiscriminate updating generates many ineffective moves.

To overcome these drawbacks of the supplementary phase, this work introduces a regenerative population strategy (RPS). RPS enhances both diversity and adaptability, giving CBSO greater robustness in complex landscapes. Instead of applying the supplementary step to every individual, RPS activates it only for a dynamically chosen subset. The number of agents selected is jointly determined by the current population diversity and the observed improvement rate in that iteration.

Population diversity (PD), denoted as , measures the spread of individuals in the search space and is therefore critical for global exploration. It is calculated by Eq. (8).

8

The improvement rate (IR), denoted as σ, quantifies the search progress of the population and is given by Eq. (9).

9

In the Eq. (9), denotes the fitness value of . After obtaining PD and IR, the composite score S of the current population is expressed as the normalized weighted sum of these two indices, as shown in Eq. (10).

10

In Eq. (14), denotes the largest PD observed so far, denotes the largest IR encountered so far. and are the weights assigned to the two indices. In this study, both weights are set to 0.5 to indicate equal importance. In RPS, the number of individuals that undergo the supplementary phase is determined by Eq. (11). Once this number is fixed, the required agents—excluding the best one—are chosen uniformly at random and moved according to Eq. (7).

11

Here, the symbol denotes the greatest integer function.

Distribution estimation strategy

A clear weakness of the CBSO algorithm is the limited exchange of information among individuals. Although CBSO uses randomly chosen agents to maintain diversity, this alone is not enough. When the landscape is complex, the lack of strong global exploration often traps the algorithm in local optima. To solve this problem, this study introduces a distribution estimation strategy (DES) that has two parts: an update part and an embedding part. In the update part, DES builds a covariance matrix from the best agents and uses it to create new ones. The matrix guides sampling along directions where the variables are strongly related, avoiding blind steps along the axes and cutting down useless searches. Moreover, the probability model covers the whole space, keeps a wide distribution in the early stages, and therefore promotes global exploration, reducing the risk of getting stuck at local peaks. The Gaussian probabilistic model is the most widely used in DES. For a D-dimensional random vector, the joint Gaussian probability density function is given by Eq. (12).

12

13

14

In DES, the dominant population consists of the agents whose fitness ranks in the top half. After the Gaussian distribution of elite set is obtained, new agents are generated by sampling with mean 0 and covariance matrix , as shown in Eq. (15).

15

Once the update phase of DES is complete, the embedding phase begins. After the CBSO update completes, the DES strategy produces additional agents. Unlike conventional hybrid methods that apply different strategies sequentially to every individual, DES merges the newly sampled agents with those generated by CBSO and then selects the best agents, based on their fitness, to advance to the next iteration, as shown in Fig. 2. This process injects richer diversity into the population and helps the algorithm escape local optima. Because DES draws samples from the entire modeled distribution—not just from agents refined by CBSO—it mitigates over-fitting. The extra agents also strengthen global exploration across the search space. The number of agents produced by DES is determined by Eq. (16).

16

Fig. 2.

The sketch for the procedure of DES.

In Eq. (12), and are adaptive control parameters whose values will be discussed in the parameter sensitivity analysis.

Implementation steps for MSIGOA

By incorporating the three proposed mechanisms into CBSO, this work obtain ECBSO. ECBSO proceeds in four stages: initialization, an FSS-based update phase, an RPS-based supplementary update phase, and the newly added DES phase. The flowchart of ECBSO is shown in Fig. 3, and its pseudocode is given in Algorithm 3.

Algorithm 3.

Enchanced connected banking system optimizer (ECBSO).

Fig. 3.

Flow chart of ECBSO algorithm.

Analysis of ECBSO complexity

Time complexity reflects the computational efficiency of an algorithm. In this section, we analyze the time complexity of both CBSO and ECBSO. For both algorithms, the complexity is governed by the population size , the number of iterations and the problem dimension . In the basic CBSO algorithm, the overall time complexity is governed by three components: population initialization, position updates, and fitness evaluations. Initialization of the agents in dimensions takes . During each iteration, every agent is updated twice, so the position-update cost for one iteration is . Since each update is followed by a fitness evaluation and each agent is evaluated twice per iteration, the fitness-computation cost per generation is . Therefore, over iterations, the total time complexity of CBSO is . For ECBSO, the FSS does not require extra position updates or fitness evaluations, so it adds no time overhead. The RPS only determines how many agents enter the supplementary phase; assuming agents participate each iteration, this adds per iteration. The DES introduces new agents, contributing an additional per iteration. Therefore, over iterations, the total time complexity of ECBSO is . In summary, the relative time complexity of CBSO and ECBSO hinges on the relationship between and the sum . When , CBSO incurs higher complexity; when , CBSO is lower. To ensure a fair comparison in the experiments that follow, this work adopts the maximum number of function evaluations as the stopping criterion.

Results and analysis on benchmark functions

This section uses a comprehensive set of benchmark functions to assess the performance of ECBSO. To highlight its strengths, this study compares it with several state-of-the-art improved algorithms. The algorithms selected for comparison cover a broad spectrum of recent improved meta-heuristics: (1) Human-based: ISGTOA, EMTLBO. (2) Evolution-based: LSHADE, APSM-jSO. (3) Swarm-based: GLS-MPA, ESLPSO. (4) Physics-based: ACGRIME, IYDSE. (5) Math-based: EPSCA, QAGO. These choices were deliberately made to benchmark ECBSO against diverse, state-of-the-art enhancement strategies rather than only classic or baseline methods. All experiments were run on Windows 11 with 32 GB of RAM and an AMD Ryzen 9 7940HX CPU at 2.40 GHz, using MATLAB R2021a. In the experiments, to mitigate the impact of randomness, all algorithms were given the same maximum number of function evaluations, set at times, and each function was solved independently 30 times. All algorithms use the random seeds specified by the CEC 2017 test suite. Given the significant influence of parameter settings on algorithm performance, the parameters for all comparison algorithms were configured as suggested by their respective source literature, as shown in Table 1.

Table 1.

Parameter settings of ECBSO and competing algorithms.

Algorithm	Parameters setting
ECBSO
CBSO
ISGTOA71
EMTLBO72
LSHADE73
APSM-jSO74
GLS-MPA75
ESLPSO76
ACGRIME77
IYDSE78
EPSCA79
QAGO80

The benchmark functions used here are from the CEC-2017 test suite, introduced at the 2017 IEEE Congress on Evolutionary Computation. This suite has 30 functions with diverse properties, including unimodal, multimodal, hybrid, and composite functions, though F2 is now omitted, leaving 29. Unimodal functions, while simple, are essential for testing an algorithm’s ability to converge on the global optimum. Multimodal functions, with their multiple peaks and valleys, assess an algorithm’s ability to locate the global optimum and escape local optima. Hybrid and composite functions, due to their complex structures, provide a comprehensive test of an algorithm’s overall performance. Table 2 summarizes the key details of the CEC-2017 test suite.

Table 2.

Detailed description of CEC2017 test functions.

Type	No	Functions name	Rang	Dimension	Min
Unimodal functions	F1	Shifted and Rotated Bent Cigar Function	[− 100,100]	10/30/50/100	100
	F3	Shifted and Rotated Zakharov Function	[− 100,100]	10/30/50/100	300
Multimodal functions	F4	Shifted and Rotated Rosenbrock’s Function	[− 100,100]	10/30/50/100	400
	F5	Shifted and Rotated Rastrigin’s Function	[− 100,100]	10/30/50/100	500
	F6	Shifted and Rotated Expanded Scaffer’s F6 Function	[− 100,100]	10/30/50/100	600
	F7	Shifted and Rotated Lunacek Bi_Rastrigin Function	[− 100,100]	10/30/50/100	700
	F8	Shifted and Rotated Non-Continuous Rastrigin’s Function	[− 100,100]	10/30/50/100	800
	F9	Shifted and Rotated Levy Function	[− 100,100]	10/30/50/100	900
	F10	Shifted and Rotated Schwefel’s Function	[− 100,100]	10/30/50/100	1000
Hybrid functions	F11	Hybrid Function 1 (N = 3)	[− 100,100]	10/30/50/100	1100
	F12	Hybrid Function 2 (N = 3)	[− 100,100]	10/30/50/100	1200
	F13	Hybrid Function 3 (N = 3)	[− 100,100]	10/30/50/100	1300
	F14	Hybrid Function 4 (N = 4)	[− 100,100]	10/30/50/100	1400
	F15	Hybrid Function 5 (N = 4)	[− 100,100]	10/30/50/100	1500
	F16	Hybrid Function 6 (N = 4)	[− 100,100]	10/30/50/100	1600
	F17	Hybrid Function 6 (N = 5)	[− 100,100]	10/30/50/100	1700
	F18	Hybrid Function 6 (N = 5)	[− 100,100]	10/30/50/100	1800
	F19	Hybrid Function 6 (N = 5)	[− 100,100]	10/30/50/100	1900
	F20	Hybrid Function 6 (N = 6)	[− 100,100]	10/30/50/100	2000
Composition functions	F21	Composition Function 1 (N = 3)	[− 100,100]	10/30/50/100	2100
	F22	Composition Function 2 (N = 3)	[− 100,100]	10/30/50/100	2200
	F23	Composition Function 3 (N = 4)	[− 100,100]	10/30/50/100	2300
	F24	Composition Function 4 (N = 4)	[− 100,100]	10/30/50/100	2400
	F25	Composition Function 5 (N = 5)	[− 100,100]	10/30/50/100	2500
	F26	Composition Function 6 (N = 5)	[− 100,100]	10/30/50/100	2600
	F27	Composition Function 7 (N = 6)	[− 100,100]	10/30/50/100	2700
	F28	Composition Function 8 (N = 6)	[− 100,100]	10/30/50/100	2800
	F29	Composition Function 9 (N = 3)	[− 100,100]	10/30/50/100	2900
	F30	Composition Function 10 (N = 3)	[− 100,100]	10/30/50/100	3000

First, a parameter sensitivity analysis was conducted to identify the optimal parameter settings for the ECBSO algorithm. Subsequently, an ablation study involving six variants of ECBSO was performed to evaluate the effectiveness of different improvement strategies and confirm their contributions to addressing the limitations of CBSO. Lastly, this work compared ECBSO with ten different types of improved algorithms to demonstrate its superiority. During the experimentation phase, all algorithms’ best values (Min), averages (Ave), standard deviations (Std), and rankings (Rank) across each CEC-2017 function were fully documented. To avoid overly lengthening the main text with detailed tables, the raw data are placed in the appendix. In the main body, this work primarily uses the Wilcoxon rank-sum test, the Friedman test, and the Nemenyi post-hoc test for result analysis. The Wilcoxon rank-sum test conducts pairwise comparisons of two algorithms’ results across all test functions to determine statistical superiority. The Friedman test assesses whether there are significant overall differences among all algorithms in the experimental set. If the Friedman test indicates significant differences, the Nemenyi post-hoc test is applied, using critical difference diagrams to identify specific algorithmic differences. All statistical tests use a significance level of α = 0.05 to ensure reliable conclusions.

Parameter sensitivity analysis

The DES contains two key parameters, b and c, that decide how many agents are produced by the DES in each iteration. The number added directly controls ECBSO’s global exploration strength and the quality of the population. Moreover, because the covariance matrix is built from the elite set, the choice of population size N also matters. To find the best values for b, c, and N, this study ran a parameter-sensitivity study. In this study N was set to 10D, 30D, 50D, and 70D, while b and c were tested from 0.1 to 0.5 in steps of 0.1. Experiments were carried out separately for each population size, and the best settings found under each size were then compared to determine the final configuration. Figure 4 shows the Friedman ranks of ECBSO under the different parameter combinations.

Fig. 4.

Friedman ranking of ECBSO with different parameters. (a) N = 10D, (b) N = 30D, (c) N = 50D, (d) N = 70D.

Figure 6 reveals that optimal parameter values for b and c are contingent on population size. When N = 10D, the configuration b = 0.5, c = 0.5 yields the best overall performance and attains the top rank on the 50D and 100D functions. For N = 30D, the combination b = 0.5, c = 0.3 achieves the lowest average rank (5.767) and leads on the 10D, 30D, and 100D instances. At N = 50D, b = 0.4 and c = 0.5 again secure the best overall result, placing ECBSO first on the 30D and 50D problems. Finally, when N is increased to 70D, the optimal setting shifts to b = 0.4 and c = 0.3, which still yields the highest overall ranking. This dependence of optimal parameters on population size stems from the fact that N governs the size of the elite subset, thereby directly influencing the fidelity of the learned probability model. Higher values of b consistently correspond to improved rankings, because they ensure a sufficient influx of DES agents early in the search. These high-quality agents enhance population diversity and promote global exploration. In contrast, small values of b delay the introduction of DES agents, weakening global search and increasing the likelihood of premature convergence. Even if additional DES agents are generated later, their distribution is conditioned on individuals that have already converged to local optima, rendering them ineffective at escaping these regions.

Fig. 6.

Nemenyi post hoc test of ECBSO with different Strategies.

Table 3 presents the Friedman-test results for ECBSO under the best-tuned parameter settings across different population sizes. The raw data for Table 3 are summarized in Tables S1–S4 in supplementary materials. It is evident that the configuration with N = 10 achieves the highest rank and is markedly superior to all others. This once again underscores the critical influence of population size on the performance of DES. Therefore, in all subsequent experiments, N, b, and c are fixed at 10D, 0.5, and 0.5, respectively.

Table 3.

Friedman test results obtained by ECBSO with different parameters.

Dimension	N = 10D b = 0.5 c = 0.5	N = 30D b = 0.5 c = 0.3	N = 50D b = 0.4 c = 0.5	N = 70D b = 0.4 c = 0.3
D = 10	1.138	1.966	3.034	3.862
D = 30	1.000	2.000	3.000	4.000
D = 50	1.000	2.069	3.000	3.931
D = 100	1.000	2.103	3.000	3.897
Average rank	1.034	2.034	3.009	3.922

Ablation experiments analysis

In this section, to quantify the contribution of each improvement strategy in ECBSO, ablation studies using six variants of ECBSO were conducted . Table 4 summarizes the configurations of these variants. In this table, the first row lists the variant labels, while the first column indicates the strategies employed; “Y” signifies inclusion, and “N” signifies exclusion of the corresponding mechanism. For instance, CBSO-F incorporates the FSS mechanism, whereas CBSO-FR excludes the DES mechanism. Such experiments play a crucial role in validating the robustness and reliability of research findings. By selectively removing specific components or factors and observing the resulting performance changes, researchers can better isolate the impact of each mechanism, thereby eliminating alternative explanations. This process helps to clarify the contribution of each component, ensuring that the conclusions drawn are valid and reproducible.

Table 4.

Details of ECBSO variants with different strategies.

Algorithm	CBSO-F	CBSO-R	CBSO-D	CBSO-FR	CBSO-FD	CBSO-RD	ECBSO
FSS	Y	N	N	Y	Y	N	Y
RPS	N	Y	N	Y	N	Y	Y
DES	N	N	Y	N	Y	Y	Y

Tables S5–S8 in supplementary materials summarize the experimental results of the ECBSO algorithm and its variants. Table 5 presents the Friedman test results for the ablation study of ECBSO, with the corresponding ranks displayed in Fig. 5. As indicated by the p-value in the last column, there is a statistically significant difference between ECBSO, CBSO, and the derived algorithms. Based on these Friedman test results, the following conclusions can be drawn: (1) ECBSO, which integrates all three improvement strategies, performs the best across functions of different dimensions, achieving average ranks of 1.448, 1.069, 1.000, and 1.000. (2) The combination of all three strategies does not hinder each other but rather collectively enhances the performance of ECBSO. (3) Each of the three strategies is effective when applied individually to CBSO, and their pairwise combinations also boost CBSO’s performance. (4) The impact of the three strategies on ECBSO, in descending order, is: FSS, DES, and RPS. (5) FSS performs better in high-dimensional scenarios, DES excels in low-dimensional ones, and RPS is insensitive to changes in dimensionality. In addition, the scalability of the ECBSO algorithm can be verified by comparing it with the basic CBSO algorithm. Scalability analysis helps to understand how algorithms adapt to increasing problem size and complexity. By systematically changing the dimensions, this work evaluated the algorithm’s ability to maintain efficiency and solution quality under growing computational demands. This evaluation is crucial for verifying real-world applicability, as scalable algorithms are better suited to the high-dimensional complex problems commonly found in industrial and engineering fields. The results in Table 5 show that the ECBSO algorithm significantly outperforms the CBSO algorithm in all dimensions, demonstrating its adaptability in high-dimensional optimization scenarios.

Table 5.

Friedman test results of ECBSO with different Strategies.

Test suite	Dimension	CBSO	CBSO-F	CBSO-R	CBSO-D	CBSO-FR	CBSO-FD	CBSO-RD	ECBSO	p-value
CEC-2017	10	7.517	4.552	5.759	5.345	3.172	4.103	4.103	1.448	8.46E−21
	30	7.966	4.069	6.862	5.862	2.586	2.828	4.759	1.069	3.87E−36
	50	7.966	4.034	6.862	6.034	2.552	2.655	4.897	1.000	5.12E−38
	100	7.966	3.966	6.828	6.138	2.483	2.655	4.966	1.000	1.34E−38
Mean rank		7.853	4.155	6.578	5.845	2.698	3.060	4.681	1.129	N/A
Overall rank		8	4	7	6	2	3	5	1	N/A

Fig. 5.

Rankings of ECBSO with different Strategies.

To quantify the contributions of the three strategies to the performance improvement of ECBSO, a Nemenyi post-hoc test was conducted based on the results in Table 5, with the results illustrated in Fig. 7. The critical difference value (CDV) was calculated according to Eq. (15), where represents the number of algorithms and represents the number of functions tested.

15

Fig. 7.

Rank sorting radar chart of ECBSO and competing algorithms.

According to the Nemenyi test criterion, there is no statistically significant difference between any pair of algorithms whose rank difference is less than the CDV. As shown in Fig. 6, there is no significant difference between ECBSO and CBSO-FD as well as CBSO-FR, on the 30D/50D/100D functions, and no significant difference between ECBSO and CBSO-FR on the 10D function. This implies that the absence of either RPS or DES alone does not significantly weaken the performance of ECBSO. There is also no significant difference between CBSO-R and CBSO across all dimensions, indicating that although RPS can enhance the performance of CBSO, the improvement is not substantial enough. On the other hand, Table 6 summarizes the results of the Wilcoxon rank-sum test based on the ablation experiments. This work recorded, for each CBSO variant, the number of functions on which it is superior to, similar to, or inferior to the basic CBSO. All variants that incorporate one or two of the proposed strategies significantly surpass the original algorithm on most test cases. Among them, the DES-based variants—CBSO-D, CBSO-FD, CBSO-RD, and ECBSO—are occasionally worse than CBSO on 10-D problems, indicating that DES can slightly impair performance at this low dimension. Nevertheless, every strategy improves CBSO on the majority of functions. We therefore conclude that the proposed enhancements are both effective and significant.

Table 6.

Wilcoxon rank sum test results of EECO with different Strategies.

vs. CBSO + / = /-	CEC-2017 test suite
	10D	30D	50D	100D
CBSO-F	25/4/0	29/0/0	29/0/0	29/0/0
CBSO-R	27/2/0	29/0/0	29/0/0	28/1/0
CBSO-D	25/3/1	28/1/0	28/1/0	27/2/0
CBSO-FR	26/3/0	29/0/0	29/0/0	29/0/0
CBSO-FD	26/1/2	29/0/0	29/0/0	29/0/0
CBSO-RD	26/1/2	27/2/0	27/2/0	28/1/0
ECBSO	27/1/1	29/0/0	29/0/0	29/0/0

Comparison analysis with improved algorithms

This section compares ECBSO with ten state-of-the-art meta-heuristics drawn from five categories: human-based (ISGTOA, EMTLBO), evolution-based (LSHADE, APSM-jSO), swarm-based (GLS-MPA, ESLPSO), physics-inspired (ACGRIME, IYDSE), and mathematics-based (EPSCA, QAGO). Detailed results of these algorithms on the CEC-2017 suite are reported in Supplementary materials, Tables S9–S12, and the corresponding ranks are visualized in Fig. 7. A radar chart is employed to provide a global view: each algorithm is depicted by a distinct colored line, and the enclosed area reflects overall performance—smaller areas denote better results. Inspection of Fig. 7 shows that ECBSO occupies the smallest region, offering a first indication of its superiority. Specifically, in the 10-dimensional benchmark, ECBSO achieved the top rank on 14 functions; when tackling 30-dimensional problems, it secured first place 20 times; for 50-dimensional functions, it emerged as the best-performing algorithm on 11 instances; and in the 100-dimensional case, it ranked first on 7 functions. In the following subsections, the outcomes of Wilcoxon rank-sum, Friedman, and Nemenyi tests, together with convergence and robustness analyses were present , all derived from the data in the appendix.

Table 7 summarizes the Wilcoxon rank-sum test results between ECBSO and the ten compared algorithms—ISGTOA, EMTLBO, LSHADE, APSM-jSO, GLS-MPA, ESLPSO, ACGRIME, IYDSE, EPSCA, and QAGO—where the symbols “ + ”, “–”, and “ = ” denote that ECBSO is significantly better, significantly worse, or not statistically different from the competitor, respectively. The outcome is visualized in Fig. 8. Across all pairwise comparisons except those involving APSM-jSO, the cumulative count of “+” symbols overwhelmingly exceeds that of “–” or “ = ”. When compared with APSM-jSO, ECBSO maintains a clear advantage on all dimensions except 100D, where the two algorithms perform similarly. These findings consistently demonstrate the superior performance of ECBSO on the benchmark suite, thereby confirming its robustness and efficiency in tackling complex optimization tasks. A detailed Wilcoxon rank-sum analysis is provided below.

Table 7.

Wilcoxon rank sum test results of ECBSO and competing algorithms.

ECBSO vs. + / = /-	Dimension	CBSO	ISGTOA	EMTLBO	LSHADE	APSM-jSO	GLS-MPA	ESLPSO	ACGRIME	IYDSE	EPSCA	QAGO
CEC-2017 test suite	10D	27/1/1	25/2/2	26/3/0	22/4/3	19/7/3	26/1/2	27/2/0	25/4/0	26/1/2	28/1/0	25/4/0
	30D	29/0/0	24/4/1	24/3/2	27/1/1	21/5/3	26/2/1	27/1/1	24/4/1	29/0/0	25/4/0	28/0/1
	50D	29/0/0	25/1/3	26/1/2	27/0/2	20/5/4	26/1/2	27/2/0	25/2/2	29/0/0	26/2/1	29/0/0
	100D	29/0/0	24/1/4	26/2/1	27/0/2	11/7/11	25/2/2	26/1/2	25/2/2	29/0/0	23/5/1	29/0/0

Fig. 8.

The number of “+/=/−” obtained by ECBSO and competing algorithms.

For 10D functions, ECBSO is superior/similar/inferior to CBSO, ISGTOA, EMTLBO, LSHADE, APSM-jSO, GLS-MPA, ESLPSO, ACGRIME, IYDSE, EPSCA, and QAGO on 27/1/1, 25/2/2, 26/3/0, 22/4/3, 19/7/3, 26/1/2, 27/2/0, 25/4/0, 26/1/2, 28/1/0, and 25/4/0 test functions. That is, when matched against each of the competing algorithms, ECBSO exhibited a statistically significant advantage on at least 19 functions.

For 30D functions, ECBSO is superior/similar/inferior to CBSO, ISGTOA, EMTLBO, LSHADE, APSM-jSO, GLS-MPA, ESLPSO, ACGRIME, IYDSE, EPSCA, and QAGO on 29/0/0, 24/4/1, 24/3/2, 27/1/1, 21/5/3, 26/2/1, 27/1/1, 24/4/1, 29/0/0, 25/4/0, and 28/0/1 test functions. That is, when matched against each of the competing algorithms, ECBSO exhibited a statistically significant advantage on at least 21 functions.

For 50D functions, ECBSO is superior/similar/inferior to CBSO, ISGTOA, EMTLBO, LSHADE, APSM-jSO, GLS-MPA, ESLPSO, ACGRIME, IYDSE, EPSCA, and QAGO on 29/0/0, 25/1/3, 26/1/2, 27/0/2, 20/5/4, 26/1/2, 27/2/0, 25/2/2, 29/0/0, 26/2/1, and 29/0/0 test functions. That is, when matched against each of the competing algorithms, ECBSO exhibited a statistically significant advantage on at least 20 functions.

For 100D functions, ECBSO is superior/similar/inferior to CBSO, ISGTOA, EMTLBO, LSHADE, APSM-jSO, GLS-MPA, ESLPSO, ACGRIME, IYDSE, EPSCA, and QAGO on 29/0/0, 24/1/4, 26/2/1, 27/0/2, 11/7/11, 25/2/2, 26/1/2, 25/2/2, 29/0/0, 23/5/1, and 29/0/0 test functions. That is, when matched against each of the competing algorithms except for APSM-jSO, ECBSO exhibited a statistically significant advantage on at least 23 functions. ECBSO and APSM-jSO have similar performance.

Pairwise comparisons consistently demonstrate ECBSO’s marked superiority over the competing algorithms. To assess overall performance differences, the Friedman test was subsequently applied; the corresponding rankings are reported in Table 8 and visualized in Fig. 9. According to the p-values in Table 8, significant differences exist between ECBSO and the reference algorithms across all dimensional settings. Specifically, ECBSO attains Friedman scores of 2.103, 1.586, 1.828, and 2.103 for the 10-, 30-, 50-, and 100-dimensional problems, respectively, whereas CBSO consistently ranks last in every case. A detailed Friedman test analysis is provided below.

Table 8.

Friedman test results of ECBSO and competing algorithms.

Test suite	Dimension	ECBSO	CBSO	ISGTOA	EMTLBO	LSHADE	APSM-jSO	GLS-MPA	ESLPSO	ACGRIME	IYDSE	EPSCA	QAGO	p-value
CEC-2017	10	2.103	9.345	6.586	7.414	6.690	5.069	5.966	6.690	6.931	8.345	7.793	5.069	1.79E−13
	30	1.586	11.138	5.207	6.276	10.138	4.138	6.621	5.690	5.345	9.414	5.345	7.103	1.18E−31
	50	1.828	11.000	5.069	6.414	10.724	3.690	7.310	5.345	5.621	8.690	5.000	7.310	1.48E−32
	100	2.103	10.931	4.759	5.897	10.897	3.207	8.138	4.724	6.172	8.310	5.034	7.828	5.20E−35
Mean rank		1.905	10.603	5.405	6.500	9.612	4.026	7.009	5.612	6.017	8.690	5.793	6.828	N/A
Overall rank		1	12	3	7	11	2	9	4	6	10	5	8	N/A

Fig. 9.

Friedman rankings of ECBSO and competing algorithms.

On the 10-dimensional benchmark functions, ECBSO, APSM-jSO, and QAGO occupy the top three positions, followed in order by GLS-MPA, ISGTOA, LSHADE/ESLPSO, ACGRIME, EMTLBO, EPSCA, IYDSE, and CBSO. Overall, ECBSO outperforms the second-ranked algorithm by a margin of 2.966 in Friedman score, establishing itself as the best-performing optimizer.

On the 30-dimensional benchmark functions, ECBSO, APSM-jSO, and ISGTOA occupy the top three positions, followed in order by ACGRIME/EPSCA, ESLPSO, EMTLBO, GLS-MPA, QAGO, IYDSE, LSHADE, and CBSO. Overall, ECBSO outperforms the second-ranked algorithm by a margin of 2.552 in Friedman score, establishing itself as the best-performing optimizer.

On the 50-dimensional benchmark functions, ECBSO, APSM-jSO, and EPSCA occupy the top three positions, followed in order by ISGTOA, ESLPSO, ACGRIME, EMTLBO, GLS-MPA/QAGO, IYDSE, LSHADE, and CBSO. Overall, ECBSO outperforms the second-ranked algorithm by a margin of 1.862 in Friedman score, establishing itself as the best-performing optimizer.

On the 100-dimensional benchmark functions, ECBSO, APSM-jSO, and ESLPSO occupy the top three positions, followed in order by ISGTOA, EPSCA, EMTLBO, ACGRIME, QAGO, GLS-MPA, IYDSE, LSHADE, and CBSO. Overall, ECBSO outperforms the second-ranked algorithm by a margin of 1.103 in Friedman score, establishing itself as the best-performing optimizer.

To quantify the magnitude of these disparities, the Nemenyi post-hoc test described in Section "Ablation experiments analysis" was performed; the results are illustrated in Fig. 10. Across the 10D, 30D, and 100D benchmarks, no statistically significant differences are observed between ECBSO and APSM-jSO. Likewise, ECBSO shows no statistically significant difference from ESLPSO and ISGTOA on the 10D problems. Significant performance gaps are observed between ECBSO and all remaining algorithms across all four-dimensional settings.

Fig. 10.

Nemenyi post hoc test of ECBSO and competing algorithms.

Figure 11 presents the convergence curves of ECBSO alongside its competing algorithms. Convergence curves are indispensable for tracking the optimization process, as they directly reveal convergence speed and ultimate accuracy while exposing common pitfalls such as premature stagnation or oscillatory behavior. By visualizing the evolution of fitness values, researchers can gauge each method’s efficiency and make informed parameter refinements to enhance performance. These plots also serve as diagnostic instruments for assessing an algorithm’s ability to adapt to varying problem complexities, rendering them integral to both algorithmic design and performance evaluation. In this section, convergence curves are provided for six representative test functions—unimodal F3, multimodal F8, hybrid F12 and F18, and composite F23 and F29—where the x-axis denotes the number of fitness evaluations and the y-axis denotes the corresponding fitness values. The results demonstrate that ECBSO attains rapid convergence and the highest-accuracy solutions on the unimodal function F3. This is attributable to the FSS module, which dynamically selects search strategies and enables ECBSO to concentrate on local exploitation when high precision is required. Moreover, on unimodal landscapes, the DES component effectively aligns the evolutionary direction, thereby accelerating convergence. When tackling the multimodal function F8, ECBSO maintains a steady search trajectory without premature stagnation—an outcome credited to the RPS mechanism. On 100-dimensional instances of F3 and F8, however, ECBSO’s performance declines slightly. This is because the high-dimensional search space prevents the elite subpopulation employed by DES from adequately representing the overall population’s evolutionary direction. On more challenging hybrid and composition functions, ECBSO consistently delivers competitive or superior optimization results, underscoring its adaptability. Such robustness in complex scenarios stems from the synergistic contributions of RPS and DES. RPS adaptively determines the number of supplementary agents based on the population state, curbing aimless exploration. Concurrently, DES employs a covariance matrix that accurately captures the population’s evolutionary trend, while its probabilistic model spans the entire search space. The initial sampling distribution is broad, inherently favoring global exploration and reducing the risk of entrapment in local optima.

Fig. 11.

Convergence curves of ECBSO and competing algorithms.

Figure 12 employs box-and-whisker plots to succinctly depict the solution distributions obtained by ECBSO and the competing algorithms on the CEC-2017 test suite. These plots simultaneously convey the median, inter-quartile range (IQR), and outliers, thereby revealing each algorithm’s central tendency, dispersion, and extreme-case performance. Outliers and box height quickly expose any performance fluctuations or failures, and the representation is more robust to extreme values than summary statistics based solely on the mean. In these diagrams, a lower and narrower box signifies convergence with both high accuracy and high stability. In this subsection, boxplots are provided for six representative test functions—unimodal F3, multimodal F8, hybrid F12 and F18, and composition F23 and F29—where the x-axis denotes the algorithms and the y-axis denotes the corresponding fitness values. As Fig. 12 clearly shows, ECBSO exhibits markedly superior robustness and stability compared with its competitors. Across all plots, ECBSO consistently displays the shortest boxes and the fewest outliers, indicating minimal variability. Collectively, the experimental evidence demonstrates that ECBSO comprehensively outperforms the compared methods on the CEC-2017 benchmark, attesting to its high reliability and exceptional adaptability in tackling global optimization problems.

Fig. 12.

Boxplots of ECBSO and competing algorithms.

Running time analysis

To guarantee both fairness and practical relevance, we measured the actual CPU-runtime of every algorithm on the same hardware and report it together with the theoretical time-complexity analysis. Table 9 lists the mean CPU time of ECBSO and its competitors on the CEC-2017 suite at 10-D, 30-D, 50-D and 100-D. According to Table 10, ECBSO ranks eighth while the original CBSO ranks second, indicating that ECBSO consumes more wall-clock time under the same number of function evaluations. This increase is mainly caused by the additional cost of updating and sampling the covariance matrix, and the gap widens as dimensionality grows because the matrix operations become more expensive when capturing evolutionary trends in high-dimensional spaces. The PM mechanism, which simultaneously probes multiple directions, also contributes to the extra runtime. On the other hand, the algorithms with larger CPU times usually achieve better solution quality—e.g., ACGRIME (the most time-consuming) holds the sixth Friedman rank, and the second- and third-most expensive methods, ISGTOA and APSM-jSO, are only outperformed by ECBSO on the CEC-2017 set—confirming that more sophisticated search operators generally require longer runs. Overall, ECBSO trades a moderate runtime increase for a substantial gain in optimization accuracy, so the additional CPU cost is acceptable for off-line or non-real-time tasks. We also recognize that the current version of ECBSO is not yet suitable for scenarios with stringent real-time requirements, and further efficiency improvements will be pursued in future work.

Table 9.

Runtime results of ECBSO and competitors.

Test suite	Dimension	ECBSO	CBSO	ISGTOA	EMTLBO	LSHADE	APSM-jSO	GLS-MPA	ESLPSO	ACGRIME	IYDSE	EPSCA	QAGO
CEC-2017	10	0.0419	0.0276	0.0582	0.0352	0.0601	0.0527	0.0172	0.0502	0.0816	0.0311	0.0299	0.0444
	30	0.3241	0.2159	0.4523	0.2492	0.3228	0.4717	0.1350	0.3789	0.6882	0.2402	0.2495	0.2992
	50	1.0569	0.7192	1.5094	0.8513	0.9709	1.6613	0.4802	1.2897	2.3459	0.8112	0.8567	0.9878
	100	5.2475	3.4046	7.2656	3.9542	4.2336	8.4966	2.5223	6.0632	11.6306	3.9394	4.1036	4.3085
Average time		1.6676	1.0918	2.3214	1.2725	1.3968	2.6706	0.7886	1.9455	3.6866	1.2555	1.3099	1.4100
Overall rank		8	2	10	4	6	11	1	9	12	3	5	7

Table 10.

Details of real-world constrained engineering optimization problems.

Problem	Name	D	g	h
RC01	Tension/compression spring design problem	3	3	0
RC02	Pressure vessel design problem	4	4	0
RC03	Three-bar truss design problem	2	3	0
RC04	Welded beam design problem	4	5	0
RC05	Gear train design problem	4	1	1
RC06	Cantilever beam design problem	5	1	0
RC07	Multiple disk clutch brake design problem	5	7	0
RC08	Step-cone pulley problem	5	8	3
RC09	Planetary Gear Train Design	9	10	1
RC10	Robot Gripper Problem	7	7	0

Results and analysis on engineering constraint optimization problem

In this section, the performance of ECBSO on constrained optimization is evaluated using ten real-world engineering design problems. These problems are taken from the CEC2020 “Real-World Constrained Optimization Problems” benchmark suite, detailed in Table 10. The benchmark suite is publicly available at https: //github.com /P-N-Suganthan /2020-RW-Constrained -Optimization. Following common practice, the original constrained problems are transformed into unconstrained ones via a penalty-function framework: whenever any constraint is violated, a large penalty value is added to the corresponding fitness value, thereby driving infeasible solutions out of the search population during the evolutionary process.

Table 11 presents a comprehensive comparison between ECBSO and the competing algorithms, including the results of the Friedman test and the Wilcoxon rank-sum test. According to Table 12, ECBSO secures the top position with a Friedman score of 2.550. The “ + / = /-” entries indicate the extent of ECBSO’s superiority: the number of “ + ” symbols consistently outweigh that of “–”. Although APSM-jSO and QAGO each outperform ECBSO on one constrained engineering problem, ECBSO surpasses them on seven and five constrained engineering problem, respectively. Figure 13 visually summarizes the ranking of ECBSO and its competitors across the engineering constrained benchmarks. The surface formed by ECBSO’s ranks is the lowest, indicating the best overall performance. This superiority stems from the synergistic effect of the three proposed mechanisms. FSS dynamically selects the most appropriate search operator at each generation. If the offspring generated by the current operator violates any constraint, a large penalty value is assigned and the individual is discarded; consequently, FSS reduces the probability of selecting that operator in the next iteration and prefers feasible-producing alternatives. DES captures the evolutionary trend by computing the covariance matrix of the elite feasible individuals, guiding the population toward promising regions and implicitly decreasing the chance of generating constraint-violating solutions. RPS continuously re-balances exploration and exploitation by measuring population diversity; when diversity drops, it injects new samples, thereby maintaining pressure toward the feasible space without premature convergence. Together, these mechanisms enable ECBSO to maintain a high proportion of feasible solutions while still refining the objective value, resulting in the consistently low ranks observed in Fig. 13. Overall, ECBSO algorithm demonstrates strong and reliable performance across all ten engineering constrained optimization problems, underscoring its potential for tackling real-world optimization tasks.

Table 11.

Statistical results of ECBSO and competing algorithms in constrained engineering problems.

No	Index	ECBSO	CBSO	ISGTOA	EMTLBO	LSHADE	APSM-jSO	GLS-MPA	ESLPSO	ACGRIME	IYDSE	EPSCA	QAGO
RC1	Best	1.2816E-02	1.2864E-02	1.3110E-02	1.2845E-02	1.3811E-02	1.2960E-02	1.2749E-02	1.3104E-02	1.3786E-02	1.2921E-02	1.3117E-02	1.2998E-02
	Mean	1.3372E-02	1.3705E-02	1.4852E-02	1.3544E-02	1.7757E-02	1.5201E-02	1.5238E-02	1.4949E-02	1.8584E-02	1.3447E-02	1.8879E-02	1.3809E-02
	Std	7.8313E-04	7.5404E-04	2.0163E-03	7.9839E-04	2.7475E-03	1.2327E-03	5.2568E-03	1.3353E-03	2.5818E-03	4.9677E-04	4.2920E-03	5.8259E-04
	Rank	1	4	6	3	10	8	9	7	11	2	12	5
RC2	Best	6.0824E + 03	8.4803E + 03	1.0595E + 04	6.4851E + 03	6.4427E + 03	6.8610E + 03	6.8434E + 03	7.5784E + 03	6.4638E + 03	1.0554E + 04	8.9199E + 03	6.6155E + 03
	Mean	6.9076E + 03	1.3187E + 04	2.5256E + 04	1.0982E + 04	8.1935E + 03	8.4906E + 03	1.4697E + 04	1.1454E + 04	7.9487E + 03	1.9122E + 04	1.6100E + 04	7.2930E + 03
	Std	4.9828E + 02	3.6504E + 03	1.0287E + 04	5.7372E + 03	1.6071E + 03	1.2343E + 03	1.1127E + 04	2.1558E + 03	1.4232E + 03	5.9985E + 03	9.0737E + 03	4.7453E + 02
	Rank	1	8	12	6	4	5	9	7	3	11	10	2
RC3	Best	2.6389E + 02	2.6407E + 02	2.6397E + 02	2.6389E + 02	2.6393E + 02	2.6389E + 02	2.6391E + 02	2.6390E + 02	2.6389E + 02	2.6404E + 02	2.6390E + 02	2.6390E + 02
	Mean	2.6397E + 02	2.6484E + 02	2.6454E + 02	2.6419E + 02	2.6446E + 02	2.6395E + 02	2.6408E + 02	2.6400E + 02	2.6424E + 02	2.6444E + 02	2.6432E + 02	2.6395E + 02
	Std	9.7197E-02	6.0850E-01	7.8925E-01	3.9160E-01	4.6806E-01	3.9420E-02	1.8921E-01	8.5548E-02	3.0963E-01	3.5266E-01	3.4870E-01	9.5738E-02
	Rank	3	12	11	6	10	1	5	4	7	9	8	2
RC4	Best	1.7252E + 00	1.9130E + 00	2.2727E + 00	1.7418E + 00	1.7795E + 00	1.7402E + 00	1.7910E + 00	1.8770E + 00	1.7112E + 00	1.9190E + 00	1.8121E + 00	1.7453E + 00
	Mean	1.7636E + 00	2.1384E + 00	2.6198E + 00	2.2635E + 00	2.1629E + 00	1.8375E + 00	2.0880E + 00	2.0194E + 00	2.2113E + 00	2.4180E + 00	2.1136E + 00	1.8741E + 00
	Std	3.3086E-02	1.3745E-01	2.5464E-01	4.3722E-01	4.3011E-01	6.1775E-02	1.9728E-01	1.4227E-01	5.2962E-01	3.2101E-01	2.2863E-01	8.0347E-02
	Rank	1	7	12	10	8	2	5	4	9	11	6	3
RC5	Best	2.3078E-11	2.3078E-11	9.9216E-10	2.3078E-11	1.1661E-10	2.7009E-12	6.6021E-10	2.7009E-12	2.3078E-11	2.7009E-12	8.8876E-10	2.7009E-12
	Mean	1.6321E-09	3.0265E-09	2.1254E-08	3.9444E-09	7.7109E-09	1.7825E-09	1.4301E-08	4.6431E-09	9.0523E-09	5.9833E-09	1.8690E-08	9.5359E-10
	Std	1.1203E-09	2.0402E-09	4.0240E-08	6.7198E-09	8.2026E-09	1.1217E-09	1.9752E-08	7.1405E-09	1.1751E-08	6.8065E-09	2.5597E-08	9.9231E-10
	Rank	2	4	12	5	8	3	10	6	9	7	11	1
RC6	Best	1.3706E + 00	1.5147E + 00	2.2300E + 00	1.3992E + 00	1.3515E + 00	1.3418E + 00	1.6682E + 00	1.6374E + 00	1.3481E + 00	1.3570E + 00	1.5329E + 00	1.3477E + 00
	Mean	1.3982E + 00	1.9453E + 00	3.1166E + 00	2.1003E + 00	1.5107E + 00	1.3742E + 00	1.9959E + 00	1.9166E + 00	1.4427E + 00	1.4149E + 00	2.2050E + 00	1.3628E + 00
	Std	2.2886E-02	2.7854E-01	4.5483E-01	5.5898E-01	1.3132E-01	2.5810E-02	2.8606E-01	1.5990E-01	6.7557E-02	3.0162E-02	4.5951E-01	1.7523E-02
	Rank	3	8	12	10	6	2	9	7	5	4	11	1
RC7	Best	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08
	Mean	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9248E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08	3.9247E + 08
	Std	6.1697E-08	6.1697E-08	6.1697E-08	6.1697E-08	6.1697E-08	1.3631E + 04	6.1697E-08	6.1697E-08	6.1697E-08	6.1697E-08	6.1697E-08	1.0831E + 03
	Rank	1	1	1	1	1	12	1	1	1	1	1	11
RC8	Best	1.6431E + 01	2.8714E + 01	1.9256E + 01	1.7821E + 01	1.6202E + 01	1.7065E + 01	1.7100E + 01	1.8444E + 01	1.6660E + 01	4.6347E + 01	1.7793E + 01	1.8382E + 01
	Mean	1.7767E + 01	1.1076E + 02	4.4381E + 01	2.6775E + 01	1.8398E + 01	1.9019E + 01	4.4727E + 01	2.0769E + 01	2.0797E + 01	1.3690E + 02	2.1949E + 01	2.7821E + 01
	Std	4.9514E-01	6.8613E + 01	3.5806E + 01	8.8469E + 00	1.7662E + 00	1.5836E + 00	5.1647E + 01	2.3505E + 00	8.5968E + 00	7.8553E + 01	5.2532E + 00	1.8471E + 01
	Rank	1	11	9	7	2	3	10	4	5	12	6	8
RC9	Best	2.3646E-01	2.8130E-01	2.6051E-01	2.4796E-01	2.3868E-01	2.4640E-01	2.4031E-01	2.5458E-01	2.3755E-01	2.4090E-01	2.3535E-01	2.3829E-01
	Mean	2.4967E-01	3.9964E-01	3.4319E-01	2.9245E-01	2.6844E-01	2.8706E-01	3.1092E-01	3.1453E-01	2.4812E-01	2.6030E-01	4.1844E-01	2.4926E-01
	Std	9.5905E-03	1.5350E-01	1.0111E-01	4.3873E-02	3.0638E-02	2.3996E-02	8.5854E-02	3.9949E-02	1.3635E-02	1.7249E-02	3.7895E-01	8.8462E-03
	Rank	3	11	10	7	5	6	8	9	1	4	12	2
RC10	Best	1.0248E-16	2.8242E-16	1.4185E-16	3.9481E-16	1.4148E-16	5.1447E + 00	1.9392E-16	4.0938E-16	1.6016E-16	5.7031E-16	2.0069E-15	4.3873E + 00
	Mean	3.2014E + 00	8.8317E-16	8.8194E-01	3.2910E + 00	4.0853E + 00	6.0552E + 00	2.5024E + 00	5.0014E + 00	3.4980E + 00	1.7500E + 00	4.6700E + 00	5.7969E + 00
	Std	2.7936E + 00	5.9000E-16	2.3401E + 00	3.7962E + 00	2.7429E + 00	6.8838E-01	3.4015E + 00	2.4684E + 00	2.2900E + 00	3.0654E + 00	2.4239E + 00	7.2434E-01
	Rank	5	1	2	6	8	12	4	10	7	3	9	11
Friedman Rank		2.550	7.150	9.150	6.550	6.650	5.400	7.450	6.350	6.250	6.850	9.050	4.600
+ / = /-		N/A	8/2/0	8/2/0	5/5/0	6/4/0	7/2/1	6/4/0	6/4/0	5/5/0	5/5/0	8/2/0	5/4/1

Fig. 13.

Ranking radar graph of ECBSO and competing algorithms.

Conclusions

Drawing on the preceding experimental evidence, this discussion addresses the merits, shortcomings, and prospective avenues for ECBSO. Ablation analyses corroborate that the FSS, RPS, and DES components are mutually complementary, collectively lifting the basic CBSO to a markedly higher level of performance. Comparative evaluations against recent, well-regarded variants further underscore ECBSO’s consistent superiority across the CEC-2017 test suite, while its convincing results on ten constrained engineering problems attest to its practical utility. Nevertheless, the algorithm still exhibits notable limitations. First, the introduction of several additional parameters has so far been validated only on the CEC-2017 benchmark; their transferability to other problem domains remains an open question. Second, the current FSS mechanism could benefit from a success-history memory: instead of randomly selecting a fallback strategy when the active one proves ineffective, future versions could probabilistically favor historically successful options. Finally, the RPS strategy randomly selects agents from the population. If it is possible to further determine whether an individual agent needs to be updated based on its current state, this should be beneficial to improving the performance of the algorithm.

Building upon the original CSBO, this study introduces ECBSO, an enhanced variant that integrates a feedback selection strategy (FSS), a regenerative population strategy (RPS), and a distribution-estimation strategy (DES). A systematic parameter-sensitivity analysis was conducted to establish the optimal parameter configuration for ECBSO. Ablation experiments confirmed the effectiveness of each proposed component. Comparisons with other advanced improved algorithms on the CEC-2017 test set and engineering constraint optimization problems demonstrate that ECBSO offers a powerful balance of exploitation and exploration, and is well-suited to practical optimization tasks. Future work will focus on extending ECBSO to the multi-objective domain, as contemporary real-world problems are often multi-objective and heavily constrained. Additionally, we will investigate its application to single-objective scenarios such as path planning, wireless sensor network coverage optimization, and cloud resource scheduling.

Author contributions

Yuchen Yin: conceptualization, methodology, writing, data testing, reviewing, software, supervision, formal analysis. Haipeng Liu: writing, data testing, reviewing. Shanshan Cai: conceptualization, methodology, writing, reviewing. Yun Ye: conceptualization, visualization, reviewing, formal analysis, supervision, project administration.

Funding

This work was supported in part by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LQN25E080011), Ningbo Natural Science Foundation (Grant No. 2024J440).

Data availability

The data is provided within the manuscript.

Competing interests

The authors declare no competing interests.