An improved human memory algorithm with multi-directional and chaotic approaches for global optimization and energy-efficient cluster head selection in WSNs

In the \(\:j-th\) dimension, the position of the best individual is represented by \(\:{X}_{best,j}\), while \(\:{X}_{ij}^{new}\) indicates the new position following the DDS. The random integer \(\:{r}_{8}\), which ranges from − 1 to 1, determines the direction in which the rime-ice particles move. A variable called \(\:\gamma\:\), which varies depending on the number of evaluations, models environmental conditions. Given the great degree of unpredictability in the particles’ movements, the distance between two particles is represented by \(\:h\), which is arbitrarily chosen between [0, 1]. As stated in the original algorithm, \(\:w\), which regulates the number of segments in the step function, is set to 5. The current and maximum number of evaluations are denoted by \(\:t\) and \(\:T\), respectively. The random number \(\:{r}_{9}\), which ranges from 0 to 1. The primary rationale for using DDS is that it reduces the risk of premature convergence. The rapid and erratic exploration in performance does not allow the algorithm to stick with the local optima. This strategy is very helpful for high-dimensional problems, where the possibility of premature convergence is higher. DDS provides a random contribution that conserves the diversity of solutions throughout the process. Therefore, AEHMO has enhanced global search capability for better convergence toward the global optimum in difficult cases.

Although DDS is mathematically inspired by drift-like motion, its role within AEHMO aligns with cognitive exploratory behavior rather than a physical metaphor. In human decision-making, periods of uncertainty often trigger exploratory deviation from familiar patterns, allowing individuals to search broadly before committing to refined choices. DDS captures this exploratory drift by enabling large, irregular movements early in the search process, while remaining subordinate to the memory recall structure that governs solution evaluation and retention.

Chaotic reverse learning strategy (CRL)

One of the major drawbacks of the original HMO algorithm is related to its convergence towards the local optimum, especially when searching within complex multi-modal search space. The core issue is that the algorithm lacks an effective mechanism for exploring diverse regions of the search space and escaping local optima. The traditional update rules in HMO are not efficient in every case to explore the global search space, which delays convergence and hence reduces performance. In this regard, CRL strategy is developed which is capable of incorporating chaotic mapping with opposition-based learning to improve the exploration capability and enhance the escaping ability of the algorithm from the local optima.

Opposition-based learning (OBL) introduces new solution vectors by creating an opposite population of the current population. This helps the algorithm to find solutions that may not be easily reachable through normal update rules. OBL is useful when there are a large number of local optima spread in the solution space, and the algorithm needs to come out of the false local optima more effectively. The formula for OBL can be written as follows:

where \(\:O{D}_{i}\left(t\right)\) is the position of the \(\:i\)^th opposite individual at iteration \(\:t\). The traditional OBL approaches create an opposite solution only at fixed positions, reducing the flexibility of the algorithm; this may affect its dynamic ability to explore new areas. From this perspective, CRL uses chaotic mapping in its opposition process to introduce structured randomness, hence allowing more flexibility and greater exploration by the algorithm. On the other hand, some works have introduced randomness in the opposition process for:

This random opposition-based learning method introduces a degree of variability, but the overall convergence rate can still be relatively slow. In this case, the CRL strategy utilizes chaotic mapping to oppose the learning strategy to enhance its exploration capability. The chaotic maps differ from random number arrays based on a probability distribution due to their dynamic properties. This results in a more rapid exploration of the solution space. The CRL formula is given as follows:

In this regard, \(\:{F}_{0}\) is a vector of random values in the range of 0–1 and \(\:{F}_{z}\) is a chaotic sequence after \(\:z\) iterations of sine mapping. This chaotic sequence follows some structured randomness, enhancing the exploration capability of this algorithm. Indeed, the proposed chaotic sequence-based opposition brings periodicity to the sine map, allowing the algorithm to probe the solution space more effectively than purely randomly generated opposition. Having too many iterations would increase the running time beyond endurance, whereas having too few iterations could result in a loss of diversity. Therefore, the parameter \(\:z\) is set to 20 as a balance to ensure that the algorithm has enough diversity to explore the solution space without significantly increasing the running time. Moreover, the proposed algorithm uses a greedy process for selection whereby the best \(\:N\) individuals from both original and opposite populations remain in the set of solutions throughout an optimization process that keeps quality as high as possible. While maintaining diversity to avoid premature convergence, the chaotic mapping utilized in the CRL strategy considerably enhances the convergence speed and exploration capability of the basic algorithm. This dynamic approach lets the algorithm provide a more efficient search mechanism and, hence, improved overall performance compared to the standard opposition-based learning methods.

From a memory-theoretic standpoint, CRL represents nonlinear and associative recall behavior, where memories are revisited in a structured yet non-repetitive manner. The chaotic mapping introduces bounded irregularity, mimicking how human recall is neither strictly deterministic nor purely random. By embedding chaotic opposition within the memory-driven framework, AEHMO enhances diversity while maintaining conceptual consistency with cognitive recall processes.

Synergistic rationale of the proposed enhancement strategies

Although AEHMO combines four strategies for enhancement, the combination of the strategies is not arbitrary, but guided by a complementary and phase-oriented design principle. Each strategy addresses one of the different limitations of the original HMO and works at different time and structure levels of the search process to guarantee a cooperative, rather than redundant, behavior.

The adaptive parameters \(\:{w}_{1}\) and \(\:{w}_{2}\) constitute a global control layer that is used to control the exploration-exploitation balance during the optimization process. By replacing the fixed variables \(\:b\) and \(\:\delta\:\) with the variable ones, this layer produces a smooth and non-linear transition from general exploration to specific exploitation to form a stable backbone for the rest of the strategies. Building upon this adaptive control, in the DDS strategy the main action is taken during the early search phase, by adding large-scale directional drift to cover more spatial area and quickly explore unexplored areas of the search space. DDS complements the \(\:{w}_{1}\) by reinforcing the early exploration without interfering with the convergence because its effect naturally fades away with the progress of iteration. The CRL strategy works on the population diversity level and provide a structured mechanism of escaping of local optima, through the chaos opposition. Unlike purely random perturbations, CRL presents deterministic but not repeating transformations, and therefore AEHMO can return to unexplored regions despite the reduced diversity of the population. This makes CRL particularly effective in situations when the influence DDS starts weakening, and the search starts to concentrate. Finally, MDMS is also intentionally limited to a small elite subset of solutions, making up a local refinement layer. By combining both Cauchy and Gaussian mutations, MDMS fills the gap between global jumps and fine-grained tuning, ensuring that high quality regions identified using DDS and CRL are efficiently exploited, rather than prematurely being fixed in position.

Collectively, these strategies form a hierarchically cooperative mechanism of non-conflicting strategies: adaptive parameters regulate the general dynamics of search, DDS and CRL ensure sufficient global coverage and diversity and MDMS stimulates intensification of exploitation in promising regions. Moreover, the selection of these four strategies was based on a limitation-driven design analysis rather arbitrary stacking of operators. Each enhancement represents a different structural deficiency that was observed in the original HMO and preliminary experiments using partial combinations of enhancements showed that the individual enhancements improved specific aspects of performance but did not consistently ensure stable convergence across both benchmark and WSN scenarios. The complete four-strategy configuration was found to have the best trade-off between exploration depth, preservation of diversity and late-stage refinement, which was confirmed by the ablation analysis in the section “Ablation experiment”.

Computational complexity

The performance of an optimization algorithm defines its applicability across various problem domains, with computational complexity being the most important factor. The pseudocode and flowchart of the proposed AEHMO are depicted in Algorithm 2 and Fig. 2. According to Algorithm 2, in the case of the AEHMO algorithm, it can be analyzed concerning its building phases. Let \(\:N\) denotes the number of solutions within the population, \(\:T\) denotes the number of maximum function evaluations, and \(\:D\) denotes the dimensionality of the problem.

• This initialization step is done once before the iteration starts. During that time, the population of solutions was created: initializing \(\:N\) solutions, each having \(\:D\) dimensions. So, the complexity in this phase will be \(\:O(N\times\:D)\), because it corresponds to generating a diverse initial population.
• The memory behavior update phase then runs for every iteration once the population has been initialized. Since an update in every individual in the population is to be dependent on its memory, this sets up a time complexity of \(\:O(T\times\:N)\), where \(\:T\) is the iterations. Also, the phase of recollective behavior, which recalls and updates solutions according to previous successes and failures, has the same complexity bound of \(\:O(T\times\:N)\) since its activity covers the population across iterations.
• Chaotic Reverse-based Learning (CRL): CRL involves a chaotic mapping process. As specified in the section “Chaotic reverse learning strategy (CRL)”, the parameter \(\:z\) (number of chaotic iterations) is set to \(\:20\). For each individual \(\:N\) in each iteration \(\:T\), the algorithm performs \(\:z\) iterations of the sine map. Thus, the CRL overhead is \(\:O(T\times\:N\times\:D\times\:z)\). Since \(\:z\) is a fixed constant (\(\:z=20\)), this remains \(\:O(T\times\:N\times\:D)\) but increases the constant factor of the runtime.
• Multi-Directional Mutation Strategy (MDMS): Unlike standard operators, MDMS is applied selectively to the elite portion of the population (the top \(\:20\mathrm{\%}-30\mathrm{\%}\)). If \(\:k\) is the fraction of the population (\(\:k\approx\:0.25\)), the overhead for Gaussian and Cauchy mutations is \(\:O(k\times\:T\times\:N\times\:D)\). This is computationally cheaper than applying mutation to the entire population.
• Dynamic Drift Search (DDS): DDS introduces a secondary position update phase to prevent local optima stagnation. It calculates a drift vector and updates the position for all \(\:N\) individuals across \(\:D\) dimensions. This introduces one additional vector-based position update per iteration, represented as \(\:O(T\times\:N\times\:D)\).

The total computational complexity of AEHMO is the summation of these parts:

Factoring out the common terms:

Although the additional strategies (CRL, MDMS, and DDS) introduce more operations per iteration-specifically the 20 iterations within CRL-they are all linear functions of \(\:T,N\), and \(\:D\). Therefore, the overall complexity class of AEHMO remains \(\:O(T\times\:N\times\:D)\). This is asymptotically equivalent to the original HMO, confirming that the significant performance gains in energy efficiency and convergence are achieved without moving to a higher order of computational complexity.

AEHMO for global optimization

This section presents a detailed evaluation of AEHMO, covering benchmark functions, experimental setup, and performance comparisons against competing algorithms. Given its representative diversity of optimization challenges, the CEC2017 benchmark suite is selected for evaluation. The goal is to investigate how effectively AEHMO solves complex optimization problems and traverses diverse fitness landscapes. The CEC2017 suite is categorized into four classes: F1 and F3 are unimodal, F4–F10 are multimodal, F11–F20 are hybrid, and F21–F30 are composite functions. These function classes collectively test the algorithm’s adaptability across scenarios that demand both broad global exploration and precise local refinement. The experiments are conducted at 10D, 30D, 50D and 100D problems for CEC2017 so that the proposed algorithm is challenged over diverse problem sizes.

Environmental settings and rival algorithms

This section describes the experimental setup used in all conducted experiments. Experiments were conducted on a system with a CPU of i7-10750 H, Windows 10 (64-bit), running at 2.60 GHz, and a capacity of 32 GB of RAM. The Matlab2020b was used as the software environment for the implementation of the algorithm. All algorithms ran with a population size of 30. The maximum number of evaluations to be applied was set to 30,000. Each algorithm was independently run 30 times on each benchmark function to ensure statistical reliability. To ensure reproducibility, all experiments were initialized using a fixed random seed (rng(42) in MATLAB), applied uniformly across all 30 independent runs and all compared algorithms. This guarantees that initial population diversity is consistent, and results are fully reproducible.

The compared algorithms include Grey Wolf Optimization Algorithm (GWO)⁴⁹, Arithmetic Optimization Algorithm (AOA)⁵⁰, IVY Algorithm⁵¹, Hiking Optimization Algorithm (HOA)⁵², Human Evolutionary Optimization Algorithm (HEOA)⁵³, Memory based Hybrid CSA (MHCSA)⁵⁴, the hybrid AOA-HHO⁵⁵, Modified Artificial Bee Colony and Particle Swarm Optimization (MHABC-PSO)⁵⁶, adaptive enhanced human memory algorithm (ASG-HMO)⁵⁷, Modified Group Teaching Optimization Algorithm (MGTOA)⁵⁸ and the original HMO. The parameter settings of each algorithm are listed in Table 2 to ensure an unbiased and comprehensive comparison.

All compared algorithms were configured using the default parameter settings reported in their respective original publications, ensuring that each algorithm operates under its recommended and validated configuration. This approach avoids any unintentional bias and is consistent with standard benchmarking practice in the metaheuristic optimization literature.

Performance indicators

In the experiments designed for global optimization, the performance of the algorithms is measured with the following set of indicators: mean (AVG), standard deviation (SD), Friedman rank (FR), and the Wilcoxon rank test. These four metrics provide a comprehensive evaluation of each algorithm’s behavior, described as follows:

• Mean (AVG) represented in Eq. (27) is one of the most important measures of central tendency, reflecting the average solution quality achieved by an algorithm across multiple runs:

(27)
\[
Mean\:=\frac{va{l}_{1}\left(x\right)+va{l}_{2}\left(x\right)+\cdots\:+{\mathrm{v}\mathrm{a}\mathrm{l}}_{M}\left(x\right)}{M}
\]

• The standard deviation (SD), calculated by Eq. (28), provides information about the dispersion or variability of the solutions supplied by the algorithm. A smaller SD reflects those solutions obtained are more similar which is defined as follows:

• Friedman rank⁵⁹ ranks algorithms based on their overall performance across a set of benchmark problems, where a lower FR indicates superior performance.
• To compare the performance of any two algorithms statistically, the Wilcoxon rank sum test is performed⁶⁰. A p-value below 0.05 indicates that the performance difference between algorithms is statistically significant and unlikely to have occurred by chance. Results show ‘R+’ to denote the number of functions where AEHMO significantly outperforms the competing algorithm, ‘R=’ to denote the number of functions where both algorithms show statistically similar performance, and ‘R−’ to denote the number of functions where AEHMO is significantly outperformed by the competing algorithm. Therefore, R− = 0 for a given competitor indicates that AEHMO never underperforms relative to that algorithm across all tested functions.

Sensitivity analysis of the parameters

A sensitivity analysis was conducted to empirically justify the selection of the three key AEHMO parameters: the chaotic iteration depth (\(\:z\)) in CRL, the MDMS mutation percentage, and the MDMS scaling factor (\(\:c\)). The analysis consistently confirmed \(\:z\) = 20, a mutation percentage of 20%–30%, and \(\:c\) = 0.8 as the optimal values, balancing exploration strength, exploitation precision, and computational efficiency. Full results and discussion are provided in Supplementary Appendix A, where Table A1 presents the effect of varying the chaotic iteration depth (\(\:z\)), Table A2 reports the impact of different MDMS mutation percentage values, and Table A3 summarizes the performance variation across tested values of the scaling factor (c).

Ablation experiment

In this section, we conduct an ablation experiment to evaluate the contribution of each single strategy introduced in order to enhance the performance of the HMO algorithm. The goal is to demonstrate, step by step, how each strategy individually enhances algorithmic performance and how their combination yields the best result in the full AEHMO configuration. In the ablation study, we consider the performance of various HMO variants on 29 benchmark functions of the CEC2017 test suite. Table 3 identifies whether each strategy is on or off in the various HMO variants. In other words, 1 means that the strategy is on while 0 means that the strategy is off.

The STD and AVG values obtained from this experiment are reported in Table B1. While the performance metrics for each individual strategy surpassed the HMO algorithm at most functions, the AHMO variant, for instance, with adaptive parameters, reduces it to an average rank of 4.52, indicating a key enhancement regarding how the balance between exploration and exploitation is performed. This allows the algorithm to smoothly transition between the global exploration phase and the local refinement phase and gives better convergence behavior compared to the original HMO, with an average rank of 5.07. This confirms that adaptability is important to allow for optimal performance over complex problem landscapes. Additionally, the DHMO variant implemented with the DDS strategy provided an even more impressive improvement; its average rank is 2.10. This strategy broadens exploration in the early stages and helps the algorithm avoid being trapped in local optima, hence making the overall search more efficient. Thus, DHMO is particularly powerful, demonstrating that the introduction of structured randomness can significantly enhance optimization results. Moderate improvements were achieved by the variants MHMO and CHMO, which had average ranks of 3.79 and 4.24, respectively.

On the other hand, Fig. 3 shows the FR results indicating the ranked performance of AEHMO to be superior among all variants with FR of 1.47, which is closely followed by the performance of DHMO with FR set to 2.10. This low performance of HMO over several functions points out that for enhancement in solving difficult optimization problems, the inclusion of diverse search strategies is essential. The results of the FR suggest that each strategy independently helps in enhancing the performance of the HMO algorithm. The combined effect of this, however, is stronger in the case of AEHMO to bring about an efficient and effective solution search mechanism. It confirms that all these strategies, while individually useful, when combined become much more powerful.

Result analysis for global optimization

This section evaluates the performance of AEHMO for global optimization using various dimensions of CEC2017. In this experiment, 10D, 30D, 50D and 100D are tested.

Result analysis using CEC2017, 10D

The performance of AEHMO on unimodal functions demonstrates strong optimization capability in 10-dimension search space as presented in Table C1. For the unimodal functions F1 and F3 AEHMO has achieved the first rank in both cases with fitness values of 1.00E + 02 and 3.00E + 02. In F1, AEHMO performed much better than all the competitors namely HMO (4.68E + 02), MGTOA (2.42E + 03), AOA (8.32E + 07), IVY (1.50E + 08), and HEOA (8.06E + 08) with an extremely low standard deviation of 2.15E-06 indicating near-perfect convergence precision. The F3 result with a standard deviation of 0.00E + 00 has demonstrated complete stability across all independent runs and has shown the effectiveness of AEHMO in exploiting smooth unimodal landscapes. These results validate that AEHMO’s adaptive parameters were successful in achieving intensive local exploitation after adequate global exploration in unimodal terrains. In the multimodal functions (F4-F10), the dominant performance was maintained by AEHMO, ranking first in four out of seven functions: F4 (4.00E + 02), F6 (6.00E + 02), F8 (8.14E + 02), F9 (9.01E + 02). The algorithm achieved the second rank in F10 after ASG-HMO (1.42E + 03), third in F7 (7.34E + 02) behind GWO (7.24E + 02) and ASG-HMO (7.30E + 02), and the fifth ranking in F5 (5.24E + 02) where GWO had the best performance (5.16E + 02). The consistently low standard deviations for F4 (4.93E-03), F6 (4.12E-01), F8 (4.92E + 00) and F9 (1.17E + 00) suggest good search behavior and consistency of solution quality over multiple runs. The superior performance in multimodal functions can be explained by the DDS, which permits extensive exploration by chaotic drift movements that prevent premature convergence to local optima in landscapes with multiple local optima. The relatively larger variance of F5 (8.36E + 00) and the lower ranking suggest that the functions with regular periodic structures can overwhelm the exploration mechanism, which may lead to occasional entrapment in the suboptimal regions.

For hybrid functions (F11-F20), AEHMO demonstrated excellent performance in this difficult category as shown in Table C1. The algorithm ranked first in nine of the 10 hybrid functions: F12 (1.41E + 03), F13 (1.31E + 03), F14 (1.42E + 03), F15 (1.50E + 03), F16 (1.62E + 03), F17 (1.72E + 03), F18 (1.81E + 03), F19 (1.90E + 03) and F20 (2.01E + 03). For example, in F12 AEHMO obtained 1.41E + 03 as compared to HMO 3.36E + 05, HEOA 4.85E + 06 and HOA 2.01E + 07 giving improvement of more than 99%. Similarly, in F18, AEHMO had a value of 1.81E + 03 and HOA had a value of 1.71E + 08. The CRL plays a critical role in this performance by introducing chaotic sine mapping that generates structured randomness allowing AEHMO to escape the local optima effectively while preserving the directed search to promising regions in complex hybrid landscapes combining shifted, rotated and expanded functions. The performance is further enhanced by the MDMS by using Gaussian and Cauchy mutations on the best 20–30% of solutions to achieve a balance between fine tuning and diversity preservation. The very low standard deviations in F18 (1.06E + 00) and F19 (4.30E-01) indicate an excellent consistency of convergence, while the second rank position of F11 with small difference from the best result (1.11E + 03 vs. 1.10E + 03) supports the good performance of AEHMO in the whole hybrid function suite.

The composite functions (F21–F30) are the most difficult optimization scenarios due to their complicated structures with various combinations of multiple basic functions with different shift and rotation transformations. AEHMO exhibited strong overall performance in this category, as shown in Table C1. AEHMO achieved the first rank in 4 of 10 composite functions, F24 (2.66E + 03), F28 (3.21E + 03), F29 (3.16E + 03), F30 (3.43E + 03) and third rank in F21, F22 (2.30E + 03), fifth in F25 (2.93E + 03) and fourth in F26 (3.09E + 03). In F30, AEHMO obtained 3.43E + 03 as compared to HMO’s 9.88E + 05, HEOA’s 1.55E + 06 and HOA’s 2.29E + 06, which is of considerable performance advantages. The low value of the standard deviation in F24 (1.24E + 02) and F30 (2.41E + 01) indicates stable convergence behavior in independent runs. However, performance of the algorithm was more varied between composite functions than unimodal, multimodal and hybrid functions with several functions having third or lower ranks. The synergistic combination of adaptive parameters, MDMS, DDS, and CRL allows successful balance between exploration and exploitation, allowing AEHMO to move through the complex compositional landscapes in which multiple basic functions generate overlapping basins of attraction.

The better than average overall performance of AEHMO with an average rank of 1.83 for all 29 CEC2017 functions in 10 dimensions shown in Fig. 4 shows AEHMO to be an effective global optimization algorithm for low dimensional problems. However, AEHMO did exhibit some significant deterioration in performance in some specific cases in composite functions which require careful analysis. In F21, AEHMO achieved the third rank with a fitness value of 2.26E + 03 (SD: 6.36E + 01), behind AOA-HHO (2.22E + 03) and MHCSA, while outperforming the remaining algorithms including ASG-HMO (2.30E + 03). This relatively high standard deviation indicates a lack of consistent convergent behavior and some convergent behavior into secondary optima and indicates that the composition function combining rotated hybrid functions is generating landscapes where the chaotic mapping parameter may need to be changed. The complex structure of the function with multiple rotation matrices and overlapping basins seems to interfere with the mechanism of adaptive balance and makes some runs move to the exploitation phases before having a sufficient global exploration. In F22, AEHMO achieved the third rank (2.30E + 03) with very low standard deviation (1.15E + 00), but MGTOA (2.25E + 03) obtained better fitness, suggesting that certain hybrid composition structures favor alternative exploration strategies. The F23 third-rank outcome (2.62E + 03) behind GWO (2.61E + 03) and AOA-HHO (2.62E + 03) seems to indicate that the composition functions benefit from the hierarchical search mechanisms instead of memory-based recall. The F25 fifth rank performance (2.93E + 03) is the most significant case of failure in composite functions, where ASG-HMO (2.92E + 03), AOA-HHO (2.92E + 03), MHCSA (2.92E + 03) and MHABC-PSO (2.93E + 03) were better than AEHMO. This composite functions with certain shift vectors result in regularly spaced local optima which can cause the DDS to spend too many iterations in non-promising areas. Similarly, for F26, AEHMO was ranked fourth (3.09E + 03) with a relatively high standard deviation (1.07E + 02) compared to those of top performers MHCSA (2.94E + 03) and AOA-HHO (2.94E + 03), indicating sensitivity to initial population distribution in asymmetric composite landscapes. The F27 fourth-rank position (3.10E + 03) indicates issues with composition functions that combine expanded functions, in which the search bias that memory-based updates generate might prevent even exploring asymmetrically distributed basins. These failure patterns indicate that, although AEHMO performs well on most optimization problems, certain composite functions remain challenging. Highly regular periodic structures, complex rotation matrices, and asymmetric basin distributions can disrupt the exploration–exploitation balance, resulting either in too much exploration in the converged regions or premature exploitation without enough global search completion in all the compositional components.

Result analysis using CEC2017, 30D

As dimensionality increased to 30, AEHMO was able to maintain an exceptional performance in unimodal functions as well as increased adaptability in multimodal landscapes as shown in Table C2. For the unimodal functions F1 and F3, AEHMO obtained first and third ranks respectively with fitness values of 2.35E + 03 and 8.52E + 03. In F1, AEHMO got first rank with a standard deviation of 1.78E + 03 that was far better than all other competitors including MHCSA with a standard deviation of 2.05E + 03, AOA-HHO with a standard deviation of 6.44E + 03, HMO with a standard deviation of 1.26E + 05, MGTOA with a standard deviation of 1.24E + 09. While the absolute fitness value did increase substantially from the 10-dimension result (1.00E + 02), AEHMO was able to retain its superior convergence capability and further maintain the largest performance gap among competitors, demonstrating an excellent scalability to higher-dimensional unimodal spaces. However, in F3, AEHMO was for the first time placed in the third rank (8.52E + 03) after MHCSA (3.02E + 03) and AOA-HHO (3.96E + 03) which represents a deviation from its first-rank position in 10 dimensions. The resulting increase of the standard deviation 1.84E + 03 shows the natural variability that comes with the increased 30-dimensional search space, which increases the difficulty of the search landscape as the dimensionality increases. For multimodal functions (F4-F10), AEHMO showed significantly better performance than 10 dimensions and ranked first in four out of seven functions: F6 (6.12E + 02), F7 (8.59E + 02), F9 (1.86E + 03) and F10 (3.78E + 03) with F5 in third position (6.56E + 02) behind ASG-HMO (5.70E + 02). The uniformly low standard deviations among F6 (5.74E + 00), F7 (3.36E + 01), F8 (1.97E + 01) and F9 (8.33E + 02) indicate that the DDS and CRL strategies become more efficient in higher dimensions by simultaneously preventing premature convergence and retaining solution diversity in complicated multimodal landscapes containing exponentially growing numbers of local optima.

The results of 30 dimensions hybrid and composite function show AEHMO’s outstanding scalability to complex high-dimensional optimization landscapes as shown in Table C2. For hybrid functions (F11-F20), AEHMO achieved the first rank in all of ten functions except for F16. Particularly good are the results for F12 (5.30E + 04), F13 (9.61E + 03), F14 (1.51E + 03), and F18 (2.63E + 03) where AEHMO gave orders of magnitude improvements over the competitors. In F12, AEHMO has obtained 5.30E + 04 whereas HMO has obtained 3.50E + 07, HEOA 1.97E + 09 and HOA 9.63E + 09, which is better than all algorithms. Similarly, in F13, AEHMO gave 9.61E + 03 while HEOA gave 1.11E + 09 and HOA gave 4.21E + 09. The very low standard deviations in F14 (1.43E + 01), F19 (1.41E + 01) and F18 (3.73E + 02) indicating stable and consistent convergence across all independent runs, which is a good sign of the stability of the algorithm in the case of complex hybrid landscapes. For composite functions (F21-F30), AEHMO got first rank in eight out of ten functions (F22 (2.30E + 03), F23 (2.73E + 03), F25 (2.89E + 03), F26 (3.91E + 03), F27 (3.20E + 03), F28 (3.18E + 03), F29 (3.80E + 03), F30 (7.36E + 03)), second rank in F24 (2.92E + 03). In F30, the performance of AEHMO was 7.36E + 03 compared with 5.17E + 06 for HMO, 2.37E + 08 for HEOA, and 8.89E + 08 for HOA which are significant performance improvement. The small standard deviation in F27 (2.33E-04) is a sign of near-perfect consistency in the convergence, while the small variance in F25 (7.07E + 00) and F28 (3.84E + 01) is a sign of the stability of AEHMO in wide range of composite structures. The enhanced efficiency of MDMS in higher dimensions by the selective mutation of elite solutions using adaptive Gaussian and Cauchy distributions makes possible the more refined balancing of exploration and exploitation as the search progresses through more extensive compositional solution spaces.

Despite the fact that the overall better performance when averaged over 30 dimensions, which has an average rank of 1.45 as shown in Fig. 5, AEHMO experienced certain patterns of failures which are different from the 10-dimension situation and deserve in-depth analysis. The largest change is in unimodal function F3 where AEHMO reached rank 3rd (8.52E + 03) from its 1 st rank in 10 dimensions, while MHCSA (3.02E + 03) and AOA-HHO (3.96E + 03) have better performance. The large standard deviation value of 1.84E + 03 indicates that the convergence behavior is not consistent from one run to the other, which implies that the structure of 30-dimensional of the function is such that the adaptive parameters may enter the exploitation phases prematurely without having attained sufficient global exploration. The shifted and rotated nature of F3, when expanded to 30 dimensions, appears to result in misleading information from gradients, which result in the memory-based recall mechanism paying attention to suboptimal regions. In case of multimodal functions, F4 and F5 are the most remarkable cases where AEHMO is ranked at the third position in both cases with fitness values of 5.05E + 02 and 6.56E + 02, respectively. For F4, MHCSA (4.97E + 02) and AOA-HHO (4.91E + 02) showed better results than AEHMO and in F5, ASG-HMO (5.70E + 02) and GWO (6.24E + 02) gave better result. The expanded structures of these functions with 30 dimensions present exponentially more local optima which can overwhelm the exploration mechanism and the algorithm may spend too many iterations in non-promising areas before finding really optimal basins. The composite function F21 is the most important failure case where AEHMO showed the poorest rank (5th with 2.43E + 03, SD: 5.99E + 01) which is significantly poorer than the best performers ASG-HMO (2.40E + 03, SD: 2.44E + 01) and GWO (2.40E + 03, SD: 1.69E + 01). This performance is a significant deterioration from the third rank in 10 dimensions and the composition function of rotated hybrid components will be a more problematic composition function as the dimensionality increases. The high variance shows that the intensity of the chaotic mapping requires dimensional scaling, as the set of parameters is less effective when the solution space increases from 10 to 30 dimensions and the optimal combination of hybrid component combinations is not always found. Additionally, F16 and F24 got 2nd rank (2.42E + 03 and 2.92E + 03) behind GWO (2.31E + 03) and ASG-HMO (2.92E + 03) respectively, with F24 having a higher standard deviation (2.90E + 01) than ASG-HMO’s 2.38E + 01, which shows consistency problems of the expanded function. The result of F16 shows that GWO’s hierarchical hunting strategy provides advantages in specific hybrid compositions that include rotated expanded functions with dimension increase. These dimensional-dependent patterns of failures suggest that although the strategies of AEHMO are generally dimension-scalable, some function characteristics, particularly the shifted structure in F3, the regular distribution of multiple optima in the expanded functions (F4, F5), and the complex rotation matrices in composite function F21, necessitate dimension-adaptive parameter tuning mechanisms, which adjust the chaotic mapping iterations, the mutation step sizes, and the drift intensity on the basis of the problem dimensionality to ensure the optimal exploration-exploitation balance in the search process in 30-dimensional spaces.

Result analysis using CEC2017, 50D

In the 50-dimensional case, AEHMO turned out to be still scalable with dimensional dependent performance variations as shown in Table C3. For unimodal function F1 and F3, AEHMO obtained the third and the first rank with fitness value of 2.80E + 06 and 3.23E + 04, respectively. The F1 third rank behind AOA-HHO (7.80E + 03) and MHCSA (8.21E + 03) represents a decline from its first-rank performance in 10D and 30D suggesting that competitor algorithms become effective in high-dimensional unimodal spaces. But AEHMO first rank achievement in F3 (3.23E + 04) with a standard deviation of 7.42E + 03 provides evidence of robust handling of the function structure. For multimodal functions (F4-F10), AEHMO obtained first rank in six out of seven functions: F4 (5.28E + 02), F5 (7.01E + 02), F6 (6.19E + 02), F7 (1.04E + 03), F9 (4.99E + 03) and F10 (5.94E + 03) with improved performance from 30D where F4 and F5 both had 3rd rank. The uniform low standard deviations in F4 (6.29E + 01), F5 (2.92E + 01), and F6 (4.15E + 00) represent the enhanced convergence stability and the AEHMO’s adaptive parameters and DDS become more efficient in exploring expanded landscapes in 50 dimensions.

For hybrid functions (F11-F20), AEHMO was ranked first six times out of ten, maintaining good performance that was seen in 30D as shown in Table C3. Particularly notable are F13 (6.03E + 03) and F14 (1.79E + 03) where AEHMO performed better than HOA by larger orders of magnitude (4.22E + 10 and 8.84E + 07 respectively). The low standard deviation in F14 (6.01E + 01) indicates near-perfect consistency in 50-dimensional hybrid spaces. For composite functions (F21-F30), AEHMO got first rank in 7 out of 10 cases, F21 (2.50E + 03), F22 (7.86E + 03), F24 (3.08E + 03), F25 (3.08E + 03), F26 (6.75E + 03), F27 (3.20E + 03), F29 (4.51E + 03).

Despite the fact that on the whole, AEHMO achieved better performance with an average rank of 1.72 in 50 dimensions as shown in Fig. 6, AEHMO did have some failures that required analysis. In F1 third rank position (2.80E + 06) is the position of degradation from first rank in both 10D and 30D scenarios, AOA-HHO (7.80E + 03) and MHCSA (8.21E + 03), which indicates that the expanded unimodal landscape of 50-dimensional function favors the algorithm using harmony search and hybrid hawk strategies which allow better population to spread. The large standard deviation value of 2.35E + 06 as compared to AOA-HHO’s 2.89E + 03 is a signal of inconsistent converging, presumably due to premature transition of adaptive parameters from exploration to exploitation before adequately exploring the vastly expanded 50-dimensional space. In multimodal functions, F8 had the third highest rank (1.11E + 03) preceded by ASG-HMO (1.00E + 03) and GWO (1.02E + 03), which suggested existence of occasional convergence variability in 50-dimensional functions. For hybrid function F11, the third-rank result (1.36E + 03) behind AOA-HHO (1.27E + 03) and MHCSA (1.30E + 03) indicates that there are some hybrid compositions with 50 dimensions that are supported by other mechanisms for diversity maintenance. The composite function failures are seen in the case of F17, F20, F23 and F28 where AEHMO got the second, fourth, fifth and fourth ranks, respectively. The F20 fourth-rank position (2.98E + 03) is the most important composite function failure, where MHCSA (2.96E + 03), ASG-HMO (2.97E + 03) and GWO (2.98E + 03) have outperformed AEHMO by small margins which indicate that there are near equivalent performances but with AEHMO showing slightly higher variance. In F23, AEHMO was ranked 5th (3.15E + 03) behind MHCSA (3.00E + 03), GWO (3.00E + 03), AOA-HHO (3.05E + 03) and ASG-HMO (3.08E + 03), showing that the landscapes formed by 50-dimensional composition functions combining Ackley and Rastrigin components are such that memory-based mechanisms have difficulty in keeping the exploration-exploitation balance reached in lower dimensions. The fact that the F28 4th rank (3.51E + 03) result is behind AOA-HHO (3.31E + 03), MHCSA (3.31E + 03) and HMO (3.40E + 03) shows that the expansion of the composition function to 50 dimensions has introduced complexity where the simpler approach of the original HMO (i.e. fixed parameters) sometimes outperforms the adaptive mechanism of AEHMO, suggesting that the parameter adaptation needs to be dimensionally scaled to avoid excessive exploration or premature convergence in ultra-high-dimensional compositional structures.

Result analysis using CEC2017, 100D

In the high-dimensional case of 100D, AEHMO proved to be continuously effective, exhibiting different performance characteristics from those observed in lower dimensions such as shown in Table C4. For unimodal functions F1 and F3, AEHMO achieved the fifth and first ranks with fitness value of 2.87E + 09 and 2.44E + 05, respectively. In F1, AEHMO’s fifth-rank position behind MHCSA (6.37E + 04), AOA-HHO (8.95E + 04), HMO (6.21E + 08), ASG-HMO (7.39E + 08) is a significant departure from former dimensional trends, suggesting that high-dimensional unimodal functions favor algorithms whose population diversity mechanisms differ from AEHMO’s memory-based approach. However, AEHMO’s first-rank performance in F3 (2.44E + 05) confirms the maintained superiority in dealing with function structure in all the dimensions tested. For multimodal functions (F4-F10), AEHMO came first for five of the functions (F4 (7.35E + 02), F5 (1.11E + 03), F6 (6.43E + 02), F7 (1.87E + 03) and F9 (2.02E + 04) and second for F8 (1.53E + 03) behind ASG-HMO (1.37E + 03) and seventh for F10 (2.41E + 04).

For hybrid functions (F11-F20), AEHMO got first rank in seven out of ten cases: F12 (2.61E + 07), F14 (2.48E + 05), F15 (5.04E + 03), F16 (5.42E + 03), F17 (4.92E + 03), F18 (6.35E + 05), and F19 (6.56E + 03), with good performance in high dimensional hybrid spaces as shown. For composite functions (F21-F30), AEHMO got the 1 st rank in four cases (F22 (1.94E + 04), F24 (3.95E + 03), F25 (3.29E + 03), F26 (1.63E + 04)), and 2nd rank at F21 (2.96E + 03).

AEHMO’s overall average rank of 1.97 in 100 dimensions as shown in Fig. 7, while slightly higher than 1.72 found in 50D and 1.45 found in 30D, confirms good scalability to high dimensional optimization with competitive advantages over most comparison algorithms. The dimensional scaling analysis shows the presence of important performance limitations approaching dimensionality 100. In unimodal function F1, the significant decline of function value from first rank in 10D, 30D to fifth rank in 100D (2.87E + 09) in which MHCSA (6.37E + 04) and AOA-HHO (8.95E + 04) obtained fitness values four orders of magnitude better, implies the adaptive parameter mechanism of AEHMO cannot manage to effectively balance exploration and exploitation when searching for a solution in high dimensional smooth landscapes, whose high standard deviation of 1.04E + 09 implies high variance. In hybrid function F11, the third-rank position (1.46E + 04) and relatively high standard deviation (4.68E + 03) in comparison with MHCSA (2.45E + 03, SD: 1.25E + 02) and AOA-HHO (2.59E + 03, SD: 2.76E + 02) shows the consistency problems in 100-dimensional compositions. The composite function failures in F20, F23 and F28 in the third, fifth and fourth ranks respectively indicates the systematic difficulties. The F23 fifth-rank result (3.90E + 03) which is behind AOA-HHO (3.33E + 03), MHCSA (3.40E + 03), ASG-HMO (3.57E + 03) and GWO (3.69E + 03) shows that the 100-dimensional compositional landscapes are exploration challenges in which the chaotic mapping intensity is not enough to traverse the exponentially expanded search space. The F28 fourth-rank position (4.45E + 03) where AOA-HHO (3.46E + 03), MHCSA (3.46E + 03), and HMO (4.01E + 03) scored the better than AEHMO suggests the existence of certain 100-dimensional compositional structures that paradoxically favor the simpler fixed parameter approach of the original HMO, and also reveals how adaptive mechanisms designed for lower dimensions, sometimes introduce unnecessary complexity or computational overhead for high-dimensional spaces. These systematic failures in unimodal, multimodal, hybrid and composite categories suggest that the current parameter settings of AEHMO especially the fixed chaotic mapping iterations and mutation step sizes and drift intensity coefficients need to have explicit dimensional-adaptive scaling mechanisms to automatically adapt the intensities of strategies based on the problem dimensionality to ensure optimal performance when moving from 50D to 100D and higher since the current linear or sigmoid based parameter adaptations become increasingly inadequate when dealing with the exponential increase in complexity of the search space and the curse of dimensionality effects.

Convergence analysis

Convergence behavior analysis across all CEC2017 function classes and dimensions (Figs. D1–D4) demonstrates AEHMO’s consistent ability to balance early exploration with late-stage exploitation throughout the 30,000 function evaluations. For unimodal functions F1 and F3, AEHMO exhibits the steepest initial descent across all dimensions, driven by the DDS strategy enabling rapid identification of promising basins in the first 5,000 FEs, followed by smooth transition to exploitation via adaptive parameter regulation around 10,000 FEs. For multimodal functions, AEHMO consistently falls below all competitors by 8,000–12,000 FEs in F4, F6, F7, and F9 across all dimensions, with the step-like descent patterns in F8 and F10 confirming that the CRL chaotic opposition mechanism successfully escapes successive local optima where competitor algorithms stagnate. Hybrid functions reveal AEHMO’s most sustained convergence behavior, with uninterrupted descent maintained beyond 15,000 FEs in F12, F13, F14, F15, F18, and F19, a phase where all competing algorithms exhibit visible plateauing. Composite functions demonstrate the strongest convergence advantage, with AEHMO maintaining a continuously widening gap over competitors in F22, F24, F25, F26, F27, F29, and F30 throughout the full evaluation budget. The primary convergence limitations emerge at 100D, where F1, F5, F10, F23, and F28 exhibit oscillatory or premature flattening behavior, suggesting that the current adaptive parameter settings encounter dimensional scaling challenges in specific high-dimensional unimodal and composite landscapes. Full convergence curves for all functions and dimensions are provided in Supplementary Appendix D.

Boxplot analysis

Boxplot analysis across all CEC2017 function classes and dimensions (Figs. E1–E4) confirms the high solution consistency and robustness of AEHMO across 30 independent runs. For unimodal and multimodal functions, AEHMO produces highly compact distributions with near-zero interquartile ranges in 10D, 30D, and 50D, outperforming competitors such as IVY, AOA, HEOA, and HOA which exhibit significantly larger box heights, heavy-tailed distributions, and frequent outliers. For hybrid functions, AEHMO consistently achieves the smallest interquartile ranges across F12, F13, F15, F18, and F19 in all tested dimensions, confirming that the MDMS elite mutation strategy effectively maintains solution quality consistency without population-wide perturbation. Composite functions reveal AEHMO’s strongest robustness characteristics, with distributions for F22, F24, F25, F26, F27, F29, and F30 collapsing to near-horizontal lines in 10D-50D, reflecting stable convergence across all runs regardless of initialization. The only notable variance increases occur at 100D for F1, F5, F10, F23, and F28, where higher dimensionality introduces initialization-dependent convergence behavior that AEHMO’s current adaptive mechanisms do not fully suppress. A detailed per-function boxplot analysis across all dimensions is provided in Supplementary Appendix E.

Statistical significance analysis

The result of the Friedman test in all the dimensional scenarios consistently indicate the statistical competitiveness of AEHMO over the comparison algorithms with the algorithm having the best mean ranks in the 10D, 30D, 50D and 100D as shown in Fig. 8. In 10-dimension optimization, AEHMO had a Friedman rank of 2.15 which is far better than all the competitors, the other best competitors were ASG-HMO (4.02), AOA-HHO (4.25) and MHCSA (4.64), while traditional algorithms such as HMO (7.53), HEOA (10.74) and HOA (10.49) showed much weaker overall performance. This ranking pattern indicates that AEHMO is consistently better than other algorithms in the low dimensional spaces of the diverse CEC2017 benchmark suite by having top or near top ranks in most of the 29 test functions. The results of the Bonferroni-Dunn post-hoc test shown in Fig. 9 indicate that the critical difference (CD) values of AEHMO algorithm at both \(\:\alpha\:\)=0.1 and \(\:\alpha\:\)=0.05 significance levels place the algorithm well below the threshold lines, showing statistically significant improvement over HMO, MGTOA, AOA, IVY, GWO, HEOA, HOA and MHABC-PSO. The dramatic performance gap between AEHMO and algorithms such as HEOA 10.74 and HOA 10.49 suggests that it is challenging for recent nature-inspired algorithms without adaptive parameter mechanisms to effectively balance exploration and exploitation in the benchmark suite.

As the dimensionality is increased to 30D, 50D and 100D AEHMO maintains its statistical advantage with Friedman ranks of 2.37, 2.87 and 2.23, respectively as shown in Fig. 9. The 30D Friedman ranks show AEHMO continuing to outperform all competing algorithms, with MHCSA (3.38), AOA-HHO (3.53) and ASG-HMO (3.93) forming the next level of competition, with HEOA (10.45) and HOA (10.88) still falling at the bottom of the rankings. The post-hoc test for 30D using the Bonferroni-Dunn confirms the statistical significance of AEHMO’s bar that is well below both CD thresholds, indicating that the observed performance differences are statistically significant. Interestingly, the 50D Friedman ranks have AEHMO with a rank of 2.87, which is slightly higher than the 30D rank (2.37), suggesting a modest reduction in relative performance with increasing dimensionality, but again, AEHMO still has significant advantages over most of its contenders. The Bonferroni-Dunn results for 50D still show AEHMO to be below CD thresholds, with AOA-HHO (3.29) and ASG-HMO (3.89) having the nearest competitive performance. A notable observation emerges in the 100D case where AEHMO achieves a surprisingly competitive Friedman rank of 2.23, which is comparable to its 10D rank of 2.15, suggesting that AEHMO scales effectively to very high-dimensional problems without significant performance degradation with MHCSA (2.65) the closest competitor with a minimum difference of 0.42 rank points. The Bonferroni-Dunn post-hoc test for 100D tells us that AEHMO and MHCSA fall within the critical difference (CD, \(\:\alpha\:\) = 0.05), indicating no statistically significant difference between them. The coexistence of AEHMO and MHCSA performance in 100D, and AOA-HHO (4.45) and ASG-HMO (4.60) preserving competitive ranks implies that high-dimensional optimization provides conditions where several well-designed algorithms with different underlying mechanisms are able to achieve similar overall performance for different function types. Despite this convergence, algorithms like HEOA (10.11–10.74), HOA (10.49–10.88), AOA (7.86–8.66) and IVY (7.90–8.39) consistently score in the bottom half in all the dimensions with their bars in all Bonferroni-Dunn charts placed significantly above both CD thresholds, indicating statistically inferior performance relative to AEHMO in all dimensional scenarios tested and that not all nature-inspired metaheuristics perform similarly on complex optimization benchmarks.

The results of the Wilcoxon rank test are used to provide pairwise statistical comparisons between AEHMO and all its competing algorithms for all the dimensional scenarios to confirm the dominance of AEHMO with overwhelmingly positive win-loss ratios as presented in Fig. 10. In optimization of 10-dimension AEHMO was found to be consistently superior to AOA (R + = 27, R-=0, R = = 2), HMO (R + = 27, R-=0, R = = 2), HEOA (R + = 26, R-=0, R = = 3), HOA (R + = 28, R-=0, R = = 1), IVY (R + = 25, R-=0, R = = 4), and MHABC-PSO (R + = 22, R-=0, R = = 7) and showed statistically significant dominance. As the dimensionality increased to 30D, the domination of AEHMO increased, exhibiting perfect or near-perfect dominance over most algorithms: AOA (R + = 28, R-=0), HEOA (R + = 29, R-=0), HMO (R + = 28, R-=0), HOA (R + = 29, R-=0), IVY (R + = 29, R-=0), and MHABC-PSO (R + = 28, R-=0), with good advantages against competitive algorithms with MGTOA (R + = 21, R-=2). The consistent pattern for all dimensions shows that AEHMO is reporting R+ values from 18 to 29 (out of maximum 29 functions) in comparison with all the competitors with R- values never exceeding 5 and mostly staying at 0–2, while algorithms such as HEOA, HOA, AOA and IVY are consistently reporting R− = 0 for all dimensions, indicating that AEHMO does not underperform relative to these methods for each function and validates the statistical significance and robustness of AEHMO’s superior performance for the entire CEC2017 benchmark suite across all tested dimensions.

The vast statistical analysis of AEHMO at various dimensional scales (10D, 30D, 50D and 100D) on the CEC2017 benchmark suite shows an outstanding algorithmic scalability and performance advantage. The 95% confidence interval analysis shown in Table F1 shows a high level of consistency in the accuracy of algorithms with AEHMO having much smaller margins of error than competitor algorithms where notable stability in the performance of the algorithm going from low dimensions to extreme high dimensions search spaces. This continued precision in spite of exponential growth in complexity of search space with increasing dimensionality provides evidence that AEHMO’s enhancement mechanisms are effective in mitigating the curse of dimensionality. The Wilcoxon signed-rank test is applied to check statistical superiority in all the dimensional scales and all pairwise comparisons are statistically significant (\(\:p\) < 0.001), indicating that AEHMO is statistically better than competing algorithms regardless of the problem’s dimensionality.

Moreover, the results of effect size analysis presented in Table F2 reveal that there is substantial practical significance in all dimensions with mean effect size \(\:r\) ranging between 0.853 and 0.873, which indicates that the effect sizes are large and performance differences are not only statistically significant but also have a large practical significance. AEHMO demonstrated superior performance compared to the established algorithms, AOA, HEOA, HMO, HOA, IVY, MGTOA and MHABC-PSO for all the dimensional levels while maintaining good performance advantages against the most competitive algorithms such as GWO, MHCSA, AOA-HHO and ASG-HMO in the whole dimensional range. The narrow CIs and consistently large effect size in all dimensional scales gives strong statistical evidence that AEHMO’s integrated enhancement strategies (adaptive parameter regulation, multi-directional mutation, DDS and CRL) exhibit strong scalability properties, supporting the robustness and reliability of AEHMO for complex real-world optimization problems involving varying dimensionality, multimodality, and challenging fitness landscapes.

Computational overhead analysis

Computational efficiency is a critical factor for the practical deployment of metaheuristic algorithms, particularly in resource-constrained scenarios such as WSN CH selection where optimization must operate under strict computational budgets. Computational time analysis across different dimensional settings (10D, 30D, 50D and 100D) on the CEC2017 benchmark suite demonstrates a balanced trade-off achieved by AEHMO in terms of computational efficiency and optimization performance. As reported in Table 4, AEHMO has a moderate execution time (0.5603, 0.9399, 1.2934 and 2.9225 s) for 10D, 30D, 50D and 100D problems, respectively, placing it within the mid-range of the compared algorithms. While AEHMO incurs higher runtime than lightweight single-strategy algorithms such as ASG-HMO, MGTOA, GWO, and HOA, its execution times remain moderate across all tested dimensions, as shown in Table 4. Notably, AEHMO exhibits significantly lower computational overhead compared to MHABC-PSO across all tested dimensions, while maintaining comparable execution times to AOA-HHO and MHCSA, while delivering substantially superior optimization performance as reflected by lower Friedman ranks.

AEHMO for engineering applications

In this section, the performance of the proposed AEHMO algorithm is investigated under realistic engineering design conditions, which include eight widely used structural and mechanical design problems: Piston Lever Design, Tension–Compression Spring Design, Tubular Column Design, Cantilever Beam Design, Corrugated Bulkhead Design, I-Beam Design, Speed Reducer Design, and Pressure Vessel Design. Constraint handling of these design tasks is performed by means of a static penalty formulation provided in Eq. (29)⁶¹:

Here, \(\:z\) is the penalized objective function where \(\:{o}_{j}\), \(\:{l}_{j}\) are positive constant penalty. In this formulation, the constraints are denoted by \(\:{U}_{j}\left(z\right)\) and \(\:{T}_{i}\left(z\right)\), and the penalty effect is governed by the parameters \(\:\alpha\:\) and \(\:\beta\:\), which respectively assume values 1 or 2. All the test problems are solved using population size 50, maximum iterations 500, and 30 independent runs to achieve a good statistical analysis.

Piston Lever Design (PLD) problem

The PLD problem is a minimization problem to minimize the amount of oil used when the piston lever changes its angle from 0^o to 45^{[o 62}. The effect of geometrical parameters \(\:H\), \(\:B\), \(\:D\) and \(\:V\) is considered for the optimization problem which has a great influence on the mechanical efficiency. This problem can be mathematically described as in the Supplementary Appendix.

AEHMO performs the best among all the other algorithms on all the statistical measures as shown in the evaluation of the PLD problem. Table 5 indicates the minimum cost with a low standard deviation for AEHMO is 8.645928 and 0.807653 respectively, meaning that there is high consistency and reliability of finding the best solutions. The best performance (global optimum) of the problem is 8.416798 and the worst-case performance is 12.058200 which is a significant improvement over all other competing algorithms. The fact that the difference between optimal and worst scores is small suggests that the algorithm is robust and less prone to be stuck in local optima and/or not converging.

Tension and compression spring design (TCSD) problem

The TCSD is concerned with the minimization of the weight of a spring subject to a number of engineering constraints⁶³. The optimization involves three key design parameters, namely wire diameter \(\:d\left({x}_{1}\right)\), mean coil diameter \(\:D\left({x}_{2}\right)\) and number of active coils \(\:N\left({x}_{3}\right)\). To maximize the performance, these variables need to be tuned to meet allowable stress, permissible deflection, and structural integrity constraints along with the provision of reliable performance under both tensile and compressive loads to ensure material efficiency and durability. This problem is a standard example in mechanical design, and it is often used for testing the robustness of algorithms under nonlinear and constrained loads. This problem can be mathematically described as in the Supplementary Appendix.

The design optimization of the TCSD problem highlights the strong optimization performance of AEHMO, as shown in Table 6. AEHMO exhibits excellent convergence with a mean fitness value of 0.012665233 and a standard deviation numerically equal to zero. This high level of stability provides strong evidence for the reliability and robustness of the proposed improvements. The best and the worst fitness values of the algorithm are equal (0.012665233), which proves that there is no performance variability, which is an important requirement for engineering applications where the optimization result should be predictable. As for the comparison of the competing algorithms, the IVY and MGTOA are relatively competitive algorithms with the mean fitness value of 0.012738194 and 0.012711070, respectively, with some fluctuation. The mean fitness of HMO is 0.014047824 and the standard deviation is 0.001622293, indicating that the convergence consistency still needs to be improved.

Tubular column design (TCD) problem

The TCD problem is to find the best design of a uniform column which will support a given compressive load \(\:P\) (pounds/inch²) with minimum total cost of manufacture⁶⁴. Design variables are the external diameter \(\:{t}_{1}\), and the wall thickness \(\:{t}_{2}\). The column is made of material with an elastic modulus of 0.85 × 10⁶ kgf/cm² and yield stress limit of 500 kgf/cm². The goal is to find optimal \(\:{t}_{1}\) and \(\:{t}_{2}\) which ensure the structural stability under compression without yielding and instability. Moreover, it is a representative problem involving a trade-off between mechanical performance and resource efficiency, as designers must balance cost-effectiveness, strength and stiffness requirements, and material savings. This problem can be mathematically described as in the Supplementary Appendix.

The results of the TCD problem, summarized in Table 7, demonstrate the strong optimization performance of AEHMO. The algorithm exhibits highly consistent convergence, with a mean cost of 26.4863615 and a numerically negligible standard deviation of 3.6450110E-15, meaning the best and worst scores were identical across all 30 independent runs. This level of consistency indicates highly stable convergence and suggests improved reliability and accuracy of the proposed enhancements for constrained structural design problems with mixed continuous and boundary variables. Performance comparisons show that despite the other algorithms might show competitive results from time to time, none can compete with the consistency of AEHMO.

Cantilever beam design (CBDP) problem

The CBDP is intended to optimally minimize the total weight of a cantilever suspension arm subject to structural constraints⁶⁵. The widths of these five segments are the optimization variables and the cross-sectional thickness is uniform. This issue is of great importance for structural optimization, which is a problem of reducing material usage without losing the mechanical performance to support loads and moments applied at the free end. It is used as a test problem for optimizing algorithms in multi-variable constrained and weight-sensitive design problems. This problem can be mathematically described as in the Supplementary Appendix.

The effectiveness of CBDP problem is analyzed and excellent optimization performance of AEHMO is presented in Table 8. AEHMO showed near-identical results across runs, with a mean cost of 1.33995636 and a very low standard deviation of 4.55626E-16, meaning that the best and worst scores were the same value in all independent runs. This high level of consistency suggests that the algorithm repeatedly converges to the same high-quality solution. These results highlight the enhanced reliability and accuracy of the proposed strategies for solving multi-variable constrained beam design problems with complex stress and deflection constraints. Comparative analysis shows performance gaps that are significant compared to competitors. The best performance is obtained by GWO with the cost value of 1.34005715 and small standard deviation value of 6.54247E-5 and IVY with the cost value of 1.34020248 is the second-best performance.

Corrugated bulkhead design (CBD) problem

The CBD problem is to reduce the weight of a structural bulkhead made by deforming a pressed steel plate, and not requiring the use of external stiffeners⁶⁶. In this formulation, the stiffness is geometrically introduced into the plate through a corrugation structure. The optimization has four primary design variables: corrugation width \(\:{x}_{1}\), depth \(\:{x}_{2}\), length \(\:{x}_{3}\) and plate thickness \(\:{x}_{4}\). The design has to satisfy six constraint conditions for structural adequacy and performance in the various load cases. This problem is a representative form-optimization task for improving structural efficiency and has been widely used to evaluate algorithm performance on multi-constraint, multi-variable engineering design problems. This problem can be mathematically described as in the Supplementary Appendix. Table 9 illustrates the excellent optimization capability of the proposed AEHMO algorithm over other methods for CBD Problem. AEHMO converged with very high consistency, with a mean cost of 6.84295801 and a numerically negligible standard deviation of 1.8225E-15, indicating that there was effectively no variation across the optimization runs. The performance of AEHMO is compared with that of competing algorithms in this challenging structural optimization problem revealing significant gaps in performance. The original HMO solution had a decent solution (mean cost = 6.88164711, std = 0.033774446), the best solution of 6.84618290 is not that far from the optimal. In contrast, the performances of HEOA and AOA were significantly worse with mean costs of 8.79781954 and 7.92043225, respectively and high standard deviations which signifies high inconsistency and convergence difficulties.

Application of AEHMO for cluster CH selection in cluster-based routing for wireless sensor networks (WSN)

The strong global optimization performance of AEHMO demonstrated in the section “AEHMO for global optimization” provides a solid foundation for its application to WSN cluster head selection. The CH selection problem is naturally a high-dimensional and multimodal combinatorial optimization problem where the solution space is exponentially large in network size and where there are multiple locally optimal CH configurations. The same capabilities that lead to AEHMO consistently outperforming competing algorithms on CEC2017, namely, adaptive exploration-exploitation regulation, multi-directional mutation for escaping local optima, and chaotic opposition-based diversity preservation, lead to superior CH selection performance. Specifically, the success of avoiding premature convergence in benchmark landscapes of high dimension is directly related to the success of the algorithm navigating the complex CH fitness landscape and finding the energy-efficient configurations consistently for different network sizes and topologies. In this section, the AEHMO algorithm is applied to the Cluster Head (CH) selection problem in cluster-based routing of WSN. The efficient CH selection is crucial to optimize energy consumption and maximize the overall network lifetime. AEHMO is applied to optimize CH selection and thereby enhance overall network performance. This section describes the network model, the energy consumption model for communication, and the performance metrics used to evaluate network efficiency.

The utilized network model

WSNs have emerged as a foundational technology for interconnected digital systems and smart applications. WSNs play a vital role in enhancing the design of internet-connected applications. The WSN is a multi-hop, self-configured network that comprises many sensor nodes; each node has the potential for wireless communication. These networks integrate several technologies, including distributed data processing, sensing systems, embedded computing, wireless communication, and modern networking, to accomplish complex monitoring and control tasks. Sensor nodes are deployed to gather environmental data, converting physical signals into electrical signals and transmitting them to a central node or sink via wireless multi-hop communication. Applications that can use WSNs include environmental observation, industrial processes, military observation, and healthcare, among many others.

The WSN network model consists of several sensors and a sink node. Sensor nodes are uniformly distributed over \(\:D\times\:D\) area. Communication between any two nodes takes place either directly via one-hop transmission or through multi-hop routes. Sensor nodes can further be divided into general nodes and source routing nodes. Each node has a unique network identifier and is deployed at a fixed location, while the sink node is positioned at the center of the network. The deployed network requires no interference from human beings afterward. All nodes have the same storage, computing power, and energy capacity. These sensor nodes can identify where they are attached within the network. Since the sink node is not energy-constrained, it possesses greater computational resources than regular sensor nodes. The sensor nodes monitor their residual energy and calculate their distances to neighboring nodes. The node can communicate with the sink node, aggregate data, and change its transmission power according to the distance for its target or recipient. Consequently, the communication links between nodes are symmetric. Each node dynamically adjusts its transmission power according to the distance to the intended recipient. Sensor nodes periodically measure the environment and transmit their data to CH. The network model, including the BS placed within the communication range of all nodes, is illustrated in Fig. 11.

Therefore, the assumptions used in the network model can be summarized as follows:

• All communications are wireless.
• Nodes are scattered randomly in the network.
• Each node has a very limited energy capacity.
• The sink node is also immobile and also has unlimited energy resources.
• CHs can directly communicate with the sink.
• A first-order radio communication model is employed to compute the radio energy dissipation for each node in each round.
• The network operates in rounds.
• Roles of the nodes switch between CH and non-CH nodes at each round start.
• TDMA is used for scheduling the communications between CHs and member nodes’ communications.

The basic energy model

The energy model is a key factor in analyzing and optimizing power consumption for each node of a network. The whole process depends on the amount of energy taken by the quantity of energy used, for example, transmission, reception, and data aggregation. The components of the energy model are described below:

• Energy consumption for data transmission:

where \(\:{E}_{TX}\left(p,d\right)\) is the energy required to transmit \(\:p\) bits over distance \(\:d\). The input parameter is \(\:{E}_{comm}\), the energy consumed per bit for transmission/reception; it is set to 50 nJ/bit. \(\:{\eta\:}_{fs}\) represents the energy loss per bit per square meter in free space, and it is set to 0.0015pJ/bit. The second term, \(\:{\eta\:}_{mp}\), accounts for the multipath fading loss per bit per meter to the fourth power when d exceeds \(\:{d}_{0}\) which is defined as follows:

• Energy Consumption for Data Reception:

where \(\:{E}_{RX}\left(p\right)\:\) is the energy used to receive a message containing \(\:p\) bits.

• Energy Consumption for Data Aggregation:

where \(\:{E}_{DA}\left(p\right)\) is the energy consumed to aggregate \(\:p\) bits of data, and \(\:{E}_{agg}\) is the energy dissipated per bit for data aggregation, set to 10 nJ/bit.

Cluster head selection in AEHMO

In the proposed scheme for WSNs, the novel AEHMO algorithm is applied for selecting the optimal CHs to enhance communication efficiency between sensor nodes and base stations. The primary objective of this method is to optimize energy management in the routing process. AEHMO selects CHs through an optimization process that accounts for energy consumption, distance to the BS, node-to-CH distance, node degree, and node centrality. The selection of CHs using AEHMO is illustrated in Fig. 12.

The process determines the optimal number of CHs, where the candidate solution size ranges from 1 to the total number of sensor nodes. The AEHMO algorithm optimizes this selection by addressing an objective function, which factors in the energy consumption, distance to BS, packet delivery ratio (PDR), and throughput.

Objective function for AEHMO-based CH selection

The fitness function employed in AEHMO serves as the core mechanism for selecting optimal CHs from the set of sensor nodes. To maintain consistency across homogeneous and heterogeneous WSN scenarios, the objective function uses the same normalized fitness components and aggregation structure throughout. In homogeneous deployments, all nodes share identical capabilities, so the heterogeneity-aware terms become constant and do not affect the optimization. In heterogeneous deployments, these terms become active while the underlying framework remains unchanged.

Let \(\:S=\left\{{s}_{1},{s}_{2},\dots\:,{s}_{m}\right\}\) denote a candidate CH configuration, where \(\:m\) represents the number of selected CHs, and let \(\:{C}_{j}\) be the set of sensor nodes assigned to \(\:\mathrm{C}\mathrm{H}{s}_{j}\). The fitness evaluation is performed over the candidate CH set and the induced cluster assignments rather than globally over all nodes. Four fitness parameters are considered: residual energy suitability, CH-to-base-station proximity, intra-cluster compactness, and a capability-aware bias term.

• The first fitness parameter, \(\:F{P}_{1}\), represents the residual energy suitability of the selected CHs. Since CHs perform energy-intensive operations including data collection, aggregation, and forwarding, nodes with higher residual energy are prioritized for CH selection. To ensure applicability to both homogeneous and heterogeneous networks, residual energy is expressed in normalized form as:

where \(\:{E}_{R}\left({s}_{j}\right)\) denotes the residual energy of the \(\:j\)-th CH and \(\:{E}_{\mathrm{m}\mathrm{a}\mathrm{x}}\left({s}_{j}\right)\) denotes its initial energy capacity. In homogeneous networks, \(\:{E}_{\mathrm{max\:}}\) is identical for all nodes, whereas in heterogeneous networks it varies across node tiers, allowing the same formulation to naturally capture capability differences.

• The second fitness parameter, \(\:F{P}_{2}\), captures the communication cost between CHs and the BS. Longer transmission distances lead to higher energy dissipation according to the adopted radio energy model. This parameter is defined as:

where \(\:\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\left({s}_{j},BS\right)\) denotes the Euclidean distance between \(\:\mathrm{C}\mathrm{H}{s}_{j}\) and the base station, and \(\:{D}_{F}\) represents the maximum possible communication distance in the deployment area. This normalized form is applicable to both fixed and mobile sink scenarios.

• The third fitness parameter, \(\:F{P}_{3}\), reflects intra-cluster compactness and accounts for the average distance between sensor nodes and their associated CH. Since intra-cluster communication occurs frequently, compact clusters significantly reduce energy consumption. This parameter is formulated as:

where \(\:\left|{C}_{j}\right|\) is the number of nodes assigned to \(\:\mathrm{C}\mathrm{H}{s}_{j}\), and \(\:\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\left({s}_{i},{s}_{j}\right)\) denotes the distance between sensor node \(\:{s}_{i}\) and its corresponding CH. Higher values of \(\:F{P}_{3}\) indicate denser and more energy-efficient clusters.

• The fourth fitness parameter, \(\:F{P}_{4}\), represents a capability-aware bias term. In homogeneous networks, where all nodes have identical initial energy and capabilities, this term becomes constant and does not influence the optimization process. In heterogeneous deployments, however, it explicitly promotes the selection of higher-capability nodes as CH s to prevent inefficient configurations where low-capability nodes are selected despite the presence of more suitable alternatives. The bias term is computed for each candidate CH as:

where \(\:\mathrm{B}\mathrm{i}\mathrm{a}\mathrm{s}\left({s}_{j}\right)\) is determined according to the capability level of node \(\:{s}_{j}\).

The overall fitness function minimized by AEHMO is defined as:

where \(\:\alpha\:,\beta\:\), and \(\:\gamma\:\) are weighting coefficients satisfying \(\:\alpha\:+\beta\:+\gamma\:=1\) and \(\:\alpha\:,\beta\:,\gamma\:\in\:\left(\mathrm{0,1}\right)\). In homogeneous scenarios, the weights are set to \(\:\alpha\:=0.40,\beta\:=0.25\), and \(\:\gamma\:=0.35\), reflecting the dominant importance of residual energy and communication efficiency, while the bias term remains inactive. Lower values of \(\:F\) correspond to superior CH configurations in terms of energy sustainability, communication cost, cluster compactness, and-when applicable-tier-aware node selection.

Cluster formation with potential function

After AEHMO identifies the optimal CH set, the remaining sensor nodes are assigned to their respective CHs using an energy- and load-aware cluster formation mechanism. At this stage, cluster formation is treated as a localized decision process rather than a global optimization procedure to ensure scalability and reduce the control overhead. Each sensor node independently determines its most suitable cluster assignment using a potential function that is a combination of several factors that are directly related to intra-cluster communication cost and cluster sustainability. The potential of assigning sensor node \(\:{s}_{i}\) to cluster head \(\:C{H}_{j}\), denoted as \(\:S{N}_{p}\), is defined as:

where \(\:{E}_{C{H}_{j}}\) represents the residual energy of the cluster head, \(\:{E}_{{s}_{i}}\) is the residual energy of the sensor node, \(\:d\left({s}_{i},C{H}_{j}\right)\) denotes the Euclidean distance between the sensor node and the CH, and \(\:{L}_{C{H}_{j}}\) is the current load of the CH measured as the number of nodes already associated with it. The parameters \(\:{\lambda\:}_{1},{\lambda\:}_{2},{\lambda\:}_{3},{\lambda\:}_{4}\) are normalized weighting coefficients satisfying \(\:\sum\:_{k=1}^{4}\:{\lambda\:}_{k}=1\). The weights used in this work are \(\:{\lambda\:}_{1}\)=0.40, \(\:{\lambda\:}_{2}\)= 0.20, \(\:{\lambda\:}_{3}\)=0.30, and \(\:{\lambda\:}_{4}\)=0.10. These values are selected based on the relative significance of each factor to the network energy consumption, stability and lifetime, and are consistent with the optimization priorities used in AEHMO-based CH selection. The CH residual energy is assigned the highest weight because CHs execute energy-intensive activities, including continuous reception of data from member nodes, aggregation and transmission of data over a long distance to the base station. Routing disruption and energy holes caused by prematurely drained CHs significantly accelerate overall network degradation. Network stability and reduced re-clustering rates are therefore direct outcomes of prioritization of CH residual energy.

Intra-cluster communication occurs far more frequently than the CH-to-BS transmission, therefore, node-to-CH distance is assigned the second highest weight. Reducing the distance between sensor nodes and their CHs significantly lowers transmission energy consumption with the first-order radio model, and per-round energy efficiency. Emphasizing this factor promotes spatially compact clusters and this improves the coverage quality and reduces the likelihood of the packet loss. The rest of the energy of the sensor node itself is assigned a moderate weight to prevent low-energy nodes from being assigned to distant or congested CHs, which could cause premature node failure. This term improves fairness in energy dissipation across the network without interfering with the membership of the cluster. The CH load is given the minimal weight since it is not a major energy driver, it is rather a correcting factor. Even when excessive CH load can increase the cost of aggregation and reception, its impact is secondary to the distance of communication and residual energy. This factor is allowed to have a relatively low weight to avoid frequent cluster reshuffling caused by minor load variations, without falling into the extreme imbalance in cluster sizes.

Each sensor node computes the potential value of each of the available CHs and selects the CH that maximizes \(\:S{N}_{p}\). When two CHs yield identical potential values, the node is assigned to the one with greater residual energy, thereby further extending cluster lifetime. By integrating energy awareness, communication efficiency, and load balancing in a single lightweight association rule, the proposed cluster formation mechanism can be seen as an enhancement of the global optimization that is adopted by AEHMO, while maintaining low computational complexity and adaptability to dynamic WSN environments.

Simulation settings and performance metrics

This section presents the experimental setup and the performance analysis of the proposed AEHMO-based CH selection approach. Various performance metrics are used to evaluate the cluster-based routing approach, including alive nodes, average energy consumption, total packets transmitted to the BS, FND, HND, and LND. A description of each metric is provided below:

• FND (First Node Death): FND refers to the number of rounds that elapse before the first node in the network dies. It indicates how long all nodes in the network are fully functional.
• HND (Half Node Death): HND represents the number of rounds at which 50% of the nodes in the network have died. Even after half of the nodes are no longer operational, the network continues to perform data transmission.
• LND (Last Node Death): LND measures the number of rounds that pass before all nodes in the network are dead, signifying the point at which the network becomes entirely inoperative.
• Alive Nodes: This metric tracks the number of nodes that remain alive during network operation. A higher number of alive nodes generally correlates with better network performance.
• Average Energy Consumption: This refers to the average energy used by each node at the termination of each round in the case of simulation. Lower energy consumption reflects more efficient use of the limited power budget, directly contributing to extended network lifetime.
• Total Packets Transmitted to the BS: The total number of packets transmitted to the BS is directly related to the number of alive nodes and their remaining energy. A higher number of alive nodes typically leads to a higher number of packets reaching the BS.

In this work, several different network scenarios are simulated by varying the number of sensor nodes in the WSN: scenario 1 with 100 nodes, and scenario 2 with 150 nodes. This enables evaluation of AEHMO’s performance across networks of varying sizes. The simulation parameters are listed in Table 10.

Ablation study

An ablation study was conducted to quantify the individual contribution of each enhancement strategy to WSN performance, evaluating each strategy in isolation across 100-node and 150-node scenarios using FND, HND, LND, and average energy consumption as metrics. The full AEHMO configuration achieved the best results across all metrics, reaching FND of 752/1150 rounds and energy consumption of 0.381/0.395 J for 100/150 nodes respectively, representing improvements of 56.3%/38.1% in FND and 7.5%/6.8% in energy reduction over the baseline HMO. The 10.9%/7.4% FND improvement of full AEHMO over the best single-strategy variant confirms clear synergistic interaction among the four strategies. Full ablation results for all variants and both network scenarios are provided in Supplementary Appendix G, Table G1.

Parameter sensitivity analysis for WSN Application

Before evaluating the performance of AEHMO in WSN cluster head selection, a sensitivity analysis was conducted to identify the most effective parameter configuration for the 150-node scenario. The analysis examined four key parameters: the chaotic iteration depth (\(\:z\)) in the CRL strategy, the MDMS mutation percentage, the MDMS scaling factor (\(\:c\)), and the fitness function weight combinations (\(\:\alpha\:,\:\beta\:,\:\gamma\:\)), using FND, HND, and LND as evaluation metrics. As shown in Table H1, the results confirmed \(\:z\) = 20 as the optimal chaotic iteration depth, achieving the best FND of 1150 rounds, HND of 1853 rounds, and LND of 3136 rounds among all tested values. Table H2 presents the effect of varying the MDMS mutation percentage, where a value of 25% yielded the best overall performance, reflecting the need for more conservative elite refinement to preserve cluster formation stability in WSN environments. Table H3 summarizes the impact of the scaling factor (\(\:c\)), where \(\:c\) = 0.7 achieved the highest network lifetime metrics. Table H4 presents the sensitivity analysis of five fitness function weight combinations, confirming that the configuration (\(\:\alpha\:\)=0.40, \(\:\beta\:\)=0.25, \(\:\gamma\:\)=0.35) strictly satisfying the physically grounded hierarchy \(\:\alpha\:\:>\:\gamma\:\:>\:\beta\:\) achieves the best performance with FND of 840 rounds, HND of 1509 rounds, and LND of 2997 rounds, while progressive violations of this ordering produce monotonically decreasing network lifetime. Across all four parameters, AEHMO demonstrated stable and robust behavior over wide parameter ranges. Full sensitivity results and discussion are provided in Supplementary Appendix H, Tables H1, H2, H3, and H4.

Alive nodes results analysis

We compare the proposed AEHMO algorithm with several state-of-the-art algorithms such as AOA, IVY, MGTOA, HMO, HEOA, GWO, HOA, MHCSA, AOA-H HO, MHABC-PSO and ASG-HMO in two different node density scenarios i.e., 100 and 150 nodes. The location of the BS is fixed at (50, 50) and performance is evaluated over up to 4000 rounds for each algorithm to evaluate each algorithm’s ability to maintain node survivability through efficient energy management.

In the case of the 100-node scenario shown in Fig. 13(a), AEHMO shows superior performance as it keeps the largest number of alive nodes during the simulation period. AEHMO keeps all 100 nodes alive over 800 rounds which is much better than its competitors (like IVY, MGTOA, HMO) which begin to experience node failures around 600–700 rounds. The red curve representing AEHMO shows a significantly slower decline than other algorithms. This increased survivability can be attributed to AEHMO’s adaptive CH selection mechanism which efficiently distributes the energy consumption within the network by dynamically choosing CHs based on residual energy, distance between nodes and CHs, distance between CHs and the BS and degree of nodes. The combination of MDMS and DDS strategies enables AEHMO to avoid premature depletion of energy in individual nodes, which can lead to premature fragmentation of the network and shorten the operational lifetime of the network.

In the 150-node scenario shown in Fig. 13(b), AEHMO demonstrates strong scalability, maintaining efficient energy management even in dense network configurations. The algorithm keeps all 150 nodes alive for more than 1000 rounds, whereas competing algorithms such as AOA, MGTOA and GWO start having nodes dying much earlier, around 800–900 rounds. More impressively, AEHMO’s performance curve demonstrates an impressive longevity with nodes surviving up to the full 3200 rounds that the simulation lasts, significantly outlasting competing algorithms that experience complete network failure before 2500 rounds. This improved performance in larger networks demonstrates AEHMO’s scalability, and the successful use of its four enhancement strategies, working in synergy. The CRL strategy avoids convergence to suboptimal CH configurations while adaptive parameters \(\:{w}_{1}\) and \(\:{w}_{2}\) ensure the balanced exploration–exploitation transitions that adapt to increasing network complexity. The balanced energy consumption mechanism in AEHMO prevents the formation of hotspots and overload of individual nodes, which is a typical failure mode in algorithms such as GWO and HEOA which do not have adaptive load balancing. These results strongly demonstrate that AEHMO provides a robust and highly scalable solution to maximize the network lifetime for different network densities in which nodes in the 150-node case were kept in operation during the entire simulation time.

Total packets transmitted to the BS result analysis

In order to measure the performance of AEHMO and other algorithms in terms of total number of packets sent to BS, two different scenarios with different number of nodes (100 and 150 nodes) were considered. The results for total packets sent to the BS show the efficiency of each algorithm to keep the communication of the network at a sustained level over the whole simulation period. Figure 14 gives a complete comparison of AEHMO with other state-of-the-art algorithms including ASG-HMO, MGTOA, GWO, AOA-HHO, HOA, HMO, AOA, MHCSA, MHABC-PSO, IVY and HEOA.

In the case with 100 nodes shown in Fig. 14(a), AEHMO has the best packet delivery performance with 16,316 successfully transmitted packets to the BS, which is about 6.5% better than the second-best algorithm MGTOA with 15,317 successfully transmitted packets. This substantial difference demonstrates the superior capability of AEHMO for sustained node operation and reliable communication based on its adaptive CH selection mechanism based on MDMS and DDS strategies. In the 150-node scenario in Fig. 14(b), AEHMO is very scalable, and achieves 38,257 packets delivered to the BS, an improvement of 4.7% over ASG-HMO (36557 packets) and 4.69% over MGTOA (36543 packets). The gap in performance is much higher if compared with lower performing algorithms like HEOA (35104 packets) and IVY (35122 packets) with AEHMO having about 9% more packets delivered. This superior performance in larger networks is mainly attributed to the CRL strategy of AEHMO, which introduces structured randomness to escape suboptimal CH configurations, and adaptive parameters \(\:{w}_{1}\) and \(\:{w}_{2}\), which dynamically change the exploration-exploitation balance during the optimization process. Unlike algorithms such as GWO (36320 packets), AOA-HHO (36315 packets), HMO (35837 packets) that use static parameter configurations, the adaptive mechanism of AEHMO helps maintain efficient routing paths throughout the network lifetime and prevents the occurrence of energy hotspots and premature network partitioning. These results clearly indicate that AEHMO extends the network lifetime and maximizes the data communication efficiency on different scales of the network, and hence is the most suitable solution for the applications that require reliable and long-term data transmission to the base station.

Network lifetime result analysis

This section discusses the network lifetime of the proposed AEHMO algorithm, which is compared against other state-of-the-art algorithms using three critical metrics: FND, HND and LND. These metrics were tested in two different scenarios with 100 and 150 node densities to fully measure the capabilities of the algorithms in prolonging the operational lifetime of the network by means of effective energy management strategies.

In the case of 100 nodes shown in Fig. 15, AEHMO achieves strong performance with FND of 752 rounds, HND of 1694 rounds and LND of 3010 rounds, which are much better than all its competitors. For FND, AEHMO, as compared to the second-best algorithm MHCSA (701 rounds), is able to delay the first node death and more than double that of weaker algorithms like IVY (291 rounds) and AOA (321 rounds). This substantial delay in FND is attributed to the DDS and MDMS strategies, which allow AEHMO to investigate the optimal choice of cluster heads, and to evenly distribute energy consumption throughout the network, which prevents any premature energy depletion of any node in the network. For HND, AEHMO’s 1694 rounds is greater than ASG-HMO (1631 rounds) and is far greater than algorithms like HEOA (1401 rounds) and GWO (1449 rounds), indicating sustained network stability. The adaptive parameters \(\:{w}_{1}\) and \(\:{w}_{2}\) play an important role in this phase as they dynamically modulate the exploration-exploitation balance in order to maintain the efficient utilization of energy over time and a higher number of operating nodes for longer periods.

In the case of 150 nodes, the scalability and robustness of AEHMO is even more pronounced with FND at 1150 rounds, 9.2% more than ASG-HMO (1115 rounds) and 81.7% more than AOA (633 rounds). The HND metric achieves 1853 rounds which is better than MHABC-PSO (1816 rounds) and AOA-HHO (1788 rounds) in terms of network stability by keeping the CH selection balanced longer. Most notably, AEHMO achieves LND at 3136 rounds, the highest for any of the algorithms and 0.8% higher than ASG-HMO (3111 rounds) and far higher than MGTOA (2993 rounds) and IVY (2641 rounds). The CRL strategy plays a key role in this accomplishment because it introduces structured randomness through the use of chaotic mapping to avoid convergence to suboptimal CH configurations which may result in rapid depletion of energy or partitioning of the network. The consistent superiority of AEHMO in all three-lifetime metrics in both network densities validates the synergistic effect of AEHMO four enhancement strategies, namely adaptive parameters, MDMS, DDS and CRL, which together ensure balanced energy distribution and prolonged network lifetime, making AEHMO well-suited for enhancing WSN sustainability.

Average energy consumption results analysis

This section presents the average energy consumption of AEHMO across various network scenarios, compared against state-of-the-art algorithms. Energy consumption is a critical metric for evaluating algorithmic efficiency of managing the available energy resources of the nodes within the network. Lower energy consumption directly translates to longer operational lifetime of the nodes, and longer network sustainability.

In the case of 100 nodes as shown in Fig. 16(a), AEHMO achieves the lowest average energy consumption at 0.381 J compared to all the other competing algorithms, and hence it is the most energy efficient algorithm. AEHMO is 4.3% lower energy consumption than the 2nd best algorithm IVY (0.398 J) and 5.9% lower energy consumption than GWO (0.405 J). The performance gap is more significant when compared to less efficient algorithms like MHCSA (0.428 J), ASG-HMO (0.426 J) and MHABC-PSO (0.423 J) where AEHMO achieves an energy consumption reduction of about 11.0%, 10.6% and 9.9% respectively. Even against the baseline HMO (0.412 J) AEHMO has shown a 7.5% improvement in energy efficiency. The DDS strategy further aids by enabling extensive searches of the solution space in early iterations that would otherwise cause the algorithm to converge to energy inefficient CH configurations that could lead to the formation of hotspots and rapid depletion of energy.

In the 150-node case shown in Fig. 16(b), AEHMO maintains its leadership in terms of the lowest average energy consumption at 0.395 J, which indicates that it has an excellent scalability as the network density increases. AEHMO achieves 4.6% less energy consumption when compared with IVY (0.414 J) and 5.5% less energy consumption when compared with AOA (0.418 J) and this difference in performance increases to 11.6% and 11.2% when compared with ASG-HMO (0.447 J) and MHCSA (0.445 J), respectively. It is worth noting that AEHMO consumes 6.8% less energy than the baseline HMO (0.424 J), for this larger network configuration. This superiority in the larger networks over a long time validates the importance of the adaptive parameters \(\:{w}_{1}\) and \(\:{w}_{2}\) used by AEHMO, which dynamically adjust the balance between exploration and exploitation throughout the optimization process, ensuring efficient energy utilization even as the complexity of networks is increasing. The CRL strategy plays an important role by introducing structured randomness through chaotic mapping, AEHMO can break out of suboptimal CH arrangements that can lead to uneven energy distribution and premature node failures, a common drawback of static algorithms such as HMO and HEOA (0.429 J). The sustained achievement of the lowest energy consumption across both network densities confirms that AEHMO’s synergistic combination of four enhancement strategies is able to offer the most energy efficient solution for the problem of WSN cluster head selection.

Result analysis for AEHMO using BS at the corner

To further evaluate the robustness of AEHMO under asymmetric network topology, an additional experiment was conducted with the BS placed at the corner of the sensing field at coordinates (100, 100) using a 100-node deployment. Under this more demanding scenario, AEHMO achieved the best performance across all compared algorithms with FND = 785, HND = 1420, and LND = 2815 rounds, outperforming the closest competitor HMO by 5.8%, 5.3%, and 4.7% in FND, HND, and LND respectively. Full results and comparative analysis are provided in Supplementary Appendix I, Table I1.

Real world heterogeneous WSN scenario validation using mobile sink strategy

To address the limitations of the scalability problems and to validate the performance of AEHMO under the real-world deployment condition, this subsection presents a comprehensive evaluation of AEHMO using a large-scale heterogeneous network scenario with 1200 sensor nodes. In contrast to the homogeneous scenarios which are simplified in previous subsections, this evaluation incorporates multi-tier heterogeneity, strategic spatial deployment, a mobile sink strategy, and complex network topologies to accurately reflect real-world WSN deployments in industrial, environmental and smart city applications.

Three tier heterogeneous network model

The heterogeneous network model generalizes the basic WSN model by adding three different levels of node capability that produce a realistic model of practical sensor deployments in which nodes have different levels of computational power, energy reserve and transmission capability. This three-tier heterogeneity (3-Tier) model consists of advanced, intermediate, and normal nodes with initial energy levels and deployment strategies.

The network with 1200 nodes is divided into three hierarchical categories based on some established principles in designing heterogeneous WSN. Let \(\:n\) = 1200 be the total number of deployed sensor nodes that are spread in the monitoring field. The inequality of the distribution is according to the inequality:

This inequality ensures that normal nodes form the majority, providing full area coverage; intermediate nodes offer supplementary support; and advanced nodes constitute a smaller but critical subset with enhanced capabilities. For the case of 3-Tier configuration, the network composition is specified with help of proportion factors \(\:k\) and \(\:{k}_{o}\):

where \(\:k\) is the proportion of advanced nodes, and \(\:{k}_{o}\) is the proportion of intermediate nodes in the network. For the 1200-node deployment, \(\:k\) is set to 0.167 and \(\:{k}_{o}\) is set to 0.25 are used resulting in:

• Advanced nodes (\(\:{n}_{adv}\)) = 200 nodes (16.7%).

• Intermediate nodes (\(\:{n}_{int}\)) = 300 nodes (25%).

• Normal nodes (\(\:{n}_{nrm}\)) = 700 nodes (58.3%).

Energy configuration of heterogeneous tier

The allocation of energy is hierarchically organized according to the following basic inequality:

This inequality reflects the realistic case where more able nodes have access to higher capacity power sources to support their extended role of CH and data aggregation point. The energy of each tier is determined as follows:

where \(\:{E}_{o}\) is the base energy assigned to normal nodes and \(\:\varPsi\:\) is the energy fraction by which intermediate nodes exceed normal nodes and \(\:\omega\:\) is the energy fraction by which advanced nodes exceed normal nodes. For the above deployment, \(\:{E}_{o}\) = 0.5 J, \(\:\varPsi\:\) = 1.0 (intermediate nodes have 100% more energy as compared to normal nodes) and \(\:\omega\:\) = 2.0 (advanced nodes have 200% more energy as compared to normal nodes). This arrangement gives a value of 0.5 J for individual node energies for normal nodes, 1.0 J for intermediate nodes, and 1.5 J for advanced nodes. The total of the network energy is computed using:.

which can be simplified to:

This hierarchical distribution of energy ensures that sophisticated nodes can be used for sustaining the CH tasks for much longer durations, intermediate nodes can act as important backups and supplementary CH services, and normal nodes are mostly sensing and data collection units with less processing overheads.

Spatial deployment topology

Unlike the homogeneous random deployment used in the previous section, this heterogeneous network uses a spatial distribution strategy that considers realistic deployment constraint, network optimization principles and practical infrastructure considerations in real-world WSN applications. The monitoring area is set up as a 300\(\:\times\:\)300 square area which represents a realistic coverage area for industrial monitoring, precision agriculture or environmental sensing applications. The position of BS is central coordinates:

This central positioning ensures the maximum transmission distance from any sensor node towards the BS is minimum and hence the energy consumption is less, and network connectivity is improved. However, in order to resolve the hotspot problem of the centralized BS architectures, moveable multiple sinks schemes are integrated, which is described in next subsection.

The deployment is based on a hierarchical zonal strategy with the aim to maximize the energy efficiency by locating more capable nodes close to the critical infrastructure and ensuring the whole area is covered through the distributed normal nodes. Advanced nodes are located in a strategic manner in a circular subsection with the BS location (150, 150) as the center. The radius of such circular zone is calculated in accordance with network density requirements and optimum CH distribution with \(\:{r}_{adv}\) is 40 m. This central location ensures that the most able nodes with the most energy reserves are located near the BS, reducing the distances over which long distance transmissions occur when inter-cluster communication is necessary, and ensuring that they will have the greatest utility as primary cluster heads throughout the operational lifetime of the network. The density of advanced nodes in circular subsection is given as:

Intermediate nodes are deployed strategically all over the sensor field area except the circular subsection which is reserved for advanced nodes. A critical constraint controls their placement in such a way that intermediate nodes have to be deployed in the vicinity of randomly selected normal nodes so that each normal node has access to at least one higher-capacity neighbor in order to provide CH services. The deployment approach makes sure that no normal node is assigned to more than two intermediate nodes which will prevent clustering imbalances and ensure energy is distributed equally. The density of the intermediate nodes is as follows:

For the deployment scenario, where \(\:Fiel{d}_{area}\) is 300\(\:\times\:\)300m². Normal nodes are randomly distributed all over the 300\(\:\times\:\)300 monitoring area (not the central circular subsection which will be used only for advanced nodes). This distribution gives high coverage of the areas with still preserving the hierarchy energy structure where normal nodes are known as the primary sensing layer that is responsible for forwards the data to their designated cluster heads. The normal nodes density is as follows:

This spatial arrangement generates realistic topology, which emulates the real-life deployments of having high-capacity nodes near the critical infrastructures (base stations, gateways, edge servers) while providing wide pervasive coverage with distributed sensing nodes. The zonal deployment strategy naturally sets up energy-efficient communication patterns because the high-capacity advanced nodes that are close to the BS can aggregate and relay data from several clusters without consuming their energy reserves too early.

Movable multiple sink strategy

One of the important problems in WSN with centralized BS architecture is the hotspot problem where the CHs near the BS get drained out of energy rapidly due to intensive relay traffic from the far clusters. To overcome this simple limitation and to enhance the network performance, movable multiple sink strategy is proposed in the deployment scenario.

In static single sink configurations, energy consumption of sensor nodes which are nearest to the BS are too high since they have to relay data from the more distant nodes, leading to premature death of nodes and network partitioning. Movable sinks in the network strategically positioned outside the network field can enhance the performance of the network in terms of reducing communication distances between the CHs and the sinks, balancing the relay load on the network, reducing congestion around the static base stations, extending the lifetime of the network by balancing the energy consumption and efficiency of data collection in large-scale deployments.

In this deployment scenario, there are four movable sinks outside of the network field, and circular trajectories. The sinks are moving on their own without any human intervention. The sinks traverse outside the network area in circular patterns. The middle of this virtual circle is the point of the center of the network \(\:Center\:=\:({c}_{x},\:{c}_{y})\). The radius of the circular trajectory is taken such that the sinks are not placed on the boundaries of the network, and the distances of communication are reasonable:

For the 300\(\:\times\:\)300 field, \(\:{r}_{c}\) is made large enough to place sinks outside of the network area. The sink locations at any instant of time are generated by means of parametric equations:

where \(\:\theta\:\) is the angular position parameter and depending on the mobility requirements and constraints of the application different rotation strategies can be used. For complete circular rotation, the parameter, \(\:\theta\:\), varies \(\:0\le\:\theta\:\le\:2\pi\:,0\le\:\theta\:\le\:\pi\:\), and \(\:0\le\:\theta\:\le\:\pi\:/2\), so the sinks can move all around the perimeter of the network. All sinks are free to move along their circular paths in either clockwise or counter-clockwise. As the network operation continues for several rounds, the sinks move to their next position as per a predefined movement schedule and collect the data from different regions of the network in turn. In the mobile sink case, it is assumed that sensor nodes and CHs know the location of sinks by periodic beacon messages transmitted by each sink at the start of each round, which is consistent with the round-based operation of LEACH like protocols. Each beacon contains the identifier of the sink and the actual coordinates in the predefined trajectory for the CHs to estimate the distances to the beacons based on RSSI or localization services and select the sink that is closest to be able to forward the data. Beaconing is performed once per round to balance location accuracy against control overhead. The additional energy costs added by sink mobility are explicitly considered in the energy model by receiving beacon packets and exchanging scarce control messages, so as to ensure that the benefits of mobility are not overestimated.

Extending AEHMO fitness function for heterogenous CH selection

In heterogeneous WSN deployments, the AEHMO fitness function defined in the section “Objective function for AEHMO-based CH selection is extended to explicitly account for node capability differences arising from the multi-tier network structure. The core fitness components-residual energy suitability (\(\:F{P}_{1}\)), CH -to-sink proximity (\(\:F{P}_{2}\)), and intra-cluster compactness (\(\:F{P}_{3}\))-remain unchanged and are evaluated using the unified formulation.

To prevent inefficient CH configurations in which low-capability nodes are selected despite the presence of more capable alternatives, an additional capability-aware bias term \(\:F{P}_{4}\) is activated. This term explicitly incorporates node-tier information into the CH selection process and is defined as:

where \(\:\mathrm{B}\mathrm{i}\mathrm{a}\mathrm{s}\left({s}_{j}\right)\) assigns higher values to nodes belonging to higher-capability tiers. In homogeneous deployments, where all nodes share identical initial energy and capability, this term reduces to a constant and does not influence the optimization outcome. The heuristic nature of metaheuristic optimization algorithms may result in suboptimal CH selection with no consideration of inherent capability difference between node tiers. For example, a normal node could be selected as CH in presence of an advanced node or an intermediate node in the same cluster, it could just happen that the optimization process has generated that configuration with good fitness based on \(\:F{P}_{1}\), \(\:F{P}_{2}\) and \(\:F{P}_{3}\) alone. However, the selection of normal node as CH in case of advanced node is fundamentally inefficient as the normal node will get exhausted much faster so that more frequent re-clustering operations are required, and network stability is low. The capability-aware bias parameter addresses this limitation by explicitly favoring higher-tier nodes in the candidate CH set:

For the 1200 node deployment with 200 advanced, 300 intermediate and 700 normal nodes, the bias values are 0.833 for advanced nodes, 0.750 for intermediate nodes and 0.417 for normal nodes. This parameter assigns higher bias values to advanced and intermediate nodes, encouraging the algorithm to leverage the energy advantage of capable nodes from the outset and not wait until energy differentiation becomes visible through \(\:F{P}_{1}\) only. Thus, the four fitness parameters are aggregated in one objective function by weighted aggregation:

where the weight coefficients must satisfy the normalization condition that \(\:\alpha\:\:+\:\beta\:\:+\:\gamma\:\:=\:1\). And each of the coefficients are bounded in such that \(\:\alpha\:\), \(\:\beta\:\), \(\:\gamma\:\) are in 0 to 1. For this heterogeneous scenario, the weights are assigned in accordance with the relative importance of each factor in large-scale networks with mobile sinks. Specifically, the value for \(\:\alpha\:\)= 0.40 (residual energy weight having the highest priority in terms of sustainability), \(\:\beta\:\)= 0.25 (CH-sink distance weight being critical for movable sink scenarios) and \(\:\gamma\:\)= 0.35 (intra-cluster compactness weight being important for communication efficiency). These weights satisfy the constraint of being a vector that adds up to 1.00, i.e., satisfy the constraint of being properly normalized. During its optimization process, AEHMO minimizes the fitness function \(\:F\), with lower values of \(\:F\) corresponding to better CH configurations in terms of energy efficiency, communications distance, clusters compactness and tier-appropriate node selection.

Multi hop communication strategies

To further enhance the energy efficiency in this large-scale heterogeneous network, AEHMO integrates with the intra-cluster and inter-cluster multi-hop routing strategies. These strategies are based on the understanding that direct long-distance transmissions are energy inefficient compared to multi-hop relaying via the intermediate nodes, especially for large 300\(\:\times\:\)300 fields where node to sink or node to CH distances can easily exceed 200 m. Despite the addition of mobile sinks and multi-hop extensions, the underlying communication and clustering structure is still LEACH-based with AEHMO only replacing the CH selection mechanism while preserving the standard setup and steady-state phases, TDMA scheduling and data aggregation model:

• Intra-cluster multi hop strategy: within a cluster, the energy consumption for communication between the cluster members (CMs) and the CH is proportional to the square of the distance for the communications below the threshold \(\:{r}_{o}\) (free-space propagation). Since, only a small fraction of nodes in a cluster are physically close to center of the cluster, many CHs elected through rotation strategies can be far from the geometric center of the cluster, this leads to increasing intra-cluster communication distances, and aggravates energy consumption for CMs. To solve this inefficiency, intra-cluster multi-hop strategy is taken. A threshold distance \(\:{d}_{intra}\) is specified in order to classify the members of the clusters:

The working principle of the communication protocol is as follows

Inner layer nodes communicate directly with CH. Outer layer nodes select the nearest inner layer node to which they send their data to CH. For an outer layer node \(\:{S}_{out}\) sending \(\:k\) bits of data to the CH through an inner layer node \(\:{S}_{in}\) where \(\:{d}_{{S}_{out}-\:{S}_{in}}\) is the distance between the two nodes and no greater than \(\:{r}_{o}\), the total energy consumption is.

The magnitude of \(\:{E}_{2-hop}\) is a function primarily of \(\:{d}_{{S}_{out}-\:{S}_{in}}\). Therefore, for intra-cluster communication, it is energy-efficient that the outer layer nodes select the inner layer node that is closest to them as a relay. This minimizes \(\:{d}_{{S}_{out}-\:{S}_{in}}\), and hence the quadratic energy term. The value of \(\:{d}_{intra}\) which is decided by network environment and algorithm designing. Larger cluster structures have larger \(\:{d}_{intra}\) value for balancing the tradeoff between overhead of relay and reduction of transmission distance. For 1200 node scenario in 300\(\:\times\:\)300 field \(\:{d}_{intra}\) is calculated using empirical optimization.

• Inter-cluster multi hop strategy: In the inter-cluster communication phase CHs need to route their aggregated data to the BS. For CHs placed at a large distance from sinks (beyond the threshold \(\:{d}_{inter}\)), direct transmission would result in an excessive energy consumption, because of the \(\:{d}^{4}\) relation from multipath propagation. Multi-hop forwarding through intermediate CHs is a more energy efficient solution. CHs select the most energy efficient relay node by measuring the distances to neighboring CHs by using received signal strength indicator (RSSI) measurement. Multi-hop forwarding is performed only if the distance between the CH and the BS (or nearest sink) is more than the inter-cluster threshold \(\:{d}_{inter}\) otherwise the CH sends the data directly. Based on distance relationship between \(\:C{H}_{i}\) and BS, and distance relationship between \(\:C{H}_{i}\) and its neighboring CHs, routing metric \(\:{R}_{t}\) for each of neighboring CHs is computed as follows:

where \(\:Rt\left(i\right)\) is the array of routing metric of neighboring CHs of \(\:C{H}_{i}\), \(\:d(i,\:BS)\) is the distance between \(\:C{H}_{i}\) and the base station (or nearest sink), \(\:d(i,\:j)\) is the distance between \(\:C{H}_{i}\) and neighboring \(\:C{H}_{j}\), \(\:d(j,\:BS)\) is the distance between neighboring \(\:C{H}_{j}\) and the base station and \(\:j\) is the index of the neighboring CH ranging from 1 to \(\:{N}_{CH}\), where \(\:{N}_{CH}\) is the total number of CHs. Based on calculated \(\:Rt\) values, \(\:C{H}_{i}\) selects the next hop node based on:

where \(\:Next\left(i\right)\:=\:j\) means that \(\:C{H}_{j}\) is selected as the relay node and \(\:Next\left(i\right)\:=\:0\) means direct transmission to the BS without any relaying. The condition \(\:max\left(Rt\right(i\left)\right)>1\) ensures that relaying through \(\:C{H}_{j}\) is geometrical favorable in comparison to direct transmission. When \(\:Rt\:>\:1\), the square distance from \(\:C{H}_{i}\) to the BS will be more than the sum of square distances via relay path so multi hop transmission will be more energy efficient.

Furthermore, the inter-cluster multi-hop strategy does not take the residual energy of relay CHs explicitly into consideration when the next hop is selected. This choice of design is justified by the strategy of CH rotation used in the network. In each round, CHs gather residual energy information from their CMs and compare their own residual energy to the initial cluster average residual energy (\(\:{C}_{initial}\)) to make sure that energy factors are well considered during CH selection. In the case of inter-cluster multi-hop routing, the relay node with the largest \(\:Rt\) is able to guarantee the most energy-efficient relay scheme from a geometric point of view. If the residual energy was further considered at this point, the approach might choose geometrically suboptimal relay paths to conserve low energy nodes which might lead to a higher overall network energy consumption. Therefore, not considering the energy aspect in relay selection is helpful to minimize information exchange overhead between CHs, without compromising the energy efficiency. For the field size of 300 m \(\:\times\:\) 300 m with movable sinks, the value of \(\:{d}_{inter}\) is obtained according to the threshold of the propagation model and network dimensions. CHs outside this threshold use multi-hop routing and CHs inside this range communicate directly with the sinks.

Results analysis for the heterogeneous WSN scenario

This subsection presents a detailed analysis of the performance of AEHMO in the large-scale heterogeneous WSN scenario (1200 nodes) with a mobile-sink strategy. The heterogeneous three-tier architecture and mobile sinks allow for a realistic testing environment to closely approximate practical WSN deployments for industrial monitoring, smart cities and environmental sensing applications.

The network survivability analysis as shown in Fig. 17 shows AEHMO’s superior performance in keeping operational nodes alive during the network lifetime. AEHMO exhibits strong node preservation, where all the 1200 nodes remain alive for a much longer time than competing algorithms. The graph shows that AEHMO maintains roughly 300 nodes in operation after 25,000 rounds and most of the competitors have network-wide node depletion before 40,000 rounds. This long survivability is attributed to AEHMO’s intelligent CH selection mechanism that considers the three-tier heterogeneity through the comparative bias parameter (\(\:F{P}_{4}\)) so that advanced nodes with higher energy reserves (1.5 J) are selected as cluster heads during the critical phases of the network. The slowly decreasing alive nodes curve of AEHMO is in sharp contrast with the rapidly declining alive-node curves of the algorithms like AOA, ASG-HMO and MHABC-PSO which indicates balanced energy consumption for the heterogeneous network topology. Notably, HEOA and HMO are relatively better than other competitors in terms of keeping nodes alive until about 36,000 and 32,000 rounds respectively, although they are still significantly inferior to AEHMO. The better performance is attributed to the synergistic combination of the adaptive parameters \(\:{w}_{1}\) and \(\:{w}_{2}\) in AEHMO which dynamically control the balance of exploration and exploitation, DDS for thorough exploration of the spatial search space for CH selection, MDMS for CH candidate refinement, and CRL for escaping suboptimal configurations which would otherwise lead to premature energy depletion in the critical areas of the network.

The total packet transmission analysis, as shown in Fig. 18, indicates that AEHMO demonstrates strong data-delivery capability in the heterogeneous mobile sink situation. AEHMO gets the highest packet delivery with 436,583 packets successfully transmitted to the base station which is closely followed by MGTOA with 428,903 packets. This represents a substantial improvement over several baseline algorithms, with AEHMO sending around 16% more packets than AOA (376,826 packets), 27% more than GWO (243,995 packets) and nearly four times that of the lowest-performing methods (HOA (107,534 packets) and AOA-HHO (111,308 packets)). Higher packet delivery is directly correlated with AEHMO’s extended network lifetime and efficient energy management as nodes that remain operational for longer periods of time can continuously sense and transmit data to the mobile sinks. The multi-hop routing strategies integrated with AEHMO’s CH selection further improves the packet delivery by optimizing the transmission paths based on the dynamic sink positions, reducing packet loss caused by long multi-hop transmission distances. The competitive performance of MGTOA implies that nature inspired algorithms with good exploration mechanisms can achieve high packet throughput in mobile sink situations, but AEHMO’s adaptive mechanisms are slightly stronger and more consistent. The middle-tier performers (MHABC-PSO: 362,115; MHCSA: 355,972; HMO: 344,795; ASG-HMO: 314,148) suggest that hybrid metaheuristic approaches provide enhanced performances over the single-strategy algorithms, but not as comprehensive enhancement mechanisms as AEHMO. The poor performance of algorithms like IVY (123,027), AOA-HHO (111,308), HOA (107,534) can be explained by the fact that they deplete energy rapidly and they are not able to adapt their CH selection strategy to the energy heterogeneous distribution and mobile sink trajectories resulting in early network partitioning and communication failures.

The average energy consumption per node analysis, as shown in Fig. 19, demonstrates strong energy efficiency of AEHMO in the large-scale heterogeneous network. AEHMO has the lowest average energy consumption at 0.6219 J per node, which is significantly better than all the other competing algorithms and is well below the overall average (0.7948 J), as indicated by the dashed reference line. This corresponds to ~ 21.8% lower energy consumption than the competitor average and shows significant improvements over high consuming algorithms such as MGTOA (0.8820 J), AOA (0.8727 J), AOA-HHO (0.8761 J) and ASG-HMO (0.8622 J). The advantage in terms of energy efficiency is more significant when it is compared to algorithms such as MHABC-PSO (0.8761 J) and HEOA (0.7438 J) with AEHMO consuming 29% and 16.4% less energy, respectively. The superior energy performance is due to several complementary mechanisms: (1) Through FP4 in the fitness function, advanced (1.5 J) and intermediate (1.0 J) nodes are preferentially selected as cluster heads, and normal nodes (0.5 J) will not be prematurely depleted of energy; (2) the intra-cluster and inter-cluster multi-hop routing schemes reduce energy costly long distance transmissions by intelligently routing data through intermediate nodes; (3) The mobile-sink strategy minimizes average CH-to-sink distance, reinforced by FP2 in the fitness function. The large gap of energy consumption between the best (AEHMO: 0.6219 J) and worst (MGTOA: 0.8820 J) performers, which is about 41.8% highlights the importance of intelligent CH selection and routing optimization for prolonging network lifetime in large-size heterogeneous WSN deployment.

The comprehensive network lifetime analysis in terms of FND, HND and LND metrics presented in Table 11 proves the outstanding performance of AEHMO in the heterogeneous mobile sink scenario. For FND, AEHMO achieves 362 rounds before the first node death, representing a substantial improvement of 87.6% as compared to the second-best performers GWO and MHCSA (both with 193 rounds), 85.6% as compared to HMO (195 rounds) and 209.4% as compared to worst performers AOA-HHO (117 rounds) and MHABC-PSO (127 rounds). This high FND improvement is a direct validation of the effectiveness of the comparative bias parameter (FP4) and the hierarchical energy configuration strategy as AEHMO manages to prevent premature depletion of low-capacity normal nodes by giving priority to advanced and intermediate nodes to perform CH duties in the critical early operation phases. The HND metric goes on to prove AEHMO’s superiority, with 1903 rounds (the point where half of the network is still operating) compared to the second-best performer AOA-HHO with 1873 rounds, MGTOA with 1791 rounds, and MHABC-PSO with 1514 rounds. The minor difference between AEHMO and AOA-HHO at HND (1.6% difference) is in stark contrast to the large FND advantage (209% improvement), suggesting that although the performance of both algorithms will eventually be similar, the early-stage energy management advantage of AEHMO gives it important operational advantages. Most importantly, the LND metric, which is total network lifetime, AEHMO managed to achieve 56,946 rounds, which is an extraordinary improvement over the second-best performer IVY (41,928 rounds), 49.3% better than HOA (38,147 rounds), 57.3% better than HEOA (36,212 rounds), and 76.8% longer than HMO (32,214 rounds). When compared to the worst performers in the large-scale scenario, AEHMO shows even more dramatic advantages: 146.5% longer LND than MGTOA (23,096 rounds), 139.3% longer than AOA-HHO (23,804 rounds) and longer than the lowest-performing AOA (21,374 rounds).

Communication overhead analysis

A communication overhead analysis was conducted across network scales ranging from 50 to 1000 nodes to evaluate the scalability of AEHMO in terms of control message exchange required for CH selection and network maintenance. AEHMO consistently achieved the lowest overhead across all network sizes, reaching 2.5%, 4.0%, 8.5%, and 12.0% at 100, 150, 500, and 1000 nodes respectively — representing reductions of up to 46.7% compared to the worst-performing competitor IVY. This efficiency is attributable to the memory-based recall mechanism that reduces redundant fitness evaluations, the DDS strategy that accelerates convergence in early iterations, and the MDMS selective mutation applied only to the top 20–30% elite solutions. Full overhead comparisons across all algorithms and network scales are provided in Supplementary Appendix J, Table J1.

Comparison between AEHMO and other state-of-the-art approaches

This section presents a comparison of the proposed AEHMO algorithm with the recent and state-of-the-art algorithms in terms of some key performance metrics: FND, HND, LND, Average Energy Consumption, Number of Transmitted Packets to BS. The comparison underlines the superiority of AEHMO in terms of prolonging network lifetime, reducing energy consumption, and increasing the number of data packets transmitted to BS. Table 12 describes the results for different metrics.

FND is an important metric that reflects the network’s early-stage energy management effectiveness. In Table 12, the FND of the proposed algorithm AEHMO has reached 1150 rounds, whereas the algorithms PSOECSM and AROA have achieved only 698 and 834 rounds respectively. These improvements in FND are mainly rooted in the fact that AEHMO’s DDS and MDMS make the selection of cluster heads wiser, while energy is distributed among nodes more evenly to delay the death of the first node. Moreover, AEHMO has also recorded the highest value of 1853 rounds regarding the HND metric, as reflected in Table 12. Other algorithms such as PSOECSM, GAPSO-H, and SWARAM recorded HND values as low as 1392 and 1412 rounds due to their inefficient energy management techniques. Regarding the LND metric, AEHMO reaches 3136 rounds, which outperforms the performance achieved by other algorithms such as PSOECSM with 2933 rounds and SWARAM with 2591 rounds. This denotes the potential of the AEHMO algorithm to extend the overall network lifetime.

These results clearly demonstrate that AEHMO outperforms all compared state-of-the-art algorithms across all metrics, and it is a highly efficient solution to extend network lifetime, reduce energy consumption, and enhance data transmission. The proposed strategies, DDS and MDMS, along with adaptive parameters and CRL play an important role in energy optimization and ensuring better performance in general.

Conclusion and future directions

This paper presented an Adaptive Enhanced Human Memory Optimization algorithm (AEHMO) for global optimization and energy-efficient CH selection in WSNs. AEHMO incorporates adaptive parameters \(\:\boldsymbol{w}1\) and \(\:\boldsymbol{w}2\), MDMS, DDS, and CRL, which collectively enhance its exploration-exploitation balance and accelerate convergence. AEHMO was rigorously evaluated on the CEC2017 benchmark suite, demonstrating strong performance on complex optimization problems. In terms of global optimization performance, AEHMO achieved the best Friedman ranks of 2.15, 2.37, 2.87, and 2.23 across 10D, 30D, 50D, and 100D problem dimensions respectively on the CEC2017 benchmark suite, consistently outperforming eleven competing algorithms including GWO, AOA, HMO, HEOA, HOA, IVY, MGTOA, MHCSA, AOA-HHO, MHABC-PSO, and ASG-HMO. The Wilcoxon rank-sum test confirmed statistically significant superiority (\(\:p\) < 0.001) across all dimensional scales, with a mean effect size r ranging between 0.853 and 0.873, validating both the statistical and practical significance of the improvements. Furthermore, AEHMO demonstrated strong engineering optimization capability, achieving perfect or near-perfect convergence consistency across five classical engineering design problems including Tension-Compression Spring, Tubular Column, Cantilever Beam, and Corrugated Bulkhead designs. Regarding CH selection in homogeneous WSN scenarios, AEHMO achieved an FND of 752 rounds, HND of 1694 rounds, LND of 3010 rounds, and the lowest average energy consumption of 0.381 J in the 100-node scenario, while in the 150-node scenario it achieved an FND of 1150 rounds, HND of 1853 rounds, LND of 3136 rounds, and average energy consumption of 0.395 J, representing improvements of up to 81.7% in FND and 4.6% lower energy consumption compared to competing algorithms. In terms of data delivery, AEHMO successfully transmitted 16,316 packets to the BS in the 100-node scenario, approximately 6.5% more than the second-best algorithm MGTOA (15,317 packets), and 38,257 packets in the 150-node scenario, approximately 4.7% more than ASG-HMO (36,557 packets).

Most notably, validation under a large-scale real-world heterogeneous WSN scenario comprising 1200 sensor nodes with a three-tier node architecture and a mobile sink strategy demonstrated AEHMO’s exceptional scalability, achieving an FND of 362 rounds, HND of 1903 rounds, and LND of 56,946 rounds. These results represent improvements in total network lifetime over the second-best performer IVY (41,928 rounds), 76.8% improvement over HMO (32,214 rounds), and 146.5% improvement over MGTOA (23,096 rounds). Additionally, AEHMO achieved the lowest average energy consumption of 0.6219 J per node, approximately 21.8% below the competitor average, and the highest packet delivery of 436,583 packets to the base station. The ablation study further confirmed that all four enhancement strategies contribute synergistically, with the full AEHMO configuration achieving 56.3% and 38.1% FND improvement over baseline HMO in the 100-node and 150-node scenarios respectively.

Despite these achievements, several limitations persist in the current study. For instance, AEHMO’s adaptability to highly dynamic and real-time network environments has not been thoroughly investigated, and parameter adaptation mechanisms may require further refinement to ensure optimal performance under diverse network conditions. Future work will focus on enhancing the scalability and adaptability of AEHMO for deployment in real-time and dynamic WSN environments. The integration of self-adjusting mechanisms and dynamic parameter tuning will improve AEHMO’s responsiveness to fluctuating network conditions and mobile WSN deployments. Another promising direction is the integration of machine learning or deep learning models with AEHMO. Such hybrid models could enhance AEHMO’s adaptability by enabling data-driven CH selection decisions that respond to real-time network dynamics. Further validation across real-world applications such as environmental monitoring, smart cities, and industrial automation will demonstrate the broader applicability and effectiveness of AEHMO. Additionally, extending AEHMO to cooperative multi-agent network designs — where node interactions are explicitly modeled — could further improve packet delivery ratios and overall energy efficiency. Furthermore, physical deployment and validation of AEHMO on real sensor mote platforms, such as Raspberry Pi or Arduino-based nodes, represent a valuable future direction that would confirm the algorithm’s practical applicability beyond simulation. Finally, applying AEHMO to broader domains such as image processing, robotics, and smart agriculture will be explored to further demonstrate its versatility as a general-purpose global optimizer.

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