1.1 Motivations

TSA has shown effectiveness in solving optimization problems, but it suffers from several key limitations that hinder its performance. Firstly, TSA tends to converge prematurely, often becoming trapped in local optima due to its limited exploration capacity [31]. Secondly, TSA’s reliance on random perturbations for seed generation introduces excessive randomness, making it difficult to control the search process effectively [32]. Lastly, TSA’s search process can lack diversity, especially at fixed locations, making it harder to solve complex problems effectively [33]. These limitations highlight the need for improvements to enhance TSA’s search capability and adaptability in complex optimization problems. To address these challenges, this study designs an adaptive framework as the core of the proposed method. This adaptive framework operates on three key components of TSA: trees, seeds, and step sizes, thereby enhancing the overall performance and robustness of the algorithm. The motivations driving this research are as follows:

• TSA struggles with balancing exploration and convergence due to its fixed seed generation, which often leads to premature convergence [27]. By redesigning the seed generation process, we introduce a dynamic method to adjust the number of seeds at different stages, addressing this issue and improving the search efficiency and convergence rate.
• TSA struggles with enhancing local exploration and accelerating convergence, particularly during the local exploration phase [34]. By introducing a nonlinear step-size adjustment mechanism and an asymptotic approximation mechanism, we improve TSA’s adaptability at different stages, which enhances convergence during the local exploration phase and ensures more efficient and stable search performance.
• TSA often lacks diversity in its search process, particularly at fixed locations, which limits its global exploration capability [35]. By introducing an adaptive migration mechanism, we enhance TSA’s ability to explore more broadly, addressing this limitation and improving its efficiency in solving complex problems.

1.2 Contribution

While the Tree-Seed Algorithm (TSA) has been the focus of our ongoing research, leading to variants such as ATSA [26] and KATSA [33], the proposed AMTSA represents a significant methodological advancement rather than an incremental improvement. Our previous works, like ATSA, introduced a double-layer framework, and KATSA utilized a k-NN strategy to enhance the search process. However, these prior models still faced challenges in achieving a robust, dynamic balance between global exploration and local exploitation, particularly in high-dimensional and complex problem landscapes. The novelty of AMTSA lies in its unique, three-part synergistic framework designed specifically to overcome these limitations. Unlike our previous approaches, AMTSA introduces:

• A Dynamic Weibull Distribution for Seed Generation: This is a fundamental departure from the more rigid or heuristic seed adjustment strategies in our prior work. By dynamically linking the seed count to the population’s real-time fitness statistics (mean and standard deviation), AMTSA can adapt its search breadth in a more principled and responsive manner.
• A GBO-Inspired Nonlinear Step-Size Mechanism: While previous variants adjusted search parameters, AMTSA is the first in our series to incorporate a sophisticated, nonlinear step-size function inspired by the Gradient-based Optimizer (GBO). This, combined with an asymptotic convergence approach, allows for a much smoother and more effective transition from an aggressive global search to a fine-tuned local search, a feature not fully realized in ATSA or KATSA.
• A Fitness-Guided Adaptive Migration Strategy: The migration mechanism in AMTSA is distinct from earlier concepts. It not only provides an escape route from local optima but does so adaptively, where the migration step size and direction are dynamically determined by the individual tree’s fitness value. This creates a more intelligent and targeted exploration capability.

In summary, AMTSA is not merely another TSA variant but a comprehensive redesign. Its true innovation lies in the integration of these three adaptive, mutually reinforcing mechanisms, which together create a more robust and efficient optimization tool specifically tailored for the high-dimensional, complex problems addressed in this paper.

In recent years, a growing emphasis has been placed on the need for scientific rigor in the design of metaheuristic algorithms. Scholars such as Sörensen [36] and Camacho-Villalón et al. [37] have published critiques of algorithms that rely heavily on natural metaphors without introducing sufficient mechanical novelty. This body of work argues that the contribution of a truly valuable new algorithm must lie in the introduction of concrete and novel mechanisms that effectively solve optimization problems, rather than simply proposing a new story. We fully concur with this perspective, and it has served as a guiding principle in the design of AMTSA. Although the Tree-Seed Algorithm (TSA) was initially inspired by the relationship between trees and seeds in nature, the core innovation of our proposed AMTSA is not founded on this metaphor. Instead, it is rooted in its concrete, quantifiable mathematical and adaptive mechanisms. The contribution of this work is not the mimicry of a natural process, but rather a framework of three synergistic, carefully engineered components designed to overcome the limitations of existing algorithms, enabling superior performance in solving complex, high-dimensional optimization problems.

2.1 A brief introduction of tree-seed algorithm

Tree-Seed Algorithm (TSA), proposed by Mustafa Servet Kiran in 2015, is a population intelligence algorithm inspired by the relationship between tree and seed [24]. TSA solves optimization problems by simulating the process of trees reproducing their offspring. The algorithm has been widely used in optimization problems such as function optimization, engineering design optimization, and resource allocation. it has attracted attention for its good global search capability and high convergence speed.The key principles of TSA are outlined below.

• Step1. Initialize each tree in the population: In TSA, each tree in the initial population is initialized using Eq (1) to generate a feasible solution as the initial tree T_i,j.(1)
  T_i,j denotes the position of the ith tree in the jth dimension. L_j,min and H_j,max are the lower and upper bounds of the jth dimension, respectively, and r_i,j is a random number uniformly distributed in the interval (0,1).
• Step2. Seed number generation mechanism: Determine the number of seeds to be generated according to Eq (2).(2)
  Among them, the fix function is used to round the elements to the integer closest to zero, low indicates the minimum number of seeds generated by each tree, and its value is ; high indicates the maximum number of seeds generated by each tree, and its value is .
• Step3. Tree Species Update Mechanism: The tree update mechanism is the core of TSA, which mainly consists of two different update formulas. The first formula Eq (3) is used for local search and the second Eq (4) is used for global search.(3)(4)
  where S_i,j is the jth dimension of the seed generated from the ith tree, T_i,j is the jth dimension of the ith tree, B_j is the jth dimension of the optimal tree position obtained so far, T_r,j is the jth dimension of a tree r randomly selected from the population, and is a random scaling factor in the range [-1, 1]. Local search focuses on refining the search scope and improving the exploitation of the algorithm, while global search enhances the exploration of the algorithm and prevents falling into local optimal solutions.
• Step4. Termination condition setting: In all experiments, the termination condition was determined by the maximum number of function evaluations MaxFEs determined by Eq (5). The updating of the number of function evaluations (FEs) followed Eq (6).(5)(6)
  where D is the dimension of the problem. ns is the number of seeds generated per tree. This termination condition ensures that the algorithm is able to stop at the right time in case of limited computational resources, avoiding ineffective search for too long.

2.2 Literature review

The Tree-Seed Algorithm (TSA) has been recognized as a potent optimization tool due to its simple structure and efficiency [38]. However, like many metaheuristics, the native TSA faces challenges in maintaining a robust balance between global exploration and local exploitation, often leading to premature convergence in complex problem landscapes. Consequently, a significant body of research has emerged, focusing on enhancing the original algorithm. These efforts can be broadly categorized into three main streams: refining tree evolution mechanisms, innovating seed generation strategies, and expanding the algorithm’s practical applications [39,40].

• Mechanisms of tree evolution: A primary research direction for improving TSA has been to enhance the mobility and exploration capability of the ’trees’ to prevent the algorithm from settling in local optima. Some studies have drawn inspiration from other successful metaheuristics. The Migration Tree-Seed Algorithm (MTSA), for instance, integrated concepts from the Grey Wolf Optimizer (GWO), employing a gravity-based learning mechanism to guide its migration strategy, thereby improving the balance between global and local search [41]. Other approaches have focused on internal mechanism design. The Triple Tree-Seed Algorithm (TriTSA), for example, designed two novel migration mechanisms based on triple learning methods and a sine-based random distribution to boost population diversity and adaptability [42]. These works underscore a clear trend: incorporating sophisticated migration strategies is a key avenue for augmenting TSA’s global search capabilities.
• Innovations in seed generation: Another major focus has been to refine the seed generation process, which is central to the algorithm’s search behavior. One group of enhancements involves introducing dynamic adaptation of key parameters. For example, STSA utilizes sine and cosine functions to dynamically adjust balancing parameters [43], while fb-TSA introduces a feedback loop to adaptively tune the search tendency (ST) value and the number of seeds (ns) based on the search progress [32]. A second group of works has focused on incorporating new information to guide seed placement. EST-TSA, for instance, leverages information from the current best solution to enhance its local search [44], and TSASC integrates the Sine Cosine Algorithm (SCA) to refine the seed position updating formula [45]. A third avenue involves hybridizing with other search strategies to increase diversity, such as LTSA employing a Lévy flight random walk [46] and DTSA designing a velocity-driven seed generation mechanism [27].
• Algorithm applications: The practical value of these algorithmic enhancements is demonstrated by the successful application of TSA and its variants across diverse and challenging domains. In engineering, CTSA has been tailored to solve constrained optimization problems by effectively combining Deb’s rules with the TSA framework [47]. In finance, a hybrid model, sinhTSA-MLP, utilized TSA to optimize a multilayer perceptron, significantly improving the accuracy of credit default risk prediction [48]. The algorithm has also proven effective in medical diagnostics, where a TSA-ANN model was developed for the accurate classification of COVID-19 cases from medical images [49]. The algorithm’s versatility is further demonstrated by DTSA’s integration of discrete operators to solve complex arrangement coding optimization tasks [27].

Despite these advances in tree evolution and seed generation, a holistic approach that simultaneously addresses adaptive step-sizing, dynamic population management, and robust migration strategies has been underexplored. Many existing methods improve one aspect of the algorithm, sometimes at the expense of another, or rely on fixed parameters that limit their adaptability across different problem types [27]. This paper aims to fill this research gap. We propose an Adaptive and Migration-enhanced Tree Seed Algorithm (AMTSA) that integrates three synergistic mechanisms to create a more balanced, robust, and efficient optimizer for complex, high-dimensional problems.

To ensure a comprehensive comparison, we acknowledge the emergence of other high-performance optimization algorithms in the computational intelligence field in recent years. Notable examples include ICSPM and ICSPM2 [50], Exploratory Cuckoo Search [51], and the Improved Salp Swarm Algorithm with HDPM (ISSA) [52]. While a direct experimental comparison with these methods was beyond the scope of the current study, benchmarking AMTSA against these promising approaches remains a valuable direction for future work.

2.3 An overview of GBO

GBO is a meta-heuristic optimization algorithm based on gradientizer [53]. Proposed by Iman Ahmadianfar, Omid Bozorg-Haddad, and Xuefeng Chu in 2020, GBO is inspired by gradient-based Newtonian methods and uses two main operators [54], Gradient Search Rule (GSR) and Local Escape Operator (LEO), and a set of vectors to explore the search space. The working principle of GBO can be briefly summarized the working principle of GBO can be briefly summarized as follows.

Step1: In the initialization phase: the GBO first generates an initial population with each individual (i.e., vector) randomly distributed in the search space. The vector representation of the population size N and dimension D is Eq (7), and the initial vector is generated by Eq (8):

(7)

(8)

Where X_min and X_max, are the lower and upper bounds of the decision variable, respectively, and rand(0,1) is a random number between [0, 1].

Step2: Gradient Search Rule (GSR): Using the gradient method in Eq (9), each individual is guided toward an improved solution.

(9)

where and are the best and worst solutions obtained during the optimization process. ε is a small value used to avoid divide-by-zero errors. is a small step size used in the numerical gradient computation. To improve the search capability of GBO and balance global exploration with local exploitation, the GSR is modified by introducing a stochastic parameter in Eq (10). In this study, is the key adaptive coefficient for balancing exploration and exploitation, and it is expressed as:

(10)

(11)

(12)

where and are used to regulate the minimum and maximum values of the parameter β range. These parameters are used to compute the adaptive parameter , which balances global exploration and local exploitation during the iterations of the algorithm. The integration of the adaptive coefficient mechanism from the GBO algorithm into TSA requires new balancing mechanisms to regulate the global and local search phases. Therefore, Sect 3.2 details the integration of the above enhancements into the TSA

Step3. Integrated position update: The position of the current vector is updated using using GSR and Directional Movement (DM).DM is given by Eq (13) and the integrated position update formula is as Eq (14).

(13)

(14)

3.1 A dynamic seed number strategy based on the Weibull distribution

The original Tree seed Algorithm(TSA) uses a completely randomized number of seeds generation strategy where the number of seeds depends on the population size. This mechanism generates the number of seeds with great randomness and lacks a dynamic and adaptive strategy, which may result in the algorithm failing to maintain good performance in different dimensions and search phases. To improve these drawbacks, we introduce an adaptive seed generation strategy with dynamic Weibull distribution, which makes the generation of the number of seeds more flexible and adaptive to the current search situation by adaptively adjusting the shape and scale parameters of the Weibull distribution to improve the search efficiency and adaptability of the algorithm. We set the scale parameter lambda and shape parameter k via Eq (15) and Eq (16), and let ns change adaptively via Eq (17).

(15)

(16)

(17)

where dmax, dmin are the upper and lower bounds of the search space, best₋obj is the value of the objective function corresponding to the minimum value of the fitness, D is the dimension of the problem, std₋obj is the standard deviation of the fitness value, mean₋obj is the mean of the fitness value, and is the upward round.

The scale parameter (lambda) directly influences the seed distribution range, with its value adjusted based on the relative volatility of the population’s fitness, represented by . A larger value of this ratio indicates higher fitness diversity and thus a larger scale parameter, resulting in more seeds for broader global exploration. Conversely, a smaller ratio leads to fewer seeds and faster convergence with more localized searches. Meanwhile, the shape parameter (k) is dynamically adjusted based on the problem’s dimensionality (D) and the difference between the population’s average fitness and the current optimal fitness (mean₋obj-best). In higher-dimensional problems, a larger k expands the seed count, promoting diversity and global exploration. If the fitness difference is large, k increases to improve global search; if small, it decreases to focus on local search and accelerate convergence. Together, these parameters ensure a balance between global exploration and local exploitation, enhancing TSA’s adaptability. The idea is shown in Fig 1.

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Fig 1. A dynamic seed number strategy based on the Weibull distribution.

https://doi.org/10.1371/journal.pone.0333304.g001

3.2 Adaptive step size with nonlinear strategy

In TSA, the step size in the seed generation equation is a uniform random number between (-1,1), which controls the search range of the seed in the neighborhood of the tree position. When , the Eq (3) of the seed generated based on the current tree is influenced by the best position, and the step size moderates the strength of this influence. However, the values of exhibit large randomness and significant jumps during the search process. This can result in insufficient fine-tuning of the search. As a result, the excavation may not adaptively adjust as needed during the search.

To solve this problem, Inspired by the GB-based improved GSR search rule in the GBO algorithm, 2 perturbation factors and are introduced to adjust the step size with a nonlinear function to enhance the search process to be more flexible. The generated solution should be able to explore the search space around its corresponding best solution. Thus, when the number of iterations reaches a later stage, the parameter values increase. This increase helps the algorithm escape from local optima. It does so by boosting the diversity of the population, which enhances the search around the best solution obtained. Then, using the method of progressive convergence (asymptotic convergence), the two perturbation factors are dynamically adjusted. This adjustment enables a smooth transition from global search to local search. The overall performance of the optimization algorithm is further improved by gradually reducing the step size. We give the perturbation factors and by Eq (18) and Eq (19), and the dynamic step size by Eq (20), and the generating seed formula by Eq (21).

(18)

(19)

(20)

(21)

where t is the current iteration number and is the maximum iteration number.

Beta ranges from 1.2 to 0.2, smoothly transitioning from exploration to exploitation. Larger beta values in early iterations enhance exploration, while smaller values in later stages focus on exploitation. The alpha nested sine function introduces oscillation, promoting diverse search paths and avoiding local optima. The idea is shown in Fig 2.

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Fig 2. Adaptive step size with nonlinear strategy.

https://doi.org/10.1371/journal.pone.0333304.g002

3.3 Adaptive migration tree strategy

The mechanism for TSA to jump out of the local optimum is necessary. Due to the incompleteness of the mechanism, seeds in TSA remain locked around their parent tree. The particle optimum that falls locally is difficult to escape from its current position. In order to solve the stagnation problem of the local optimum, the introduction of a migration mechanism is necessary. To solve this problem, we design a migration mechanism. This mechanism adaptively determines the step size and direction. It provides an opportunity for the tree seeds that have fallen into a local optimum to escape from their current position.

This migration mechanism is generalized in nature, all parent trees are compared for migration to decide whether to migrate to that position or not. The step size is determined by a dynamic step factor with an adaptive function, the direction is determined by a random direction vector with an adaptive function, and the final migration formula consists of the step size, direction, and Cauchy mutation strategy. In order to achieve the problem of jumping out of the local optimum. Eq (22) is given the adaptive function which dynamically adjusts the individual’s step size according to the fitness. Individuals with lower fitness have larger step sizes to encourage more exploration, while individuals with higher fitness have smaller step sizes and move more cautiously.

(22)

The adaptive function is the basis for determining the step size and direction, but in order to ensure the limitation of the step size, a nonlinearly varying dynamic change factor generated from the sine function is added. Eq (23) gives the factor

(23)

where r is a uniformly distributed random number taking values between [0, 1].

When traversing each parent tree, the step final value (Eq (24))) and the effect of direction (Eq (25)) are determined by the fitness of that parent tree compared with the fitness of the randomly matched individuals. And the parent tree after the update is completed is generated by Eq (26).

(24)

(25)

(26)

(27)

where target₋index is the randomly selected target parent tree and is the position of the parent tree after migration. The idea is shown in Fig 3.

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Fig 3. Adaptive migration tree strategy.

https://doi.org/10.1371/journal.pone.0333304.g003

3.4 AMTSA: A novel tree seed algorithm

The algorithm is improved to address the original defects of the original Trees seed Algorithm (TSA). An adaptive seed generation strategy with dynamic Weibull distribution is introduced, which adaptively adjusts the shape and scale parameters of the Weibull distribution and determines the number of seeds to be generated through different dimensions and search stages.It significantly improves the global search capability during the exploration phase. Additionally, it enhances the fast response and optimization search capability during the convergence phase. Inspired by the GSR search rule in the GBO algorithm, the original step size is nonlinearly improved to realize the smooth transition from global search to local search, and to improve the mining accuracy and adaptive ability of the optimization algorithm. Finally, an adaptive migration mechanism is designed to generate a migration strategy by integrating global search and adaptive directional step size, giving each tree will jump out of the local optimum and mine better points. These strategies these methods together promote a more dynamic, adaptive and efficient optimization algorithm that optimizes the original shortcomings of the TSA algorithm. The flowchart of the algorithm is shown in Fig 4.

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Fig 4. The flowchart for the proposed AMTSA.

https://doi.org/10.1371/journal.pone.0333304.g004

3.5 Time complexity analysis of AMTSA

To evaluate the efficiency of the AMTSA algorithm, it is essential to analyze its time complexity, which reflects how the algorithm’s computational cost grows with respect to the problem size. The AMTSA algorithm consists of several major stages, including initialization, iterative updates, seed production, and migration mechanisms. In this section, we provide a detailed analysis of the time complexity involved in each phase of the algorithm.

In the initialization phase, the algorithm generates random positions for the trees within a defined search space. Since the tree population consists of N trees, each having D dimensions, the time complexity for this operation is . Additionally, the objective function value for each tree is computed, which also requires time, as each evaluation involves processing a D-dimensional vector. Therefore, the total time complexity of the initialization phase is .

The iterative update phase, which is the core of AMTSA, involves several key steps within each generation. This phase iterates for a maximum of generations, and within each iteration, multiple operations are performed, including adaptive weight calculations, seed generation, fitness evaluations, and updates to tree positions. The adaptive weight calculation is computationally light, with a time complexity of O(1), as it involves basic arithmetic operations. Seed production, on the other hand, is more computationally demanding. Each tree produces a variable number of seeds, ns, based on a Weibull distribution. The number of seeds generated for each tree is bounded, and the average number of seeds is approximately O(1). For each seed, the fitness function is evaluated, which takes O(D) time. Thus, the total complexity for seed production and fitness evaluation across all trees is .

In the migration phase, which is designed to help the algorithm escape local optima, each tree adjusts its position dynamically based on its fitness and the positions of other trees. This phase includes random number generation, position updates, and fitness comparisons. Since these operations are performed for each of the N trees and each involves operations that are linear with respect to the problem dimension D, the time complexity for the migration phase is .

Finally, the overall complexity of the algorithm is dominated by the iterative update phase. As the algorithm performs iterations, each requiring time, the total time complexity of AMTSA is . This means that the computational cost of the algorithm increases linearly with both the number of trees N and the problem dimension D, as well as the number of generations .

In summary, the time complexity of AMTSA is primarily driven by the iterative update phase, which is . This reflects the typical behavior of evolutionary algorithms, where the time complexity grows with the size of the population and the dimensionality of the search space, as well as the number of iterations required for convergence.

Algorithm 1 The pseudo-code of the AMTSA.

1: Step 1. Initialize the population

2: Set the initial number of zones and problem dimensions;

3: Set the ST parameter and termination criteria;

4: Evaluate each tree’s fitness against the target function;

5: Creation1:

6: Generate initial positions (and any auxiliary data) via Sine

  chaotic mapping;

7: End Creation1

8: Step 2. Seed-based search

9: for each tree do

10:   Compute based on iterations;

11:   Compute current fitness mean and standard deviation;

12:   Derive Weibull parameters (shape k, scale λ) from fitness

  data;

13:   Determine number of seeds for this tree via Weibull

  distribution;

14:   for each seed do

15:    for each dimension do

16:     if then

17:      Creation2:

18:      Compute dynamic step size (Eq (20));

19:      Generate new tree position (Eq (21));

20:     else

21:      Generate new tree position (Eq (21));

22:      End Creation2

23:     end if

24:    end for

25:   end for

26:   Select the best seed; if it outperforms the tree, replace

  the tree with that seed;

27: end for

28: Step 3. Migratory perturbation

29: Design perturbation term (Eq (23));

30: for each tree do

31:   Randomly pick a target individual;

32:   if some condition then

33:    Compute step size (Eq (25));

34: else

35:    Compute step size (Eq (25));

36:   if

37:   Determine direction via Normal-distributed random vector

  weighted by fitness;

38:   Update tree position (Eq (27));

39: end for

40: Compare new vs. old positions; keep the better one;

41: Step 4. Termination check

42: If stopping criteria not met, go to Step 2;

43: Step 5. Report

44: Output the best-found solution.

4.1 Experimental fundamentals

All experiments were implemented in the MATLAB 2016a programming environment and executed on a computer with an Intel Core i7-11800H processor, running a 64-bit Windows 10 operating system. To ensure a rigorous and fair comparison, all experiments were designed to strictly adhere to the official guidelines of the IEEE CEC 2014 benchmark competition. The 30 benchmark functions provided by the competition were evaluated at 30, 50, and 100 dimensions. For statistical significance, the final results reported for each function are the average of 30 independent runs. Following the competition’s standard protocol, the stopping criterion for all algorithms was uniformly set to a maximum number of fitness evaluations (MaxFEs), defined as D×10,000, where D is the problem dimension. This ensures that every algorithm is allocated an identical computational budget proportional to the problem’s difficulty. Furthermore, to address the critical issue of fair parameterization, the settings for each comparative algorithm, including their population size strategies, were configured based on the recommendations from their respective original publications or the official CEC 2014 report. This avoids the bias of using a single, fixed setting for all algorithms.

A comprehensive list detailing the specific parameter settings for every algorithm is provided in Table 1. This standardized protocol guarantees that all algorithms are compared on a level playing field, making the performance evaluation both reproducible and credible.

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Table 1. The initial parameters of comparative algorithms.

https://doi.org/10.1371/journal.pone.0333304.t001

4.2 Comparative algorithms

To comprehensively evaluate the performance of the proposed AMTSA, it was benchmarked against a large and diverse suite of state-of-the-art and classic optimization algorithms. The comparison suite was carefully selected to assess AMTSA’s performance in multiple contexts. The algorithms are grouped into three categories:

High-Performance and CEC Benchmark Algorithms. To validate AMTSA against the highest standards, we included several algorithms known for their top-tier performance in rigorous academic competitions, particularly the IEEE CEC 2014 benchmark. This group includes LSHADE [55], JADE [56], as well as other CEC competition winners and highly regarded methods such as GaAPPADE [57], MVMO-SH [58], and the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [59].

Established and Novel Heuristic Algorithms. To position AMTSA in the broader context of metaheuristics, it was also compared against other respected optimizers. This set includes Differential Evolution (DE) [60,61], Genetic Algorithm (GA) [62], and Bat Algorithm (BA) [63].

TSA Variants. To demonstrate the improvements over the original algorithm and provide a comprehensive comparison within its own family, AMTSA is compared with a wide range of TSA variants. This includes the original Tree-Seed Algorithm (TSA) [24], recent notable versions such as EST-TSA [44], fb-TSA [32], STSA [43], and MTSA [64]. The comparison is also extended to our prior works, ATSA [26] and KATSA [33].

4.3 Evaluation metrics

To analyze the algorithm’s behavior, this study assesses population diversity, the balance between exploration and exploitation, and convergence performance.

Population Diversity Evaluation. The assessment of population diversity is a key metric for evaluating efficiency, as it directly affects the depth of exploration in the search space [12]. This paper represents population diversity using the dispersion between individuals and the centroid of the group, calculated using Eqs (28) and (29).

(28)

(29)

Exploration and Exploitation Analysis. The balance between exploration and exploitation is a cornerstone of performance in metaheuristic algorithms. Exploration refers to the algorithm’s ability to search broadly across the entire solution space to discover globally promising regions, while exploitation involves refining the search in the vicinity of known good solutions to find the local optimum precisely. A successful algorithm must transition smoothly from exploration to exploitation during its run. To quantitatively measure these two behaviors, this study adopts a widely recognized method based on population diversity. The diversity, which reflects the spread of individuals in the population, is calculated using the average distance of individuals from their population centroid. The diversity at iteration t, denoted as , is calculated using Eqs (30) and (31). A high diversity value indicates that the population is widely dispersed, signifying a state of exploration. Conversely, a low diversity value suggests that the population has converged around promising areas, which is characteristic of exploitation. Therefore, we define the percentage of exploration and exploitation at iteration t as follows:.

(30)

(31)

where is the population diversity at the current iteration t, and is the maximum diversity recorded throughout the entire optimization process. This metric allows for a dynamic assessment of the algorithm’s behavior, providing clear insights into how it balances the search process over time.

Convergence Curve. The convergence curve is used to visually assess an algorithm’s efficiency and convergence speed. It plots the best fitness value found so far against the number of iterations. A steeper curve indicates faster convergence toward the optimal solution.

5.1 Qualitative analysis

Qualitative analysis of the algorithm is one of the criteria to detect the performance of the algorithm, we conducted convergence behavior analysis, population diversity analysis and exploration mining analysis experiments for this algorithm to observe the performance of the algorithm. Among them, we chose the single-peak function F1 to evaluate the exploitation ability of the algorithm and the multi-peak function F8 to evaluate the exploration ability of the algorithm, as shown in Fig 5 below.

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Fig 5. Unimodal and Multimodal functions.

https://doi.org/10.1371/journal.pone.0333304.g005

5.1.1 Convergence behavior analysis.

In the case where the algorithm was subjected to a convergence behavior analysis, a total of four tests were performed with the following specific features.

• Figs 6(a) and 5.1.1(a) show the optimization process of the AMTSA algorithm. In these figures, the black dots represent the area covered by the current seed, while the red dots indicate the best position found, i.e., the optimal solution. The clustering phenomenon of black dots around the red dots indicates that AMTSA is moving towards convergence during the gradual optimization process.
• The Figs 6(b) and 5.1.1(b) demonstrate the convergence of AMTSA, clearly reflecting its ability to rapidly approach the optimal solution. The significant decreases in the convergence curves not only highlight the algorithm’s efficiency in the optimization search process, but also demonstrate its keen ability to explore the problem space. This fast convergence property gives AMTSA a significant advantage in complex optimization tasks.
• The Figs 6(c) and 7(c) monitor the change in the first dimension, providing an important insight into the behavior of the algorithm and effectively preventing premature convergence to a local optimum. Experimental evidence shows that the AMTSA algorithm is able to efficiently steer the search process to avoid falling into a local optimum, thus ensuring that it explores a wider solution space. This ability significantly improves its performance in complex optimization problems.
• The Figs 6(d) and 7(d) demonstrate the convergence trend of the mean over multiple iterations. In these figures, the significant decrease in the curve indicates that AMTSA performs well in terms of overall convergence, further validating its effectiveness in optimization tasks. This result not only reflects the stability of the algorithm, but also emphasizes its reliability and advantages in dealing with complex problems.

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Fig 6. The qualitative analysis of AMTSA in unimodal function (F1) (search history (a), average fitness history (b), trajectory history (c), convergence curve (d)).

https://doi.org/10.1371/journal.pone.0333304.g006

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Fig 7. The qualitative analysis of AMTSA in multimodal function (F8) (search history (a), average fitness history (b), trajectory history (c), convergence curve (d)).

https://doi.org/10.1371/journal.pone.0333304.g007

5.1.2 Population diversity analysis.

In optimization algorithms, population diversity is a key metric for evaluating efficiency, as it directly affects the depth of exploration in the search space. Higher population diversity helps avoid local optima and maintains the algorithm’s global search capability. Thus, maintaining appropriate population diversity is crucial for ensuring the algorithm’s robustness and strong global search performance.

To analyze the population diversity dynamics of AMTSA, Fig 8 contrasts its behavior on different types of test functions: the unimodal functions F1 (Fig 8a) and F2 (Fig 8b), and the complex multimodal function F8 (Fig 8c). When addressing simple unimodal problems such as F1 and F2, the population diversity of AMTSA decays rapidly. This phenomenon is not a deficiency in the algorithm’s exploratory capability but is rather a direct manifestation of its efficient exploitation strategy. As the function landscape is simple with a unique global optimum, the algorithm quickly locates the target region, prompting the population to converge efficiently, which naturally leads to a decrease in diversity. In sharp contrast, when faced with the complex multimodal function F8, which features numerous local optima, AMTSA demonstrates its powerful exploration capability. As shown in Fig 8(c), AMTSA proactively increases and persistently maintains a level of population diversity far higher than that of TSA from the early stages of the search. This sustained high diversity, driven by the dynamic seed generation strategy, is the key mechanism that enables the algorithm to perform global exploration and avoid premature convergence. In summary, AMTSA’s population diversity is not static but is intelligently regulated according to the problem’s complexity. It prioritizes convergence efficiency on simple problems while emphasizing global exploration on complex ones. This adaptive mechanism is the core reason for its robust performance across diverse optimization tasks.

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Fig 8. Comparative analysis of exploration-exploitation dynamics on diverse benchmark functions.

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5.1.3 Exploration and exploitation analysis.

To evaluate AMTSA’s search dynamics, its exploration and exploitation behaviors were analyzed on both a simple unimodal function (F1) and a complex multimodal function (F8). The results, compared against the standard TSA, are presented in Fig 10. As shown in Fig 9(a) for the unimodal function F1, both AMTSA and TSA exhibit a rapid transition to full exploitation. This behavior is expected and efficient for a function with a single global optimum, as it allows the algorithms to quickly converge without allocating unnecessary resources to prolonged global exploration. However, the superiority of AMTSA’s adaptive strategy becomes evident on the complex multimodal function F8, as depicted in Fig 9(b). While the standard TSA displays a slow, gradual shift from exploration to exploitation, AMTSA maintains a high level of exploration for a significantly longer duration (approximately 350 iterations). This sustained exploratory phase, driven by its adaptive mechanisms, allows AMTSA to thoroughly survey the rugged search landscape and effectively evade the numerous local optima. Following this crucial phase, AMTSA makes a decisive transition to full exploitation to refine the best-found solution.

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Fig 9. Comparative analysis of exploration-exploitation dynamics on diverse benchmark functions.

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Fig 10. Convergence curves of the AMTSA and TSA and its recent variants in 30 dimensions.

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In summary, these results confirm that AMTSA possesses a more sophisticated control over its search dynamics. It intelligently allocates computational effort by prioritizing rapid convergence on simple problems and sustained exploration on complex ones, which is key to its robust performance.

5.2 Quantitative analysis

In this section, we demonstrate the superior performance of AMTSA through three well-designed experiments. Sect 4.3.1 provides a rigorous comparison of AMTSA with multiple TSA variants, clearly revealing its strengths. Sect 4.3.2 then extends the comparison to include additional metaheuristics, highlighting AMTSA unique ability to cope with complex problems. Finally, Sect 4.3.3 provides box-and-line plots generated based on the results of 30 rounds of experiments, providing an intuitive view of AMTSA stability over multiple iterations. Together, these experimental results support AMTSA position as an efficient and reliable optimization algorithm.

5.2.1 Comparative Experiment 1: AMTSA versus EST-TSA, MTSA, TSA, STSA, and fb-TSA.

The first comparative experiment aims at evaluating the basic TSA [24] and its latest variants, including EST-TSA [44], fb-TSA [32], STSA [43], and MTSA [64], to demonstrate their respective advantages. Tables 2–4 provide comparative evaluations of AMTSA with these algorithms on 30, 50 and 100 dimensions. In addition, the convergence curves are shown in detail in Figs 10–12, providing a visual basis for understanding the performance of the different algorithms. These comparisons not only reveal the superiority of AMTSA in high-dimensional spaces, but also emphasize its unique performance in terms of convergence speed and stability.

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Fig 11. Convergence curves of the AMTSA and TSA and its recent variants in 50 dimensions.

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Fig 12. Convergence curve of the AMTSA and TSA and its recent variants in 100 dimensions.

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Table 2. The mean values for the AMTSA, TSA, MTSA, fb-TSA, EST-TSA, and STSA in 30 dimensions.

https://doi.org/10.1371/journal.pone.0333304.t002

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Table 3. The mean values for the AMTSA, TSA, MTSA, fb-TSA, EST-TSA, and STSA in 50 dimensions.

https://doi.org/10.1371/journal.pone.0333304.t003

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Table 4. The mean values for the AMTSA, TSA, MTSA, fb-TSA, EST-TSA, and STSA in 100 dimensions.

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Tables 2–4 show the average optimal values obtained through 30 experimental iterations, each of which was performed for 500 iterations, which provides a valuable perspective for evaluating the convergence performance of the algorithms. These aggregated data not only help to identify the superiority of different algorithms, but also provide a basis for further performance analysis, reflecting the differences in the performance of the algorithms in the optimization process. An in-depth analysis of these results enables a more comprehensive understanding of the potential and advantages of AMTSA in practical applications.

In addition, the convergence process over 500 iterations was recorded in detail for each experimental iteration to reveal the performance dynamics. By computing the average of 30 locally optimal solutions at each iteration point, we obtained average convergence curves that demonstrate the subtle performance changes of the algorithm during the optimization process. The slopes of these curves, in turn, provide a quantitative assessment of the speed of convergence, further enhancing the understanding of the algorithm’s efficiency. This in-depth analysis allows us to more clearly identify the performance characteristics at different stages, providing a valuable reference for improving the algorithm.

The results show that designing an adaptive migration mechanism enhances the ability to jump out of local optima. In addition, the improvement of the position update formulation (Eq (21)) significantly increases the convergence speed compared to the previously proposed TSA variant.

Based on multiple comparisons and experimental data, we conclude that the algorithm designed in this study outperforms other TSA algorithms and their variants in terms of optimization.

5.2.2 Comparative Experiment 2: AMTSA versus established and novel heuristic optimization algorithms.

Tables 5–7 present the detailed performance results of AMTSA over 30 experiments, with systematic comparisons against other algorithms, recording the mean optimal values and overall rankings. These statistical data clearly indicate that AMTSA demonstrates unique advantages in handling complex, particularly multimodal, problems.

To more intuitively illustrate the dynamic convergence process, and in response to the valuable reviewer feedback on providing a comprehensive presentation, we have revised the convergence curves shown in Figs 13–15. The revised figures now feature a deliberately chosen set of representative functions intended to provide a more balanced and objective overview of algorithm performance. In addition to cases where AMTSA is dominant, we have specifically included functions where it was not the top performer at each dimension: F5 and F25 were added at 30 dimensions (Fig 13); F12 and F25 at 50 dimensions (Fig 14); and F12 and F30 at 100 dimensions (Fig 15).

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Fig 13. Convergence curves of the AMTSA and TSA and its recent variants in 30 dimensions.

https://doi.org/10.1371/journal.pone.0333304.g013

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Fig 14. Convergence curve of the AMTSA and TSA and its recent variants in 50 dimensions.

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Fig 15. Convergence curves of the AMTSA and TSA and its recent variants in 100 dimensions.

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A comprehensive analysis of these selected curves reveals that while AMTSA shows excellent performance in most scenarios, the newly included plots provide a more complete picture. On certain functions, other state-of-the-art algorithms are highly competitive. For example, on the complex composite function F25, CMA-ES demonstrates more stable convergence in the later stages. Similarly, in high-dimensional tests on F12 and F30, algorithms like LSHADE also achieve very competitive results. This indicates that while our proposed adaptive migration mechanism and GBO-inspired nonlinear step-size strategy significantly enhance overall performance, the choice of the optimal algorithm remains problem-dependent, consistent with the No Free Lunch theorem. Together, these experimental results confirm the effectiveness of AMTSA as an efficient and robust optimization tool and position it as a valuable option for solving complex optimization problems.

The mean values for AMTSA and 11 comparative algorithms in 30 dimensions.

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Table 5. The mean values for AMTSA and 11 comparative algorithms in 30 dimensions.

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Table 6. The mean values for AMTSA and 11 comparative algorithms in 50 dimensions.

https://doi.org/10.1371/journal.pone.0333304.t006

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Table 7. The mean values for AMTSA and 11 comparative algorithms in 100 dimensions.

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5.3 Parameter sensitivity analysis

The proper setting of algorithmic parameters is crucial for its performance and robustness. This section aims to evaluate the dependence of the AMTSA algorithm on key internal parameters through a systematic parameter sensitivity analysis, providing empirical support for the effectiveness of the parameter values selected in this paper. We primarily focus on three key parameters: the search tendency parameter ST, the calculation constant for the shape parameter k in the dynamic Weibull distribution (Eq (16)), and the dynamic range of the β parameter in the adaptive step size strategy (Eq (18)). All experiments were conducted on CEC 2014 benchmark functions with a dimension of 100. Each set of experiments was independently run 30 times, and the average best fitness value was recorded.

The ST parameter plays a crucial role in balancing global exploration and local exploitation within the Tree Seed Algorithm. To evaluate the sensitivity of AMTSA to ST values, we tested the performance of ST=0.05,0.1,0.3,0.5,0.8 on F1 (unimodal), F2 (unimodal), F4 (multimodal), F5 (multimodal), F17 (composition function), and F18 (composition function) with a dimension of 100. Table 8 presents the average best values of AMTSA for different ST values on these test functions. As can be seen from Table 8, when ST is set to 0.1, AMTSA achieves the best average optimal values across all tested functions, indicating that ST=0.1 provides an ideal balance between exploration and exploitation for AMTSA. Both smaller and larger ST values may lead to a decrease in algorithm performance, such as premature convergence to local optima or slowed convergence speed. The F5 function, being a constant function, is unaffected by the ST parameter, further validating the reasonableness of the experimental setup. This result strongly supports the decision to adopt ST=0.1 as the default parameter in this paper and demonstrates the good robustness of AMTSA to the ST parameter.

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Table 8. Sensitivity Analysis of AMTSA to Different ST Parameters on 100 Dimensions (Mean Best Values).

https://doi.org/10.1371/journal.pone.0333304.t008

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Table 9. Average runtime (in seconds) of compared algorithms over 30 runs.

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Additionally, we performed a sensitivity analysis on the constant term 10 () included in the calculation of parameter k in Eq (16) and the dynamic range of parameter β in Eq (18). As shown in Fig 16, the sensitivity analysis for the constant C_k in Eq (16) reveals that when C_k is 10, the convergence curves of AMTSA on F1, F4, and F17 are the smoothest, and the final convergence accuracy is the highest. This indicates that C_k = 10 helps the algorithm maintain an appropriate population diversity and search efficiency at different search stages. Similarly, the sensitivity analysis of the dynamic range for parameter β in Eq (18) intuitively demonstrates that when the range is set to [0.2,1.2], AMTSA achieves the fastest convergence speed on F2, F5 and F18, and the stability of the final solution is optimal. This range design ensures a smooth transition between global exploration in the early stage and local exploitation in the later stage, effectively avoiding premature convergence and local optima, thereby significantly enhancing the overall performance of the algorithm. In summary, the parameter sensitivity analysis conducted in this section shows that the AMTSA algorithm has good robustness to the selected key parameters. The experimental results support the parameter choices adopted in this paper, proving that these parameter values enable AMTSA to achieve an effective balance between exploration and exploitation and maintain stable high performance on different types of complex optimization problems. Although the current parameters have been optimized through systematic analysis, future work could further explore a more complete self-adaptive parameter adjustment mechanism to further improve the algorithm’s generality and adaptability across a wider range of problem domains.

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Fig 16. Convergence curves for the sensitivity analysis of the ST parameter in AMTSA on representative CEC 2014 functions at 100 dimensions.

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5.4 Ablation study of AMTSA components

To scientifically validate the effectiveness of each new component in AMTSA, an ablation study was conducted. This study was designed to analyze the contribution of three core mechanisms: the dynamic Weibull distribution-based seed strategy (DS), the nonlinear step-size adjustment (NS), and the adaptive migration (AM). We compared the performance of the full AMTSA against the original TSA (as a baseline) and three variants with a single mechanism removed (AMTSA-noDS, AMTSA-noNS, and AMTSA-noAM). The comparison was performed on functions F1, F4, F17, and F30 at 30 dimensions under identical experimental conditions.

The convergence curves for each algorithm are presented in Fig 17. It is evident from the figure that compared to the complete AMTSA, removing any single component (DS, NS, or AM) leads to a significant degradation in convergence speed and solution accuracy on most test functions. This result provides strong evidence that all three of our proposed innovations make indispensable contributions to the algorithm’s performance, and their synergy is key to AMTSA’s ability to efficiently solve complex problems.

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Fig 17. Convergence curves for the ablation study of AMTSA components on representative benchmark functions.

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5.5 Computational efficiency and scalability analysis

In addition to solution quality, the practical utility of an optimization algorithm is also determined by its computational efficiency. To assess this, we conducted an empirical runtime comparison to evaluate the practical computational cost and scalability of AMTSA. The experiment was designed to measure the average wall-clock time required for AMTSA and a representative set of comparative algorithms–including its baseline TSA, the classic DE, and the high-performance JADE and LSHADE–to complete their standard search budget. All tests were executed on the same machine to ensure a fair comparison. We measured the average CPU time in seconds over 30 independent runs for each algorithm to complete the full MaxFEs = evaluations on several benchmark functions at dimensions D = 30, 50, and 100.

The results of this analysis are presented in Table 9. As expected, the computationally lightweight DE algorithm was the fastest in all scenarios. The results show that AMTSA incurs a moderate computational overhead compared to its baseline, TSA, which is attributable to the additional calculations required by its three adaptive mechanisms. However, this modest increase in runtime is justified by the significant improvement in optimization accuracy, as demonstrated in our main experimental results. Importantly, AMTSA’s runtime is highly competitive with other state-of-the-art algorithms like JADE and LSHADE, indicating that its efficiency is well within the range of modern high-performance optimizers. Furthermore, by observing the increase in runtime as the dimension D grows from 30 to 100, it is evident that AMTSA exhibits good scalability. Its computational cost increases at a rate comparable to the other established algorithms, confirming its suitability for tackling high-dimensional problems.

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Table 10. Result of Wilcoxon’s test for AMTSA and other algorithms.

https://doi.org/10.1371/journal.pone.0333304.t010

5.6 Statistical experiments

To rigorously evaluate the performance of the proposed AMTSA, a statistical comparison was conducted using the Wilcoxon’s Signed-Rank Test [65], with the results presented in Table 10. In this analysis, AMTSA was systematically benchmarked against the ten state-of-the-art and classic algorithms featured in Comparative Experiment 2. The table details the p-values computed at significance levels of α=0.1 and α=0.05, and indicates whether the null hypothesis–that there is no significant difference between the paired algorithms–was rejected. This non-parametric statistical test is particularly well-suited for this analysis as it does not assume a normal distribution of the results, thus providing a more accurate and robust reflection of the true performance differences.

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Table 11. Performance metrics of each algorithm at different thresholds under test images.

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The statistical results unequivocally demonstrate the significant superiority of the AMTSA algorithm. As evidenced in Table 10, AMTSA achieved a statistically significant advantage over all ten comparative algorithms across all tested dimensions (D = 30, 50, and 100) . In every paired comparison, the calculated p-value was substantially lower than the strictest significance level of α=0.05, leading to the consistent rejection of the null hypothesis. This outcome not only confirms the superior performance and stability of AMTSA but also provides strong statistical validation for its effectiveness. In summary, the AMTSA algorithm is not only a high-performing optimizer but is also statistically proven to be a significant advancement over a wide range of established and high-performance metaheuristics, underscoring its substantial value and potential for wide-ranging applications in complex optimization problems.

6.1 AMTSA-optimized multi-threshold lung cancer CT image segmentation model

Lung cancer CT image segmentation aims to accurately distinguish lung nodules from the background. Multi-threshold segmentation methods partition the image into different regions by setting multiple gray-level thresholds, thereby enabling fine segmentation. However, traditional multi-threshold segmentation algorithms often encounter issues such as high computational complexity and susceptibility to local optima in practical applications, leading to imprecise threshold selection and, consequently, limiting both segmentation performance and clinical reliability.

To address these challenges, this paper proposes a multi-threshold lung cancer CT image segmentation method optimized using the AMTSA algorithm. The proposed approach first constructs a fitness function based on the gray-level features of CT images and adopts the Minimum Symmetric Cross-Entropy (MSCE) criterion as the evaluation standard, thereby transforming the multi-threshold segmentation problem into a threshold optimization task solved by AMTSA. By fully leveraging the adaptive tree migration mechanism and the seed generation strategy based on the dynamic Weibull distribution inherent in AMTSA, the algorithm iteratively optimizes within a complex search space to obtain the optimal threshold set, achieving precise segmentation of lung nodules and the background. Experimental results indicate that this method significantly outperforms traditional approaches in segmentation performance metrics, such as the Structural Similarity Index (SSIM) and Peak Signal-to-Noise Ratio (PSNR), offering more efficient and accurate image analysis support for early lung cancer diagnosis.

To ensure the fairness and reproducibility of our SVM-based recognition experiments, all comparative optimization algorithms sought the optimal hyperparameters for the SVM under identical conditions. Specifically, all algorithms operated within a unified, predefined search space. The search range for the SVM’s regularization parameter, C, was set to [0.1, 100], while the range for the RBF kernel coefficient, gamma, was set to [0.001, 1]. This standardized setup ensures that the final performance differences directly reflect the efficiency of the optimization algorithms themselves, rather than any variance in their search domains.

6.1.3 Lung cancer CT image segmentation experiments and results analysis.

Lung Cancer CT Image Dataset

In evaluating the performance of the AMTSA algorithm in multi-threshold lung cancer CT image segmentation, this study compares it with other optimization algorithms, including the Sine Cosine Algorithm (SCA) [66], Social Network Search (SNS) [67], Harris Hawks Optimization (HHO) [68], Grey Wolf Optimizer (GWO) [41], and Dynamic Bat Optimization (DBO) [69]. To comprehensively assess the performance of these algorithms, metrics such as the Feature Similarity Index (FSIM), Structural Similarity Index (SSIM), and Mean Squared Error (MSE) are used for system evaluation.

For the multi-threshold image segmentation application, we utilized the Lung PET-CT-Dx dataset, publicly available from The Cancer Imaging Archive (TCIA) [cite]. The original dataset contains DICOM images with a resolution of 512x512 pixels. For our experiments, we selected four representative CT images that feature typical solid and ground-glass nodules. The datasets for this retrospective study were accessed between October 2024 and January 2025. The authors did not have access to any information that could be used to identify individual participants during or after data collection, as all data were fully anonymized by the source institutions. Fig 18 shows the data preprocessing steps.

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Fig 18. Example of lung cancer CT image preprocessing results.

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Evaluation Metrics

To quantitatively evaluate the performance of the segmentation algorithms, we employed two widely-recognized metrics: the Peak Signal-to-Noise Ratio (PSNR) and the Structural Similarity Index (SSIM) [70]. PSNR primarily measures the fidelity of the segmented image against the original, while SSIM assesses the preservation of structural information. Higher values for both metrics indicate superior segmentation performance. Fig 19 shows the optimal thresholds under different algorithmic optimisation.

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Fig 19. Visual comparison of segmentation results for a sample CT image using different algorithms and threshold levels.

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Fig 20 shows the convergence curves of each algorithm on different images. By observing the convergence curves, it is evident that the AMTSA algorithm demonstrates exceptional fast convergence ability. Compared to other algorithms, the AMTSA algorithm typically approaches the optimal solution in a very small number of iterations, sometimes achieving segmentation performance that other algorithms can only attain after multiple iterations. This indicates that AMTSA can find high-quality segmentation solutions quickly and effectively with lower computational cost, making it suitable for applications with high efficiency requirements.

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Fig 20. Convergence curves on different images.

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To further evaluate the performance of different algorithms in image segmentation, this study uses Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM) as evaluation metrics, representing image fidelity and structural similarity, respectively. Tables 11 and 12 show the PSNR and SSIM values of six algorithms (AMTSA, DBO, HHO, GWO, SCA, SNS) on two lung cancer CT test images, using different numbers of thresholds (2, 10, 20).

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Table 12. Performance metrics of each algorithm at different thresholds under test images.

https://doi.org/10.1371/journal.pone.0333304.t012

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Table 13. Performance metrics of each algorithm at different thresholds under test images.

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An analysis of the performance metrics in Tables 11 and 12 indicates that the AMTSA algorithm demonstrates competitive performance in the task of lung cancer CT image segmentation. Specifically, regarding the PSNR metric, AMTSA generally excels at high threshold levels (e.g., 20 thresholds), achieving PSNR values of 21.5721 and 21.6521 on the two test images, respectively. This suggests the algorithm’s strong potential for preserving image fidelity. It is noteworthy, however, that under certain settings (e.g., 10 thresholds on Test Image 1), other algorithms such as SCA exhibited comparable or even superior performance. The results for the SSIM metric present a more varied picture. For instance, on Test Image 1, the SNS and DBO algorithms achieved higher SSIM values than AMTSA at 20 thresholds, suggesting they had an advantage in maintaining the structural similarity of that particular image. Concurrently, AMTSA obtained the highest SSIM score (0.6093) for Test Image 2 at the 20-threshold level.

Taken together, these findings suggest that no single algorithm achieved absolute superiority across all metrics and test images. AMTSA shows a particular strength in the PSNR metric, especially at higher threshold levels, which may indicate a favorable trade-off between image fidelity and segmentation detail. While its SSIM performance was not universally the best, its consistently high PSNR values underscore its potential as an effective optimization tool for image segmentation.

6.2 AMTSA-optimized lung cancer recognition model

This study uses a subset of the lung cancer histopathological image dataset, focusing specifically on 300 lung cancer images. The original dataset contains 25,000 histopathological images, divided into five categories. All images are 768x768 pixels in size and are in JPEG file format. The dataset originates from a HIPAA-compliant and validated source, consisting of 750 lung tissue images (250 benign lung tissue, 250 lung adenocarcinomas, and 250 lung squamous cell carcinomas). The dataset was then augmented to 25,000 images using the Augmentor package. For the purposes of this study, only lung cancer images were selected, including 100 lung benign tissue images, 100 lung adenocarcinoma images, and 100 lung squamous cell carcinoma images, totaling 300 images. These 300 images were used to train and evaluate the AMTSA-optimized lung cancer recognition model. The selection of these specific categories helps to focus the study on lung cancer classification, providing support for the development of accurate and effective diagnostic models.

To address the potential for overfitting due to the limited size of the 300-image dataset and to ensure a robust evaluation of the model’s generalization performance, we employed a 10-fold stratified cross-validation strategy. The dataset, consisting of 100 images from each of the three classes (benign, adenocarcinoma, and squamous cell carcinoma), was partitioned into 10 equal-sized folds. For each fold, the class distribution was kept consistent with the overall dataset (stratification). The cross-validation process consisted of 10 iterations. In each iteration, one unique fold was reserved as the test set, while the remaining 9 folds were used to train the AMTSA-SVM model. The final performance metrics reported in Table 13 (Accuracy, Recall, etc.) represent the average values and standard deviations computed across all 10 folds. This rigorous validation method ensures that our results are a reliable estimate of the model’s performance on unseen data.

Early and accurate diagnosis of lung cancer is crucial for improving patient survival rates, and recognition based on lung CT images plays a key role in this process. Support Vector Machine (SVM) has demonstrated certain advantages in lung image classification. However, traditional SVM suffers from limitations in parameter selection and model optimization, which can negatively impact its classification performance. The AMTSA algorithm is known for its excellent global search and local optimization capabilities, enabling efficient optimization in complex search spaces. By incorporating AMTSA into the parameter optimization process of SVM, the goal is to leverage its strengths to enhance the performance of SVM in lung cancer recognition. Specifically, based on the feature data of lung CT images, a fitness function is constructed, transforming the SVM parameter selection problem into an optimization task for AMTSA. Through iterative searching with AMTSA, the optimal parameter combination for SVM is identified, thereby achieving precise lung cancer recognition. This model construction approach integrates the classification power of SVM with the optimization capability of AMTSA, offering the potential to significantly improve the accuracy and efficiency of lung cancer recognition. This model provides more reliable technical support for early lung cancer diagnosis, contributing to advancements in both research and clinical applications

6.2.1 Feature extraction from lung CT images.

Early and accurate diagnosis of lung cancer depends on the effective extraction of representative features from CT images. Due to the limitations of traditional handcrafted feature extraction methods, this study employs Convolutional Neural Networks (CNNs) for feature extraction. CNNs are capable of automatically learning complex features from images, particularly suited for capturing high-level spatial and texture information, which is crucial for effective lung cancer detection.

In this study, we use the pre-trained EfficientNetB3 model, which is based on deep convolutional neural networks and has demonstrated excellent performance in various computer vision tasks. To improve the efficiency and accuracy of feature extraction, the EfficientNetB3 model pre-trained on ImageNet is utilized, with the classification layer removed and global average pooling (GAP) applied to aggregate features. This approach converts each image into a fixed-length feature vector, facilitating subsequent classification and analysis.

The process is as follows: first, each input lung CT image is resized to 300x300 pixels and preprocessed, including normalization, to meet the input requirements of the EfficientNetB3 model. Let the input image be , and the image is then preprocessed using the function , resulting in the normalized image . The image is then passed through the EfficientNetB3 network, where the image is processed through convolutional layers to extract features:

(32)

where is the feature map extracted by the convolutional layers. Global average pooling is then applied to reduce the feature map to a fixed-length vector:

(33)

The resulting feature vector is a 1536-dimensional vector containing high-level information such as texture, shape, and edges of the image. These features are then used as input for training and evaluation of the AMTSA-optimized lung cancer recognition model.

By utilizing CNN for feature extraction, this approach not only preserves key information from the images but also enhances the model’s generalization ability, allowing it to process various types of lung CT images and providing strong support for subsequent lung cancer recognition.

6.2.2 Lung cancer recognition process based on AMTSA-optimized SVM.

Support Vector Machine (SVM) has shown certain advantages in classifying small lung image datasets. However, traditional SVM suffers from limitations in parameter selection and model optimization, which affect its classification performance. The AMTSA algorithm (Adaptive and Migration-enhanced Tree Seed Algorithm) excels in global search and local optimization, enabling efficient optimization in complex search spaces.

By integrating AMTSA into the parameter optimization process of SVM, the goal is to enhance the performance of SVM in lung cancer recognition by leveraging AMTSA’s optimization capabilities. Specifically, based on the feature data of lung CT images, a fitness function is constructed to transform the SVM parameter selection problem into an optimization task for AMTSA. Through iterative searching with AMTSA, the optimal SVM parameter combination is identified, achieving precise lung cancer recognition.

This model construction approach effectively integrates the classification power of SVM with the optimization ability of AMTSA, promising to significantly improve the accuracy and efficiency of lung cancer recognition, thus providing more reliable technical support for early lung cancer diagnosis. The specific process flow is shown in Fig 21.

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Fig 21. Lung cancer recognition process based on AMTSA-optimized SVM.

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