Scientific approach to problem solving-inspired optimization of stacking ensemble learning for enhanced civil engineering informatics

Abstract

This study introduces the Scientific Approach to Problem Solving-inspired Optimization (SAPSO) algorithm, a novel metaheuristic specifically designed for applications in civil engineering informatics. SAPSO imitates the structured process of scientific inquiry—covering problem review, hypothesis formulation, data collection, and analysis—to systematically explore complex search spaces. This approach enables SAPSO to reliably identify global optima. The algorithm’s performance was extensively tested against eleven leading metaheuristic algorithms using the IEEE Congress on Evolutionary Computation benchmark suites from 2020 (CEC 2020) and 2022 (CEC 2022). The comparison included the Artificial Bee Colony, Cultural Algorithm, Genetic Algorithm, Differential Evolution, Artificial Gorilla Troops Optimizer, Grey Wolf Optimizer, Particle Swarm Optimization, Red Kite Optimization Algorithm, Symbiotic Organisms Search, Teaching–Learning-Based Optimization, and Whale Optimization Algorithm. Statistical analysis with the Wilcoxon rank-sum test confirmed SAPSO’s superior results across these benchmarks. Additionally, this study presents a stacked ensemble machine learning framework called the SAPSO-Weighted Features Stacking System (SAPSO-WFSS), which combines SAPSO with two predictive models: a Radial Basis Function Neural Network and Least Squares Support Vector Regression. SAPSO is used to optimize both feature weights and model hyperparameters. Experiments on five diverse civil engineering case studies show that SAPSO-WFSS provides high accuracy, with Mean Absolute Percentage Error values as low as 2.4%, outperforming traditional methods. These findings demonstrate SAPSO’s potential as a powerful tool for improving prediction reliability in infrastructure maintenance and solving complex optimization problems in civil engineering.

Introduction

Optimization plays a vital role across many fields, allowing for notable reductions in manufacturing efforts while improving overall productivity (Talatahari et al. 2021; Alimoradi et al. 2022; Pan et al. 2022). One of the earliest classical optimization problems—maximizing the area of a parallelogram inscribed within a triangle—was introduced by Euclid in the fourth century BCE, highlighting the long-standing importance of optimization in mathematical research.

While gradient-based methods have traditionally been dominant in the field of mathematical optimization, applying them directly to real-world problems has become increasingly impractical because of the growing complexity and non-linearity in modern systems (Talatahari et al. 2021; Eslami et al. 2022; Kutlu Onay 2023). In particular, finding gradients becomes very difficult when working with implicit or non-differentiable objective functions, often resulting in subpar results when using conventional gradient-based techniques (Hu et al. 2024; Wang et al. 2024).

In response to the limitations of traditional optimization methods, researchers have put significant effort into developing innovative strategies, especially in the form of metaheuristic algorithms (Tawhid and Ibrahim 2022; Jia et al. 2024a, b). These algorithms are often inspired by natural phenomena that have evolved over thousands of years, and they convert complex adaptive behaviors into simple heuristic rules, which together create powerful computational frameworks.

By systematically analyzing these heuristic principles, researchers can uncover the core logic of their operation and leverage their built-in strengths. This understanding enables the effective integration of intelligence-inspired techniques into various applications, including modeling, simulation, and optimization tasks.

The main goal in creating a new metaheuristic optimizer is to enhance the efficiency of search processes in complex problem spaces and to handle intricate challenges more effectively. The success of these optimizers heavily depends on the performance of their operators, which must be able to produce diverse, high-quality solutions tailored to each problem’s specific features. These operators need to accurately imitate strategic search behaviors across different landscapes, ensuring both adaptability and robustness. This highlights the growing need to design a new generation of metaheuristic algorithms that can provide quick and accurate solutions across a broad range of optimization tasks (Jia et al. 2024a, b; Ouyang et al. 2024).

This paper introduces a new metaheuristic algorithm called the Scientific Approach to Problem Solving-inspired Optimization (SAPSO). Unlike previous methods, SAPSO draws inspiration from both natural processes and human cognitive behavior, mimicking the structured reasoning used in scientific inquiry. It alternates between exploration and exploitation phases, maintaining a dynamic yet balanced optimization process. This balance is achieved through a unique algorithmic framework that switches between exploration activities—such as reviewing problems and formulating hypotheses—and exploitation activities—such as gathering data, analyzing it, and interpreting results. These phases are guided by an activity-switching mechanism that adaptively directs the algorithm’s behavior during the optimization process.

Furthermore, this study explores the integration of the SAPSO optimizer with advanced stacked ensemble learning models, thereby broadening its use within civil engineering informatics. By using SAPSO to optimize both feature weighting and model hyperparameters within stacked frameworks, the proposed method improves the accuracy of predictive analytics. This combined approach not only enhances the precision of civil engineering models but also provides a solid foundation for empirical validation across various engineering applications.

The case studies in this research show significant improvements in predictive performance through the use of the SAPSO-weighted feature stacking ensemble system (SAPSO-WFSS). These models leverage SAPSO’s strengths in both feature selection and parameter tuning, resulting in better forecasting capabilities. Consequently, SAPSO-WFSS becomes a valuable and adaptable tool for tackling complex prediction tasks in civil engineering practice.

The rest of this paper is organized as follows. Section 2 gives an overview of how metaheuristic algorithms are classified. Section 3 discusses the main components and design principles of the proposed SAPSO optimizer. Section 4 tests the algorithm’s effectiveness using benchmark functions. Section 5 introduces a new ensemble learning framework enhanced by SAPSO, while Section 6 explores its application in five real-world civil engineering scenarios. Finally, the conclusion reviews the main findings and suggests directions for future research.

Related works

Categories of metaheuristic optimization algorithms

Metaheuristic optimization algorithms can be generally divided into four main categories: human-based algorithms, swarm intelligence algorithms, physics- and chemistry-inspired algorithms, and evolutionary algorithms (Fig. 1) (Chou and Truong 2021; Li et al. 2024). Among these, evolutionary algorithms (EAs) are population-based methods that use principles of biological evolution, such as selection, crossover, mutation, and elimination. These processes help generate better solutions over successive iterations.

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Fig. 1

Classification of nature-inspired metaheuristic algorithms

Prominent examples of EAs include the Genetic Algorithm (GA) (Holland 1992), Evolutionary Strategies (ES) (Rudolph 2012), and Differential Evolution (DE) (Storn and Price 1997). Other notable algorithms in this category include Biogeography-Based Optimization (BBO) (Simon 2008), Lagrange Elementary Optimization (LEO) (Aladdin and Rashid 2023), and Enterprise Development (ED) (Truong and Chou 2024). Recent hybrid evolutionary approaches, such as the Hybrid Gazelle Optimization Algorithm with Differential Evolution (HGOADE) (Biswas et al. 2025) and the Oppositional-Based Learning and Laplacian Crossover Augmented Material Generation Algorithm (MGA-OBL-LP) (Mehta et al. 2025), further expand the capabilities of EAs for solving complex optimization problems.

Swarm Intelligence (SI) algorithms are characterized by features such as decentralized control, emergent behavior, and self-organization. These algorithms imitate the collective behavior seen in natural animal groups—including horses, insects, fish, and birds—to help guide the search process in complex optimization landscapes.

Widely used SI algorithms include Particle Swarm Optimization (PSO) (Kennedy and Eberhart 1995), Ant Colony Optimization (ACO) (Dorigo et al. 2006), Artificial Bee Colony (ABC) (Karaboga and Basturk 2007), Cuckoo Search (CS) (Gandomi et al. 2013), Bat Algorithm (BA) (Yang and Hossein Gandomi 2012), and Firefly Algorithm (FA) (Yang 2010). More recent developments in the field include the Jellyfish Search Optimizer (JS) (Chou and Truong 2021), Arctic Tern Optimizer (ATO) (Chou and Molla 2024), and hybrid approaches such as Threefry and Philox with Opposition-Based PSO Ranked Inertia Weight (ORIW-PSO-TF and ORIW-PSO-P) (Hassan et al. 2021), as well as the Improved Fire Hawks Optimizer (iFHO) (Ashraf et al. 2023). Additionally, the FOX-inspired Tree-Seed Algorithm (FOX-TSA) (Aula and Rashid 2024) demonstrates how hybridization can further improve the adaptability and performance of SI-based optimization methods.

Physics- and chemistry-inspired algorithms draw from natural processes governed by physical and chemical laws. These algorithms mimic phenomena like thermodynamics, gravity, atomic interactions, and molecular dynamics to direct optimization approaches. A broad range of algorithms in this category imitate specific scientific principles or behaviors.

Notable examples include Simulated Annealing (SA) (Kirkpatrick et al. 1983), Gravitational Search Algorithm (GSA) (Rashedi et al. 2009), and Chemical Reaction Optimization (CRO) (Lam and Li 2012). Other prominent algorithms are the Big Bang–Big Crunch (BBBC) algorithm (Erol and Eksin 2006), Charged System Search (CSS) (Kaveh and Talatahari 2010), and its variant, the Magnetic Charged System Search (MCSS) (Kaveh et al. 2013). Additional examples include Ray Optimization (RO) (Kaveh and Khayatazad 2012), Atom Search Optimization (ASO) (Zhao et al. 2019), Vortex Search Algorithm (VSA) (Doğan and Ölmez 2015), Water Evaporation Optimization (WEO) (Kaveh and Bakhshpoori 2016), and the Lightning Search Algorithm (LSA) (Shareef et al. 2015). These algorithms, inspired by physics and chemistry, provide diverse methods for balancing exploration and exploitation, making them effective for solving a wide range of complex optimization problems.

The last category of metaheuristic algorithms discussed in this study includes human-based algorithms. These are inspired by various aspects of human behavior, including physical actions and cognitive processes such as reasoning, learning, and social interaction. These algorithms imitate how humans solve problems, adapt to environments, and make decisions—often through iterative, experience-driven strategies.

Prominent examples in this category include the Teaching–Learning-Based Optimization (TLBO) algorithm (Rao et al. 2011), the Ideology Algorithm (IA) (Huan et al. 2017), and the Socio-Evolution and Learning Optimization (SELO) algorithm (Kumar et al. 2018). Other notable methods are the Cognitive Behavior Optimization Algorithm (COA) (Li et al. 2016), Human Mental Search (HMS) (Mousavirad and Ebrahimpour-Komleh 2017), and the Cultural Algorithm (CA) (Omran 2016).

Additionally, this category includes innovative methods such as the Forensic-Based Investigation (FBI) algorithm (Chou and Nguyen 2020), Poor and Rich Optimization (PRO) (Samareh Moosavi and Bardsiri 2019), Student Psychology-Based Optimization (SPBO) (Das et al. 2020), Learner Performance-Based Behavior Algorithm (LPB) (Rahman and Rashid 2021), Pilgrimage Walk Optimization (PWO) (Chou and Liu 2023), and the Age of Exploration-Inspired Optimizer (AEIO) (Chou et al. 2025). These algorithms highlight the richness and flexibility of human behavioral patterns as a basis for solving complex optimization challenges.

The No Free Lunch (NFL) Theorem highlights the need for continuous innovation in metaheuristic algorithm design, as no single optimization method can perform best across all problem types (Wolpert and Macready 1997). Because of the diverse structure, complexity, and features of real-world and engineering optimization problems, expecting any one algorithm to be universally effective is unrealistic. Therefore, enhancing existing methods and developing new approaches are essential for progress in the field.

In this context, the Scientific Approach to Problem-Solving-inspired Optimization (SAPSO) algorithm offers an exciting new direction by mimicking the structured, iterative nature of scientific research. SAPSO integrates human cognitive patterns and decision-making logic into its optimization process, using a dynamic activity-switching mechanism to balance exploration and exploitation. This innovative framework enhances SAPSO’s adaptability and problem-solving capabilities, especially in complex or previously untested scenarios. Therefore, SAPSO makes a meaningful contribution to the evolving field of metaheuristic optimization methods.

Utilizing metaheuristics in applied mechanics and engineering informatics

Applied mechanics and engineering fields encounter many complex challenges, such as evaluating the shear capacity of reinforced concrete walls, estimating bridge scour depth, determining the peak friction angle of fiber-reinforced soil, and improving construction productivity (Truong and Chou 2022). These issues are inherently heterogeneous and highly variable, often exhibiting nonlinear behavior and unpredictable results (Cheng et al. 2022; Bangyal et al. 2023; Zamir et al. 2024).

Addressing such complexity requires advanced predictive tools. In response, researchers have developed integrated models that combine machine learning with algorithms of matching complexity to enhance modeling accuracy, computational efficiency, and predictive robustness (Chou et al. 2022a; Zamani et al. 2022). Central to this effort is the optimization of model parameters, a task widely recognized as essential across scientific disciplines (Khatir et al. 2024). As a result, the development of integrated machine learning frameworks supported by nature-inspired metaheuristic algorithms has become a key research focus. These hybrid approaches provide robust solutions for overcoming the limitations of traditional methods in managing complex engineering estimation tasks (Chou et al. 2021).

Machine learning (ML) ensemble models are a powerful tool in predictive analytics, combining multiple base learners to improve performance, stability, and generalization (Wakjira et al. 2021). These base learners can include models like Radial Basis Function Neural Networks (RBFNN) and Least Squares Support Vector Regression (LSSVR), both frequently used in engineering settings. Ensemble methods are particularly effective for addressing complex, real-world engineering problems because they reduce prediction variance and help lower generalization errors (Kotu and Deshpande 2019).

Among the most common ensemble strategies are bagging, boosting, and stacking (Wakjira et al. 2021). In particular, the stacking technique combines the outputs of multiple base learners into a strong and flexible predictive framework by training a meta-learner on their combined predictions (Zhang et al. 2021). In this study, the stacking model is further improved by adding feature weighting, where individual features are assigned specific weights based on their importance. This leads to the development of the Weighted Feature Stacking System (WFSS), which enhances the overall effectiveness of the ensemble by refining the input space before prediction.

Although stacking ensemble methods offer significant improvements in predictive accuracy over individual base models, their performance heavily relies on proper hyperparameter tuning (Cao et al. 2022a, b). In this study, we address this issue by introducing SAPSO, a new and flexible metaheuristic optimization algorithm capable of adapting to different problem domains. SAPSO is employed to continuously optimize the hyperparameters of the Weighted Feature Stacking System (WFSS), improving its predictive power.

To validate the proposed approach, the SAPSO-optimized WFSS was tested on a series of real-world civil engineering problems. Its performance was compared to traditional machine learning methods and well-established design standards. The results consistently demonstrated better predictive accuracy, confirming the effectiveness of the optimization strategy. After these validations, SAPSO was seamlessly integrated with the WFSS framework, forming a unified, high-performing hybrid system.

Development of SAPSO algorithm

Inspiration

Research is a systematic process that involves collecting, analyzing, and interpreting data to generate new knowledge or enhance existing understanding. It includes gathering information, critically evaluating it, and drawing conclusions to test hypotheses or answer specific research questions. Although the definition and methods of research can vary across disciplines and settings, their primary goal remains consistent: to address significant societal issues and deepen our understanding of complex phenomena (Adu and Miles 2023; Yadav 2023; Reis et al. 2024).

The main goals of research include discovering new information, establishing empirical facts, testing theories, and solving practical problems. Research also aims to expand knowledge in a specific field and generate evidence to support informed, data-driven decisions. Its importance lies in its ability to foster the development of new ideas, theoretical frameworks, and practical insights (Adu and Miles 2023; Yadav 2023; Reis et al. 2024).

Additionally, research provides a foundation for policy improvements, strategic planning, and best practices in various fields. It helps stakeholders better understand and tackle complex social, economic, and environmental issues, while also fostering innovation through new products, technologies, and processes (Adu and Miles 2023; Yadav 2023; Reis et al. 2024).

A flowchart of the research process, as outlined by Thomas et al. (2022), illustrates the systematic steps from identifying a research problem to formulating hypotheses or research questions. This process begins with recognizing an area of concern, followed by extensive reading, critical reflection, and a comprehensive literature review to synthesize existing findings and contextual knowledge (Thomas et al. 2022).

A crucial part of any thorough research is using clear operational definitions that precisely describe key terms. These definitions ensure clarity and consistency by identifying observable phenomena and enabling empirical testing of hypotheses. With the research framework in place, the study advances through careful planning and the implementation of appropriate methods.

Once the data have been collected and analyzed, the results are systematically presented and interpreted within the context of existing theories, concepts, and prior studies. Finally, these interpretations are linked back to the original research assumptions or questions, completing the cycle of inquiry.

Accordingly, the research cycle can be divided into four consecutive stages (Fig. 2): (1) reviewing and defining the problem, (2) formulating hypotheses, (3) collecting data, and (4) analyzing and interpreting results. In the final stage, outcomes are evaluated based on whether the hypotheses are accepted or rejected. Together, these stages comprise the essential steps of the scientific method as used in systematic problem-solving. Each iteration of this process yields a measurable research performance value, reflecting the rigor and effectiveness of the inquiry and contributing to the broader growth of knowledge.

Building upon this principle, understanding which research activity most significantly impacts overall performance requires evaluating and comparing these individual performance values. Inspired by this need, we propose a new optimization technique—the Scientific Approach to Problem Solving-inspired Optimization (SAPSO) algorithm. SAPSO draws directly from the structured methodology of scientific research, translating its iterative logic into a computational framework capable of addressing complex optimization problems.

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Fig. 2

Research cycle

Algorithmic design

Population initialization

Similar to most metaheuristic algorithms, the SAPSO optimizer starts by randomly creating an initial population with a uniform distribution, ensuring a wide variety of potential solutions across the search space. Each individual in the population represents a researcher’s proposed solution, with each feature dimension corresponding to a specific aspect of that researcher’s abilities or decision variables. This representation allows SAPSO to explore different solution strategies simultaneously. The initialization of the population at time t = 1 is mathematically defined as follows:

1

Here, denotes the solution at each step for each researcher, indicates a randomly generated value between 0 and 1, and and represent the lower and upper bounds of the given problem, respectively.

Search iteration

Step 1: Reviewing and defining the problem

Literature reviews serve several key functions in the research process. They typically start the inductive reasoning stage, enabling scholars to thoroughly explore and clarify specific phenomena by synthesizing existing knowledge on a particular topic. In this process, researchers systematically compare and contrast previous studies, carefully examining their theoretical frameworks, problem statements, methodologies, and findings to identify trends, gaps, and areas of agreement.

A key challenge in conducting an effective literature review is creating meaningful connections among a diverse range of work. To do this, scholars usually assess and compare essential research elements—including study participants, measurement tools, experimental interventions, research designs, statistical methods, and results. By examining these components with one another, researchers can draw informed and contextually relevant conclusions (Thomas et al. 2022; Adu and Miles 2023).

Conducting a literature review is a crucial step in accurately identifying and articulating the research problem. After thoroughly reviewing the existing body of knowledge and placing the proposed study within its broader academic and practical context, researchers often refine the research problem, questions, and hypotheses into precise and targeted forms. When multiple potential research directions are possible, the first step is to define a focused and specific topic—a process usually started by reviewing article abstracts and, when needed, examining key parts of relevant sources (Thomas et al. 2022; Adu and Miles 2023).

Even a brief engagement with a few influential studies can spark new ideas and uncover unresolved questions in the literature. This iterative interaction with prior research lays the groundwork for a solid investigation and ensures the study addresses a genuine knowledge gap (Thomas et al. 2022; Adu and Miles 2023).

Engaging in discussions with a faculty advisor or an experienced graduate student can be beneficial during the early stages of research development. These conversations help identify potential problems, clarify unclear ideas, and prevent researchers from wasting time on ineffective strategies. Once the research problem is clearly defined, conducting a thorough literature review becomes an essential next step. To support this process, the research community has created various methods that systematically use existing literature to shape and improve research questions.

In the context of the SAPSO algorithm, this conceptual process is modeled computationally. A subset of ​ randomly selected researchers (i.e., candidate solutions) is identified to compute vector effects, thereby expanding the pool of potentially promising solutions. This emulation reflects the intellectual diversity typically encountered during collaborative literature review. Figure 3 illustrates this step using a scenario where . The mathematical representation of this stage is formalized in Eq. (2), marking the beginning of the SAPSO optimizer’s emulation of problem review and refinement.

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Fig. 3

Simulating the step of reviewing and defining the problem

2

Here, represents a randomly generated number within the range [−1,1]. The term denotes the newly computed solution at step t, while refers to the randomly selected solution from the set in the d^th dimension, where ​ is the population size and indicates the number of solutions influencing the new solution. The parameter is randomly selected from the set , where is the number of dimensions in the search space. Experimental results indicate that setting provides optimal performance within a limited computational timeframe.

Step 2: Formulating the hypothesis

A hypothesis is a predictive statement that describes the expected outcome of a study. Before starting research, investigators must specify their objectives. These proposed ideas or guesses are usually based on theoretical frameworks, previous empirical findings, or sometimes the researcher’s personal experience and observations. However, it is essential to note that the latter source is generally considered the least reliable because it lacks the scientific rigor inherent to scientific inquiry and is susceptible to biases from non-systematic knowledge gathering (Thomas et al. 2022; Adu and Miles 2023).

In any rigorous study, each subproblem must be explicitly formulated as an experimental hypothesis. Distinct hypothesis formulations may represent different subpopulations or solution candidates, each capturing a unique perspective on the research question. To construct the overarching hypothesis, SAPSO combines the most refined or promising hypothesis with an additional set of randomly selected hypotheses. This process introduces diversity while preserving high-quality candidates, enhancing the algorithm’s exploratory capacity. The mechanism underlying this formulation is mathematically defined in Eq. (3) and visually illustrated in Fig. 4.

3

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Fig. 4

Formulating the hypothesis step

Here, denotes a randomly generated number within the range [−1,1]. The term represents the newly computed solution at the current step, while refers to the randomly selected solution from the set in the d^th dimension, where ​ is the population size. Additionally, takes values from 1 to , where is the number of dimensions in the search space, and denotes the current best solution. Experimental results indicate that setting yields optimal performance within a limited computational timeframe.

Step 3: Gathering the data

Indeed, Step 2—formulating the hypothesis—can only be effectively completed once the researcher identifies suitable strategies for data collection, as these strategies are essential for assessing the validity of the proposed hypothesis. To ensure successful problem solving, it is necessary to evaluate the accuracy of measurement instruments rigorously, the implementation of experimental controls, and the overall objectivity and precision of the data collection process (Thomas et al. 2022; Adu and Miles 2023).

In many cases, collecting data can be relatively simple and require only routine effort. However, designing and validating the data collection strategy—ensuring methodological rigor and internal validity—remains one of the most critical and intellectually demanding parts of the research process (Thomas et al. 2022; Adu and Miles 2023).

One of the most complex and intellectually demanding stages of the research process involves developing a solid methodological strategy. The chosen approach must be carefully crafted to maximize both internal validity and external validity, as these factors significantly impact the credibility and relevance of the study’s results. These types of validity are influenced by the underlying research design and the controls put in place throughout the investigation.

Internal validity is the extent to which the observed outcomes can be confidently linked to the experimental interventions rather than to external factors. To ensure this, researchers need to reduce potential biases or confounding variables carefully. Conversely, external validity deals with how well the results apply to larger, real-world settings (Thomas et al. 2022).

Balancing these two forms of validity is especially difficult in behavioral and social science research, where controls needed for internal validity often limit the natural conditions required for external validity (Thomas et al. 2022).

To replicate this step within the SAPSO framework, two distinct search scenarios are analyzed. In Scenario 1, the search focuses on the current optimal solution, emphasizing local refinement and exploitation. In Scenario 2, the algorithm investigates alternative solutions randomly, encouraging diversity and exploration of the broader search space.

A comparative evaluation determines which scenario to pursue. Specifically, if rand₁ < rand₂, Scenario 1 is selected; otherwise, Scenario 2 is used. This probabilistic mechanism allows the algorithm to switch between intensification and diversification strategies dynamically. Eqs. (4) and (5) show the mathematical formulations for both scenarios, as illustrated in Fig. 5.

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Fig. 5

Formulating the experiment step

Scenario 1, when rand₁< rand₂

4

Scenario 2, when rand₁≥ rand₂

5

In this context, rand₁ and rand₂ are random values within the range [0, 1], while rand(0,1) denotes a randomly generated number within the same range. The term represents the new solution at the current iteration, and refers to the best solution found in the previous iteration. Additionally, , , and denote the first, second, and third randomly selected solutions, respectively, where in the d^th dimension. Here, is the population size, and , where D is the number of dimensions in the search space.

Step 4: Analyzing and interpreting results

This stage of the research process poses significant challenges for beginner researchers, especially those at the master’s level. While it typically involves statistical analysis, many beginners often lack adequate training in statistics, which leads to discomfort, confusion, or a sense of overwhelm. In addition to technical skills, conducting thorough data analysis and interpretation requires not only methodological knowledge but also practical experience and critical thinking skills—areas that can be particularly difficult for those new to academic research.

The greatest challenge is in assessing and interpreting results, where the researcher must decide if the data support or oppose the study’s original hypothesis. This step requires accuracy, objectivity, and the skill to place findings within the broader theoretical and empirical context (Thomas et al. 2022; Adu and Miles 2023).

By comparing their findings with those reported in the existing literature, researchers can identify meaningful connections and place their results within a broader theoretical framework. While the problem formulation phase mainly uses deductive reasoning, this stage focuses on inductive thinking, where the goal is to generate new insights from existing knowledge. The investigator aims to either support or expand existing theories by combining their outcomes with prior research conclusions.

In the context of the SAPSO algorithm, each outcome is shaped by a combination of influences from the current best solution (denoted as ) and the mean value of all other solutions (denoted as ). This collaborative influence guides the refinement of solutions. The procedural flow of this integration is depicted in Fig. 6 and formally described in Eq. (6).

6

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Fig. 6

Formulation of the analysis and interpretation step

where

7

In this context, refers to a random number within the range [−1, 1]. The term represents the new solution at time t, while corresponds to the best solution at time t–1. Additionally, denotes the central point of the other solutions that influence the new solution. Here, represents the total number of solutions, and d, where D is the number of dimensions in the search space.

Mechanism of switching activities

In the SAPSO framework, the researcher simulates a structured and iterative research process by concentrating on one phase at a time. This progression is guided by an activity-switching mechanism that dynamically manages the transition among four key stages at each time step t: (1) reviewing and defining the problem, (2) formulating the hypothesis, (3) collecting data, and (4) analyzing and interpreting results. This mechanism is mathematically expressed by the function c(t), as outlined in Eq. (8).

Each value of c(t) corresponds to a specific stage in the research cycle:

• c(t) = 1: Problem review and definition.

• c(t) = 2: Hypothesis formulation.

• c(t) = 3: Data collection.

• c(t) = 4: Analysis and interpretation of results.

The SAPSO process begins with problem review when c(t) = 1, transitions to hypothesis formulation at c(t) = 2, continues with data collection at c(t) = 3, and culminates in analysis and interpretation when c(t) = 4. The detailed pseudocode governing this activity-switching process is provided in Fig. 7, while Fig. 8 offers a visual illustration of step selection during each iteration t.

8

where is the maximum number of iterations.

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Fig. 7

Pseudocode of the activity switching mechanism

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Fig. 8

Activity-switching mechanism over iterations in the SAPSO algorithm

Boundary condition

For each newly generated solution, it is essential to verify that it respects the variable boundaries to ensure feasibility. In this study, linear constraints are assumed, and a feasible starting point or initial population is used when evaluating boundary conditions. This approach guarantees that all candidate solutions remain within the defined search space throughout the optimization process.

To handle boundary violations, the method utilizes linear equations, which allow for the elimination of certain variables by expressing them as linear combinations of the remaining variables (Koziel and Michalewicz 1999). Specifically, if a newly generated solution component violates at least one constraint—either the lower bound or the upper bound —the value of is corrected using a linear inequality and replaced with an adjusted value

9

Pseudocode

Building on the core components and mechanisms described earlier, Fig. 9 presents the combined pseudocode of the SAPSO optimizer. In this framework, Steps 1 and 2 mainly focus on the exploration phase, during which the algorithm systematically explores the solution space through activities like problem review and hypothesis development.

As the optimization process advances, the activity-switching mechanism gradually moves to Steps 3 and 4, prioritizing exploitation. These later stages—data collection and analysis—aim to refine the search around promising areas of the solution space, thereby enhancing convergence toward optimal solutions. This adaptive shift between exploration and exploitation is a key feature of SAPSO, enabling it to maintain both diversity and precision throughout the search process.

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Fig. 9

Pseudocode of the SAPSO optimizer

Benchmarking metaheuristics by mathematical functions

IEEE CEC 2020 & 2022

To thoroughly assess the performance of the proposed SAPSO optimizer, a set of 30 benchmark functions from the IEEE CEC 2020 test suite was used, with dimensionalities ranging from 10 to 20 (Biswas and Suganthan 2020). These functions cover a wide range of optimization landscapes, intended to carefully evaluate the algorithm’s ability in both exploration and exploitation.

Specifically, CFa1 is a unimodal function, ideal for testing local search performance. CFa2 and CFa3 are multimodal functions, challenging the optimizer’s ability to avoid local optima and find the global solution. CFa4 represents a flat landscape with a single optimum, designed to evaluate convergence in low-gradient environments. Functions CFa5 through CFa7 are hybrid functions that combine features of multiple types, while CFa8 to CFa10 are composition functions that simulate highly complex, real-world scenarios. A detailed description of these benchmark functions can be found in Table A.1 (Supplementary Information).

Along with the CEC 2020 benchmarks, the SAPSO optimizer was also tested on 24 benchmark functions from the CEC 2022 test suite, with problem sizes set at 10 and 20 (Kumar et al. 2021). These functions offer a wide range of optimization problems aimed at assessing the robustness and flexibility of metaheuristic algorithms. The suite includes:

• CFb1 to CFb5: Unimodal functions, appropriate for evaluating convergence accuracy in smooth, single-optimum landscapes.

• CFb6 to CFb8: Hybrid functions that combine multiple search landscapes to test the optimizer’s versatility.

• CFb9 to CFb12: Composition functions, recognized for their complexity and similarity to real-world problem structures.

A comprehensive summary of all 24 functions is given in Table A.2 (Supplementary Information).

Algorithm assessment

A total of thirty benchmark functions from CEC 2020 and twenty-four from CEC 2022 were selected to evaluate the performance of the SAPSO algorithm thoroughly. To provide a rigorous and meaningful assessment, SAPSO’s performance was compared against eleven established metaheuristic algorithms, including:

• Artificial Bee Colony (ABC) (Karaboga and Basturk 2007).

• Cultural Algorithm (CA) (Omran 2016).

• Genetic Algorithm (GA) (Holland 1992).

• Differential Evolution (DE) (Storn and Price 1997).

• Artificial Gorilla Troops Optimizer (GTO) (Abdollahzadeh et al. 2021).

• Grey Wolf Optimizer (GWO) (Mirjalili et al. 2014).

• Particle Swarm Optimization (PSO) (Kennedy and Eberhart 1995).

• Red Kite Optimization Algorithm (ROA) (Archana et al. 2024).

• Symbiotic Organisms Search (SOS) (Cheng and Prayogo 2014).

• Teaching–Learning-Based Optimization (TLBO) (Rao et al. 2011).

• Whale Optimization Algorithm (WOA) (Mirjalili and Lewis 2016).

To promote fairness and reduce the impact of stochastic variance, each optimizer was independently run thirty times on each benchmark function. This repeated assessment was conducted using standardized mathematical functions, following best practices for minimizing randomness in metaheuristic comparisons (Chou and Truong 2021).

Wilcoxon’s rank-sum test

The optimization performance of the SAPSO algorithm was statistically evaluated by comparing it with other metaheuristic algorithms using the nonparametric Wilcoxon rank-sum test, a reliable method suitable for comparing paired samples without assuming normality (Derrac et al. 2011). The analysis was performed at a 1% significance level (α = 0.01) to ensure high confidence in the results.

In this context, the performance of SAPSO is represented by the population mean . In contrast, the performance of each comparison algorithm is denoted as . The hypotheses for the statistical test are formulated as follows:

10

This formulation allows a thorough assessment of SAPSO’s superiority across benchmark functions, based on observed performance distributions.

Computational time

All optimization algorithms used in this study were implemented in MATLAB R2016a and run on a Windows PC with an Intel Core i5-7500 CPU (3.40 GHz) and 8 GB of RAM. To evaluate the computational efficiency of each optimizer, the time taken to solve each benchmark problem was systematically recorded. This metric offers an additional way to compare performance, supplementing solution quality with insights into the computational resource requirements.

Statistical results in the mathematical test

For all mathematical benchmark tests, the population size, number of iterations, and maximum evaluation limit were consistently set to 50, 10,000, and 500,000, respectively. To ensure fairness and consistency in comparison, all other internal parameters for each metaheuristic algorithm were kept at their default values. The SAPSO optimizer, along with eleven competing metaheuristic algorithms, was run thirty times per test case to account for stochastic variability and to allow for statistically reliable comparisons.

The statistical performance results for the CEC 2020 benchmark functions are shown in Fig. 10a and b, while the p-values from the Wilcoxon rank-sum test, comparing SAPSO with each competing algorithm, are displayed in Fig. 10c. Similarly, the performance metrics for the CEC 2022 benchmark functions are shown in Fig. 11a and b, with the associated p-values summarized in Fig. 11c. A complete overview of all Wilcoxon rank-sum test results is provided in Tables 1 and 2.

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Fig. 10

a Absolute error of the mean values of the 12 metaheuristic algorithms on the CEC2020 functions. b Standard deviations of the 12 metaheuristic algorithms on the CEC2020 functions. c P-value of Wilcoxon’s rank-sum test between SAPSO and the compared algorithms on the CEC2020 functions

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Fig. 11

a Absolute error of the mean values of the 12 metaheuristic algorithms on the CEC2022 functions. b Standard deviations of the 12 metaheuristic algorithms on the CEC2022 functions. c P-value of Wilcoxon’s rank-sum test between SAPSO and the compared algorithms on the CEC2022 functions

Table 1. Results from Wilcoxon rank sum tests for solving CEC-2020 functions

Fun.	C	SAPSO vs.	ABC	CA	DE	GA	GTO	GWO
CFa1	U	+	3	3	3	3	3	3
		≈	0	0	0	0	0	0
CFa2-CFa3	M	+	6	6	6	6	6	6
		≈	0	0	0	0	0	0
CFa4	E	+	3	3	3	3	3	3
		≈	0	0	0	0	0	0
CFa5-CFa10	H	+	17	18	18	18	18	18
		≈	1	0	0	0	0	0
Total		+	29	30	30	30	30	30
		≈	1	0	0	0	0	0
Total CPU time (Sec.)		210.74	446.72	3039.13	695.82	815.48	326.60	216.79

Fun.		SAPSO vs.	PSO	ROA	SOS	TLBO	WOA
CFa1	U	+	3	3	3	3	3
		≈	0	0	0	0	0
CFa2-CFa3	M	+	6	6	6	6	6
		≈	0	0	0	0	0
CFa4	E	+	3	3	3	3	3
		≈	0	0	0	0	0
CFa5-CFa10	H	+	18	18	18	18	18
		≈	0	0	0	0	0
Total		+	30	30	30	30	30
		≈	0	0	0	0	0
Total CPU time (Sec.)		210.74	527.29	239.55	253.34	235.57	212.54

“C” represents characteristics; “M” stands for multimodal; “U” denotes unimodal; “E” represents expanded; “H” signifies hybrid composite functions; “+” indicates inferior performance compared to SAPSO; while “≈” indicates no significant difference in performance between the compared algorithm and SAPSO

Table 2. Results from Wilcoxon rank sum tests for solving CEC-2022 functions

Fun.	C	SAPSO vs.	ABC	CA	DE	GA	GTO	GWO
CFb1	U	+	2	2	2	2	2	2
		≈	0	0	0	0	0	0
CFb2-CFb5	M	+	7	8	8	8	8	8
		≈	1	0	0	0	0	0
CFb6-CFb12	H	+	14	14	14	14	13	14
		≈	0	0	0	0	1	0
Total		+	23	24	24	24	23	24
		≈	1	0	0	0	1	0
Total CPU time (Sec.)		166.25	506.14	2518.59	346.87	620.81	185.05	169.94

Fun.			PSO	ROA	SOS	TLBO	WOA
CFb1	U	+	2	2	1	2	2
		≈	0	0	1	0	0
CFb2-CFb5	M	+	8	7	7	8	8
		≈	0	1	1	0	0
CFb6-CFb12	H	+	14	14	14	14	14
		≈	0	0	0	0	0
Total		+	24	23	22	24	24
		≈	0	1	2	0	0
Total CPU time (Sec.)		166.25	1039.33	172.27	220.04	190.72	176.24

“C” represents characteristics; “M” stands for multimodal; “U” denotes unimodal; “E” represents expanded, and “H” signifies hybrid composite functions; “+” indicates inferior performance compared to SAPSO; while “≈"means no significant difference in performance between the compared algorithm and SAPSO

Capabilities of the SAPSO algorithm

This section explains the three main parts of the strong SAPSO optimization algorithm: (1) exploring the search area, (2) using promising solutions, and (3) moving toward the global best. Each part is vital for balancing search variety with solution improvement, ensuring both flexibility and accuracy during the optimization process.

Exploring the search space

Multimodal functions, which contain many local optima, serve as effective benchmarks for assessing the exploration abilities of optimization algorithms (Askari et al. 2020). In this study, the performance of the proposed SAPSO optimizer was evaluated using 14 multimodal functions from the CEC benchmark suite, including CFa2 and CFa3 at multiple dimensional levels, as well as CFb2 through CFb5 in both 10- and 20-dimensional settings (as summarized in Tables 1 and 2).

The Wilcoxon rank-sum test results, shown in Tables 1 and 2, indicate that SAPSO outperformed:

• The Artificial Bee Colony (ABC) algorithm in 13 out of 14 cases.

• The Cultural Algorithm (CA), Differential Evolution (DE), Genetic Algorithm (GA), Artificial Gorilla Troops Optimizer (GTO), Grey Wolf Optimizer (GWO), Particle Swarm Optimization (PSO), Teaching–Learning-Based Optimization (TLBO), and Whale Optimization Algorithm (WOA) in all 14 cases.

• The Red Kite Optimization Algorithm (ROA) and Symbiotic Organisms Search (SOS) in 13 out of 14 cases.

These findings clearly show that the SAPSO optimizer has better exploration ability, effectively navigating complex, multimodal search spaces more efficiently than the competing algorithms.

Leveraging promising solutions

Unimodal functions are essential for evaluating the exploration abilities of optimization algorithms because they feature a single global optimum and few local distractions (Askari et al. 2020). In this study, five high-scale unimodal benchmark functions were used—specifically, CFa1 tested at multiple dimensional levels and CFb1 examined in two different dimensions—to assess the local search performance of the proposed SAPSO optimizer, along with 11 comparison algorithms (as shown in Tables 1 and 2).

The Wilcoxon rank-sum test results, summarized in Tables 1 and 2, reveal the following outcomes for SAPSO:

• Outperformed ABC, CA, DE, GA, GTO, GWO, PSO, ROA, TLBO, and WOA in all 5 test cases (5/5).

• Outperformed SOS in 4 out of 5 cases (4/5).

These results highlight the strong exploitation ability of the SAPSO optimizer, showing its effectiveness in precisely converging toward the global optimum in unimodal problem landscapes.

Converging towards the best solution

Across the entire suite of benchmark tests, the SAPSO optimizer demonstrated superior performance compared to eleven well-established metaheuristic algorithms. Specifically, SAPSO outperformed:

• ABC in 52 out of 54 cases,

• CA, DE, GA, GWO, PSO, TLBO, and WOA in all 54 cases,

• GTO and ROA in 53 out of 54 cases, and.

• SOS in 52 out of 54 cases.

These results are statistically validated by the Wilcoxon rank-sum test p-values, confirming SAPSO’s significant advantage in solution quality.

In addition to accuracy, computational efficiency was also evaluated. As shown in Tables 1 and 2, SAPSO achieved the lowest CPU times across all 54 benchmark problems, requiring only 210.74 s for the 30 CEC 2020 functions and 166.25 s for the 24 CEC 2022 functions.

Moreover, SAPSO consistently delivered optimal or near-optimal solutions across various problem types—including unimodal, multimodal, separable, non-separable, expanded, and hybrid composite functions—demonstrating its robustness, flexibility, and computational efficiency.

The convergence curves shown in Fig. 12—which display representative unimodal and multimodal benchmark functions—provide a visual comparison of the proposed SAPSO optimizer against eleven other optimization algorithms. These curves consistently illustrate SAPSO’s superior convergence performance across various problem types.

In particular, for the unimodal functions (CFa1 and CFb1 in Fig. 12), SAPSO showed rapid and sustained convergence toward the global optimum. The smooth and steep progress of the curves indicates not only the optimizer’s efficiency in exploitation but also its ability to maintain momentum throughout the search process, effectively outperforming all competing methods in both speed and accuracy.

The time control mechanism within the SAPSO optimizer is crucial for allowing the algorithm to escape local optima effectively. This ability is clearly shown by the experimental results on multimodal functions—specifically CFa2, CFb3, and CFb5, as depicted in Fig. 12. The figure demonstrates how SAPSO adjusts its balance between exploration and exploitation throughout the evaluation process.

In the early phases of the optimization, SAPSO focuses on exploration, especially on complex functions like the expanded function CFa4 and the hybrid composite functions—CFa8, CFa9, CFb7, CFb9, and CFb12. As the process continues, the algorithm gradually shifts to exploitation, honing in on the most promising areas.

The convergence curves demonstrate SAPSO’s capability to prevent premature convergence, improve solution quality, and speed up progress toward the global optimum, thereby emphasizing the effectiveness of its adaptive strategy in complex search environments.

The analysis and visualization results confirm that the SAPSO optimizer successfully achieves a perfect balance between exploration and exploitation—a crucial factor in high-performance metaheuristic optimization. This balance results directly from SAPSO’s algorithmic design, which switches between exploratory phases—involving problem review and hypothesis development—and exploitative phases, including data collection, analysis, and interpretation of results.

The transition between these phases is controlled by the activity-switching mechanism, which dynamically adjusts the optimizer’s focus based on the current state of the search process. This adaptive coordination allows SAPSO to systematically explore the search space while gradually increasing focus on promising areas, ensuring both diversity and efficient convergence throughout the optimization.

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Fig. 12

Convergence performance between SAPSO and selected optimizers on CEC 2020 and CEC 2022 benchmark functions

Crafting informatics solutions

Stacking ensemble model

To improve prediction accuracy and reduce generalization errors, ensemble machine learning techniques combine the outputs of multiple base learners to produce more reliable predictions (Kotu and Deshpande 2019). Among the most common ensemble methods are stacking, boosting, and bagging (Wakjira et al. 2021).

As illustrated in Fig. 13, stacking is a flexible and robust learning framework that integrates the predictions of several weak base learners to construct a stronger composite model (Zhang et al. 2021). Formally, stacking can be expressed as a function , where each base learner contributes an estimated model used in the ensemble process. The meta-learner then combines these individual outputs to produce the final predictive model , as described in Eq. (11).

A general schematic of this stacking framework, including the role of base and meta-learners, is presented in Fig. 13.

11

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Fig. 13

General framework of the stacking ensemble model

Regression with weighted features

In the Weighted Feature Stacking System (WFSS), each instance is individually adjusted by assigning a weight to each feature, allowing for a more detailed representation of the input data. These feature-weighted instances are then used to train the stacking ensemble, helping the model focus more on the most informative features during training and thereby improving overall predictive performance.

Each instance in the dataset is individually modified by assigning feature-specific weights, resulting in the development of the Weighted Feature Stacking System (WFSS). This weighting method enables the model to focus more on the most relevant features during learning. The adjusted, feature-weighted instances are then used to train the stacking ensemble, which enhances its predictive accuracy and robustness. The mathematical formulation of this method appears in Eq. (12), and the conceptual framework is shown in Fig. 14.

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Fig. 14

Principle of the weighted feature-based regression system

Given dataset , now that is , and is calculated by,

12

Here, represents the weight of the feature, d denotes the total number of features, refers to the parameter of the machine learning algorithm, and indicates the total number of parameters.

13

where

and is the weighted feature vector.

Metaheuristic-optimized weighed feature stacking system

In stacking-based machine learning, different base classifiers are used to enhance overall performance. However, it cannot be easy to achieve balanced results across all base and meta classifiers, mainly when relying on a fixed parametric optimization approach. This rigidity can hinder the overall effectiveness of the ensemble model.

To address this limitation, this paper presents a new hybrid stacking ensemble framework that improves prediction accuracy by fine-tuning the hyperparameters of both base and meta learners using a metaheuristic optimization algorithm. This adaptive tuning process allows the ensemble system to adjust its internal setup for peak performance dynamically. The structure of the optimized stacking framework is shown in Fig. 15.

The hyperparameters of the stacking system play a critical role in determining its predictive accuracy. For the base learners, the parameters include for ML₁ (LSSVR), and for ML₂ (RBFNN). For the meta-learner, the configuration depends on the selected model: either and for ML_c (LSSVR) or and for ML_c (RBFNN). In addition, the feature space is fine-tuned using weights (where ), which are assigned to individual features. The optimization of these hyperparameters—specifically, or —constitutes a complex multi-dimensional optimization problem. This challenge is effectively addressed using the SAPSO algorithm, which dynamically searches for the optimal configuration to maximize the model’s performance.    

The resulting integrated framework is known as the SAPSO-Optimized Weighted Feature Stacking System (SAPSO-WFSS). A detailed flowchart of the SAPSO-WFSS framework is shown in Fig. 15, illustrating the complete optimization and learning process.

The dataset was initially divided, with 90% designated for training and the remaining 10% kept for testing. Within the training set, the SAPSO-WFSS model was further trained on 70% of the data, while the remaining 30% was used for hyperparameter validation. This stratified method ensures solid model development while avoiding overfitting.

The objective function employed to guide the SAPSO-based optimization of the WFSS model is formally presented in the following equation, capturing the trade-off between prediction accuracy and model generalization.

14

where for LSSVR as a meta-learner, or for RBFNN as a meta-learner.        

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Fig. 15

Framework of SAPSO-WFSS

The original stacking system (SS) was trained on a learning dataset enhanced with weighted features, where both the feature weights and hyperparameters were jointly optimized using the SAPSO algorithm. This simultaneous optimization allowed the model to customize its learning process for better predictive performance.

After training, the performance of the enhanced stacking system was tested on the held-out test dataset to evaluate its ability to generalize. As shown in Fig. 15, this integration of the SAPSO optimizer with the stacking framework led to the creation of the SAPSO-WFSS hybrid intelligent system, representing a strong and adaptable approach to machine learning optimization.

Application in civil engineering informatics

Case study background

In this study, the forecasting performance of the proposed SAPSO-WFSS model was assessed using five benchmark datasets sourced from the literature. These datasets cover a wide range of civil engineering applications, including:

• Dataset 1: Elastic modulus of recycled aggregate concrete.

• Dataset 2: Bearing capacity of axially loaded piles.

• Dataset 3: Shear capacity of reinforced concrete walls.

• Dataset 4: Deflection of reinforced concrete beams.

• Dataset 5: Construction productivity.

Detailed information for each dataset is provided in Tables B.1 to B.10 (Supplementary Information), while Table 3 offers a summarized overview of their key features, including sample size, input variables, and output targets. This multi-domain evaluation highlights the versatility and robustness of the SAPSO-WFSS framework across various engineering prediction tasks.

Table 3. Sources of datasets in the literature

Dataset	Area	Data source	Data description	Sample size
Dataset 1	Construction and building materials	Golafshani and Behnood (2018a, b)	Elastic modulus of recycled aggregate concrete	400
Dataset 2	Engineering structure - structural foundation	Pham et al. (2020a, b, c)	Bearing capacity in axial piles	472
Dataset 3	Engineering structure - reinforced concrete walls	Chou et al. (2022b)	Shear capacity of reinforced concrete walls	492
Dataset 4	Engineering structure - reinforced concrete beams	Nguyen et al. (2023)	Long-term deflection of reinforced concrete beams	217
Dataset 5	Construction management	Khan (2005) and Wang (2005)	Productivity of formwork installation	220

Elastic modulus of recycled aggregate concrete

With the ongoing growth of the construction industry, the demand for aggregates—a key component of concrete—remains consistently high. At the same time, the demolition of aging infrastructure creates large amounts of crushed concrete, raising significant environmental concerns, especially regarding the reduction of available landfill space.

Recent research shows that using recycled and repurposed concrete aggregates from demolished structures—rather than relying only on non-renewable virgin materials—can greatly enhance resource sustainability. This approach not only saves natural resources but also reduces the environmental impact of traditional landfill disposal, supporting more sustainable construction methods.

Elasticity is a key mechanical property in the concrete industry, indicating a material’s ability to deform elastically under applied stress (Fig. 16). In practice, when natural aggregate concrete (NAC) and recycled aggregate concrete (RAC) are made with the same water-to-cement ratio (w/c), RAC usually shows a lower elastic modulus than NAC (Rahal 2007).

Numerous researchers have proposed empirical equations to estimate the elastic modulus of concrete based on other parameters, such as compressive strength (Behnood et al. 2015). However, these formulations are primarily derived from experimental data on NAC, raising valid concerns about their applicability to RAC. This gap highlights the need for new predictive models specifically designed for estimating the elastic modulus of RAC (.

The input parameters used in this study to model are summarized in Tables B.1 and B.2, based on datasets from Golafshani and Behnood (2018a, b) and Cheng and Gosno (2021).

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Fig. 16

Visualization of RAC sample

Bearing capacity in axial piles

In the design of pile foundations, an essential factor is the precise estimation of axial pile bearing capacity (P_u) (Drusa et al. 2016). Pile load tests are widely considered the most dependable method for evaluating this capacity, as they are firmly based on the theoretical principles that govern driven pile behavior (Birid 2018). However, despite their accuracy, these tests are often costly and time-consuming, especially for small- to medium-sized enterprises (Birid 2018).

As a result, researchers have made a concerted effort to develop more economical and time-efficient alternatives. One widely studied approach involves using in-situ test data to predict pile-bearing capacity. Among these, the Standard Penetration Test (SPT) has become one of the most commonly used techniques because of its practicality and accessibility (Bouafia and Derbala 2002; Kozłowski and Niemczynski 2016).

Traditional methods for assessing the mechanical properties of piles have mainly depended on key parameters like pile diameter, pile length, soil type, and the number of Standard Penetration Test (SPT) blows recorded within each soil layer. However, these methods often yield inconsistent and unreliable results, primarily because they selectively include certain input variables and overlook other influential factors (Pham et al. 2020a, b, c).

This variability highlights the need for a standardized method that systematically identifies and includes the most relevant and comprehensive set of parameters. Creating such a process is vital for improving the accuracy and reliability of pile capacity predictions in geotechnical engineering.

The dataset used in this study, shown in Fig. 17 (Cao et al. 2022a, b), comes from 472 field tests on precast reinforced concrete piles conducted by Pham et al. (2020a, b, c) in Ha Nam Province, Vietnam. These piles featured square cross-sections with closed tips and were installed using a continuous hydraulic jack-in mechanism.

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Fig. 17

Visualization of the equipment arrangement along with geotechnical details at the testing site

After installation, the piles were allowed to settle for at least seven days, as recommended by the original researchers. Then, vertical loads were applied gradually at intervals of about 6, 12, and 24 h, reaching 100%, 150%, and 200% of the specified design load, respectively. This staged loading method provided a controlled and dependable evaluation of the axial bearing capacity.

A comprehensive summary of the input parameters used in this analysis is available in Tables B.3 and B.4.

Shear capacity of reinforced concrete walls

Reinforced concrete (RC) shear walls (SWs) are vital structural elements designed to withstand lateral forces, primarily those generated by seismic activity (Cüneyt Aydin and Bayrak 2021). Their function in improving the lateral stiffness and strength of buildings has made them an essential part of modern seismic design.

Empirical evidence from recent earthquake events consistently shows that buildings equipped with shear walls perform better during seismic activity compared to those without (Gallardo et al. 2021). As shown in Fig. 18, a shear wall acts as a vertical component that can resist in-plane shear forces, bending moments, and axial loads, thus helping to maintain the overall stability and integrity of structural systems (Chou et al. 2022b).

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Fig. 18

Required design inputs and shear capacity of the reinforced concrete shear wall

The flexural and shear capacities of shear walls (SWs) are covered in modern structural design standards, including the American Concrete Institute (ACI) 318 − 19 and Eurocode 2 (EC-2), both of which are highly regarded for their engineering thoroughness and practical use. While flexural capacity is clearly explained through flexural theory, the shear capacity provisions in the ACI 318 − 19 code are considered somewhat basic. They may lack the detail required for contemporary applications (Tran et al. 2017).

Research has shown that ACI 318 − 19 often provides a lower safety margin and fails to properly account for the behavior of high-strength concrete shear walls, which could compromise safety in advanced design scenarios. In contrast, Eurocode 8, which covers seismic design, includes conservative shear design provisions, leading to overly cautious estimates that might result in uneconomical designs (Chandra et al. 2018).

A more advanced and accurate way to estimate the peak shear strength of shear walls could provide a valuable alternative to the overly simple rules in current building codes. Although rational design methods, like the truss model (Chandra et al. 2018) and the softened strut-and-tie method (Hwang et al. 2001), have been suggested, these approaches involve complex analyses that can be difficult for practicing structural engineers.

This complexity emphasizes the need for a practical approach that balances accuracy and usability, allowing engineers to make reliable shear strength predictions without a heavy computational load. To help develop such a model, the relevant input parameters for this study are summarized in Tables B.5 and B.6 (Chou et al. 2022b).

Long-term deflection of reinforced concrete beams

In the design and assessment of the long-term serviceability of reinforced concrete (RC) structural components (Fig. 19), particular focus is given to accurately estimating long-term deflection (Gribniak et al. 2013; Lee et al. 2019; Jia et al. 2022; Nguyen et al. 2023). Over a structure’s lifespan, the horizontal deflection of RC elements gradually increases due to the cumulative effects of both internal and external factors.

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Fig. 19

Cross-sectional shapes of RC beams

Key contributing factors include environmental conditions, elastic deformation under service loads, creep, shrinkage, and sustained loading (Aghayere 2019). These influences interact in complex ways, making accurate deflection prediction essential—especially in the design of precision-engineered, long-span RC beams with small cross-sectional dimensions. Ensuring the reliability of such predictions is crucial for maintaining structural performance, safety, and serviceability over time.

Current formulas for predicting long-term deflection in reinforced concrete (RC) members often lack precision because they overlook important geometric parameters and the inherent mechanical properties of the structural elements (Gilbert 1999). Consequently, these empirical models are usually only suitable for simplified RC configurations, typically with uniform geometry and standard loading conditions (Gribniak et al. 2013).

To address the inherent limitations of these models, many design codes incorporate broad safety margins, including variability buffers that may reach up to 62% of the actual deflection value (Gribniak et al. 2013). While these margins aim to compensate for prediction uncertainties, they can result in overly conservative designs, reducing structural efficiency and material optimization.

Furthermore, most traditional methods are mainly aimed at estimating the immediate deflection of reinforced concrete (RC) beams, providing limited understanding of their long-term performance. Although some recent studies have incorporated geometric parameters into design code formulas (Marí et al. 2010), the overall accuracy of these models remains insufficient.

Relying solely on simple linear models limits the ability to accurately capture the complex, nonlinear interactions among factors such as creep, shrinkage, and sustained loading that influence long-term deflection. Therefore, there is a significant need for more advanced models, especially those using robust nonlinear methods, to provide reliable deflection estimates during the early design stages (Nguyen et al. 2023).

A comprehensive overview of the input parameters used in this study is presented in Tables B.7 and B.8 (Nguyen et al. 2023).

Construction productivity

During the construction phase, site productivity plays a crucial role in overall project efficiency (Fig. 20). However, its natural variability—affected by factors such as site size and measurement location—presents significant challenges for construction managers to predict accurately. Additionally, a wide range of factors can influence task-level productivity, including labor skill levels, the variety of materials and tools used, the complexity of sequential tasks, and current site conditions.

[See PDF for image]

Fig. 20

3D-view of slab formwork

The influence of each of these factors can vary greatly depending on the specific context and characteristics of a project. To handle this complexity, many models and analytical methods have been proposed to predict labor productivity by systematically analyzing construction workflows and their underlying variables. A summary of the input parameters used in this study is included in Tables B.9 and B.10 (Khan 2005; Wang 2005).

To analyze the relationship between multiple factors and crew productivity, Oral and Oral (2010) used two-dimensional mapping methods with a Self-Organizing Map (SOM). Their model achieved a Mean Absolute Percentage Error (MAPE) of 25.05%, providing initial insights into productivity prediction in specific conditions (Oral and Oral 2010).

Building on this foundation, Cheng et al. (2021) introduced a more advanced method that utilized artificial intelligence (AI) and inference modeling to predict productivity in building projects. Their integrated model combined a Least Squares Support Vector Machine (LSSVM), Symbiotic Organisms Search (SOS) algorithm, and a Feature Selection (FS) mechanism. The results were highly encouraging, with a Root Mean Square Error (RMSE) of 0.0721 m²/labor hour, a Mean Absolute Error (MAE) of 0.0563 m²/labor hour, an R-squared value of 0.979, and an MAPE of 3.67%.

More recently, Truong and Chou (2022) developed the Fuzzy Adaptive Jellyfish Search-Optimized Stacking (FAJS-SS) model to improve labor productivity forecasting further. This hybrid approach showed exceptional predictive accuracy, achieving an R-squared value of 0.984, an MAPE of 2.79%, an RMSE of 0.009 m²/labor-hour, and an MAE of 0.045 m²/labor-hour.

Evaluation and validation

Performance metrics

To evaluate the predictive effectiveness of the proposed techniques, the study uses five well-known performance metrics. These include the correlation coefficient (R), which measures the strength and direction of the linear relationship between predicted and actual values; the mean absolute error (MAE), which quantifies the average size of prediction errors; the root mean square error (RMSE), which highlights larger errors due to its squared terms; and the mean absolute percentage error (MAPE), which expresses prediction accuracy as a percentage. Together, these metrics offer a comprehensive assessment of model performance across both absolute and relative error measures.

The Mean Absolute Percentage Error (MAPE) measures the average absolute percentage difference between predicted and actual values, with lower MAPE values showing better predictive accuracy. As a commonly used metric, MAPE is especially helpful for comparing the performance of predictive models across different scales.

The Root Mean Square Error (RMSE) measures the spread of prediction errors by giving more weight to larger deviations due to squaring residuals. In contrast, the Mean Absolute Error (MAE) offers a more straightforward way to see the average size of prediction errors, giving insight into typical error magnitude.

The correlation coefficient (R) measures the strength and direction of the linear relationship between predicted and observed values. In summary, lower values of MAPE, MAE, and RMSE indicate higher model accuracy, while a higher R value signifies stronger predictive performance.

Furthermore, the Synthesis Index (SI), introduced by Truong and Chou (2022), is used as a combined metric to assess the overall predictive accuracy of both the proposed and comparison models. The SI is calculated by averaging the values of the individual performance metrics—MAPE, MAE, RMSE, and R—providing a comprehensive measure of a model’s forecasting ability.

This aggregated index supports a more balanced comparison across various evaluation criteria, helping to identify models that show consistent performance across different aspects of predictive accuracy. The mathematical formulas for each of these metrics, including the SI, are shown in Equations (15) through (19).

15

16

17

18

19

In the equations provided, the forecasted values are denoted by y’, while the observed (actual) values are represented by y. The term P_i refers to the i^th power of the performance metric, and m denotes the total number of performance measures considered in the evaluation.

To incorporate the correlation coefficient (R) into Eq. (19)—which is used to compute the Synthesis Index (SI)—it must first be converted into an error-based form. This is done by transforming it into (1 − R), aligning it with the minimization goal of other error-based metrics like MAPE, MAE, and RMSE.

Cross-fold validation

K-fold cross-validation is a commonly used method for evaluating the predictive performance of machine learning models while reducing bias caused by random splitting of training and testing data. This method involves dividing the dataset into k equal parts or folds, with the model being trained on (k – 1) folds and tested on the remaining fold. This process is repeated k times, with each fold serving once as the test set (Kohavi 1995).

To further improve the reliability of the evaluation, stratification is commonly used—ensuring that the distribution of response variables within each fold mirrors that of the original dataset. In this study, ten-fold cross-validation was employed to assess the consistency and generalization performance of the proposed model. This choice is supported by prior research (Kohavi 1995), which suggests that ten-fold provides an effective balance between estimation bias and variance.

Results and discussion

The parameter configurations for the original stacking system (SS), the proposed SAPSO-SS and SAPSO-WFSS, as well as for several established metaheuristic algorithms used to optimize SS for applications in applied mechanics and engineering, are detailed in Tables C.1 and C.2 (Supplementary Information). To evaluate performance comprehensively, the proposed SAPSO-WFSS_LSSVR system was benchmarked against a range of comparative models, including SS_LSSVR, SS_RBFNN, SAPSO-SS_LSSVR, SAPSO-SS_RBFNN, SAPSO-WFSS_LSSVR, SAPSO-WFSS_RBFNN, GA-WFSS_LSSVR, PSO-WFSS_LSSVR, SOS-WFSS_LSSVR, and TLBO-WFSS_LSSVR, in addition to other models reported in the literature. The results, presented in Tables 4, 5, 6, 7 and 8; Fig. 21, provide a comprehensive performance comparison.

For the five datasets analyzed, the SAPSO-WFSS_LSSVR model achieved Mean Absolute Percentage Errors (MAPEs) of 4.8556%, 6.1588%, 9.7784%, 12.3837%, and 2.3865%, respectively. Notably, the model exhibited a Synthesis Index (SI) of 0.000 across all five cases, emphasizing its exceptional accuracy, robustness, and effectiveness in handling complex prediction tasks in engineering and applied mechanics domains.

Table 4. Assessment of predictive accuracy in dataset 1 for the elastic modulus of recycled aggregate concrete

Model/System	Author	MAE (GPa)	RMSE (GPa)	MAPE (%)	R	SI (Rank)
ANN	Golafshani and Behnood (2018a, b)	1.5175	2.3463	6.0679	0.9156	0.10 (10)
SVR		1.6994	2.7471	6.7915	0.8870	0.16 (12)
GP	Golafshani and Behnood (2018a, b)	2.1857	2.9595	8.8990	0.8643	0.22 (15)
ABCP		2.0557	2.6294	8.4772	0.8953	0.16 (14)
BBP		2.0056	2.6399	8.3133	0.8945	0.16 (13)
SPOT	Cheng and Gosno (2021)	1.6000	2.1000	6.5800	0.9300	0.08 (9)
SSLSSVR	This study	1.9695	0.3764	8.0593	0.8787	0.13 (11)
SSRBFNN		13.4947	11.8854	60.9833	0.7728	1.00 (16)
SAPSO-SSLSSVR		1.7299	0.3271	6.9495	0.9113	0.07 (7)
SAPSO-SSRBFNN		1.7081	0.3030	6.7597	0.9085	0.08 (8)
SAPSO-WFSSLSSVR		1.2589	0.1531	4.8556	0.9477	0.00 (1)
SAPSO-WFSSRBFNN		1.4205	0.3028	5.5948	0.9358	0.03 (6)
GA-WFSSLSSVR		1.2676	0.2149	4.9809	0.9481	0.00 (2)
PSO-WFSSLSSVR		1.3152	0.1788	5.2182	0.9476	0.00 (3)
SOS-WFSSLSSVR		1.3722	0.3277	5.2849	0.9361	0.02 (5)
TLBO-WFSSLSSVR		1.3285	0.2246	5.3282	0.9469	0.01 (4)

ANN stands for Artificial Neural Network, SVR denotes Support Vector Regression, GP refers to Genetic Programming, ABCP represents Artificial Bee Colony Programming, BBP stands for Biogeography-Based Programming, SPOT denotes Symbiotic Polyhedron Operation Tree, SS refers to a baseline Stacking System, LSSVR stands for Least Squares Support Vector Regression, RBFNN denotes Radial Basis Function Neural Network, WFSS refers to the Weighted Feature Stacking System, GA stands for Genetic Algorithm, PSO denotes Particle Swarm Optimization, SOS represents Symbiotic Organisms Search, and TLBO refers to Teaching-Learning-Based Optimization. Bold values indicate the performance measures of the best model.

Table 5. Assessment of predictive accuracy in dataset 2 for axial pile bearing capacity

Model/System	Author	MAE (kN)	RMSE (kN)	MAPE (%)	R	SI (Rank)
GA-DLNN	Pham et al. (2020a, b, c)	75.927	95.118	–	0.9607	(*)
ANN	Pham et al. (2020a, b, c)	3.190	116.366	–	0.8994	(*)
RF		2.924	98.161	–	0.9306	(*)
SA–GP	Yong et al. (2021)	10.265	13.6689	9.159	0.981	0.12 (5)
IMNNIM	Cao et al. (2022a, b)	67.98	90.92	7.24	0.9644	(*)
SSLSSVR	This study	79.9027	17.6400	8.3454	0.9401	0.87 (10)
SSRBFNN		4639.6623	4429.1268	363.8987	0.3595	1.00 (11)
SAPSO-SSLSSVR		68.4862	9.1113	7.2077	0.9647	0.38 (9)
SAPSO-SSRBFNN		64.2420	10.4948	6.6530	0.9700	0.20 (8)
SAPSO-WFSSLSSVR		57.9853	6.9485	6.1588	0.9747	0.00 (1)
SAPSO-WFSSRBFNN		60.4631	9.4132	6.4273	0.9736	0.09 (3)
GA-WFSSLSSVR		60.9791	8.5905	6.4758	0.9727	0.11 (4)
PSO-WFSSLSSVR		61.2630	8.2685	6.5285	0.9714	0.13 (6)
SOS-WFSSLSSVR		61.7609	11.1746	6.5335	0.9712	0.16 (7)
TLBO-WFSSLSSVR		59.4422	7.6585	6.3354	0.9731	0.06 (2)

GA-DLNN refers to the Genetic Algorithm–Deep Learning Neural Network; ANN stands for Artificial Neural Network; RF represents Random Forest; SA-GP denotes the Simulated Annealing–Genetic Programming (tree-based) model; and IMNNIM stands for Intelligent Multivariate Neural Network Inference Model. Bold values indicate the performance measures of the best model.

Table 6. Assessment of predictive accuracy in dataset 3 for shear capacity of reinforced concrete walls

Model/System	Author	MAE (kN)	RMSE (kN)	MAPE (%)	R	SI (Rank)
ACI 318 − 19 provision	Chou et al. (2022b)	233.755	373.725	34.736	0.9236	1.00 (13)
XGBoost	Feng et al. (2021)	92.3	48.79	15.89	0.9889	0.15 (10)
JS-XGBoost	Chou et al. (2022b)	59.50	94.36	15.16	0.9899	0.13 (8)
SSLSSVR	This study	78.2728	21.6719	21.0151	0.9561	0.26 (11)
SSRBFNN		114.2872	58.3278	19.9402	0.8885	0.51 (12)
SAPSO-SSLSSVR		65.7342	11.6297	17.2580	0.9819	0.12 (7)
SAPSO-SSRBFNN		64.4603	19.0034	14.3865	0.9677	0.13 (9)
SAPSO-WFSSLSSVR		50.2267	13.0068	9.7784	0.9865	0.00 (1)
SAPSO-WFSSRBFNN		51.8114	15.3866	11.7546	0.9884	0.02 (3)
GA-WFSSLSSVR		51.4512	13.8745	10.9742	0.9861	0.02 (2)
PSO-WFSSLSSVR		52.4363	16.7073	11.4372	0.9791	0.04 (6)
SOS-WFSSLSSVR		54.5907	10.9995	11.8958	0.9856	0.03 (4)
TLBO-WFSSLSSVR		57.4272	9.9534	12.2502	0.9835	0.04 (5)

ACI stands for the American Concrete Institute; XGBoost refers to Extreme Gradient Boosting; and JS-XGBoost represents the Jellyfish Search–XGBoost model. Bold values indicate the performance measures of the best model.

Table 7. Assessment of predictive accuracy in dataset 4 for long-term Deflection of reinforced concrete beams

Model/System	Author	MAE (mm)	RMSE (mm)	MAPE (%)	R	SI (Rank)
ACI 318 − 83 Building Code	Araújo (2005), Pham et al. (2020a, b, c)	8.368	13.629	31.949	0.931	1.00 (13)
WFR-FBI-LSSVR	Nguyen et al. (2023)	4.09	7.86	15.21	0.9529	0.26 (10)
Bagging ensemble LR	Pham et al. (2020a, b, c)	4.597	8.190	16.749	0.972	0.27 (11)
Stacking ensemble MLP+SMOreg+LR model		4.466	8.686	15.523	0.970	0.26 (9)
SSLSSVR	This study	6.5751	2.8739	30.3738	0.8713	0.77 (12)
SSRBFNN		28329.1494	28303.7763	341402.9358	0.3493	(*)
SAPSO-SSLSSVR		4.2962	1.7159	15.4082	0.9346	0.18 (8)
SAPSO-SSRBFNN		3.9049	1.4021	14.8540	0.9675	0.04 (4)
SAPSO-WFSSLSSVR		3.7243	1.4297	12.3837	0.9650	0.00 (1)
SAPSO-WFSSRBFNN		4.2633	1.5545	14.4315	0.9584	0.09 (6)
GA-WFSSLSSVR		3.9310	1.5414	11.7090	0.9647	0.01 (2)
PSO-WFSSLSSVR		4.2278	1.6373	12.9629	0.9499	0.09 (7)
SOS-WFSSLSSVR		3.7070	1.6189	12.8612	0.9549	0.04 (5)
TLBO-WFSSLSSVR		3.7818	1.3564	12.0568	0.9560	0.02 (3)

ACI stands for the American Concrete Institute; WFR refers to Wrapper-Based Feature Refinement; FBI denotes the Forensic-Based Investigation algorithm; LSSVR represents the Least Squares Support Vector Regression model; LR stands for Linear Regression; MLP refers to Multilayer Perceptron networks; and SMOreg denotes Support Vector Regression. Bold values indicate the performance measures of the best model.

Table 8. Assessment of predictive accuracy in dataset 5 for construction productivity

Model/System	Author	MAE (m2/labor.h)	RMSE (m2/labor.h)	MAPE (%)	R	SI (Rank)
SOM	Oral and Oral (2010)	–	–	25.05	–	(*)
SOS-LSSVM-FS	Cheng et al. (2021)	0.0563	0.0721	3.67	0.979	0.77 (10)
FAJS-SS	Truong and Chou (2022)	0.0450	0.009	2.79	0.984	0.15 (4)
SSLSSVR	This study	0.0723	0.0162	4.8351	0.9629	0.99 (11)
SSRBFNN		0.0726	0.0141	4.8932	0.9620	1.00 (12)
SAPSO-SSLSSVR		0.0520	0.0111	3.3135	0.9794	0.36 (9)
SAPSO-SSRBFNN		0.0504	0.0127	3.2552	0.9809	0.32 (8)
SAPSO-WFSSLSSVR		0.0380	0.0101	2.3865	0.9871	0.00 (1)
SAPSO-WFSSRBFNN		0.0480	0.0178	3.0347	0.9823	0.28 (7)
GA-WFSSLSSVR		0.0440	0.0178	2.7868	0.9783	0.26 (6)
PSO-WFSSLSSVR		0.0447	0.0177	2.7948	0.9858	0.17 (5)
SOS-WFSSLSSVR		0.0442	0.0114	2.8176	0.9853	0.14 (3)
TLBO-WFSSLSSVR		0.0402	0.0102	2.5696	0.9868	0.05 (2)

SOM stands for Self-Organizing Maps; SOS refers to Symbiotic Organisms Search; LSSVM denotes the Least Squares Support Vector Machine; FS represents Dynamic Feature Selection; and FAJS-SS denotes the Fuzzy Adaptive Jellyfish Search-Optimized Stacking System. Bold values indicate the performance measures of the best model.

[See PDF for image]

Fig. 21

Mean absolute percentage errors of the compared prediction methods

The superior accuracy of the proposed model stems from three main factors. First, the model uses a feature-weighting mechanism that adjusts the importance of each input variable by assigning optimal weights, which improves the model’s ability to identify relevant patterns in the data.

Second, using a stacking ensemble framework—a technique consistently shown in the literature to outperform individual models—greatly enhances predictive robustness and generalization performance.

Third, as shown in Tables 4, 5, 6, 7 and 8, the application of the SAPSO algorithm plays a crucial role by effectively tuning the hyperparameters of the WFSS model. This optimization step ensures that the model performs at its best, ultimately boosting its exceptional predictive accuracy across various engineering datasets.

Concluding remarks and future research directions

This study presents the Scientific Approach to Problem Solving-inspired Optimization (SAPSO) algorithm—a new metaheuristic framework that mimics the iterative and organized steps of scientific research. The SAPSO source code is included in Appendix D (Supplementary Information). What sets SAPSO apart from existing methods is its explicit incorporation of the cyclical research stages—namely, problem review, hypothesis development, data gathering, and result analysis—within a flexible activity-switching system. This design allows SAPSO to effectively balance exploration and exploitation, improving its performance in complex, high-dimensional optimization problems.

SAPSO’s capabilities were thoroughly validated through empirical benchmarking on 54 large-scale optimization problems from the CEC 2020 and CEC 2022 competitions. These benchmark functions cover a broad range of optimization challenges, including unimodal, multimodal, separable, non-separable, expanded, and hybrid composite functions. Comparative evaluations against 11 state-of-the-art metaheuristic algorithms—including ABC, CA, DE, GA, GTO, GWO, PSO, ROA, SOS, TLBO, and WOA—consistently showed SAPSO’s superior performance. This was statistically confirmed using the Wilcoxon rank-sum test, which verified SAPSO’s significant outperformance in most test cases, highlighting its robustness, adaptability, and scalability.

Further expanding its usefulness, the study introduced a stacked ensemble learning framework—the SAPSO-weighted feature stacking system (SAPSO-WFSS)—with two configurations: SAPSO-WFSS_RBFNN and SAPSO-WFSS_LSSVR. In this method, SAPSO is employed to optimize both feature weights and hyperparameters, boosting the ensemble’s predictive capacity. When tested on five real-world civil engineering case studies, the SAPSO-WFSS_LSSVR model achieved MAPE values of 4.8556%, 6.1588%, 9.7784%, 12.3837%, and 2.3865%, respectively. These results confirm SAPSO-WFSS’s ability to deliver highly accurate forecasts, especially in complex and data-heavy engineering environments.

In summary, SAPSO signifies a notable advancement in metaheuristic algorithm development, driven by its unique blend of scientific reasoning with computational logic. Its versatility, strong convergence properties, and adaptability across both optimization and predictive modeling tasks establish SAPSO as a powerful tool in the evolving fields of computational optimization and machine learning.

Despite its proven effectiveness, there are several promising paths to improve SAPSO further.

• Adaptive Parameter Control: The current version uses fixed control parameters. Future versions could include adaptive or self-tuning mechanisms that adjust parameters dynamically based on problem complexity, thereby enhancing generalization to non-stationary or heterogeneous search spaces.

• Hybrid Integration with AI Frameworks: Embedding SAPSO within neural networks, deep learning architectures, or reinforcement learning models could enhance its learning abilities, allowing for more intelligent and context-aware decision-making during optimization.

• Cross-Algorithm Hybridization: Integrating SAPSO with other nature-inspired algorithms (e.g., DE, PSO, GWO) can improve its exploration–exploitation balance and lead to better performance across various problem domains.

• Broadening Benchmarking and Real-World Applications: Conducting systematic evaluations against new optimization algorithms across a broader range of industrial and engineering applications would better highlight SAPSO’s relative strengths and reveal possible limitations in specific use cases.

• Computational Acceleration: Due to the high computational cost of large-scale problems, implementing SAPSO on GPU-enabled platforms or utilizing parallel/distributed computing can significantly decrease execution times and improve scalability.

• Theoretical Foundations: A more in-depth theoretical analysis of SAPSO’s convergence properties, particularly regarding its activity-switching mechanism and search dynamics, would provide stronger guarantees and help guide the design of future hybrid algorithms.

This research presents SAPSO as a strong, adaptable, and innovative metaheuristic framework that connects scientific inquiry with algorithmic optimization. Its successful integration with ensemble machine learning systems shows excellent potential for addressing upcoming challenges in civil engineering informatics, as well as other data-heavy fields like environmental modeling, biomedical systems, and industrial design.

Acknowledgements

The authors would like to thank the National Science and Technology Council (grant nos. 113-2811-E-011-017-MY3 and 110-2221-E-011-080-MY3, Taiwan) for their financial support of this research.

Author contributions

JSC and DNT wrote the main manuscript text and prepared all figures and tables. Both authors reviewed the manuscript.

Data availability

All data generated or analyzed during this study are included in this published article (and its Supplementary Information files).

Declarations

Conflict of interest

We declare no known conflicts of interest associated with this publication and confirm that no significant financial support for this work has influenced its outcome.

Replication of results

The datasets, codes, and replication of results that are generated and analyzed in this study are available from the corresponding author upon reasonable request.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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