Portia spider algorithm: an evolutionary computation approach for engineering application

Meta-heuristic algorithms, when compared to traditional optimization methods, distinguish themselves by their simplicity in terms of both comprehension and implementation. Traditional optimization techniques, such as integer programming (Williams 2009), linear programming (Dantzig 2002), mixed programming (Hooker and Osorio 1999), and various constrained optimization methods (Gautier and Granot 1994), are typically designed for well-structured problems. Such problems possess clear structural information, established parameters, and a uniquely identifiable global optimum. These traditional methods are amenable to rigorous evaluation through computational complexity and convergence theories. They are particularly adept at addressing problems characterized by a single extremum. However, their performance can be less than satisfactory in scenarios dominated by multi-extremum challenges.

Conversely, meta-heuristic algorithms, through their astute design, manage to strike a nuanced balance between evading local optima and ensuring convergence to a viable solution. This design feature significantly augments the likelihood of pinpointing the global optimum in an array of optimization contexts. One of the salient characteristics of meta-heuristics is their relative indifference to initial conditions, which reinforces the consistency of optimization outcomes. Their inherent robustness, coupled with domain independence, renders them indispensable in a myriad of practical applications.

Given the pivotal role of optimization across diverse fields, there has been an upswing in the emphasis on metaheuristic techniques (Spall 2005). Various domains have leveraged these methods (Boussaïd et al. 2013; Parejo et al. 2012; Zhou et al. 2011) that operate by generating and refining random solutions iteratively [9]. In stark contrast to traditional methods that necessitate well-defined mathematical models, metaheuristics optimize by varying inputs and scrutinizing the corresponding outputs to maximize or minimize objective functions. This attribute ensures they are less susceptible to ensnarement in local optima compared to their traditional counterparts. Another distinguishing feature of metaheuristics is their adaptability. In a broader context, metaheuristics are categorized according to the extent of random choices they engender during each optimization iteration. Additionally, they can be classed by their foundational inspiration, which could be grounded in swarm intelligence, evolutionary principles, physics, mathematics, or human concepts (Fig. 1).

Evolutionary algorithms, a subset of meta-heuristic techniques, emulate the evolutionary mechanisms observed in nature. Among these, the genetic algorithm (GA) stands out as one of the most renowned. Introduced by Holland (1992), the GA has since garnered extensive attention across various domains, serving as a go-to method for optimization and search problems. Similarly inspired by natural processes, Simon (2008) unveiled the biogeography-based optimization (BBO). This approach taps into the mathematical principles underlying the habitat distribution of life forms. Simon drew parallels between BBO and the evolution of GA and artificial neural networks, particularly emphasizing its prowess in addressing optimization challenges. Storn and Price (1997) contributed to the field in the form of an innovative approach - differential evolution (DE) that promises efficient minimization of complex, potentially nonlinear, and non-differentiable continuous space functions. Their method stands out not only for its rapid convergence and superior certainty over other prevalent global optimization techniques but also for its inherent simplicity and adaptability to parallel computing. Adding to the tapestry of evolutionary algorithms, Jaderyan and Khotanlou (2016) introduced the virulence optimization algorithm (VOA). Inspired by the intricate infection mechanisms employed by viruses, this method involves identifying the fittest viruses, cloning them to amplify infection rates, and judiciously evading already infected zones. When tested against 11 benchmark functions, the VOA showcased its potential as a formidable resource for handling complex optimization conundrums.

Numerous techniques have drawn inspiration from mathematical concepts and physical phenomena, as evident in recent advancements in optimization methods. Rezaei et al. (2023) presented the geometric mean optimizer (GMO), a parameter-free meta-heuristic method grounded in the geometric mean operator. Their research indicated that the GMO excels at tackling a diverse array of optimization challenges, surpassing other contemporary meta-heuristic algorithms in multiple benchmark tests. Similarly, Abualigah et al. (2021) introduced the arithmetic optimization algorithm (AOA). This innovative meta-heuristic method uses mathematical arithmetic operators to optimize across varying search spaces. Through comprehensive evaluation on test functions and applied engineering design tasks, AOA demonstrated superior performance, convergence behavior, and computational complexity compared to eleven other renowned optimization algorithms. Chickermane and Gea (1996) proposed the generalized convex approximation (GCA), a cutting-edge method for structural optimization. By leveraging design sensitivity data, they constructed a series of convex subproblems, which in turn showcased enhanced convergence rates when assessed against standard test problems.

Shifting focus to algorithms inspired by physical phenomena, Mirjalili et al. (2016) introduced the multi-verse optimizer (MVO). This nature-inspired algorithm derives its concepts from cosmological phenomena such as white holes, black holes, and wormholes. Through mathematical modeling of these cosmic events, MVO effectively employs exploration, exploitation, and local search techniques. Kaveh and Mahdavi (2014) have put forward the colliding bodies optimization (CBO) meta-heuristic algorithm, a method underpinned by the principles of one-dimensional collisions. When applied to truss designs with discrete dimension variables, this approach – devoid of the need for parameter tuning and offering a lucid formulation—demonstrated its efficacy in structural optimization. Another intriguing method is the gravitational search algorithm (GSA) introduced by Rashedi et al. (2009). This strategy, inspired by gravity and mass interactions, adopts masses as searcher agents. The research underscores its formidable performance in solving diverse nonlinear functions, especially when pitted against well-established heuristic search methods.

A Kaveh and Siamak Talatahari (2010) proposed the charged system search (CSS) algorithm, which employs principles from electrostatics and Newtonian mechanics. This algorithm demonstrates superior performance in optimization tasks amidst discontinuous or non-convex territories, circumventing the need for gradient information or the smoothness of the search area. Abedinpourshotorban et al. (2016) presented the electromagnetic field optimization (EFO). This physics-inspired optimization technique integrates the golden ratio to dictate attraction-repulsion forces among electromagnetic particles within a tri-field structure. Remarkably, its superiority concerning precision and speed of convergence was evident on the 30 high-dimensional CEC 2014 benchmark problems, outperforming other leading optimization algorithms. Kashan (2015) introduced optics-inspired optimization (OIO) which models numerical optimization processes on the reflective behaviors of concave and convex mirrors, yielding a competitive and parameter-efficient method for solving complex optimization problems.

Human concepts have increasingly become a source of inspiration for metaheuristic techniques, leading to the development of novel optimization methods that mimic various human activities and thought processes. Reynolds (1994) presented a computational model for cultural evolution, which employs dual inheritance within a population of individuals characterized by specific behavioral traits and a collective “belief space”. This model is accelerated by GA and version spaces, which serve to enhance learning rates within the structure of the cultural algorithm. Rao et al. (2011) introduced the teaching–learning-based optimization (TLBO). This method, which is inspired by the pedagogical effect of a mentor on learners, proves effective for mechanical design problems. In a series of test cases and real-world applications, TLBO showcased performance superiority over other population-based optimization algorithms. Gandomi (2014) unveiled the interior search algorithm (ISA), a unique optimization technique that takes inspiration from the art and craft of interior design. Noteworthy is its simplicity, requiring tuning of just one parameter. The ISA stands out for its capability to solve optimization challenges, surpassing established algorithms in benchmark tests.

In a blend of music, Lee and Geem (2005) presented the harmony search (HS) method. Drawing inspiration from the musical process of seeking harmony, this method sets itself apart by not relying on gradient information. The research evidences the robustness and effectiveness of HS in addressing diverse engineering optimization problems, often delivering solutions that may surpass those derived from traditional algorithms. Sadollah et al. (2013) have proposed the mine blast algorithm (MBA), an innovative population-based optimization method influenced by the explosive dynamics of mine blasts. The results obtained from their research underscore the algorithm’s efficiency and its exceptional performance in addressing both constrained optimization and engineering design tasks, especially when juxtaposed against well-established optimization methods. Kashan (2014) introduced the league championship algorithm (LCA). This method simulates a sporting championship environment, where artificial teams vie for dominance in a league. Teams adapt their strategies or solutions based on game results, while also modeling player transfers at season’s end. Empirical analysis conducted on a variety of benchmark functions showcases its promising performance, hinting at its potential suitability for practical applications in the future.

In recent years, metaheuristic techniques that draw inspiration from animal behavior have garnered significant attention from researchers, as highlighted in Table 1. Kennedy and Eberhart (1995) presented the concept of optimizing nonlinear functions through the particle swarm optimization (PSO) technique. Their research delved into the evolution of different paradigms within this approach, executed one such paradigm, and subjected it to benchmark tests. They further identified potential applications, notably in nonlinear function optimization and neural network training. Gandomi et al. (2013) presented the cuckoo search (CS) metaheuristic optimization algorithm, which is enriched by Lévy flights. Through comprehensive testing on benchmark nonlinear constrained optimization tasks and structural engineering design challenges, CS demonstrated a distinct edge over other cutting-edge techniques. The authors also emphasized its unique search features, positioning it as a valuable subject for future research endeavors. Seyyedabbasi and Kiani (2023) introduced the sand cat swarm optimization (SCSO) algorithm. Drawing inspiration from the survival strategies of sand cats, SCSO proves adept at solving optimization challenges by skillfully balancing exploration and exploitation phases. The algorithm’s exceptional performance, evident in benchmark tests and intricate engineering design problems, places it a notch above other metaheuristic competitors.

Chen et al. (2022) unveiled the egret swarm optimization algorithm (ESOA), an inventive meta-heuristic technique influenced by the hunting behaviors of two egret species. Across an array of benchmark functions and engineering challenges, ESOA showcased its marked effectiveness and robustness, often overshadowing other optimization algorithms. Abualigah et al. (2021) brought to light the Aquila optimizer (AO), a metaheuristic method that mirrors the predatory strategies of Aquila birds. Their work delineates four distinct methods that reflect the bird’s hunting strategies. The AO algorithm underwent rigorous validation, with tests on renowned functions, the intricate CEC2017 and CEC2019 functions, and real-world engineering scenarios. Mirjalili et al. (2017) introduced the salp swarm algorithm (SSA) and its multi-objective counterpart (MSSA), inspired by salps’ swarming patterns, and validated their effectiveness on mathematical and applied engineering design tasks, showcasing their ability to efficiently converge to optimal solutions and approximate Pareto fronts. Bansal et al. (2014) introduced the spider monkey optimization (SMO) method, an innovative approach to numerical optimization inspired by the adaptable hunting behaviors of spider monkeys, animals known for their fission-fusion social dynamics.

The behavior of spiders in addressing optimization challenges has also inspired the development of metaheuristics and is applied in various fields. Cuevas et al. (2013) introduced the social spider optimization (SSO) technique, inspired by the cooperative interactions of social spiders, featuring distinct evolutionary operators for male and female agents, and exhibits high proficiency in locating global optima on various benchmark functions. Ouadfel and Taleb-Ahmed (2016) explored the effectiveness of the flower pollination (FP) and SSO method in image segmentation through multilevel thresholding, demonstrating that while both outperform the PSO and BAT algorithms, SSO maintains superior stability and efficiency across various threshold numbers, making it a compelling choice for complex image thresholding tasks. Ewees et al. (2017) presented an enhanced adaptive neuro-fuzzy inference system (ANFIS) optimized using the SSO algorithm for forecasting biochar yield from manure pyrolysis, showcasing superior performance over classical ANFIS and other methodologies such as artificial bee colony, PSO, and least square-support vector machine. Nguyen and Vo (2020) introduced an improved SSO technique that enhances the solution generation process for optimal reactive power dispatch problems, demonstrating reduced computational steps and parameters, faster simulation times, and consistently superior solution quality when benchmarked against standard SSO and other leading methods.

In an independent study, James and Li (2015) unveiled a novel social spider algorithm (SOSA) for global optimization. This algorithm uniquely harnesses web vibrations, which social spiders use to locate prey, deviating from conventional swarm intelligence strategies and showing superior results in benchmark assessments. SOSA has been applied in a variety of fields. For instance, Elsayed et al. (2016) presented a modified version of SOSA to address the non-convex economic load dispatch task, which includes real-world restrictions like valve point impacts and prohibited operating areas. The enhanced algorithm demonstrates improved performance across four benchmark systems, surpassing the original SOSA and displaying a competitive advantage in the existing literature. El-Bages and Elsayed (2017) presented an SOSA model for static transmission expansion planning that provides cost-effective solutions, validated on benchmark systems with varying degrees of complexity. Baş and Ülker (2020) introduced a binary version of SOSA, augmented with similarity measures and logic gates, and their results indicate superior performance in solving unimodal, multimodal, and uncapacitated facility location problems when compared with traditional algorithms.

The evolution and enhancement of existing algorithms have become a focal point for researchers in recent years. The thrust to augment existing algorithms for emerging challenges has driven significant academic interest. Kaveh and Talatahari (2010a) brought forth an advanced version of the ant colony optimization (ACO), termed the improved ACO. Crafted for constrained engineering design challenges, this approach deftly manages both continuous and discrete problems. Central to its design is the sub-optimization mechanism (SOM), inspired by finite element principles, which efficiently trims down pheromone matrices, decision vectors, function evaluations, and optimization durations without jeopardizing the likelihood of pinpointing optimal solutions. Chakraborty et al. (2009) embarked on refining the harmony search (HS) algorithm, which finds its muse in musical improvisation. Their method hybridizes HS with the differential evolution (DE) algorithm, aiming to mitigate slow and premature convergence challenges on intricate fitness terrains. Performance metrics of this hybrid were juxtaposed against classical HS, global best HS, and a renowned DE variant. Assessments were made across diverse benchmark functions and practical optimization challenges, based on criteria like precision, computational pace, and consistency in reaching optima.

Son and Nguyen Dang (2023b) showcased a composite model named the hybrid multi-verse optimizer (hDMVO), which integrates MVO and the sine cosine algorithm (SCA). This model is adept at managing discrete time-cost trade-off (TCTO) quandaries in construction project orchestration. Its prowess shines through benchmark evaluations and its knack for devising superior solutions in large-scale TCTO scenarios for intricate projects. Abd Elaziz et al. (2017) unveiled an augmented SCA, enriched with opposition-based learning (OBL). This enhancement broadens the exploration horizon of the search domain, leading to heightened precision and overall performance. Its efficacy is evident from benchmark evaluations and complex engineering tasks, underscoring the value of this synergistic approach. Pham, Trang, et al. (2023) articulated a proficient scheduling optimization technique for ready-mix concrete (RMC) truck dispatches. At its core is a novel hybrid swarm intelligence algorithm fusing the grey wolf optimizer (GWO) with the dragonfly algorithm (DA). The resultant algorithm excels in performance compared to its standalone counterparts and heralds a leap in multi-independent batch plant cooperation for refined RMC deliveries in construction sectors.

In recent years, optimization techniques have found widespread applications across diverse fields. Pham, Nguyen Dang, et al. (2023) introduced an enhanced SCA that integrates roulette wheel selection with OBL, demonstrating superior performance over traditional optimization algorithms in various engineering optimization contexts. Son and Nguyen Dang (2023a) presented the MVO model as an effective tool for addressing time–cost optimization issues in construction project management, surpassing other techniques in small-scale applications. Kumar et al. (2023) unveiled the multi-objective MVO, a dual-archive metaheuristic tailored for complex structural optimization, and illustrated its superior performance in real-world applications over leading algorithms such as MOEA/D and NSGA-II through comprehensive evaluations employing established performance metrics. Aye et al. (2023) introduced an innovative surrogate-aided optimization approach for enhancing airfoil shapes, incorporating both CFD and XFoil simulations. This method demonstrated superior performance in comparison to conventional surrogate-assisted techniques.

Nonut et al. (2022) advocated for the application of metaheuristics in the system identification of fixed-wing UAVs by optimizing aerodynamic and stability derivatives to minimize R-squared errors, with the L-SHADE algorithm proving to be the most effective method through extensive statistical analysis. Singh et al. (2022) unveiled an improved follow-the-leader (iFTL) technique, inspired by the foraging behaviors of sheep. Comprehensive testing within the Comparing Continuous Optimisers framework, alongside a suite of benchmark functions and truss design problems, revealed that iFTL outperforms fourteen well-established optimization algorithms in terms of performance and stability. In another study, Kumar et al. (2022) enhanced the efficacy of discrete meta-heuristics for truss design by incorporating both a random mutation search phase and selection based on simulated annealing into five pre-existing algorithms. The resultant modifications led to improved search diversification and intensification, yielding superior optimization results in complex structural engineering challenges. Kumar, Tejani, Pholdee, Bureerat, et al. (2022) introduced a groundbreaking multi-objective TLBO method. This novel approach incorporates non-dominated sorting and an external archive, adeptly mitigating early convergence and preventing local optima entrapment in multi-objective contexts.

Recent trends in algorithmic research spotlight both the inception of new algorithms and the refinement of pre-existing ones. The burgeoning interest in this area owes much to the No Free Lunch (NFL) theorem (Wolpert and Macready 1997), which posits that no single optimization technique can universally address all optimization challenges.

In light of the NFL theorem, this study introduces the Portia spider algorithm (PSA), inspired by the predatory strategies of the Portia spider. The PSA’s efficacy was rigorously assessed using a suite of benchmarks: 23 classical test cases, 29 CEC2017 benchmark test functions, and five practical engineering optimization tasks. The algorithm’s performance was then benchmarked against a collection of established meta-heuristic algorithms.

The primary attributes and contributions of the PSA include:

The subsequent sections of this study are structured as follows: Sect. 2 introduces a comprehensive framework for the PSA. In Sects. 3, 23 classical and 29 CEC2017 test functions were employed to probe the convergence properties of the PSA. Section 4 evaluates the real-world applicability of the PSA through five engineering optimization scenarios. Finally, Sect. 5 summarizes the findings, spotlighting the study’s novel contributions, and signposting avenues for prospective research in this domain.

In the context of optimization, especially when considering evolutionary algorithms and metaheuristics, demonstrating algorithmic efficiency relies on the application of predefined test cases. Given the intrinsically stochastic attributes of these methodologies, the attainment of optimal outcomes mandates the deployment of a meticulously curated test function suite. It remains quintessential to ascertain that any discerned enhancements are not merely manifestations of stochastic variance. Presently, the field is devoid of a universally endorsed benchmarking paradigm, thus driving scholars to delve into an extensive assortment of test instances. Considering this, the present investigation encompasses an eclectic compilation of test functions, integrating 23 classical test cases and the CEC2017 test functions, with diverse traits, to facilitate an exhaustive appraisal of algorithmic prowess.

The objective of this section is to thoroughly evaluate the effectiveness of the PSA by applying it to five practical engineering optimization challenges, each distinguished by multiple inequality constraints. The overarching goal is to ascertain the algorithm’s adeptness in handling these constraints throughout the entire optimization process.

Portia spiders are known for their ability to solve novel problems. For instance, when faced with an obstacle that hinders a direct path to their prey, these spiders can choose a detour, even if it leads to a temporary loss of sight of the target. Furthermore, they demonstrate a capacity for learning from past experiences and adjusting tactics accordingly.

This article introduces a nature-inspired model based on the hunting behaviours of the Portia spider, termed the Portia spider algorithm (PSA). Within this framework, the stalking and striking behaviours of the Portia spider inform the exploration phase, while the tactics of invading and imitating guide the exploitation phase.

Our extensive evaluations, which range from 23 classical test functions to 29 CEC2017 benchmarks and five engineering optimization problems, highlight the robustness and adaptability of PSA. Significantly, when compared to other meta-heuristic algorithms, PSA exhibits potential advantages, underscoring its potential in computational optimization. PSA is well-equipped to adapt and traverse complex optimization landscapes, marking its distinctive position in the field of evolutionary algorithms. Subsequent studies could explore further refinements to PSA, leveraging its inherent strengths and customizing it to address specific domain challenges.

Although PSA offers innovative approaches to optimization, it may encounter limitations such as slow convergence rates, particularly when applied to high-dimensional or intricate optimization challenges. To enhance its performance, integrating techniques like Lévy flights, mutation, and additional evolutionary operators could be beneficial. These methods can introduce new diversity into the population of solutions and allow the algorithm to escape local optima more effectively.

Moreover, hybridizing the PSA with other stochastic optimization algorithms could further improve its efficacy. By combining the strengths of PSA with other proven strategies, it is possible to create a more robust algorithm that leverages the advantages of each component method. This could lead to improved convergence speeds, better solution quality, and increased reliability across a wider range of optimization problems.

Such enhancements would be particularly useful in complex optimization landscapes, where the balance between exploration (diversification) and exploitation (intensification) is crucial for finding global optima. The adaptive capabilities of hybrid algorithms can be particularly adept at navigating these challenges, offering a powerful tool for complex optimization tasks.