Advancing Engineering Solutions with Protozoa-Based Differential Evolution: A Hybrid Optimization Approach

Abstract

This paper presents a novel Hybrid Artificial Protozoa Optimizer with Differential Evolution (HPDE), combining the biologically inspired principles of the Artificial Protozoa Optimizer (APO) with the powerful optimization strategies of Differential Evolution (DE) to address complex and engineering design challenges. The HPDE algorithm is designed to balance exploration and exploitation features, utilizing innovative features such as autotrophic and heterotrophic foraging behaviors, dormancy, and reproduction processes alongside the DE strategy. The performance of HPDE was evaluated on the CEC2014 benchmark functions, and it was compared against two sets of state-of-the-art optimizers comprising 23 different algorithms. The results demonstrate HPDE’s good performance, outperforming competitors in 24 functions out of 30 from the first set and 23 functions from the second set. Additionally, HPDE has been successfully applied to a range of complex engineering design problems, including robot gripper optimization, welded beam design optimization, pressure vessel design optimization, spring design optimization, speed reducer design optimization, cantilever beam design optimization, and three-bar truss design optimization. The results consistently showcase HPDE’s good performance in solving these engineering problems when compared with the competing algorithms.

1. Introduction

Optimization is a fundamental task across all scientific and engineering disciplines, where it serves to determine the optimal parameter values within systems to achieve desired outcomes efficiently [1]. As technologies evolve and systems become increasingly complex, optimization challenges have intensified, requiring more sophisticated approaches [2,3]. Traditional methods often face limitations, including susceptibility to converging on local optima, difficulties in handling high-dimensional or unknown search spaces, and typically operating with single-solution approaches. These challenges highlight the need for developing advanced algorithms that can navigate complex landscapes more effectively [4].

Metaheuristic algorithms have emerged as powerful tools to address these complex optimization challenges [5]. They provide flexible, high-level strategies designed to explore large and intricate search spaces systematically and efficiently. Unlike traditional optimization techniques, metaheuristics do not rely on gradients or derivatives, which makes them applicable to a wider range of problems, including those that are non-differentiable, discontinuous, or stochastic. This capability is especially critical in real-world applications where the objective functions and constraints may not be precisely defined or may change over time [6].

The development of metaheuristic algorithms is often inspired by natural and social phenomena. Evolutionary Algorithms (EAs) [7] simulate the process of natural selection, where the fittest individuals are selected for reproduction in order to produce offspring of the next generation. Swarm Intelligence (SI) algorithms [8] are inspired by the social behavior of animals, such as birds flocking and fish schooling, and are particularly noted for their robustness and ability to converge rapidly to a good solution. Physics-based methods [9] use metaphors from physics, such as the laws of gravity and motion, to guide the search process, while human-inspired algorithms [10] often simulate human decision-making or social behavior.

These algorithms are particularly valued for their versatility and robustness, allowing them to perform well across a diverse array of problem types and environments. They are capable of balancing exploration (diversifying the search across the global landscape to avoid local optima) with exploitation (intensifying the search around promising areas) [11], which is crucial for achieving near-optimal solutions efficiently. Additionally, the scalability of metaheuristics makes them suitable for solving large-scale problems that are beyond the reach of conventional optimization methods [12].

Moreover, metaheuristics are often hybridized with other optimization techniques to form even more powerful algorithms [13]. For example, combining the global search capability of a metaheuristic with the local search capabilities of a more deterministic method can yield a hybrid algorithm that leverages the strengths of both approaches. This is particularly effective in refining solutions to very high accuracies, which are often required in engineering and industrial applications [14].

Metaheuristic algorithms, inspired by natural phenomena, operate through various mechanisms, yet they fundamentally rely on two core concepts: exploration (diversification) and exploitation (intensification). As stated by Eiben and Schippers, “exploration and exploitation are the two cornerstones of problem-solving by search” [15]. Exploration involves creating diverse solutions to thoroughly investigate the global search space, while exploitation focuses on refining the search around promising areas to find optimal solutions. These two components are inherently conflicting, where an emphasis on exploration can weaken exploitation and vice versa. Despite its importance, achieving the optimal balance between exploration and exploitation remains a challenging task, and no researcher has set a standard for this process [16]. Furthermore, the “No Free Lunch” (NFL) theorem asserts that no single heuristic can consistently outperform others across all problem domains [17]. This theory implies that the performance of any algorithm will average out when applied to different problems, suggesting that an algorithm effective for certain problems may not perform as well for others [18]. This understanding has driven the continuous development and proposal of new algorithms, each seeking to better balance exploration and exploitation for improved optimization performance across varied problem sets [18].

The main research contributions of this paper are as follows:

• A new hybrid optimization algorithm named Hybrid Protozoa Differential Evolution (HPDE) is designed, combining the strengths of the Artificial Protozoa Optimizer (APO) and Differential Evolution (DE) to enhance the optimization process.
• The HPDE algorithm’s mathematical models are based on the survival behaviors of protozoa and the evolutionary strategies of DE. Specifically, APO’s mechanisms of foraging, dormancy, and reproduction are integrated with DE’s mutation and crossover strategies. Autotrophic foraging and dormancy contribute to exploration, while heterotrophic foraging, reproduction, and DE’s evolutionary strategies enhance exploitation.
• The HPDE is implemented and evaluated using unimodal, multimodal, hybrid, and composition functions from the 2022 IEEE Congress on Evolutionary Computation benchmark. Experimental results verify that HPDE outperforms 23 state-of-the-art algorithms.
• The effectiveness of HPDE is further demonstrated by addressing challenging real-world problems, including Robot Gripper and six engineering design problems
• HPDE consistently outperformed leading algorithms across a variety of optimization challenges.
• Key results include: First, we achieved the lowest error rates across the majority of CEC2014 benchmark functions, with significant advantages in multimodal and high-dimensional problems. Second, we demonstrated good robustness and reliability, as evidenced by low standard deviations and standard errors in its results. Third, we achieved top rankings in engineering design problems with optimal or near-optimal solutions that outperformed competitors like APO, GWO, and WOA.

Motivation of This Work

Recent advancements in metaheuristic optimization have introduced several adaptive algorithms, each with unique mechanisms for enhancing search performance. Among these, the Artificial Protozoa Optimizer (APO) [19] has gained attention due to its biological inspiration drawn from protozoa survival behaviors, including foraging, dormancy, and reproduction. These behaviors enable APO to adapt its search process dynamically, allowing it to balance the exploration and exploitation phases more effectively. Such adaptive mechanisms are not entirely unique to APO; other algorithms also incorporate adaptive behaviors. However, APO’s distinct process of switching between foraging and dormancy phases provides a flexible strategy for avoiding premature convergence and local optima traps, which are common issues in optimization.

One primary motivation for selecting APO lies in its versatility and robustness when tackling complex optimization tasks. Although Differential Evolution (DE) algorithms have exhibited substantial success in a variety of applications, they often face limitations with highly nonlinear and multimodal functions, which can result in slow convergence and suboptimal solutions [20]. To address these shortcomings, we propose a novel hybrid algorithm that combines APO with DE, termed the Hybrid Protozoa Differential Evolution (HPDE). The integration of APO’s adaptive mechanisms with DE’s mutation and crossover strategies aims to enhance the optimizer’s global convergence, allowing it to balance exploration and exploitation more effectively while reducing stagnation in the search process.

Unlike many traditional algorithms, such as the Genetic Algorithm (GA) [21] and Particle Swarm Optimization (PSO) [22], which often suffer from premature convergence [23], APO inherently incorporates adaptive dormancy and foraging behaviors that reduce the problem of premature convergence. While bio-inspired algorithms like the Grey Wolf Optimizer (GWO) [24] and Whale Optimization Algorithm (WOA) [25] have achieved commendable balances between exploration and exploitation, they frequently require extensive parameter tuning, which can be computationally costly and dependent on specific problem characteristics [26]. Additionally, these algorithms often exhibit sensitivity to initial conditions and parameter settings, resulting in inconsistent performance across diverse problem domains [27].

The proposed HPDE algorithm leverages APO’s adaptability to dynamically alter its search behavior, addressing many of the limitations noted in traditional and bio-inspired algorithms. By integrating APO’s adaptive mechanism with DE’s robust search capabilities, HPDE demonstrates an improved convergence rate and enhanced search capability over a range of benchmark functions and engineering design problems.

Table 1. Comparative summary of optimization algorithms.

Acronym	Algorithm Name	Descreption	Disadvantages	Year
SSOA	Synergistic Swarm Optimization [28]	Combines multiple Swarm behaviors for enhanced optimization	Increased computational complexity	2024
FLO	Frilled Lizard Optimizer [29]	Mimics the foraging behavior of frilled lizards	May require parameter tuning	2024
CPO	Chinese Pangolin Optimizer [30]	Inspired by the digging and defense behaviors of Chinese pangolins	Potentially sensitive to parameters	2024
FVIM	Four Vector Optimizer [31]	Utilizes four vector operations for optimization	May get trapped in local optima	2024
SHIO	Success History Intelligent Optimizer [32]	Adapts search based on historical successes	May have complex parameter settings	2022
ZOA	Zebra Optimizer [33]	Simulates herd behavior and movement patterns of zebras	May converge slowly in some cases	2022
DOA	Dingo Optimizer [34]	Models cooperative hunting strategies of dingoes	May need careful parameter adjustment	2021
ROA	Remora Optimizer [35]	Inspired by symbiotic relationships of remoras	May have slow convergence in exploitation	2021
AO	Aquila Optimizer [36]	Mimics hunting strategies of eagles	May require parameter tuning	2021
CHIMP	Chimp Optimizer [37]	Simulates intelligence and hunting behavior of chimpanzees	May need careful parameter tuning	2020
STOA	Sooty Tern Optimizer [38]	Inspired by migration and hunting of sooty terns	May converge prematurely	2019
SOA	Seagull Optimizer [39]	Models soaring and attacking behavior of seagulls	May require improvement in convergence speed	2019
SCSO	Sand Cat Optimizer [40]	Simulates hunting strategies of sand cats	May need parameter adjustment	2023
MVO	Multi-Verse Optimizer [41]	Based on multiverse concepts in physics	May require improvement in convergence speed	2016
WOA	Whale Optimizer [25]	Mimics bubble-net feeding of humpback whales	May converge prematurely	2016
SCA	Sine Cosine [27]	Utilizes sine and cosine functions for search movements	May lack exploitation efficiency	2016
MFO	Moth-Flame Optimizer [42]	Simulates moth navigation using transverse orientation	May get trapped in local optima	2015
GWO	Grey Wolf Optimizer [24]	Mimics leadership and hunting in grey wolves	May need convergence speed improvement	2014
PSO	Particle Swarm Optimizer [22]	Models social behavior of bird flocking	Prone to premature convergence	1995
SA	Simulated Annealing [43]	Based on annealing process in metallurgy	Slow convergence, sensitive to cooling schedule	1983
GA	Genetic Algorithm [21]	Inspired by natural selection and genetics	May have slow convergence, premature convergence	1960

provides a comparative summary of optimization algorithms that, like the bio-inspired APO, draw on natural or biological processes to guide the search process, detailing each algorithm’s main contributions, advantages, and limitations.

Furthermore, while some hybrid algorithms attempt to integrate multiple optimization strategies to enhance performance, they often introduce increased computational complexity and may still struggle with maintaining population diversity, thereby risking convergence to suboptimal solutions. Another notable limitation is the inadequate handling of constraints in many algorithms, which restricts their applicability to real-world engineering problems that inherently involve complex, multidimensional constraints [3].

Addressing these gaps necessitates the development of an optimizer that not only leverages the exploratory capabilities of bio-inspired algorithms but also incorporates robust mechanisms for exploitation to refine solutions effectively. To address these challenges, we propose the Hybrid Artificial Protozoa Optimizer combined with Differential Evolution (HPDE). This hybrid approach synergistically integrates the Artificial Protozoa Optimizer (APO) [19] that has the diverse foraging and reproduction behaviors with the DE, renowned for its efficient mutation and crossover operations that bolster exploitation. By combining these strengths, HPDE aims to achieve a good balance between exploration and exploitation, reduce the likelihood of premature convergence, and maintain population diversity more effectively. Additionally, the hybrid framework is designed to handle both continuous and discrete optimization problems with constraints, thereby broadening its applicability to a wider range of real-world scenarios. Through this integration, HPDE aspires to overcome the limitations observed in existing algorithms, offering enhanced convergence speed, improved solution quality, and greater robustness across diverse optimization landscapes.

The primary advantage of our proposed work (HPDE) compared with other optimization algorithms lies in its enhanced ability to balance exploration and exploitation during the search process. By integrating the bio-inspired mechanisms of the APO with the robust mutation and crossover operations of DE, our method leverages the strengths of both algorithms to achieve good performance. Specifically, the HPDE demonstrates improved convergence speed and solution quality, as evidenced by its performance on the CEC2014 benchmark suite consisting of 30 functions. Comparative experiments show that the HPDE consistently outperforms the compared state-of-the-art algorithms, including ChOA, GWO, FOX, WOA, MVO, DOA, MFO, RIME, DBO, WSO, RTH, SHIO, COA, OHO, SCA, PSO, SHO, SDE, and RSA. The hybrid approach effectively combines the exploratory capabilities of APO with the exploitative efficiency of DE, resulting in an optimizer that is robust across diverse optimization problems, including continuous and discrete spaces with constraints.

2. Literature Review

In the diverse landscape of computational optimization, a plethora of innovative metaheuristic algorithms have been developed, each drawing inspiration from various natural behaviors and phenomena. These algorithms, designed to solve complex optimization problems, emulate the adaptive strategies of animals, the dynamic interactions of social insects, and the physical principles observed in the natural world.

Evolution and genetics-inspired optimizers include Evolution Strategies (ES) [44] and Genetic Programming (GP) [45]. These algorithms are based on the principles of biological evolution and genetic variation, respectively, simulating the process of natural selection and genetic operations to optimize complex systems [46]. Moreover, the bird-inspired optimizers are represented by the Falcon Optimization Algorithm (FOA) [47], Greylag Goose Optimization (GGO) [48], Northern Goshawk Optimization (NGO) [49], and Artificial Hummingbird Algorithm (CSA) [50]. Furthermore, the aquatic animal-inspired optimizers feature the Beluga Whale Optimization (BWO) [51], which is inspired by the social and hunting behaviors of beluga whales, and the Jellyfish Search (JS) [52], which mimics the passive drifting mechanism of jellyfish. Additionally, the Whale Optimization Algorithm (WOA) [25] models the bubble-net hunting strategy of humpback whales.

In addition, the plant-inspired optimizer includes the Invasive Weed Optimization algorithm (IWO) [53], which models the spreading and reproductive strategies of invasive weed species, adapting these strategies to solve optimization problems. On the other hand, the disease model-inspired optimizer is the Liver Cancer Algorithm (LCA) [54], which draws from the growth patterns and characteristics of liver cancer cells to develop robust search mechanisms in optimization landscapes.

Moreover, the mammal-inspired optimizers feature the Puma Optimizer (PO) [55], (BMO) [56], Grey Wolf Optimizer (GWO) [24], Adaptive Fox Optimization (AFO) [57], and Honey Badger Algorithm (HBA) [58]. Insect-inspired optimizers include Ant Colony Optimization (ACO) [59], which emulates the pheromone-laying and path-finding behavior of ants; Aphid–Ant Mutualism (AAM) [60], which models the mutualistic relationship between aphids and ants; and Artificial Bee Colony (ABC) [61], inspired by the foraging behavior of honeybees. Ant Lion Optimizer (ALO) [62] draws inspiration from the predatory behavior of antlions, which create traps to capture prey. Other physics-based optimizers include the Artificial Electric Field Algorithm (AEFA) [63], Black Hole Algorithm (BH) [64], Electromagnetic Field Optimization (EFO) [65], and Gravitational Search Algorithm (GSA) [66]. These algorithms utilize principles from physics, such as electric fields, black hole dynamics, electromagnetic principles, and gravitational forces, to guide the search process in optimization tasks.

Primate-inspired optimizers include the Artificial Gorilla Troops Optimizer (RIME) [67], which mimics the social structure and collaborative behavior of gorilla troops in their natural habitat. In addition, the activity- and sport-inspired optimizers feature the Alpine Skiing Optimization (ASO) [68], which draws inspiration from the strategic and dynamic movements involved in alpine skiing. Futhermore, the Swarm Intelligence optimizers include Particle Swarm Optimization (PSO) [69], which emulates the social behavior of birds and fish, adapting it to solve optimization problems effectively.

In addition, the natural process and physics-based optimizers feature the Snow Ablation Optimizer (SAO) [70], String Theory Algorithm (STA) [71], Water Cycle Algorithm (WCA) [72], Atom Search Optimization (ASO) [73], Chemical Reaction Optimization (CRO) [74], and Thermal Exchange Optimization (TEO) [75]. The fitness and distance-inspired optimizers include Distance-Fitness Learning (DFL) [76], which leverages the correlation between distance and fitness to inform optimization; materials and chemical structure-based optimizers include the Crystal Structure Algorithm (CryStAl) [77], Equilibrium Optimizer (EO) [78], Henry Gas Solubility Optimization (HGSO) [79], and Nuclear Reaction Optimization (NRO) [80].

The Farmland Fertility Algorithm (FFA) is inspired by the process of enhancing soil fertility in agriculture, modeling optimization as a balance between exploration and exploitation to improve solutions iteratively [81]. The African Vultures Optimization Algorithm (AVOA) mimics the collaborative hunting strategies of vultures, leveraging their unique soaring and keen sight to focus on promising regions in the search space [82]. The Mountain Gazelle Optimizer (MGO) is based on the fast and evasive movements of gazelles in mountainous terrain, using their speed and agility to avoid local optima while searching for better solutions [83]. The Artificial Gorilla Troops Optimizer (GTO) imitates the social intelligence and group behaviors of gorillas, balancing leadership and individual contributions to enhance convergence toward optimal solutions.

In addition, Differential Evolution (DE) algorithms have seen extensive development, especially in addressing complex engineering optimization problems. Researchers have thus developed various DE enhancements through hybrid methodologies, advanced constraint-handling mechanisms, and adaptive control strategies, some of which are listed below.

Cantú et al. [84] examined constraint-handling techniques for DE, particularly suited for process engineering problems characterized by non-convex and discontinuous constraints. This study demonstrated that the gradient-based repair technique was highly effective, underscoring the importance of appropriate constraint-handling mechanisms in optimizing DE’s performance in complex, constrained environments.

Nguyen et al. [85] introduced a Classification-assisted Differential Evolution (CaDE) approach, where an AdaBoost classifier discards infeasible solutions early in the evaluation process. By reducing unnecessary fitness evaluations, this machine learning integration enhances DE’s computational efficiency, making it particularly valuable for constrained engineering tasks that benefit from such strategic filtering.

Kizilay et al. [86] presented a Q-Learning-assisted DE variant (DE–QL), which dynamically adapts mutation strategies and crossover rates. By integrating Q-Learning as a guiding mechanism, DE–QL effectively balances exploration within feasible and infeasible regions, demonstrating its utility for constrained engineering problems that require nuanced constraint navigation.

Samal et al. [87] developed a Modified Differential Evolution (MDE) that dynamically adjusts the scaling factor and crossover ratio, improving DE’s performance on multimodal functions. Experimental results show that MDE outperforms well-established optimization algorithms, such as Particle Swarm Optimization (PSO) and Cuckoo Search, emphasizing the effectiveness of adaptive parameter tuning in navigating complex, multimodal landscapes.

Zhang et al. [88] proposed a hybrid Cuckoo Search and Differential Evolution algorithm (CSDE) to address premature convergence issues by dividing the population into subgroups, allowing Cuckoo Search and DE to operate independently while sharing information. This hybrid model exemplifies how combining complementary optimization techniques can improve convergence rates and solution accuracy, especially in engineering problems requiring a balance between global and local search.

Dragoi and Curteanu [89] provided an extensive review of DE applications in chemical engineering, highlighting DE’s versatility in handling varied constraints and objectives within model and process optimization. This review underscores DE’s potential across engineering fields and its adaptability to optimize complex processes.

In a similar context, Zuo and Gao [90] introduced a dual-population DE variant, ADPDE, where a dynamic population division based on individual potential enhances convergence speed and solution accuracy. By adopting distinct mutation strategies within subgroups, ADPDE achieves robust performance on constrained optimization tasks, demonstrating the utility of population management in DE.

Tang and Wang [91] extended DE’s capability by integrating it into a Whale Optimization Algorithm (WOAAD) based on an atom-like structure. WOAAD’s design incorporates DE-inspired modifications, effectively addressing local convergence issues, which proves advantageous for optimizing engineering design problems requiring precision and adaptive convergence.

Alshinwan et al. [92] proposed a hybrid approach, combining Prairie Dog Optimization with DE (PDO–DE), demonstrating its applicability in both engineering design and network intrusion detection. By enhancing PDO’s search mechanisms with DE’s mutation and crossover operators, PDO–DE strikes an effective balance between exploration and exploitation, underscoring DE’s adaptability across different optimization domains.

De Melo and Carosio [93] explored a Multi-View Differential Evolution (MVDE) approach, in which multiple mutation strategies are applied to generate different population views at each iteration. A winner-takes-all approach merges these views, balancing exploration and exploitation effectively. MVDE’s competitive performance on constrained engineering problems illustrates the benefits of combining varied mutation strategies within DE.

In advancing mutation schemes, Mohamed et al. [94] proposed Enhanced Directed Differential Evolution (EDDE), an algorithm that utilizes both high-quality and low-quality population members to balance exploration and exploitation. EDDE’s robustness and efficiency in constrained domains highlight the importance of mutation scheme customization in optimizing DE’s performance.

Zeng et al. [95] developed a Competitive Mechanism-Integrated Multi-Objective Whale Optimization Algorithm with Differential Evolution (CMWOA) for multi-objective optimization. By incorporating a competitive mechanism, CMWOA uses a refined crowding distance calculation for improved population density depiction and guides population updates more effectively. DE’s adaptive parameters further diversify the population, with testing on multiple benchmark functions demonstrating CMWOA’s ability to outperform other methods in convergence and accuracy. CMWOA’s application to real-world problems further verifies its practicality in diverse optimization settings, highlighting DE’s flexibility in hybrid configurations.

Despite the clear success of these strategies in tackling diverse optimization tasks, several deficiencies remain. Many existing approaches confront the problem of stagnation in local minima, often due to inadequate exploration mechanisms or static parameter settings. Furthermore, hybrid methods sometimes lack well-defined theoretical guidelines for combining distinct search strategies or for adjusting parameters adaptively, thereby limiting scalability and robustness in more complex or higher-dimensional tasks. Real-world engineering applications, in particular, can pose dynamic constraints that many algorithms are neither explicitly designed to handle nor systematically tested against. These gaps motivate the introduction of improved hybrid and bio-inspired metaheuristics that pursue a better exploration–exploitation trade-off, integrate adaptive parameter tuning, and scale effectively to large, constrained, or time-critical problems.

In this context, the present study proposes a new hybrid optimization method, termed Hybrid Protozoa Differential Evolution (HPDE), which combines the survival behaviors observed in protozoa with the evolutionary strengths of DE. Protozoa-inspired foraging, dormancy, and reproduction mechanisms (as modeled by the Artificial Protozoa Optimizer, APO) are merged with DE’s well-known mutation and crossover strategies. Autotrophic foraging and dormancy bolster the exploration capabilities of the algorithm, while heterotrophic foraging and reproduction interact synergistically with DE’s evolutionary operators to reinforce exploitation. This fusion is carefully formulated to overcome local minima entrapment and to maintain a flexible, adaptive framework for parameter adjustments in real-world scenarios.

2.1. Overview of Solving Highly Nonlinear and Complex Engineering Design Optimization Problems Using Metaheuristic Tools

Recent work has focused on improving optimization algorithms for complex engineering problems. Akl et al. proposed an improved Harris Hawks Optimization (IHHO), incorporating logarithmic and exponential strategies to avoid local optima [96]. Yu et al. introduced the Improved Adaptive Grey Wolf Optimization (IAGWO), enhancing convergence speed and accuracy with velocity and Inverse Multiquadratic Function adjustments [97]. Liu et al. developed the PRPCSO algorithm, combining Padé Approximation with intelligent population reduction [98]. Garg et al. proposed the LX-TLA, using the Laplacian operator to improve Teaching Learning Algorithm performance [99]. Gopi and Mohapatra introduced OLCA, utilizing opposition-based learning for global optimization [100]. Finally, Wang et al. presented LGJO, which leverages chaotic mapping and Gaussian mutation for industrial applications [101].

Furthermore, Moustafa et al. proposed the Subtraction-Average-Based Optimizer (SAOA), enhanced with cooperative learning, which outperformed GWO and PSO in power system applications [102]. El-Shorbagy and Elazeem’s Convex Combination Search Algorithm (CCS) effectively balanced exploration and exploitation in multimodal tests and engineering challenges [103]. Givi et al.’s Red Panda Optimization (RPO), inspired by red panda behaviors, surpassed several state-of-the-art algorithms in benchmarks and engineering problems [104]. Similarly, Gharehchopogh et al. introduced the Chaotic Quasi-Oppositional Farmland Fertility Algorithm (CQFFA), improving exploration and convergence with chaotic maps and learning mechanisms [105].

Pan et al.’s Gannet Optimization Algorithm (GOA) demonstrated success in large-scale constrained optimization [106], while Tang et al.’s hybrid PSODO improved global search and convergence speed [91]. Recent advancements in optimization algorithms have shown significant improvements in solving complex engineering problems. Hu et al. proposed the ACEPSO, which introduced adaptive population grouping and co-evolved mechanisms to enhance diversity and avoid local optima [107]. Ewees introduced a harmony-driven method, GBOHS, integrating Harmony Search (HS) with a gradient-based optimizer to improve convergence and accuracy in global optimization and feature selection [108]. Pan et al. improved the Gannet Optimization Algorithm (GOA) by incorporating parallel and compact strategies, enhancing memory efficiency, and avoiding local solutions in engineering tasks [109]. Che and He enhanced the Seagull Optimization Algorithm (SOA) by introducing mutualism and commensalism mechanisms, improving the algorithm’s exploitation capabilities in complex engineering challenges [110].

2.2. Overview of Optimization Problems

Optimization problems are fundamental in various fields of science and engineering, where the goal is to find the best solution among all feasible solutions. These problems arise in numerous applications such as machine learning, operations research, engineering design, finance, logistics, and many others. The primary objective is to optimize a performance criterion, which is represented mathematically by an objective function [111].

A general optimization problem can be formulated as shown in Equation (1):

$$\begin{array}{cccc}\hfill & \underset{\mathbf{x}}{\mathrm{minimize}}\hfill & \hfill & f\left(\mathbf{x}\right)\hfill \\ \hfill & \mathrm{subject}\mathrm{to}\hfill & \hfill & \mathbf{x}\in \mathcal{S},\hfill \end{array}$$

(1)

where $f\left(\mathbf{x}\right)$ is the objective function to be minimized, $\mathbf{x}=[x_1,x_2,\dots ,x_{\dim }]$ is the vector of decision variables, and $\mathcal{S}$ represents the feasible solution space defined by the constraints of the problem.

The feasible solution space $\mathcal{S}$ is typically defined by a set of constraints, which may include.

Boundary constraints: variables are bounded within lower and upper limits as shown in Equation (2):

$${\mathbf{x}}_{min}\le \mathbf{x}\le {\mathbf{x}}_{max},$$

(2)

where ${\mathbf{x}}_{min}$ and ${\mathbf{x}}_{max}$ are vectors containing the lower and upper bounds for each decision variable.

Equality constraints: functions that must be satisfied exactly as shown in Equation (3):

$$h_j\left(\mathbf{x}\right)=0,j=1,2,\dots ,n_e,$$

(3)

Here, $h_j\left(\mathbf{x}\right)$ are equality constraint functions.

Inequality constraints: equations that impose upper or lower limits as shown in Equation (4):

$$g_k\left(\mathbf{x}\right)\le 0,k=1,2,\dots ,n_i,$$

(4)

Here, $g_k\left(\mathbf{x}\right)$ are inequality constraint functions.

In many real-world applications, the optimization problem involves complex, nonlinear, and high-dimensional objective functions with multiple local minima or maxima. These characteristics make it challenging to find the global optimum using traditional optimization methods [112].

To address these challenges, metaheuristic algorithms have been developed. In this research, we focus on applying and analyzing the hybrid Protozoa Optimizer (PO) with DE for solving continuous optimization problems as formulated in Equation (1); we also focus on applying HPDE to solve engineering problems.

2.3. Overview of Protozoa Optimizer

The artificial Protozoa Optimizer (APO) [19] is a bio-inspired optimization algorithm that mimics the survival behaviors of protozoa, particularly focusing on foraging, dormancy, and reproduction. This optimizer is designed to solve continuous optimization problems by balancing exploration and exploitation through mathematical models derived from protozoa’s biological activities.

Foraging behavior in protozoa involves both autotrophic and heterotrophic mechanisms to obtain nutrients. The mathematical models for these modes are defined as follows.

2.4. APO Mathematical Models

This section introduces the mathematical framework of APO. The solution set is represented by a population of protozoa, where each protozoan occupies a position in a multidimensional space defined by $dim$ variables.

2.4.1. Notations and Nomenclature

The notations and symbols utilized in the model are summarized as follows. The population size is denoted by $ps$, while $dim$ represents the number of decision variables. The dimension index, indicated by $di$, ranges from 1 to $dim$. The notation $np$ stands for the number of neighbor pairs within the model, and f refers to the foraging factor. Weight factors in autotrophic and heterotrophic modes are represented by $w_a$ and $w_h$, respectively.

The proportion fraction of dormancy and reproduction is denoted as $pf$, while $p_{ah}$ and $p_{dr}$ represent the probability of autotrophic and heterotrophic behavior and the probability of dormancy and reproduction, respectively. A random number within the range $[0,1]$ is noted as $rand$. The current iteration number is indicated by $iter$, with $ite{r}_{max}$ representing the maximum iteration count permissible.

The position of the ith protozoan is denoted by $X_i$, whereas $X_{min}$ and $X_{max}$ denote the lower and upper bounds of the decision variables, respectively. The mapping vectors used in foraging and reproduction are represented by $M_f$ and $M_r$. An index vector within dormancy and reproduction processes is denoted by $D_{r_{\mathrm{index}}}$, while a random vector containing elements within $[0,1]$ is represented by $Rand$.

The ceiling and flooring functions are represented as $[·]$ and $\lfloor ·\rfloor$, respectively. The fitness function is denoted by $f(·)$, and $\mathrm{sort}(·)$ is used to arrange fitness values in ascending order. Additionally, $\text{rand\_perm}(n,l)$ generates a vector of l unique integers selected randomly within the range 1 to n. Lastly, the Hadamard product is represented by the symbol ⊙. This notation provides a clear and concise reference for interpreting the model’s various parameters and operations.

2.4.2. Foraging Mechanism

The foraging behavior in protozoa, central to this optimization model, is driven by both internal and external factors. Internal factors are related to each protozoan’s individual foraging characteristics, while external factors capture environmental influences, such as interactions with neighboring protozoa.

2.4.3. Autotrophic Mode

In autotrophic mode, protozoa synthesize nutrients when exposed to light, prompting movement toward regions with optimal light conditions. Protozoa in areas of high light intensity will move to positions with reduced intensity and vice versa. Assuming that the light conditions around a given protozoan, say the jth protozoan, are favorable for photosynthesis, the mathematical model guiding the movement of the ith protozoan toward this target is as shown in Equation (5) [19]:

$$X_i^{\mathrm{new}}=X_i+f·(X_j-X_i)+{\displaystyle \frac{1}{np}}\sum _{k=1}^{np}{w}_a·(X_{k_{-}}-X_{k_{+}})\odot M_f$$

(5)

Here, $X_i^{\mathrm{new}}$ and $X_i$ represent the updated and current positions of the ith protozoan, respectively. $X_j$ is the position of the selected protozoan acting as a target. The terms $X_{k_{-}}$ and $X_{k_{+}}$ denote neighboring protozoa chosen based on rank relative to i. Specifically, $X_{k_{-}}$ refers to a protozoan with a rank lower than i, while $X_{k_{+}}$ has a rank higher than i. This structure enables adaptive movement within the foraging process.

The spatial positions and ranking of protozoa are specified as shown in Equations (6) and (7) [19]:

$$X_i=[x_i^1,x_i^2,\dots ,x_i^{dim}]$$

(6)

$$X_i=\mathrm{sort}\left(X_i\right)$$

(7)

The foraging factor f dynamically adjusts based on the iteration count as shown in Equation (8) [19]:

$$f=\mathrm{rand}·(1+cos\left({\displaystyle \frac{iter}{ite{r}_{max}}}·\pi \right))$$

(8)

Additional parameters defining neighborhood structure and weight are given in Equations (9) and (10) [19]:

$$n{p}_{\max }=⌈{\displaystyle \frac{ps-1}{2}}⌉$$

(9)

$$w_a=e^{-\left|{\displaystyle \frac{f\left(X_{k_{-}}\right)}{f(X_{k_{+}}+\mathrm{eps})}}\right|}$$

(10)

The mapping vector $M_f$, which dictates the selection of dimensions during foraging, is defined in Equation (11) [19]:

$$M_f\left[di\right]=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}di\mathrm{is}\mathrm{in}\text{rand\_perm}(\dim ,⌈{\displaystyle \frac{1}{ps}}⌉)\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(11)

2.4.4. Heterotrophic Mode

In low-light environments, protozoa can absorb organic nutrients directly from the surroundings. Given that $X_{\mathrm{near}}$ represents a nutrient-rich location nearby, the heterotrophic mode governs the protozoan’s movement toward this point using the model as shown in Equation (12) [19]

$$X_i^{\mathrm{new}}=X_i+f·(X_{\mathrm{near}}-X_i)+{\displaystyle \frac{1}{np}}\sum _{k=1}^{np}{w}_h·(X_{i-k}-X_{i+k})\odot M_f$$

(12)

The location of $X_{\mathrm{near}}$ is determined as in Equation (13) [19]:

$$X_{\mathrm{near}}=(1\pm \mathrm{Rand}·\left(1-{\displaystyle \frac{iter}{ite{r}_{max}}}\right))\odot X_i$$

(13)

The weight factor $w_h$ for the heterotrophic mode is calculated as in Equation (14) [19]:

$$w_h=e^{-\left|{\displaystyle \frac{f\left(X_{i-k}\right)}{f\left(X_{i+k}\right)+\mathrm{eps}}}\right|}$$

(14)

2.5. Dormancy

Dormancy is an adaptive survival response activated during unfavorable conditions, where a protozoan in a dormant state is replaced by a newly generated one, maintaining the population size. The dormancy model is defined as it is in Equation (15) [19]:

$$X_i^{\mathrm{new}}=X_{min}+Rand\odot (X_{max}-X_{min})$$

(15)

Here, $X_{min}$ and $X_{max}$ represent the lower and upper bounds, respectively, defined as shown in Equation (16) [19]:

$$X_{min}=[l{b}_1,l{b}_2,\cdots ,l{b}_{dim}],X_{max}=[u{b}_1,u{b}_2,\cdots ,u{b}_{dim}]$$

(16)

2.6. Reproduction

Reproduction is modeled as binary fission, where a protozoan divides into two identical copies. This division is simulated by duplicating the protozoan and applying a perturbation. The reproduction model is shown in (17) [19]:

$$X_i^{\mathrm{new}}=X_i\pm rand·(X_{min}+Rand\odot (X_{max}-X_{min}))\odot M_r$$

(17)

The mapping vector for reproduction, $M_r$, specifies the dimensions involved in the perturbation:

$$M_r\left[di\right]=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}di\mathrm{is}\mathrm{in}\text{rand\_perm}(\dim ,rand)\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(18)

2.7. Overview of Differential Evolution

Differential Evolution (DE) is a population-based optimization algorithm where a population of candidate solutions iteratively evolves toward optimal solutions by applying mutation, crossover, and selection operators. The DE process is defined mathematically through several equations, each governing a specific stage of the evolution process.

The DE algorithm begins with the initialization of a population of $ps$ candidate solutions, where each candidate is represented by a vector $X_i$ with $dim$ decision variables. Let $X_i^{\left(g\right)}=\{x_{i,1}^{\left(g\right)},x_{i,2}^{\left(g\right)},\cdots ,x_{i,dim}^{\left(g\right)}\}$ denote the position of the ith individual in the population at generation g. Each candidate vector is initialized randomly within the bounds specified by $X_{min}$ and $X_{max}$.

• Mutation

The mutation process creates a mutant vector for each target vector $X_i^{\left(g\right)}$ by combining the position vectors of other randomly selected candidates. As shown in Equation (19), this mutation strategy enables the exploration of new solutions based on the diversity within the population.

$$V_i^{\left(g\right)}=X_{r_1}^{\left(g\right)}+F·(X_{r_2}^{\left(g\right)}-X_{r_3}^{\left(g\right)}),$$

(19)

where $V_i^{\left(g\right)}$ represents the mutant vector, $X_{r_1}^{\left(g\right)}$, $X_{r_2}^{\left(g\right)}$, and $X_{r_3}^{\left(g\right)}$ are randomly selected individuals from the population such that $r_1,r_2,r_3\ne i$. The parameter F is a scaling factor (typically within the range $[0,2]$) that controls the amplification of the differential variation between individuals.

• Crossover

To increase diversity, DE applies a crossover operator that combines elements of the mutant vector $V_i^{\left(g\right)}$ and the target vector $X_i^{\left(g\right)}$ to produce a trial vector $U_i^{\left(g\right)}$. The crossover is typically defined as shown in Equation (20), this crossover strategy helps maintain diversity by blending solutions from different vectors.

$$U_{i,j}^{\left(g\right)}=\left\{\begin{array}{cc}{V}_{i,j}^{\left(g\right)}\hfill & \mathrm{if}ran{d}_j\le C_r\mathrm{or}j=j_{\mathrm{rand}},\hfill \\ X_{i,j}^{\left(g\right)}\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(20)

where j represents the dimension index, $C_r$ is the crossover probability (within $[0,1]$), $ran{d}_j$ is a random number generated for each dimension j, and $j_{\mathrm{rand}}$ is a randomly chosen index that ensures at least one component from the mutant vector $V_i^{\left(g\right)}$ is included in the trial vector $U_i^{\left(g\right)}$.

• Selection

The selection step determines whether the trial vector $U_i^{\left(g\right)}$ should replace the target vector $X_i^{\left(g\right)}$ in the next generation. This decision is based on the fitness values of the vectors, where the fitness function $f(·)$ evaluates the objective to be minimized. The selection process is defined as follows:

$$X_i^{(g+1)}=\left\{\begin{array}{cc}{U}_i^{\left(g\right)}\hfill & \mathrm{if}f\left(U_i^{\left(g\right)}\right)\le f\left(X_i^{\left(g\right)}\right),\hfill \\ X_i^{\left(g\right)}\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(21)

As shown in Equation (21), the trial vector $U_i^{\left(g\right)}$ replaces the target vector $X_i^{\left(g\right)}$ if it yields a lower fitness value, thus ensuring that better solutions are retained in the population for subsequent generations.

• Termination

The algorithm iterates through the mutation, crossover, and selection steps until a predefined stopping criterion is met, typically either a maximum number of generations $ite{r}_{max}$ or an acceptable error threshold. Let $iter$ denote the current iteration; the termination condition can be represented. As indicated in Equation (22), the algorithm concludes when either the maximum iteration count is reached or the target accuracy is achieved.

$$iter\ge ite{r}_{max}\mathrm{or}|f\left(X_{\mathrm{best}}\right)-f_{\mathrm{target}}|\le ϵ,$$

(22)

where $X_{\mathrm{best}}$ is the best solution found, $f_{\mathrm{target}}$ is the target fitness, and $ϵ$ is a small threshold value.

3. Proposed HPDE Optimization Algorithm

In this section, the details of the newly hybrid algorithm are presented. However, the APO primarily focuses on mimicking biological behaviors to explore and exploit the search space. However, its exploration capabilities might sometimes be limited, especially in high-dimensional or complex landscapes. By integrating DE’s mutation and crossover operations, the hybrid algorithm benefits from enhanced exploration capabilities, allowing it to escape local optima and explore the search space more effectively.

Maintaining diversity in the population is crucial for avoiding premature convergence. APO’s dormancy and reproduction forms introduce diversity by reinitializing or modifying protozoa positions. The addition of DE operations further enhances diversity by generating new candidate solutions through the combination of multiple individuals. This hybrid approach helps in preserving population diversity and improving the algorithm’s ability to find global optima. Furthermore, APO adapts its behavior based on the current state of the population and the optimization process. The incorporation of DE introduces an additional adaptive layer, where DE operations are applied with a certain probability. This probabilistic application of DE ensures that the hybrid algorithm can dynamically balance exploration and exploitation based on the optimization stage, leading to more efficient convergence. The hybrid algorithm combines the APO’s biologically inspired mechanisms and DE’s evolutionary strategies. APO’s foraging forms (autotroph and heterotroph) and DE’s mutation and crossover operations complement each other, resulting in a more robust optimization process. The hybridization ensures that the strengths of both approaches are utilized effectively, leading to improved overall performance.

3.1. Mathematical Model of HPDE

• Initialization

The initial population of protozoa is generated randomly within the bounds ${\mathbf{X}}_{min}$ and ${\mathbf{X}}_{max}$, as shown in Equation (23):

$${\mathbf{x}}_i^0={\mathbf{X}}_{min}+{\mathbf{r}}_i·({\mathbf{X}}_{max}-{\mathbf{X}}_{min}),i=1,2,\dots ,ps$$

(23)

where ${\mathbf{r}}_i$ is a vector of uniformly distributed random numbers in the interval [0, 1].

• Fitness Evaluation

The fitness of each protozoa is evaluated using the objective function f, as shown in Equation (24):

$$f_i^0=f\left({\mathbf{x}}_i^0\right),i=1,2,\dots ,ps$$

(24)

• Main Loop (iter = 1 to iter_max)

Sort by fitness: the population is sorted by fitness in ascending order, as shown in Equation (25):

$$\{{\mathbf{x}}_i,f_i\}\leftarrow \mathrm{sort}\left(\{{\mathbf{x}}_i,f_i\}\right),i=1,2,\dots ,ps$$

(25)

Proportion fraction (pf) the proportion fraction $pf$ is calculated as shown in Equation (26):

$$pf=p{f}_{max}·r,r\sim U(0,1)$$

(26)

where $U(0,1)$ is a uniform random variable. $p{f}_{max}$ is the maximum proportion fraction, a predefined parameter that sets the upper limit for $pf$, and r is a random number drawn from a uniform distribution in the interval $[0,1]$, denoted as $r\sim U(0,1)$.

The role of $p{f}_{max}$ in the algorithm is to act as a control parameter that determines the maximum possible value of the proportion fraction $pf$, effectively controlling the maximum proportion of the population that can participate in a specific behavior during each iteration. By adjusting $p{f}_{max}$, one can balance the exploration and exploitation capabilities of the algorithm: a higher $p{f}_{max}$ allows more protozoa to engage in behaviors like reproduction, enhancing exploration, while a lower $p{f}_{max}$ focuses the algorithm on exploitation by limiting the number of protozoa undergoing certain behaviors. The multiplication by a random variable r introduces stochasticity, allowing $pf$ to vary between 0 and $p{f}_{max}$ in each iteration, which helps prevent premature convergence and maintains diversity in the population. Selecting $p{f}_{max}$ is typically performed through empirical tuning or recommendations from the literature, with common values ranging between 0.05 and 0.3, but the optimal value can depend on the specific problem and characteristics of the search space.

Random indices for dormancy/reproduction: Random indices for protozoa entering dormancy or reproduction forms are selected as shown in Equation (27):

$$\mathcal{R}=\{i_1,i_2,\dots ,i_k\},k=\lceil ps·pf\rceil$$

(27)

where $ps$ is the population size, $pf$ is the proportion fraction determining the fraction of the population to undergo dormancy or reproduction, and $\lceil ·\rceil$ denotes the ceiling function. The set $\mathcal{R}$ contains k unique random indices $i_j$ selected from the population, where each $i_j$ satisfies $1\le i_j\le ps$. These indices correspond to the protozoa that will engage in dormancy or reproduction behaviors during the current iteration of the algorithm.

• Dormancy/Reproduction Forms

Dormancy/reproduction probability: The probability $p_{dr}$ of a protozoa being in dormancy or reproduction form is given by, as shown in Equation (28):

$$p_{dr}={\displaystyle \frac{1}{2}}\left(1+cos\left(\pi \left(1-{\displaystyle \frac{i}{ps}}\right)\right)\right),i\in \mathcal{R}$$

(28)

• Dormancy Form

If a protozoan is in dormancy form, its new position is randomly reinitialized, as shown in Equation (29):

$${\mathbf{x}}_i^{\prime }={\mathbf{X}}_{min}+{\mathbf{r}}_i·({\mathbf{X}}_{max}-{\mathbf{X}}_{min})$$

(29)

where ${\mathbf{x}}_i^{\prime }$ is the updated position vector of the i-th protozoan after dormancy; ${\mathbf{X}}_{min}$ and ${\mathbf{X}}_{max}$ are the minimum and maximum bounds of the search space, respectively; ${\mathbf{r}}_i$ is a random vector associated with the i-th protozoan, where each element is independently drawn from a uniform distribution in the interval $[0,1]$; and the symbol · denotes element-wise multiplication between vectors.

• Reproduction Form

If a protozoan is in reproduction form, it follows the reproduction mechanism. As shown in Equation (30), the mapping vector is

$${\mathbf{M}}_r=[m_{r1},m_{r2},\dots ,m_{rd}],m_{rj}=\left\{\begin{array}{cc}1,\hfill & \mathrm{with}\mathrm{probability}p,\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(30)

where ${\mathbf{M}}_r$ is the reproduction mapping vector determining which dimensions will be modified during reproduction; $m_{rj}$ corresponds to the j-th dimension and is set to 1 with probability p and 0 otherwise; d denotes the dimensionality of the problem (number of decision variables); and p is the probability of selecting a dimension for updating.

The reproduction form is defined as it is in Equations (31) and (32):

$${\mathbf{x}}_i^{\prime }={\mathbf{x}}_i+\lambda ·\left({\mathbf{X}}_{min}+{\mathbf{r}}_i·({\mathbf{X}}_{max}-{\mathbf{X}}_{min})\right)·{\mathbf{M}}_r$$

(31)

where $\lambda$ is a random factor of $\pm 1$.

$$X_{{\mathrm{new}}_i}=X_i\pm \mathrm{rand}·(X_{min}+\mathrm{Rand}·(X_{max}-X_{min}))·M_r$$

(32)

where ${\mathbf{x}}_i^{\prime }$ is the updated position vector of the i-th protozoan after reproduction; ${\mathbf{x}}_i$ is the current position vector of the i-th protozoan; $\lambda$ is a random factor that takes the value $+1$ or $-1$ with equal probability; ${\mathbf{r}}_i$ is a random vector where each element is drawn from a uniform distribution in $[0,1]$; ${\mathbf{M}}_r$ is the reproduction mapping vector from Equation (30); and the symbols · denote element-wise multiplication.

• Foraging Forms

Foraging factor: the foraging factor f is calculated as shown in Equation (33):

$$f=r·\left(1+cos\left(\pi {\displaystyle \frac{iter}{ite{r}_{max}}}\right)\right),r\sim U(0,1)$$

(33)

where f is the foraging factor influencing the step size during foraging; r is a random number drawn from a uniform distribution over $[0,1]$, denoted $r\sim U(0,1)$; $iter$ is the current iteration number; $ite{r}_{max}$ is the maximum number of iterations; cos denotes the cosine function; and $\pi$ is the mathematical constant pi.

Autotroph/heterotroph probability: the probability $p_{ah}$ of protozoa being in autotroph or heterotroph form is given by Equation (34):

$$p_{ah}={\displaystyle \frac{1}{2}}\left(1+cos\left(\pi {\displaystyle \frac{iter}{ite{r}_{max}}}\right)\right)$$

(34)

where $p_{ah}$ is the probability of a protozoan being in autotroph or heterotroph form; $iter$ is the current iteration number; $ite{r}_{max}$ is the maximum number of iterations; and the function oscillates between 0 and 1 over iterations.

• Autotroph Form

If a protozoan is in an autotrophic form, it follows the autotrophic mechanism.

The weight factor is defined as it is in Equation (35):

$$w_a=exp\left(-\left|{\displaystyle \frac{f_{km}}{f_{kp}+ϵ}}\right|\right),km,kp\mathrm{are}\mathrm{neighbor}\mathrm{indices}$$

(35)

where $w_a$ is the weight factor influencing the movement in autotroph form; $f_{km}$ and $f_{kp}$ are the fitness values of protozoa at neighbor indices $km$ and $kp$, respectively; $ϵ$ is a small constant to avoid division by zero; $|·|$ denotes the absolute value; and exp denotes the exponential function.

The effect of phototaxis is

$${\mathbf{e}}_{pn}=w_a·({\mathbf{x}}_{km}-{\mathbf{x}}_{kp})$$

(36)

where ${\mathbf{e}}_{pn}$ is the phototaxis effect vector for the n-th neighbor pair; ${\mathbf{x}}_{km}$ and ${\mathbf{x}}_{kp}$ are the positions of neighbor protozoa at indices $km$ and $kp$, respectively; $w_a$ is the weight factor from Equation (35); and · denotes element-wise multiplication.

The new position for the autotroph form is

$${\mathbf{x}}_i^{\prime }={\mathbf{x}}_i+f·\left({\mathbf{x}}_j-{\mathbf{x}}_i+{\displaystyle \frac{1}{np}}\sum _{n=1}^{np}{\mathbf{e}}_{pn}\right)·{\mathbf{M}}_f$$

(37)

where ${\mathbf{x}}_i^{\prime }$ is the updated position vector of the i-th protozoan in autotroph form; ${\mathbf{x}}_i$ is the current position vector of the i-th protozoan; f is the foraging factor from Equation (33); ${\mathbf{x}}_j$ is the position vector of a randomly selected protozoan j; $np$ is the number of neighbor pairs considered; ${\sum }_{n=1}^{np}{\mathbf{e}}_{pn}$ is the summation of phototaxis effects from all neighbor pairs; ${\mathbf{M}}_f$ is the foraging mapping vector determining which dimensions are updated; and · denotes element-wise multiplication.

• Heterotroph Form

If a protozoan is in a heterotrophic form, it follows the heterotrophic mechanism.

As shown in Equation (38), the weight factor is

$$w_h=exp\left(-\left|{\displaystyle \frac{f_{imk}}{f_{ipk}+ϵ}}\right|\right)$$

(38)

where $w_h$ is the weight factor influencing the movement in heterotroph form; $f_{imk}$ and $f_{ipk}$ are the fitness values of protozoa at indices $imk$ and $ipk$, respectively; $ϵ$ is a small constant to avoid division by zero; $|·|$ denotes the absolute value; and exp denotes the exponential function.

The effect of chemotaxis is

$${\mathbf{e}}_{pn}=w_h·({\mathbf{x}}_{imk}-{\mathbf{x}}_{ipk})$$

(39)

where ${\mathbf{e}}_{pn}$ is the chemotaxis effect vector for the n-th neighbor pair; ${\mathbf{x}}_{imk}$ and ${\mathbf{x}}_{ipk}$ are the position vectors of neighbor protozoa at indices $imk$ and $ipk$, respectively; $w_h$ is the weight factor from Equation (38); and · denotes element-wise multiplication.

Furthermore, the nearby position is applied as shown in Equation (40):

$${\mathbf{x}}_{\mathrm{near}}=\left(1+\lambda ·r·\left(1-{\displaystyle \frac{iter}{ite{r}_{max}}}\right)\right)·{\mathbf{x}}_i$$

(40)

where ${\mathbf{x}}_{\mathrm{near}}$ is the nearby position vector for the i-th protozoan; $\lambda$ is a random factor taking the value $+1$ or $-1$ with equal probability; r is a random number drawn from a uniform distribution over $[0,1]$; $iter$ is the current iteration number; $ite{r}_{max}$ is the maximum number of iterations; and · denotes element-wise multiplication.

Moreover, the new position for the heterotroph form is applied as shown in Equation (41):

$${\mathbf{x}}_i^{\prime }={\mathbf{x}}_i+f·\left({\mathbf{x}}_{\mathrm{near}}-{\mathbf{x}}_i+{\displaystyle \frac{1}{np}}\sum _{n=1}^{np}{\mathbf{e}}_{pn}\right)·{\mathbf{M}}_f$$

(41)

where ${\mathbf{x}}_i^{\prime }$ is the updated position vector of the i-th protozoan in heterotroph form; ${\mathbf{x}}_i$ is the current position vector of the i-th protozoan; f is the foraging factor defined previously; ${\mathbf{x}}_{\mathrm{near}}$ is the nearby position vector from Equation (40); $np$ is the number of neighbor pairs considered; ${\sum }_{n=1}^{np}{\mathbf{e}}_{pn}$ is the summation of chemotaxis effects from all neighbor pairs; ${\mathbf{M}}_f$ is the foraging mapping vector determining which dimensions are updated; and · denotes element-wise multiplication.

• Applying Differential Evolution (DE) Operations

The DE mutation and crossover operations are applied as shown in Equation (42):

$$\mathrm{mutant}=\mathbf{a}+F·(\mathbf{b}-\mathbf{c}),\mathbf{a},\mathbf{b},\mathbf{c}\in \mathrm{randomly}\mathrm{selected}\mathrm{vectors}$$

(42)

where mutant is the mutant vector generated in the mutation operation; $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are randomly selected distinct vectors from the current population; F is the mutation scaling factor controlling the differential variation; and · denotes scalar multiplication.

The crossover operation is defined as shown in Equation (43):

$${\mathrm{trial}}_j=\left\{\begin{array}{cc}{\mathrm{mutant}}_j,\hfill & \mathrm{if}{r}_j<CR,\hfill \\ {\mathbf{x}}_{ij},\hfill & \mathrm{otherwise}\hfill \end{array}\right.j=1,2,\dots ,dim$$

(43)

where ${\mathrm{trial}}_j$ is the j-th component of the trial vector; ${\mathrm{mutant}}_j$ is the j-th component of the mutant vector from Equation (42); ${\mathbf{x}}_{ij}$ is the j-th component of the current vector ${\mathbf{x}}_i$; $r_j$ is a random number drawn from a uniform distribution over $[0,1]$; $CR$ is the crossover probability controlling the rate of crossover; $dim$ is the dimensionality of the problem.

The fitness evaluation for the trial vector is performed as shown in Equation (44):

$$f_{\mathrm{trial}}=f\left(\mathrm{trial}\right)$$

(44)

where $f_{\mathrm{trial}}$ is the fitness value (objective function evaluation) of the trial vector trial; $f(·)$ denotes the objective function being minimized.

If $f_{\mathrm{trial}}<f_i$, the selection operation is applied as shown in Equation (45):

$${\mathbf{x}}_i=\mathrm{trial},f_i=f_{\mathrm{trial}}$$

(45)

where ${\mathbf{x}}_i$ is updated to the trial vector if it has a better fitness value; $f_i$ is the fitness value of the i-th vector in the current population.

Boundary control: the new positions of the protozoa are ensured to be within the boundaries, as shown in Equation (46):

$${\mathbf{x}}_i^{\prime }=max\left(min\left({\mathbf{x}}_i^{\prime },{\mathbf{X}}_{max}\right),{\mathbf{X}}_{min}\right),i=1,2,\dots ,ps$$

(46)

where ${\mathbf{x}}_i^{\prime }$ is the updated position vector of the i-th protozoan after boundary control; ${\mathbf{X}}_{min}$ and ${\mathbf{X}}_{max}$ are the minimum and maximum bounds of the search space, respectively; $min(·)$ and $max(·)$ are element-wise minimum and maximum functions; $ps$ is the population size.

Finally, the population is updated based on the fitness of the new positions, as shown in Equation (47):

$${\mathbf{x}}_i=\left\{\begin{array}{cc}{\mathbf{x}}_i^{\prime },\hfill & \mathrm{if}f\left({\mathbf{x}}_i^{\prime }\right)<f\left({\mathbf{x}}_i\right),\hfill \\ {\mathbf{x}}_i,\hfill & \mathrm{otherwise}\hfill \end{array}\right.,i=1,2,\dots ,ps$$

(47)

where ${\mathbf{x}}_i$ is the updated position vector of the i-th protozoan in the population; ${\mathbf{x}}_i^{\prime }$ is the new position vector after the update mechanisms; $f\left({\mathbf{x}}_i^{\prime }\right)$ and $f\left({\mathbf{x}}_i\right)$ are the fitness values of the new and current positions, respectively; $ps$ is the population size.

3.2. Discussion on Hybrid APO with DE Algorithm Design and Pseudocode

The design of the Hybrid Artificial Protozoa Optimizer with Differential Evolution (HPDE) algorithm is centered around integrating the strengths of both the Artificial Protozoa Optimizer (APO) and Differential Evolution (DE). APO mimics the biological behaviors of protozoa, which include foraging, reproduction, and dormancy. However, while APO excels in exploration, its exploitation capabilities can be limited, especially in complex and high-dimensional search spaces. To address this limitation, DE is incorporated into the hybrid algorithm to enhance its search capabilities and to improve the balance between exploration and exploitation.

In the design of HPDE, the APO’s natural behaviors are preserved and enhanced by DE’s mutation and crossover strategies. The DE operations are not applied uniformly but with a certain probability, allowing the algorithm to adapt dynamically to the optimization process. This ensures that the algorithm can intensify the search in promising regions of the search space while maintaining diversity in the population to avoid premature convergence.

The process begins with the initialization of the population, followed by the evaluation of fitness. The population is then sorted based on fitness values, ensuring that the best solutions are prioritized. The algorithm iteratively updates the population, applying APO’s dormancy and reproduction mechanisms, as well as DE’s mutation and crossover operations.

One of the key innovations in HPDE is the use of proportion fractions and random indices to introduce diversity into the population. This is particularly important in maintaining a broad search space and preventing the algorithm from getting trapped in local optima. Additionally, the algorithm adapts the application of DE operations based on the current state of the population, ensuring a balance between exploration and exploitation throughout the optimization process.

The pseudocode reflects this hybrid approach, with APO’s biological behaviors forming the core of the algorithm, while DE operations are integrated strategically to enhance performance. By combining the strengths of both APO and DE, HPDE is capable of addressing a wide range of optimization problems, from benchmark functions to real-world engineering design challenges.

3.3. Discussion on Hybrid APO with DE Algorithm Design and Pseudocode

The design of the Hybrid Artificial Protozoa Optimizer with Differential Evolution (HPDE) algorithm consists of different stages, including initialization, fitness evaluation, exploration through APO’s biological behaviors, and exploitation enhanced by DE’s mutation and crossover operations.

Initialization: In the first stage, the population of protozoa is initialized randomly within the given bounds. This initialization sets the stage for the search process by distributing potential solutions across the search space. Each protozoan represents a candidate solution, and their positions are determined based on uniform random distribution. This diversity in the initial population ensures a broad exploration of the search space from the outset.

Fitness Evaluation: Following initialization, the fitness of each protozoan is evaluated using the objective function. This step assigns a fitness value to each candidate solution, which reflects its quality in terms of the optimization goal. The population is then sorted based on these fitness values, prioritizing the best solutions for subsequent stages.

Exploration: The next stage focuses on exploration, driven primarily by APO’s biologically inspired mechanisms. APO simulates the behaviors of protozoa, including dormancy, reproduction, and foraging. Dormancy and reproduction introduce diversity into the population by reinitializing or modifying protozoa positions, thereby preventing premature convergence. Foraging, on the other hand, allows protozoa to move within the search space based on autotroph and heterotroph behaviors, guiding the search toward promising regions.

Exploitation: To enhance exploitation, the DE algorithm is integrated into the hybrid approach. DE’s mutation and crossover operations are applied to generate new candidate solutions by combining existing ones. These operations are not applied uniformly across the population but are introduced with a certain probability, allowing the algorithm to adapt to the current optimization stage. This probabilistic application ensures that the algorithm can focus on intensifying the search in areas where promising solutions have been identified while still maintaining sufficient diversity in the population.

Population Update and Convergence: After applying both APO and DE operations, boundary control is performed to ensure that the solutions remain within the defined bounds. The population is then updated based on the fitness of the newly generated solutions. This process iterates until the maximum number of iterations is reached or another stopping criterion is satisfied.

The pseudocode (see Algorithm 1) demonstrates the flow of the algorithm. Starting with the initialization of the population, the algorithm progresses through fitness evaluation, exploration via APO, and exploitation through DE, with careful management of population diversity throughout the process.

The HPDE algorithm effectively combines the exploration strengths of APO with the exploitation capabilities of DE, resulting in a robust optimization method. The algorithm’s design ensures that it can handle a wide range of optimization problems, from standard benchmark functions to complex real-world engineering challenges. By maintaining a balance between exploration and exploitation, HPDE is able to avoid local optima and achieve good performance across different problem domains In addition, the flowchart of HPDE is provided in Figure 1.

Algorithm 1 Pseudocode and algorithm steps of HPDE

1: Input: Objective function f, dimension $dim$, population size $ps$, maximum iterations $ite{r}_{max}$, lower bound ${\mathbf{X}}_{min}$, upper bound ${\mathbf{X}}_{max}$ 2: Output: Best solution ${\mathbf{x}}_{\mathrm{best}}$, best fitness value $f_{\mathrm{best}}$ 3: Initialize population ${\mathbf{x}}_i\in [{\mathbf{X}}_{min},{\mathbf{X}}_{max}]$ for $i=1,2,\dots ,ps$    ▹ See Equation (23) 4: Evaluate fitness $f_i=f\left({\mathbf{x}}_i\right)$ for $i=1,2,\dots ,ps$        ▹ See Equation (24) 5: Sort population by fitness             ▹ See Equation (25) 6: for$iter=1$ to $ite{r}_{max}$ do 7:     Calculate proportion fraction $pf$    ▹ See Equation (26) 8:     Select random indices for dormancy/reproduction $\mathcal{R}$    ▹ See Equation (27) 9:     for each $i\in \mathcal{R}$ do 10:         Calculate dormancy/reproduction probability $p_{dr}$        ▹ See Equation (28) 11:         if random value < $p_{dr}$ then 12:            Perform dormancy form         ▹ See Equation (29) 13:         else 14:            Perform reproduction form     ▹ See Equations (30) and (31) 15:         end if 16:     end for 17:     for each $i\notin \mathcal{R}$ do 18:         Calculate foraging factor f        ▹ See Equation (33) 19:         Calculate autotroph/heterotroph probability $p_{ah}$    ▹ See Equation (34) 20:         if random value < $p_{ah}$ then 21:            Perform autotroph form         ▹ See Equations (35)–(37) 22:         else 23:            Perform heterotroph form         ▹ See Equations (38)–(41) 24:         end if 25:     end for 26:     for each $i=1$ to $ps$ do 27:         if random value < 0.2 then 28:            Apply Differential Evolution (DE)     ▹ See Equations (42)–(45) 29:         end if 30:     end for 31:     Ensure boundary control         ▹ See Equation (46) 32:     Update population based on fitness         ▹ See Equation (47) 33: end for 34: return Best solution ${\mathbf{x}}_{\mathrm{best}}$, best fitness value $f_{\mathrm{best}}$

3.4. Exploration and Exploitation Features of HPDE Algorithm

Exploration and exploitation are fundamental concepts in metaheuristic optimization algorithms, and they are crucial for the performance of the HPDE Hybrid algorithm. Exploration refers to the algorithm’s ability to search the global solution space broadly, ensuring diverse candidate solutions are considered. Exploitation, on the other hand, focuses on intensively searching around the best solutions found so far, refining them to achieve optimality. The HPDE Hybrid algorithm integrates mechanisms to balance these two aspects effectively.

Exploration in the HPDE Hybrid algorithm is primarily achieved through the foraging behavior of protozoa and the dormancy forms. During the foraging process, protozoa move through the solution space based on autotrophic and heterotrophic mechanisms. The autotrophic mode, where protozoa move towards suitable light conditions, ensures diverse sampling of the search space by allowing protozoa to explore new regions. This is modeled by Equations (35)–(37). The heterotrophic mode, where protozoa absorb organic matter, also contributes to exploration by moving protozoa towards nutrient-rich areas, modeled by Equations (38)–(41). Additionally, the dormancy form, as described in Equation (29), allows protozoa to be reinitialized to new random positions, injecting further diversity into the population.

Exploitation is facilitated by the reproduction forms and the selective application of Differential Evolution (DE) operations. The reproduction form, detailed in Equations (30) and (31), allows protozoa to create new solutions by perturbing their current positions, focusing the search around promising areas. This mechanism ensures that good solutions are refined over iterations. The DE operations, involving mutation, crossover, and selection as described by Equations (42)–(45), further enhance exploitation. The mutation operation generates new candidate solutions by combining existing ones, while crossover blends these candidates to create trial solutions. The selection process ensures that only solutions with improved fitness values are retained in the population. These DE operations intensively exploit the solution space around the best-found solutions, driving the search towards optimality.

The balance between exploration and exploitation is dynamically managed throughout the algorithm’s iterations. The proportion fraction ($pf$), as calculated by Equation (26), determines the ratio of protozoa undergoing dormancy or reproduction versus those in foraging modes. This adaptive mechanism ensures that exploration is emphasized in the early stages of the search, while exploitation becomes more prominent as the search progresses and the algorithm converges towards optimal solutions.

3.5. Solving Single-Objective Optimization Problems Using HPDE

Single-objective optimization problems typically involve one minimum point, which could either be the sole global optimum with no local points or include several local points with one global optimum [113]. To resolve such a problem using HPDE, it must first be mathematically formulated as displayed in Equation (48):

$$X=min\left(f\right(x\left)\right)$$

(48)

In this equation, $f\left(x\right)$ represents the objective function to be minimized, while n denotes the total number of variables. The variable bounds are defined as $X_{\min }\le x\le X_{\max }$, with the inequality constraints being $g\left(x\right)\le 0$ and equality constraints as $h\left(x\right)=0$.

The process of solving a single-objective optimization problem using HPDE starts with the initial step, which encompasses setting up the algorithm parameters. Then, the fitness value is computed using the formulated single-objective problem, and these values are stored and arranged. The top two values are selected and placed in temporary variables. In the fourth step, the iterative process begins with entry into the while loop. In this phase, the particle moves toward the single global optimum point. The first and second-best positions are updated, and with each iteration, the fitness function is calculated, and the boundary limits of the search space are verified. Once the loop concludes, the results are reported, and the global optimum point is identified.

4. Evaluation of HPDE in Solving Various Engineering Design Problems

This section delves into the application of HPDE for solving various engineering design problems. It provides a comprehensive comparison of HPDE’s effectiveness against a broad spectrum of other optimization algorithms. The evaluation focuses on the ability of these algorithms to handle complex, real-world engineering challenges, showcasing HPDE’s strengths and potential areas for improvement. Through detailed analysis and empirical results, this section highlights the advantages of using HPDE in engineering optimization, underscoring its robustness, efficiency, and accuracy in finding optimal solutions.

4.1. Robot Gripper Problem

The Robot Gripper problem models a robotic gripper’s mechanisms as shown in Figure 2, aiming to optimize its design and functionality by configuring physical dimensions and angles to maximize grip efficiency and stability. The mathematical model defines several parameters and constraints to simulate realistic operational conditions.

A mathematical model for the Robotic Gripper optimization problem is given in Appendix A.

The results of the Robot Gripper optimization problem are shown in

Table 2. Robot Gripper optimization problem.

Optimizer	Best	x1	x2	x3	x4	x5	x6	x7
HPDE	2.54714	150	149.881	200	1.86E-07	149.7948	101.1292	2.302793
APO	2.653614	149.1163	145.9238	200	3.051837	109.4974	103.0503	2.124082
CSA	2.554382	149.9889	149.8681	199.6885	1.68E-18	140.3496	101.3016	2.261251
WSO	2.616181	148.6972	148.1965	199.9947	0.35152	80.33606	103.518	1.954732
GWO	3.115494	149.9551	134.6845	198.9066	14.5216	141.7656	125.8555	2.49649
COOT	3.116414	148.1836	122.3659	181.9864	25.6268	146.6658	103.1818	2.704846
SA	2.557381	150	149.8016	200	0.073718	138.4248	101.6757	2.258888
HGS	2.992719	150	149.2177	200	0.003539	148.1204	129.04	2.465132
ChOA	3.434886	150	149.6112	200	0	14.72397	123.5176	1.694239
SCA	4.289317	150	150	200	0	10.67676	100	1.60428
WOA	3.730008	149.9999	149.7655	144.5789	0	149.7574	109.8456	2.700656
AVOA	4.124148	149.8944	127.8498	155.084	21.12102	64.67494	128.1202	2.133409
HHO	4.487444	149.9995	149.8802	113.5487	0	10	101.1621	1.545976
DBO	4.289317	150	150	200	0	26.35673	100	1.702452
PSO	5.587274	73.47576	39.34421	177.1839	13.60019	47.21313	209.8807	2.843062
BO	5.498976	27.81725	49.13919	159.5358	10.02408	53.56557	138.5473	1.760682
AO	5.73049	149.9376	130.2317	141.849	9.274051	104.9349	183.8283	2.607121
HHO	4.010135	150	144.0767	169.966	3.061461	18.85633	155.0149	1.823084
FOX	7.05106	149.9991	124.0087	131.3446	0.216875	90.06024	206.0321	2.484178
RIME	5.348892	143.2657	107.797	165.8416	37.03908	122.388	161.3981	3.128836
BWO	4.289317	150	150	200	0	150	100	2.401751
COA	4.260015	147.7469	94.43728	196.9958	49.24896	147.7469	148.5583	3.092834
GA	9.77E+21	10	10	10	10	10	10	10

. The HPDE optimizer achieves a relatively competitive result with a best value of 2.54714. Compared with other optimization algorithms, HPDE demonstrates good performance, illustrating its effectiveness in optimizing the engineering design. Algorithms such as APO and CSA, with best values of 2.653614 and 2.554382, respectively, offer close competition but still fall slightly behind HPDE. Notably, some algorithms, like GA, perform significantly worse, with a best value reaching as high as $9.77\times {10}^{21}$, indicating substantial inferiority vs. the optimal solution found by HPDE.

The HPDE optimizer showcases an effective performance rank that is first among all tested optimizers, as indicated in

Table 3. Robot Gripper optimization problem.

Optimizer	Mean	Std	Min	Max	Time	Rank
HPDE	2.68537	0.226658	2.54714	3.087935	35.68552	1
APO	2.797473	0.163347	2.653614	3.052574	62.74848	2
CSA	3.100101	0.434225	2.554382	3.581738	26.7366	3
WSO	3.122543	0.363452	2.616181	3.606743	21.82177	4
GWO	3.381012	0.271715	3.115494	3.701685	30.42672	5
COOT	3.381756	0.215337	3.116414	3.71352	30.46024	6
SA	3.50365	0.576583	2.557381	4.008731	49.29251	7
HGS	4.04267	0.907644	2.992719	5.310951	28.77036	8
ChOA	4.075728	0.509109	3.434886	4.688536	55.29998	9
SCA	4.289317	0	4.289317	4.289317	59.01137	10
WOA	4.695308	0.900598	3.730008	6.043087	28.91176	11
AVOA	4.779911	0.686398	4.124148	5.639517	30.12082	12
HHO	5.977274	1.529106	4.487444	8.496856	67.05441	13
DBO	6.162695	1.938555	4.289317	8.578634	43.3856	14
PSO	9.599868	2.335064	5.587274	11.2361	42.4945	15
BO	11.92929	7.875758	5.498976	25.53762	71.98759	16
AO	18.40973	13.26528	5.73049	37.74424	64.9251	17
HHO	18.44778	30.37468	4.010135	72.77344	67.544	18
FOX	85.05408	131.3719	7.05106	317.3963	28.77017	19
RIME	2.34E+16	5.23E+16	5.348892	1.17E+17	54.54848	20
BWO	9.3E+21	2.08E+22	4.289317	4.65E+22	43.898	21
COA	1.34E+22	2.28E+22	4.260015	5.25E+22	85.68558	22
GA	1.07E+26	2.16E+26	9.77E+21	4.92E+26	38.78996	23

, with a mean value of $2.68537$, a minimum (Min) of $2.54714$, and a maximum (Max) of $3.087935$. When compared with other algorithms, HPDE not only achieves the lowest minimum value but also maintains a lower mean value than its competitors, such as APO and CSA, which have higher mean values of $2.797473$ and $3.100101$, respectively. Noteworthy is HPDE’s standard deviation of $0.226658$, signifying consistent performance across trials. This optimizer also excels in computational efficiency with a relatively moderate computational time of $35.68552$ s, outperforming several other algorithms like APO and CSA in terms of both effectiveness and efficiency.

4.2. Welded Beam Design Optimization

In the field of structural engineering, the welded beam design presents an archetypal problem that seeks to find an optimal balance between cost-efficiency and stringent safety standards. The design must adhere to multiple constraints without sacrificing cost-effectiveness [114].

The welded beam design problem, as illustrated in Figure 3, is an optimization problem in structural engineering. The mathematical model for the weld beam problem is shown in Appendix B.

In the welded beam design optimization problem, the HPDE exhibits an outperforming performance with a best value of $1.673507$, as illustrated in

Table 4. Welded beam design optimization comparison results.

Optimizer	Best	x1	x2	x3	x4
HPDE	1.673507	0.197983	3.369775	9.190433	0.198903
GWO	1.677278	0.200132	3.32519	9.163809	0.200329
AVOA	1.68819	0.183774	3.647123	9.189539	0.19894
COOT	1.677198	0.193109	3.452122	9.189441	0.198944
CSA	1.671121	0.197956	3.354014	9.192021	0.198832
ChOA	1.777303	0.197795	3.664357	8.928664	0.213358
SCA	1.815314	0.192197	3.615917	10	0.196785
DBO	1.851948	0.146324	4.552712	10	0.19542
HGS	1.677809	0.195285	3.394495	9.232798	0.198642
HHO	1.714181	0.19685	3.574442	9.09864	0.202935
AO	1.886445	0.180003	4.437627	9.953933	0.195663
PSO	2.226236	0.869779	1.228095	4.257463	0.398624
FOX	1.948645	0.266171	2.764965	7.823685	0.274511
COA	1.976053	0.1	7.478441	9.213166	0.198886
BWO	2.175329	0.1	8.174456	8.762248	0.223053
BO	2.173346	0.286457	2.148416	0.556314	0.198662
RIME	1.964253	0.101483	7.170883	9.307058	0.198604
SA	2.61181	0.425809	1.745779	7.012998	0.425809
WOA	2.850973	0.328609	2.720556	5.74579	0.546603
GA	4.983052	0.1	0.1	0.1	0.1

. This result places HPDE as one of the top performers among the listed algorithms, narrowly outperforming the CSA, which has a best of $1.671121$, and followed by GWO with $1.677278$. The variables optimized by APO, including $x1$, $x2$, $x3$, and $x4$, are well within competitive ranges when compared with others like GWO and CSA, ensuring a balanced design solution. Notably, HPDE performs significantly better than other advanced algorithms such as AVOA, COOT, and HGS, indicating its effectiveness in handling the complexities of beam design. Algorithms such as GA and WOA, on the other hand, show much higher best values, reflecting less optimal solutions in this specific engineering context.

In the welded beam design optimization problem, the performance of the HPDE is notably good, as evidenced in

Table 5. Welded beam design optimization statistical comparison results.

Optimizer	Mean	Std	Min	Max	Time	Rank
HPDE	1.674353	0.000774	1.673507	1.675425	0.509625	1
GWO	1.679066	0.002004	1.677278	1.682505	0.259323	2
AVOA	1.793266	0.118804	1.68819	1.925414	0.267854	3
COOT	1.798715	0.150512	1.677198	2.03391	0.244342	4
CSA	1.819832	0.121314	1.671121	1.928418	0.231074	5
ChOA	1.847255	0.04044	1.777303	1.879054	0.370165	6
SCA	1.862978	0.029324	1.815314	1.891756	0.244342	7
DBO	1.94309	0.069837	1.851948	2.029321	0.277223	8
HGS	1.979432	0.245941	1.677809	2.252595	0.242702	9
HHO	2.087893	0.246008	1.714181	2.359761	0.566038	10
AO	2.204788	0.286563	1.886445	2.575639	0.504042	11
PSO	2.269761	0.033228	2.226236	2.29754	1.335465	12
FOX	2.285226	0.296845	1.948645	2.707909	0.255454	13
COA	2.345034	0.230845	1.976053	2.559688	0.586764	14
BWO	2.424774	0.265848	2.175329	2.812801	0.289148	15
BO	2.560528	0.300971	2.173346	2.892572	1.635307	16
RIME	2.808909	0.710027	1.964253	3.825404	0.460184	17
SA	3.25223	0.552228	2.61181	4.099335	0.442409	18
WOA	3.930812	0.966242	2.850973	4.950146	0.254136	19
GA	6.502565	2.072872	4.983052	10.03831	63.05204	20

. The algorithm achieves an outperforming mean best value of $1.674353$, with a minimal standard deviation of $0.000774$, indicating a high level of consistency across runs. The minimal and maximal best values reported for APO are $1.673507$ and $1.675425$, respectively, which are significantly better than those achieved by most competitors. For instance, the closest contender, GWO, has a slightly higher mean best of $1.679066$ and a wider best range, reflecting less consistency. Notably, HPDE also ranks first among all tested algorithms, emphasizing its efficiency in exploring and exploiting the design space. This is further complemented by its relatively short computational time of $0.509625$ s, which is efficient compared with other algorithms, particularly in contrast to GA, which exhibits a much higher mean best of $6.502565$ with a significantly longer computational time of $63.05204$ s.

4.3. Pressure Vessel Design Optimization

Optimizing the design of pressure vessels is a critical endeavor in mechanical engineering, focusing on determining optimal dimensions that ensure structural integrity, safety, and cost-effectiveness. This section describes the mathematical formulation for the pressure vessel design optimization problem [115].

The objective of the optimization is to minimize the cost associated with the materials and manufacturing of the pressure vessel, constrained by the vessel’s design and performance requirements [116].

The pressure vessel design optimization problem ( See Figure 4) involves the cylindrical and hemispherical parts of a pressure vessel. The objective is to minimize the total cost, including material, forming, and welding, subject to constraints on the vessel’s physical dimensions and operational safety.

In the pressure vessel design optimization problem, HDPE demonstrates a strong performance with a best score of $5885.363$, as detailed in

Table 6. Pressure vessel design optimization comparison results.

Optimizer	Best Score	x1	x2	x3	x4
HDPE	5885.363	12.45088	6.154508	40.3202	199.992
SHIO	6109.215	12.79151	6.554225	40.70604	196.8767
FVIM	5927.99	12.72971	6.333602	41.20091	188.151
RTH	5888.662	12.48178	6.169751	40.42027	198.6035
COA	6080.027	14.02655	6.934578	45.40416	139.6319
CSA	5894.422	12.51667	6.190974	40.53311	197.1072
MFO	5981.271	13.29076	6.56963	43.04002	165.3082
RIME	5946.184	12.98271	6.425251	42.04245	177.3318
WSO	5885.912	12.45382	6.155922	40.32681	199.901
RSO	9442.609	19.32286	11.85051	59.69156	61.08678
SCA	6303.989	13.33922	6.727022	40.98908	191.001
GWO	5928.282	12.52592	6.32199	40.5147	197.5136
WOA	5898.948	12.57686	6.216747	40.72816	194.3899
PSO	6044.525	12.58352	6.826001	40.62535	196.5722
RIME	5885.333	12.4507	6.154387	40.31962	200
BWO	6298.223	15.505	7.664127	50.21048	96.68401
FOX	5941.533	12.90076	6.40964	41.77709	180.6599
MVO	6065.602	13.66465	6.769669	44.01309	154.4982
COOT	5885.348	12.45073	6.154388	40.31963	199.9999
ChOA	9613.718	15.25987	19.45149	40.34886	200

. This score is notably lower than most of the other competing algorithms, including SHIO and FVIM, which register scores of $6109.215$ and $5927.990$, respectively, indicating a more optimized solution by MPA. Additionally, HDPE closely rivals the scores of RTH and CSA and slightly outperforms WSO, which has a nearly identical score of $5885.912$. Among the rest, algorithms like RSO and ChOA exhibit significantly higher scores of $9442.609$ and $9613.718$, reflecting less optimal solutions compared with MPA. The low variability in the components optimized by HDPE (x1 through x4) compared with the higher variability seen in COA and MFO also highlights HDPE’s capability to consistently converge to efficient solutions.

In the pressure vessel design optimization problem, HDPE exhibits outstanding efficiency and consistency, achieving a minimal score of $5885.363$ with a mean very close to the minimum at $5885.421$ and an extremely narrow range extending up to a maximum of $5885.505$, as shown in

Table 7. Pressure vessel design optimization statistical comparison results.

Optimizer	Min	Mean	Max	Std
HDPE	5885.363	5885.421	5885.505	0.06051
SHIO	6109.215	6734.693	7419.164	618.9455
FVIM	5927.99	6002.53	6273.525	151.672
RTH	5888.662	6176.939	6423.756	217.2817
COA	6080.027	6280.306	6636.188	216.628
CSA	5894.422	5991.594	6203.661	133.1088
MFO	5981.271	6759.414	7319.001	685.775
RIME	5946.184	6404.484	7127.972	561.4591
WSO	5885.912	5886.467	5886.829	0.386159
RSO	9442.609	14771.4	20188.88	4282.191
SCA	6303.989	7303.449	7953.458	862.6954
GWO	5928.282	5960.048	6051.649	51.61522
WOA	5898.948	6325.344	6712.783	374.9533
PSO	6044.525	6754.617	7391.859	667.2422
RIME	5885.333	6248.188	6685.679	390.0295
BWO	6298.223	6572.598	6752.056	179.1332
FOX	5941.533	6457.356	7244.257	538.2857
MVO	6065.602	6309.868	6494.06	196.3284
COOT	5885.348	5995.756	6212.356	139.8568
ChOA	9613.718	15549.21	25397.64	6761.925

. This slight variation is underscored by an impressively low standard deviation of $0.06051$, signifying HDPE’s robust performance and ability to consistently find near-optimal solutions. Compared with other algorithms such as SHIO, which has a wider spread in results indicated by a standard deviation of $618.9455$, and even CSA with a deviation of $133.1088$, HDPE maintains good precision in optimization.

4.4. Spring Design Optimization

The optimization of tension/compression spring design is a quintessential problem in mechanical engineering that entails determining optimal dimensions to meet specific performance criteria while minimizing material usage. This section delineates the mathematical formulation of the optimization problem associated with the tension/compression spring design [117].

The design optimization problem is characterized by the objective of minimizing the spring’s weight, subject to constraints arising from the spring’s physical and geometrical properties. The design variables are constrained within predefined bounds to ensure practical feasibility and adherence to engineering standards [118].

The spring design optimization problem, illustrated in Figure 5, involves determining the optimal dimensions of a helical spring to achieve desired mechanical properties and performance. The objective can include minimizing the weight or material cost while ensuring that the spring can withstand specified loads without failing due to excessive stress or deformation. The mathematical model of the spring design problem is shown in Appendix D.

The objective is to find the set of design variables within the specified bounds that minimize the objective function while satisfying all the imposed constraints. This optimization problem is fundamental in the design and manufacturing of mechanical springs, facilitating the development of efficient, reliable, and cost-effective components [119].

In the spring design optimization problem, HDPE demonstrates an exceptional ability to refine the design to a near-optimal configuration with a best score of $0.012665$, as depicted in

Table 8. Spring design optimization comparison results.

Optimizer	Best Score	x1	x2	x3
HDPE	0.012665	0.051757	0.358354	11.19372
SHIO	0.012702	0.051	0.339952	12.36581
FVIM	0.012688	0.051	0.340346	12.33284
RTH	0.012673	0.051044	0.341404	12.24668
COA	0.012702	0.052279	0.370834	10.53267
CSA	0.012668	0.051899	0.361762	11.00061
MFO	0.012674	0.051	0.340366	12.31619
RIME	0.012907	0.051	0.336476	12.7483
WSO	0.012665	0.051791	0.359186	11.14589
SCA	0.013179	0.051	0.334209	13.16073
GWO	0.012686	0.051	0.340197	12.33711
WOA	0.012674	0.051	0.340366	12.31619
PSO	0.012691	0.051	0.340337	12.33682
RIME	0.012674	0.051003	0.340427	12.31205
BWO	0.012675	0.051	0.340355	12.31741
FOX	0.012667	0.052007	0.364411	10.85178
MVO	0.012851	0.051	0.33745	12.64155
COOT	0.012665	0.051762	0.358485	11.18613
ChOA	0.014101	0.051	0.318916	15

. This score positions HDPE as a leading optimizer, matched only by HDPE and marginally surpassed by the COOT, both delivering a score of $0.012665$ as well. The variables optimized by HDPE show it effectively balances the trade-offs between wire diameter, mean coil diameter, and active coils to achieve an efficient design. When compared with other algorithms like SHIO and FVIM, which record slightly higher scores of $0.012702$ and $0.012688$, respectively, HDPE illustrates good optimization efficiency. Particularly, HDPE’s performance is starkly better than that of RSO, which shows an anomalously high best score of 1122611, indicating a significant deviation from optimal design values.

In the spring design optimization problem, HDPE showcases outstanding optimization performance, achieving an extremely low minimum score of $0.012665$, a mean score of $0.012666$, and a maximum score of $0.012667$, as shown in

Table 9. Spring design optimization statistical comparison results.

Optimizer	Min	Mean	Max	Std
HDPE	0.012665	0.012666	0.012667	6.84E-07
SHIO	0.012702	0.013018	0.014065	5.86E-04
FVIM	0.012688	0.012728	0.012816	5.18E-05
RTH	0.012673	0.012725	0.012931	1.15E-04
COA	0.012702	0.013185	0.013594	4.37E-04
CSA	0.012668	0.012741	0.012840	6.90E-05
MFO	0.012674	0.013639	0.014310	7.37E-04
RIME	0.012907	0.013584	0.014083	4.65E-04
WSO	0.012665	0.012666	0.012666	1.57E-07
SCA	0.013179	0.013492	0.013945	2.81E-04
GWO	0.012686	0.012790	0.013150	2.02E-04
WOA	0.012674	0.013295	0.014227	8.41E-04
PSO	0.012691	0.012797	0.013041	1.42E-04
RIME	0.012674	0.012735	0.012844	8.03E-05
BWO	0.012675	0.012830	0.013232	2.37E-04
FOX	0.012667	0.013591	0.015045	9.73E-04
MVO	0.012851	0.017147	0.018500	2.41E-03
COOT	0.012665	0.012724	0.012845	7.56E-05
ChOA	0.014101	0.026870	0.077392	2.82E-02

. The exceptionally low standard deviation of $6.84\times {10}^{-7}$ indicates an extraordinary level of consistency and precision in finding the optimal solution across different runs. Compared with other algorithms, HDPE stands out for its ability to maintain near-perfect consistency; for instance, WSO, although matching the minimum score, demonstrates a slightly lesser consistency with a standard deviation of $1.57\times {10}^{-7}$. Notably, other algorithms, such as SHIO and FVIM, show higher variability in their results with standard deviations of $0.000586$ and $0.0000518$, respectively, reflecting a less stable performance.

4.5. Cantilever Beam Design Optimization

The design optimization of a cantilever beam focuses on finding the optimal dimensions to minimize the beam’s weight while ensuring it withstands specific loading conditions without surpassing the allowable deflection or stress limits. This section presents the mathematical formulation for the cantilever beam design optimization problem.

The objective of the optimization problem is to minimize the total weight of the cantilever beam, subject to constraints that ensure the beam’s structural integrity under prescribed loading conditions [120].

The cantilever beam design optimization problem, illustrated in Figure 6, involves a beam that is fixed at one end and free at the other. This setup is frequently used in engineering to evaluate structural integrity under various loads. The figure shows a beam divided into five sections, each with a constant thickness labeled $X_1$.

The optimization aims to determine the dimensions of the beam that minimize its weight while conforming to the deflection constraint, striking a balance between material efficiency and structural robustness. This optimization process is vital in ensuring the economic and structural efficiency of cantilever beam designs in various engineering applications [121]. The mathematical model of the cantilever beam is described in Appendix E.

In the Cantilever Beam Design Optimization challenge, the HDPE algorithm demonstrates exceptional precision, achieving one of the best scores of $1.339956$, as shown in

Table 10. Cantilever beam design optimization comparison results.

Optimizer	Best	x1	x2	x3	x4	x5
HDPE	1.339956	6.016017	5.309175	4.49433	3.501474	2.152664
APO	1.339956	6.016	5.309202	4.494349	3.501442	2.152667
CSA	1.339957	6.011903	5.310016	4.494671	3.504511	2.15257
FOX	1.339966	6.029922	5.298097	4.499491	3.496631	2.149675
GWO	1.339986	6.008206	5.314082	4.486563	3.495365	2.169925
AVOA	1.339999	6.04725	5.282769	4.486281	3.5107	2.147349
HGS	1.340061	5.999341	5.354903	4.482618	3.512575	2.125902
SMA	1.340038	6.059368	5.279012	4.472343	3.50423	2.16002
COOT	1.340079	6.063441	5.266629	4.51831	3.480422	2.146828
DBO	1.340301	6.004986	5.292142	4.583626	3.464709	2.133728
SA	1.340087	6.071833	5.31073	4.49119	3.461369	2.140635
HHO	1.34029	6.047777	5.296493	4.556366	3.481764	2.096603
AO	1.340614	6.083929	5.298421	4.441267	3.443507	2.217077
BWO	1.343529	6.201372	5.374855	4.531323	3.336224	2.087143
ChOA	1.354911	5.872229	5.762381	4.294268	3.63985	2.144588
SCA	1.384963	6.10422	5.618964	5.307663	3.067786	2.096291
GA	1.374527	5.915852	4.839187	5.072629	3.900756	2.299245
WOA	1.358327	5.579113	6.166904	4.481649	3.417939	2.122457
COA	1.380957	5.805822	5.860765	5.116042	2.866586	2.481502
RIME	1.377588	6.22025	6.422443	3.855527	3.42287	2.155643
BO	1.498807	6.997957	7.96097	4.537444	21.82205	15.10797
PSO	1.583287	4.849851	1.787312	2.774397	4.041981	3.876916

. This score is matched only by the Arctic Puffin Optimization (APO), suggesting a high level of efficacy in these algorithms for optimizing complex structural designs. Both HDPE and APO provide virtually identical and optimal beam dimensions, indicating their effective search capabilities in the solution space. In comparison, other algorithms like FOX and GWO achieve slightly higher scores, such as $1.339966$ and $1.339986$, respectively, pointing to a marginal but noticeable difference in optimization quality. Notably, the variance in results is significant across algorithms, with BWO, ChOA, and BO recording much higher best scores of $1.343529$, $1.354911$, and $1.498807$, respectively, illustrating the challenges faced in beam optimization.

In the cantilever beam design optimization problem, the HDPE algorithm achieves remarkable success, as evidenced in

Table 11. Cantilever beam design optimization statistical comparison results.

Optimizer	Mean	Std	Min	Max	Time	Rank
HDPE	1.339956	1.33E-13	1.339956	1.339956	0.460858	1
APO	1.339956	7.11E-10	1.339956	1.339956	1.108432	2
CSA	1.33997	9.55E-06	1.339957	1.33998	0.5681	3
FOX	1.339986	1.78E-05	1.339966	1.340004	0.601993	4
GWO	1.340089	0.000104	1.339986	1.340231	0.583104	5
AVOA	1.340102	0.000137	1.339999	1.340309	0.567114	6
HGS	1.340171	0.000141	1.340061	1.340417	0.513557	7
SMA	1.340177	9.5E-05	1.340038	1.340267	0.943501	8
COOT	1.340371	0.000306	1.340079	1.34086	0.567638	9
DBO	1.340504	0.000193	1.340301	1.340743	0.626738	10
SA	1.341671	0.001895	1.340087	1.344698	0.967117	11
HHO	1.342434	0.001321	1.34029	1.343388	1.209469	12
AO	1.342838	0.001561	1.340614	1.344942	1.067942	13
BWO	1.349759	0.003702	1.343529	1.353033	0.660017	14
ChOA	1.363435	0.00746	1.354911	1.371509	1.05763	15
SCA	1.399397	0.014482	1.384963	1.417514	0.483168	16
GA	1.404958	0.04426	1.374527	1.481258	0.569801	17
WOA	1.411489	0.032075	1.358327	1.437995	0.496565	18
COA	1.417339	0.053371	1.380957	1.50469	1.33091	19
RIME	1.606955	0.445104	1.377588	2.399886	0.990892	20
BO	1.830452	0.228787	1.498807	2.128332	5.04746	21
PSO	1.983959	0.238377	1.583287	2.164886	4.087807	22

. HDPE demonstrates exceptional accuracy and consistency, achieving the best performance with a minimum, mean, and maximum score, all tightly clustered at $1.339956$. The incredibly small standard deviation of $1.33\times {10}^{-13}$ further emphasizes the algorithm’s precision and stability across multiple runs. This performance is good when compared with other top contenders, such as the Arctic Puffin Optimization (APO), which, while matching the minimum and maximum values, shows a slightly higher standard deviation of $7.11\times {10}^{-10}$ and longer computation time. The other algorithms, like CSA and FOX, show slightly less optimized results with minimum values of $1.339957$ and $1.339966$, respectively, and higher standard deviations, indicating less consistency in achieving optimal results. The HDPE ranks first in terms of performance but also excels in efficiency with a relatively quick computation time of $0.460858$ s.

5. Comparative Analysis of HPDE

This section outlines the experiment conducted to assess the computational performance of the HPDE optimizer on established benchmark optimization problems. The outcomes of HPDE are comprehensively reviewed, analyzed, and contrasted with other prominent metaheuristic algorithms that have shown notable performance in academic studies.

5.1. IEEE Congress on Evolutionary Computation CEC2014 Benchmark Functions

The CEC2014 benchmark suite, introduced during the IEEE Congress on Evolutionary Computation in 2014, is an essential collection of test functions designed for the comprehensive evaluation of evolutionary algorithms and other optimization techniques. This suite provides a broad spectrum of functions characterized by various complexities and properties such as unimodality, multimodality, separability, and non-separability, catering to the need for assessing the adaptability and efficiency of algorithms under diverse operational conditions.

Designed to simulate real-world scenarios, the CEC2014 functions are crafted to challenge optimization algorithms on multiple fronts, including their capacity to effectively locate global optima, handle complex, high-dimensional spaces, and demonstrate robustness across various problem structures. The suite’s diverse array of functions ensures that it is an excellent tool for benchmarking algorithms intended for a wide range of practical applications, from engineering optimizations to complex data analysis tasks.

Each function in the suite targets specific strengths and vulnerabilities of optimization algorithms, evaluating attributes such as convergence rate, accuracy, escape from local optima, and sensitivity to initialization parameters. The suite encompasses both continuous and discrete problem types, reflecting the versatile application landscape of modern computational problems.

The functions are organized into distinct categories like unimodal, multimodal, and hybrid, each offering unique challenges that help in assessing the comprehensive performance of algorithms across different types of landscapes. Additionally, the suite includes composition functions that combine elements from several basic functions to create more intricate and challenging optimization environments.

5.2. Parameters for Algorithm Benchmarking

Ensuring consistent evaluation of optimization algorithms in benchmarking environments for CEC2014. Standard parameters are employed across these benchmarks to create a uniform testing ground where the efficacy of various evolutionary algorithms can be assessed. The details of these parameters are summarized in

Table 12. Parameter settings for CEC benchmark functions.

Parameter	Value
Population Size	30
Maximum Function Evaluations	1000
Dimensionality (D)	10
Search Range	${[-100,100]}^D$
Rotation	Applied to all rotated functions
Shift	Applied to all shifted functions

, whereas the parameter settings for compared algorithms are presented in

Table 13. Parameter settings for compared algorithms.

Algorithm	Parameter
CHoA	$r1=\mathrm{random}(0,1)$, $r2=\mathrm{random}(0,1)$, $r3=\mathrm{random}(0,1)$, $r4=\mathrm{random}(0,1)$
GWO	Convergence constant $a=[2,0]$
FOX	$c_{\mathrm{Max}}=1$, $c_{\mathrm{Min}}=0.00004$
WOA	Convergence constant $a=[2,0]$, $b=1$
MVO	$WE{P}_{\mathrm{Max}}=1$, $WE{P}_{\mathrm{Min}}=0.2$
DOA	Not provided
MFO	a linearly decreases from −1 to −2
RIME	$w=0.9$, $c=0.1$
DBO	$U=0.05$, $V=0.05$, $L=0.05$
WSO	a linearly decreases from −1 to −2

.

A population size of 30, along with a dimensionality of 10, balances the computational load with the complexity necessary for effective algorithm testing. Limiting the function evaluations to 1000 ensures that the algorithms can demonstrate their convergence capabilities without undue computational burden. The search parameter of ${[-100,100]}^D$ provides a comprehensive and consistent field for algorithmic operations.

Rotational and shifting manipulations are incorporated into the functions to elevate their complexity, thereby simulating more intricate optimization scenarios. By omitting noise, the focus remains squarely on the algorithms’ ability to navigate complex environments, enhancing the assessment of their robustness and efficacy. This structured approach ensures that comparisons between different algorithms are fair and highlight their respective strengths and weaknesses within a broad spectrum of benchmark settings.

This study involved a comprehensive assessment and comparison of the effectiveness of a wide range of optimization algorithms.

6. Quantitative Analysis of HPDE Performance

To assess the performance of hybrid HPDE in comparison with other optimizers, we applied key statistical tools: the mean, standard error of the mean, and the standard deviation. The mean, a central measure, captures the average result achieved by the algorithms over multiple trials, providing a comprehensive view of their standard operational levels. Additionally, the standard deviation assesses the spread of results around this mean, illuminating the consistency and reliability of the algorithms across different instances. These statistics are essential for appraising the algorithms’ stability and their performance predictability.

6.1. Statistical Results of HPDE with 51 Independent Runs on the IEEE CEC2014

The HPDE optimizer’s performance on the CEC2014 benchmark functions showcases significant strengths, as shown in

Table 14. Statistical results of HPDE with 51 independent runs on the CEC2014 (F1–F15) 30 Agent with 1000 FEs (first set of optimizers).

Fun	Analysis	HPDE	APO	ChOA	GWO	FOX	WOA	MVO	DOA	MFO	RIME	DBO	WSO
F1	Mean	1.13E+02	4.41E+04	7.75E+04	5.96E+07	2.72E+05	8.51E+06	2.10E+05	1.43E+05	1.50E+06	6.11E+02	4.76E+07	6.32E+02
	Std	2.47E+01	5.82E+04	7.97E+04	9.32E+07	5.97E+05	6.75E+06	1.26E+05	1.15E+05	1.19E+06	7.35E+02	2.94E+07	1.01E+03
	SEM	7.80E+00	1.84E+04	2.52E+04	2.95E+07	1.89E+05	2.13E+06	4.00E+04	3.65E+04	3.78E+05	2.32E+02	9.29E+06	3.19E+02
	Rank	1	4	5	12	8	10	7	6	9	2	11	3
F2	Mean	2.00E+02	7.56E+02	1.24E+03	7.95E+09	2.75E+03	5.51E+05	6.63E+03	4.22E+03	6.88E+03	2.07E+02	3.43E+09	2.09E+02
	Std	9.06E-11	1.24E+03	1.64E+03	1.98E+09	2.63E+03	2.30E+05	3.91E+03	4.88E+03	4.76E+03	1.46E+01	1.52E+09	1.17E+01
	SEM	2.87E-11	3.92E+02	5.17E+02	6.25E+08	8.33E+02	7.27E+04	1.24E+03	1.54E+03	1.51E+03	4.61E+00	4.82E+08	3.71E+00
	Rank	1	4	5	12	6	10	8	7	9	2	11	3
F3	Mean	3.00E+02	3.00E+02	3.49E+02	1.80E+04	8.29E+02	4.17E+04	3.64E+02	8.17E+02	1.48E+04	3.00E+02	1.08E+04	3.00E+02
	Std	0.00E+00	5.61E-02	4.18E+01	7.28E+03	7.24E+02	1.64E+04	3.69E+01	5.62E+02	7.75E+03	2.29E-01	2.36E+03	2.04E-02
	SEM	0.00E+00	1.77E-02	1.32E+01	2.30E+03	2.29E+02	5.17E+03	1.17E+01	1.78E+02	2.45E+03	7.23E-02	7.47E+02	6.44E-03
	Rank	1	3	5	11	8	12	6	7	10	4	9	2
F4	Mean	4.21E+02	4.35E+02	4.35E+02	1.86E+03	4.32E+02	4.35E+02	4.30E+02	4.19E+02	4.25E+02	4.22E+02	1.84E+03	4.14E+02
	Std	1.10E+01	7.82E-07	1.23E-13	1.40E+03	1.61E+01	2.27E+01	1.36E+01	1.65E+01	1.58E+01	1.63E+01	6.18E+02	1.76E+01
	SEM	3.47E+00	2.47E-07	3.88E-14	4.42E+02	5.09E+00	7.19E+00	4.32E+00	5.23E+00	4.99E+00	5.16E+00	1.95E+02	5.58E+00
	Rank	3	9	8	12	7	10	6	2	5	4	11	1
F5	Mean	5.18E+02	5.18E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02
	Std	6.33E+00	5.37E+00	6.66E-02	7.76E-02	2.36E-02	9.84E-02	1.56E-02	1.82E-02	1.35E-01	6.14E-02	8.93E-02	8.84E-02
	SEM	2.00E+00	1.70E+00	2.11E-02	2.45E-02	7.46E-03	3.11E-02	4.94E-03	5.76E-03	4.28E-02	1.94E-02	2.82E-02	2.80E-02
	Rank	1	2	9	10	3	8	5	4	6	7	11	12
F6	Mean	6.00E+02	6.00E+02	6.01E+02	6.10E+02	6.10E+02	6.07E+02	6.02E+02	6.05E+02	6.04E+02	6.05E+02	6.06E+02	6.01E+02
	Std	6.19E-01	4.26E-01	1.37E+00	2.66E+00	1.48E+00	1.85E+00	1.49E+00	2.28E+00	1.70E+00	2.18E+00	1.11E+00	1.03E+00
	SEM	1.96E-01	1.35E-01	4.34E-01	8.43E-01	4.68E-01	5.84E-01	4.70E-01	7.21E-01	5.37E-01	6.88E-01	3.51E-01	3.26E-01
	Rank	1	2	4	11	12	10	5	8	6	7	9	3
F7	Mean	7.00E+02	7.00E+02	7.00E+02	8.45E+02	7.00E+02	7.01E+02	7.00E+02	7.00E+02	7.01E+02	7.00E+02	8.88E+02	7.00E+02
	Std	1.24E-02	1.31E-02	6.64E-02	8.14E+01	3.30E-01	4.64E-01	1.53E-01	1.35E-01	1.21E+00	1.88E-01	4.96E+01	4.11E-02
	SEM	3.92E-03	4.15E-03	2.10E-02	2.57E+01	1.04E-01	1.47E-01	4.84E-02	4.27E-02	3.82E-01	5.96E-02	1.57E+01	1.30E-02
	Rank	1	2	4	11	8	10	7	5	9	6	12	3
F8	Mean	8.00E+02	8.00E+02	8.06E+02	8.92E+02	8.39E+02	8.39E+02	8.14E+02	8.07E+02	8.20E+02	8.21E+02	8.61E+02	8.04E+02
	Std	1.10E-09	3.94E-05	2.47E+00	1.43E+01	1.55E+01	1.27E+01	2.32E+00	4.80E+00	7.39E+00	4.39E+00	9.53E+00	1.44E+00
	SEM	3.48E-10	1.25E-05	7.82E-01	4.51E+00	4.90E+00	4.02E+00	7.33E-01	1.52E+00	2.34E+00	1.39E+00	3.01E+00	4.55E-01
	Rank	1	2	4	12	10	9	6	5	7	8	11	3
F9	Mean	9.03E+02	9.03E+02	9.13E+02	9.71E+02	9.58E+02	9.58E+02	9.16E+02	9.24E+02	9.29E+02	9.33E+02	9.55E+02	9.08E+02
	Std	1.17E+00	9.88E-01	4.60E+00	1.73E+01	1.54E+01	1.92E+01	5.46E+00	7.28E+00	1.06E+01	1.16E+01	5.55E+00	2.97E+00
	SEM	3.71E-01	3.12E-01	1.45E+00	5.47E+00	4.87E+00	6.07E+00	1.73E+00	2.30E+00	3.34E+00	3.68E+00	1.76E+00	9.40E-01
	Rank	1	2	4	12	10	11	5	6	7	8	9	3
F10	Mean	1.04E+03	1.04E+03	1.14E+03	2.00E+03	2.20E+03	1.53E+03	1.31E+03	1.19E+03	1.30E+03	1.54E+03	2.55E+03	1.56E+03
	Std	8.95E+00	1.01E+01	8.90E+01	3.83E+02	2.66E+02	3.29E+02	2.42E+02	1.12E+02	1.44E+02	2.80E+02	1.82E+02	4.05E+02
	SEM	2.83E+00	3.20E+00	2.82E+01	1.21E+02	8.40E+01	1.04E+02	7.66E+01	3.53E+01	4.54E+01	8.86E+01	5.74E+01	1.28E+02
	Rank	1	2	3	10	11	7	6	4	5	8	12	9
F11	Mean	1.24E+03	1.29E+03	1.59E+03	2.51E+03	2.53E+03	2.05E+03	1.75E+03	1.93E+03	2.16E+03	2.02E+03	2.66E+03	1.38E+03
	Std	1.16E+02	9.07E+01	3.23E+02	3.43E+02	3.52E+02	3.14E+02	2.82E+02	2.17E+02	3.06E+02	2.36E+02	1.02E+02	3.53E+02
	SEM	3.65E+01	2.87E+01	1.02E+02	1.08E+02	1.11E+02	9.92E+01	8.93E+01	6.87E+01	9.69E+01	7.46E+01	3.22E+01	1.12E+02
	Rank	1	2	4	10	11	8	5	6	9	7	12	3
F12	Mean	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03
	Std	4.00E-02	3.29E-02	1.15E-01	4.84E-01	6.83E-01	4.60E-01	1.34E-01	2.49E-01	1.88E-01	1.84E-01	2.92E-01	3.86E-01
	SEM	1.26E-02	1.04E-02	3.63E-02	1.53E-01	2.16E-01	1.45E-01	4.25E-02	7.89E-02	5.94E-02	5.83E-02	9.22E-02	1.22E-01
	Rank	1	3	4	10	9	8	2	6	7	5	12	11
F13	Mean	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03
	Std	1.65E-02	1.38E-02	1.61E-02	1.19E+00	2.74E-01	2.00E-01	7.63E-02	1.95E-01	1.01E-01	7.11E-02	6.16E-01	6.08E-02
	SEM	5.22E-03	4.38E-03	5.08E-03	3.75E-01	8.65E-02	6.32E-02	2.41E-02	6.18E-02	3.21E-02	2.25E-02	1.95E-01	1.92E-02
	Rank	1	2	3	11	9	10	5	8	7	6	12	4
F14	Mean	1.40E+03	1.40E+03	1.40E+03	1.43E+03	1.40E+03	1.40E+03	1.40E+03	1.40E+03	1.40E+03	1.40E+03	1.44E+03	1.40E+03
	Std	3.81E-02	5.87E-02	6.07E-02	1.53E+01	2.36E-01	1.68E-01	4.70E-02	1.84E-01	2.58E-01	1.66E-01	7.05E+00	7.11E-02
	SEM	1.20E-02	1.86E-02	1.92E-02	4.85E+00	7.45E-02	5.30E-02	1.49E-02	5.82E-02	8.15E-02	5.24E-02	2.23E+00	2.25E-02
	Rank	1	5	3	11	10	7	2	9	8	6	12	4
F15	Mean	1.50E+03	1.50E+03	1.50E+03	1.68E+04	1.51E+03	1.51E+03	1.50E+03	1.50E+03	1.50E+03	1.50E+03	3.61E+03	1.50E+03
	Std	1.10E-01	1.11E-01	2.41E-01	1.81E+04	2.21E+00	2.70E+00	5.40E-01	1.11E+00	6.79E-01	7.24E-01	1.64E+03	6.74E-01
	SEM	3.48E-02	3.51E-02	7.62E-02	5.73E+03	7.00E-01	8.54E-01	1.71E-01	3.51E-01	2.15E-01	2.29E-01	5.19E+02	2.13E-01
	Rank	1	2	3	12	9	10	4	8	5	7	11	6

and

Table 15. Statistical results of HPDE with 51 independent runs on the CEC2014 (F16–F30) 30 Agent with 1000 FEs (first set of optimizers).

Fun	Analysis	HPDE	APO	ChOA	GWO	FOX	WOA	MVO	DOA	MFO	RIME	DBO	WSO
F16	Mean	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03
	Std	4.41E-01	4.60E-01	7.25E-01	3.50E-01	5.10E-01	3.32E-01	4.34E-01	3.32E-01	3.70E-01	5.24E-01	2.26E-01	2.55E-01
	SEM	1.39E-01	1.45E-01	2.29E-01	1.11E-01	1.61E-01	1.05E-01	1.37E-01	1.05E-01	1.17E-01	1.66E-01	7.15E-02	8.07E-02
	Rank	1	2	3	11	12	8	6	7	9	5	10	4
F17	Mean	1.72E+03	1.81E+03	3.45E+03	6.91E+05	2.88E+03	5.84E+04	6.39E+03	6.73E+03	4.65E+04	2.13E+03	1.89E+05	1.82E+03
	Std	1.46E+01	7.74E+01	1.95E+03	3.14E+05	6.87E+02	8.28E+04	5.41E+03	3.34E+03	6.12E+04	2.53E+02	1.59E+05	6.40E+01
	SEM	4.62E+00	2.45E+01	6.16E+02	9.93E+04	2.17E+02	2.62E+04	1.71E+03	1.05E+03	1.93E+04	8.00E+01	5.04E+04	2.02E+01
	Rank	1	2	6	12	5	10	7	8	9	4	11	3
F18	Mean	1.80E+03	1.80E+03	5.84E+03	2.81E+05	1.92E+03	2.28E+04	1.51E+04	6.23E+03	1.20E+04	1.84E+03	1.57E+04	1.81E+03
	Std	3.50E-01	1.92E+00	3.25E+03	4.05E+05	5.42E+01	1.47E+04	9.37E+03	5.60E+03	1.16E+04	2.71E+01	6.57E+03	8.88E+00
	SEM	1.11E-01	6.08E-01	1.03E+03	1.28E+05	1.71E+01	4.66E+03	2.96E+03	1.77E+03	3.66E+03	8.56E+00	2.08E+03	2.81E+00
	Rank	1	2	6	12	5	11	9	7	8	4	10	3
F19	Mean	1.90E+03	1.90E+03	1.90E+03	1.94E+03	1.91E+03	1.91E+03	1.90E+03	1.90E+03	1.90E+03	1.90E+03	1.93E+03	1.90E+03
	Std	4.37E-01	4.28E-01	8.00E-01	2.71E+01	1.35E+00	1.30E+00	6.97E-01	7.12E-01	9.13E-01	1.66E+00	1.46E+01	9.01E-01
	SEM	1.38E-01	1.35E-01	2.53E-01	8.57E+00	4.28E-01	4.11E-01	2.20E-01	2.25E-01	2.89E-01	5.26E-01	4.60E+00	2.85E-01
	Rank	2	1	4	12	10	9	6	5	7	8	11	3
F20	Mean	2.00E+03	2.00E+03	2.12E+03	2.52E+05	2.36E+03	7.19E+03	2.18E+03	2.84E+03	1.17E+04	2.02E+03	6.79E+03	2.01E+03
	Std	3.86E-01	3.03E-01	9.84E+01	3.47E+05	2.89E+02	4.31E+03	4.24E+02	1.44E+03	1.18E+04	1.89E+01	2.18E+03	5.22E+00
	SEM	1.22E-01	9.60E-02	3.11E+01	1.10E+05	9.14E+01	1.36E+03	1.34E+02	4.56E+02	3.72E+03	5.97E+00	6.90E+02	1.65E+00
	Rank	2	1	5	12	7	10	6	8	11	4	9	3
F21	Mean	2.10E+03	2.10E+03	2.39E+03	7.97E+05	2.68E+03	7.24E+04	5.28E+03	5.74E+03	3.98E+03	2.40E+03	6.27E+04	2.12E+03
	Std	1.90E-01	1.67E+00	2.04E+02	1.75E+06	3.27E+02	1.31E+05	2.50E+03	2.13E+03	2.19E+03	2.46E+02	4.63E+04	2.79E+01
	SEM	6.02E-02	5.27E-01	6.45E+01	5.55E+05	1.03E+02	4.13E+04	7.91E+02	6.74E+02	6.93E+02	7.77E+01	1.46E+04	8.81E+00
	Rank	1	2	4	12	6	11	8	9	7	5	10	3
F22	Mean	2.20E+03	2.20E+03	2.22E+03	2.57E+03	2.46E+03	2.30E+03	2.31E+03	2.31E+03	2.24E+03	2.23E+03	2.38E+03	2.24E+03
	Std	6.27E+00	6.20E+00	9.37E+00	1.50E+02	1.54E+02	7.16E+01	1.09E+02	6.90E+01	1.16E+01	7.73E+00	6.34E+01	2.96E+01
	SEM	1.98E+00	1.96E+00	2.96E+00	4.75E+01	4.87E+01	2.26E+01	3.45E+01	2.18E+01	3.66E+00	2.44E+00	2.01E+01	9.37E+00
	Rank	1	2	3	12	11	7	9	8	6	4	10	5
F23	Mean	2.62E+03	2.63E+03	2.63E+03	2.71E+03	2.50E+03	2.63E+03	2.63E+03	2.63E+03	2.63E+03	2.50E+03	2.50E+03	2.63E+03
	Std	4.79E-13	4.26E-11	6.06E-13	9.43E+01	0.00E+00	3.05E+00	1.03E-03	5.61E-04	5.27E+00	0.00E+00	0.00E+00	2.00E-06
	SEM	1.52E-13	1.35E-11	1.92E-13	2.98E+01	0.00E+00	9.65E-01	3.24E-04	1.77E-04	1.67E+00	0.00E+00	0.00E+00	6.34E-07
	Rank	4	6	5	12	1	10	9	8	11	1	1	7
F24	Mean	2.51E+03	2.51E+03	2.53E+03	2.60E+03	2.58E+03	2.57E+03	2.53E+03	2.54E+03	2.53E+03	2.57E+03	2.59E+03	2.51E+03
	Std	1.71E+00	4.71E+00	2.69E+01	1.75E+01	2.35E+01	2.88E+01	6.56E+00	2.64E+01	1.15E+01	3.66E+01	2.03E+01	5.85E+00
	SEM	5.41E-01	1.49E+00	8.49E+00	5.54E+00	7.42E+00	9.12E+00	2.07E+00	8.35E+00	3.62E+00	1.16E+01	6.42E+00	1.85E+00
	Rank	2	1	5	12	10	9	4	7	6	8	11	3
F25	Mean	2.64E+03	2.65E+03	2.69E+03	2.70E+03	2.70E+03	2.69E+03	2.69E+03	2.69E+03	2.70E+03	2.68E+03	2.70E+03	2.66E+03
	Std	3.75E+01	3.54E+01	2.39E+01	7.28E-01	1.52E+00	1.18E+01	3.32E+01	1.11E+01	8.76E+00	2.92E+01	7.60E-01	2.63E+01
	SEM	1.19E+01	1.12E+01	7.57E+00	2.30E-01	4.80E-01	3.72E+00	1.05E+01	3.52E+00	2.77E+00	9.22E+00	2.40E-01	8.33E+00
	Rank	1	2	6	12	10	7	5	8	9	4	11	3
F26	Mean	2.70E+03	2.70E+03	2.70E+03	2.70E+03	2.70E+03	2.70E+03	2.70E+03	2.70E+03	2.70E+03	2.70E+03	2.71E+03	2.70E+03
	Std	2.17E-02	1.84E-02	1.31E-02	3.20E+00	2.12E-01	1.41E-01	6.23E-02	1.20E-01	1.24E-01	6.70E-02	1.40E+01	5.03E-02
	SEM	6.86E-03	5.82E-03	4.15E-03	1.01E+00	6.70E-02	4.46E-02	1.97E-02	3.80E-02	3.94E-02	2.12E-02	4.42E+00	1.59E-02
	Rank	1	3	2	11	10	9	5	8	7	6	12	4
F27	Mean	2.85E+03	2.96E+03	2.98E+03	3.18E+03	2.90E+03	3.11E+03	3.04E+03	3.06E+03	3.07E+03	2.80E+03	2.76E+03	2.94E+03
	Std	1.59E+02	1.38E+02	1.50E+02	7.08E+01	0.00E+00	1.58E+02	1.24E+02	1.29E+02	1.29E+02	1.03E+02	1.86E+01	1.64E+02
	SEM	5.04E+01	4.37E+01	4.75E+01	2.24E+01	0.00E+00	5.01E+01	3.92E+01	4.08E+01	4.09E+01	3.27E+01	5.87E+00	5.20E+01
	Rank	3	6	7	12	4	11	8	9	10	2	1	5
F28	Mean	3.18E+03	3.18E+03	3.19E+03	3.86E+03	3.00E+03	3.35E+03	3.25E+03	3.28E+03	3.19E+03	3.00E+03	3.27E+03	3.22E+03
	Std	3.50E+00	4.18E+00	3.23E+01	4.68E+02	0.00E+00	1.65E+02	6.43E+01	4.44E+01	1.12E+01	0.00E+00	1.99E+02	1.21E+02
	SEM	1.11E+00	1.32E+00	1.02E+01	1.48E+02	0.00E+00	5.21E+01	2.03E+01	1.40E+01	3.53E+00	0.00E+00	6.28E+01	3.83E+01
	Rank	4	3	5	12	1	11	8	10	6	1	9	7
F29	Mean	3.13E+03	3.21E+03	3.80E+05	1.22E+06	3.42E+03	4.04E+03	2.00E+05	4.01E+03	4.05E+03	3.28E+03	5.13E+03	3.13E+03
	Std	5.71E+00	1.03E+02	7.98E+05	2.34E+06	3.17E+02	5.06E+02	6.21E+05	7.06E+02	5.82E+02	1.81E+02	3.67E+03	1.45E+01
	SEM	1.81E+00	3.25E+01	2.52E+05	7.41E+05	1.00E+02	1.60E+02	1.96E+05	2.23E+02	1.84E+02	5.72E+01	1.16E+03	4.59E+00
	Rank	1	3	11	12	5	7	10	6	8	4	9	2
F30	Mean	3.48E+03	3.50E+03	3.73E+03	3.44E+04	4.66E+03	5.05E+03	4.05E+03	4.71E+03	3.98E+03	3.81E+03	5.08E+03	3.59E+03
	Std	2.26E+01	2.32E+01	3.53E+02	4.52E+04	6.15E+02	9.79E+02	5.11E+02	6.09E+02	2.51E+02	3.21E+02	1.46E+03	8.82E+01
	SEM	7.15E+00	7.34E+00	1.12E+02	1.43E+04	1.94E+02	3.10E+02	1.62E+02	1.92E+02	7.92E+01	1.02E+02	4.60E+02	2.79E+01
	Rank	1	2	4	12	8	10	7	9	6	5	11	3

, consistently ranking among the top performers with low mean values that indicate an efficient approach to the global optimum. Its robustness is evidenced by low standard deviations and standard errors, highlighting its reliability and consistency across multiple runs. The successful hybridization of the Artificial Protozoa Optimizer (APO) and Differential Evolution (DE) in HPDE enhances its capabilities; APO contributes diverse explorative strategies, while DE enhances exploitation, making HPDE good in balancing exploration and exploitation compared with algorithms like the Grey Wolf Optimizer (GWO) and the Whale Optimization Algorithm (WOA). Additionally, HPDE competes well against more established algorithms like Genetic Algorithms, excelling in unimodal functions through effective exploitation and demonstrating strong performance in multimodal and composite functions by avoiding premature convergence and dynamically adapting strategies.

The error measurement results, as shown in

Table 16. Error measurement comparison results of HPDE with the first group of optimizers over CEC2014 (F1–F30) benchmark functions (first set of optimizers).

Fun	HPDE	APO	ChOA	GWO	FOX	WOA	MVO	DOA	MFO	RIME	DBO	WSO
F1	1.32E+01	4.40E+04	7.74E+04	5.96E+07	2.72E+05	8.51E+06	2.10E+05	1.43E+05	1.50E+06	5.11E+02	4.76E+07	5.32E+02
F2	3.38E-11	5.56E+02	1.04E+03	7.95E+09	2.55E+03	5.50E+05	6.43E+03	4.02E+03	6.68E+03	6.84E+00	3.43E+09	8.99E+00
F3	0.00E+00	1.78E-02	4.93E+01	1.77E+04	5.29E+02	4.14E+04	6.45E+01	5.17E+02	1.45E+04	9.34E-02	1.05E+04	6.63E-03
F4	2.13E+01	3.48E+01	3.48E+01	1.46E+03	3.20E+01	3.54E+01	3.03E+01	1.92E+01	2.50E+01	2.22E+01	1.44E+03	1.43E+01
F5	1.80E+01	1.83E+01	2.02E+01	2.04E+01	2.00E+01	2.01E+01	2.00E+01	2.00E+01	2.01E+01	2.01E+01	2.04E+01	2.04E+01
F6	3.67E-01	3.82E-01	1.24E+00	9.88E+00	1.01E+01	6.94E+00	1.91E+00	5.29E+00	4.02E+00	5.02E+00	6.21E+00	8.66E-01
F7	1.03E-02	1.45E-02	6.25E-02	1.45E+02	4.47E-01	1.03E+00	3.09E-01	1.96E-01	5.30E-01	2.22E-01	1.88E+02	6.20E-02
F8	3.93E-10	1.96E-05	5.77E+00	9.17E+01	3.91E+01	3.88E+01	1.40E+01	6.77E+00	2.04E+01	2.07E+01	6.14E+01	3.99E+00
F9	2.57E+00	3.13E+00	1.29E+01	7.11E+01	5.77E+01	5.80E+01	1.62E+01	2.45E+01	2.93E+01	3.28E+01	5.51E+01	7.56E+00
F10	3.57E+01	3.85E+01	1.44E+02	9.96E+02	1.20E+03	5.27E+02	3.08E+02	1.87E+02	2.99E+02	5.36E+02	1.55E+03	5.64E+02
F11	1.43E+02	1.95E+02	4.85E+02	1.41E+03	1.43E+03	9.48E+02	6.50E+02	8.30E+02	1.06E+03	9.21E+02	1.56E+03	2.78E+02
F12	1.68E-01	2.47E-01	2.49E-01	1.27E+00	1.09E+00	9.90E-01	1.73E-01	3.25E-01	3.25E-01	2.68E-01	1.55E+00	1.36E+00
F13	4.30E-02	4.62E-02	5.50E-02	3.45E+00	3.83E-01	4.46E-01	1.80E-01	3.64E-01	2.84E-01	2.11E-01	4.14E+00	1.65E-01
F14	2.17E-01	2.89E-01	2.57E-01	3.08E+01	3.96E-01	3.36E-01	2.21E-01	3.58E-01	3.47E-01	2.91E-01	3.84E+01	2.77E-01
F15	5.13E-01	5.43E-01	9.64E-01	1.53E+04	5.65E+00	6.07E+00	1.27E+00	2.11E+00	1.43E+00	1.88E+00	2.11E+03	1.43E+00
F16	1.72E+00	1.77E+00	1.91E+00	3.75E+00	3.82E+00	3.32E+00	3.04E+00	3.12E+00	3.41E+00	2.64E+00	3.45E+00	2.27E+00
F17	1.75E+01	1.09E+02	1.75E+03	6.90E+05	1.18E+03	5.67E+04	4.69E+03	5.03E+03	4.48E+04	4.31E+02	1.87E+05	1.25E+02
F18	4.88E-01	2.66E+00	4.04E+03	2.79E+05	1.22E+02	2.10E+04	1.33E+04	4.43E+03	1.02E+04	3.71E+01	1.39E+04	1.39E+01
F19	8.42E-01	7.00E-01	1.71E+00	3.56E+01	6.34E+00	5.78E+00	2.45E+00	2.31E+00	2.60E+00	2.61E+00	2.80E+01	1.34E+00
F20	4.87E-01	4.82E-01	1.24E+02	2.50E+05	3.61E+02	5.19E+03	1.79E+02	8.44E+02	9.72E+03	2.48E+01	4.79E+03	7.25E+00
F21	6.36E-01	1.59E+00	2.86E+02	7.94E+05	5.80E+02	7.03E+04	3.18E+03	3.64E+03	1.88E+03	2.99E+02	6.06E+04	1.79E+01
F22	2.81E+00	3.22E+00	1.76E+01	3.68E+02	2.59E+02	1.05E+02	1.14E+02	1.07E+02	4.09E+01	2.65E+01	1.80E+02	3.74E+01
F23	3.20E+02	3.29E+02	3.29E+02	4.10E+02	2.00E+02	3.34E+02	3.29E+02	3.29E+02	3.34E+02	2.00E+02	2.00E+02	3.29E+02
F24	1.09E+02	1.08E+02	1.25E+02	2.01E+02	1.84E+02	1.74E+02	1.25E+02	1.40E+02	1.34E+02	1.66E+02	1.87E+02	1.12E+02
F25	1.43E+02	1.51E+02	1.88E+02	2.00E+02	2.00E+02	1.93E+02	1.86E+02	1.93E+02	1.98E+02	1.82E+02	2.00E+02	1.59E+02
F26	1.00E+02	1.00E+02	1.00E+02	1.05E+02	1.00E+02	1.00E+02	1.00E+02	1.00E+02	1.00E+02	1.00E+02	1.06E+02	1.00E+02
F27	1.53E+02	2.59E+02	2.80E+02	4.81E+02	2.00E+02	4.10E+02	3.39E+02	3.59E+02	3.72E+02	1.02E+02	5.69E+01	2.37E+02
F28	3.79E+02	3.78E+02	3.89E+02	1.06E+03	2.00E+02	5.54E+02	4.53E+02	4.82E+02	3.90E+02	2.00E+02	4.71E+02	4.18E+02
F29	2.25E+02	3.11E+02	3.77E+05	1.22E+06	5.18E+02	1.14E+03	1.97E+05	1.11E+03	1.15E+03	3.82E+02	2.23E+03	2.29E+02
F30	4.81E+02	5.01E+02	7.32E+02	3.14E+04	1.66E+03	2.05E+03	1.05E+03	1.71E+03	9.80E+02	8.08E+02	2.08E+03	5.89E+02

for the HPDE optimizer over CEC2014 benchmark functions (F1–F30), underscore HPDE’s good performance. Demonstrating remarkably low mean errors, HPDE outperforms competitors like GWO, DBO, and APO, especially in complex multimodal and high-dimensional scenarios such as functions F17, F18, and F29. For instance, in F17, where other algorithms show errors ranging from hundreds to hundreds of thousands, HPDE maintains a much lower error, illustrating its effective global optimization capability and ability to avoid local minima. In simpler function scenarios like F1 through F6, HPDE often achieves near-zero errors, showcasing outstanding exploitation capabilities. This consistent performance across various function types, from unimodal to multimodal and separable to non-separable, indicates HPDE’s adaptability and reliability. These attributes mark HPDE as a versatile and efficient optimization tool suitable for a broad range of applications requiring precise and robust solutions, outshining other algorithms that exhibit higher variability and less consistent performance across the benchmark suite.

HPDE optimizer showcases good performance across two sets of comparative analyses over the CEC2014 benchmark functions (F1–F30), as shown in

Table 17. Statistical results of HPDE with 51 independent runs on the CEC2014 (F1–F15) 30 Agent with 1000 FEs second group of optimizers (second set of optimizers).

Fun	Analysis	HPDE	RTH	SHIO	ChOA	COA	OHO	ChOA	SCA	PSO	SHO	SDE	WOA	RSA
F1	Mean	1.13E+02	1.00E+02	6.91E+06	4.55E+05	9.18E+04	3.94E+08	1.76E+07	8.59E+06	4.83E+06	1.17E+07	1.08E+02	1.83E+03	1.17E+08
	Std	2.47E+01	2.46E-01	3.65E+06	2.21E+05	6.45E+04	1.73E+08	1.26E+07	4.47E+06	2.16E+06	4.40E+06	4.98E+00	1.89E+03	5.24E+07
	SEM	7.80E+00	7.79E-02	1.15E+06	7.00E+04	2.04E+04	5.46E+07	3.98E+06	1.41E+06	6.84E+05	1.39E+06	1.58E+00	5.97E+02	1.66E+07
	Rank	3	1	8	6	5	13	11	9	7	10	2	4	12
F2	Mean	2.00E+02	2.00E+02	9.30E+07	3.76E+02	2.17E+03	7.14E+09	4.08E+09	5.90E+08	3.48E+08	3.19E+06	5.28E+02	2.08E+02	6.46E+09
	Std	9.06E-11	1.28E-11	2.94E+08	1.36E+02	2.22E+03	1.17E+09	2.17E+09	2.65E+08	4.37E+08	3.61E+06	3.09E+02	2.39E+01	9.90E+08
	SEM	2.87E-11	4.04E-12	9.30E+07	4.29E+01	7.02E+02	3.70E+08	6.87E+08	8.39E+07	1.38E+08	1.14E+06	9.77E+01	7.55E+00	3.13E+08
	Rank	2	1	8	4	6	13	11	10	9	7	5	3	12
F3	Mean	3.00E+02	3.00E+02	1.10E+04	7.34E+03	5.17E+03	1.24E+04	2.01E+04	3.28E+03	6.82E+03	6.14E+03	3.00E+02	3.00E+02	8.75E+03
	Std	0.00E+00	9.72E-11	3.60E+03	2.75E+03	3.65E+03	1.56E+03	5.28E+03	1.93E+03	4.56E+03	3.19E+03	1.59E-01	2.10E-02	3.33E+03
	SEM	0.00E+00	3.07E-11	1.14E+03	8.71E+02	1.15E+03	4.95E+02	1.67E+03	6.12E+02	1.44E+03	1.01E+03	5.02E-02	6.64E-03	1.05E+03
	Rank	1	2	11	9	6	12	13	5	8	7	4	3	10
F4	Mean	4.21E+02	4.25E+02	4.44E+02	4.35E+02	4.18E+02	3.23E+03	1.22E+03	4.58E+02	4.21E+02	4.59E+02	4.05E+02	4.28E+02	1.18E+03
	Std	1.10E+01	1.61E+01	3.15E+01	2.71E-01	1.73E+01	1.10E+03	6.51E+02	1.40E+01	1.30E+01	3.44E+01	1.07E+01	1.47E+01	4.94E+02
	SEM	3.47E+00	5.11E+00	9.95E+00	8.57E-02	5.49E+00	3.47E+02	2.06E+02	4.43E+00	4.11E+00	1.09E+01	3.39E+00	4.64E+00	1.56E+02
	Rank	4	5	8	7	2	13	12	9	3	10	1	6	11
F5	Mean	5.18E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02	5.20E+02
	Std	6.33E+00	1.23E-01	7.20E-02	1.75E-01	6.99E-02	8.16E-02	3.18E-02	8.56E-02	1.13E-01	3.35E-02	7.16E-02	8.13E-02	9.87E-02
	SEM	2.00E+00	3.89E-02	2.28E-02	5.53E-02	2.21E-02	2.58E-02	1.01E-02	2.71E-02	3.56E-02	1.06E-02	2.26E-02	2.57E-02	3.12E-02
	Rank	1	5	13	7	3	9	4	12	11	2	10	6	8
F6	Mean	6.00E+02	6.06E+02	6.04E+02	6.00E+02	6.05E+02	6.11E+02	6.09E+02	6.07E+02	6.05E+02	6.05E+02	6.02E+02	6.04E+02	6.10E+02
	Std	6.19E-01	1.44E+00	1.45E+00	7.55E-01	2.46E+00	3.97E-01	1.17E+00	1.42E+00	1.06E+00	1.43E+00	1.02E+00	1.35E+00	8.94E-01
	SEM	1.96E-01	4.55E-01	4.59E-01	2.39E-01	7.77E-01	1.25E-01	3.71E-01	4.50E-01	3.34E-01	4.52E-01	3.24E-01	4.26E-01	2.83E-01
	Rank	1	9	5	2	7	13	11	10	6	8	3	4	12
F7	Mean	7.00E+02	7.00E+02	7.01E+02	7.00E+02	7.00E+02	8.98E+02	7.86E+02	7.11E+02	7.06E+02	7.07E+02	7.00E+02	7.00E+02	7.84E+02
	Std	1.24E-02	1.24E-01	1.05E+00	2.76E-01	3.41E-02	5.78E+01	3.26E+01	3.02E+00	4.21E+00	9.33E+00	9.45E-02	1.80E-01	2.58E+01
	SEM	3.92E-03	3.92E-02	3.30E-01	8.74E-02	1.08E-02	1.83E+01	1.03E+01	9.55E-01	1.33E+00	2.95E+00	2.99E-02	5.70E-02	8.17E+00
	Rank	1	3	7	6	2	13	12	10	8	9	4	5	11
F8	Mean	8.00E+02	8.19E+02	8.27E+02	8.19E+02	8.03E+02	8.70E+02	8.33E+02	8.42E+02	8.18E+02	8.08E+02	8.09E+02	8.08E+02	8.72E+02
	Std	1.10E-09	7.59E+00	9.20E+00	1.07E+01	2.60E+00	4.88E+00	1.08E+01	5.99E+00	8.66E+00	1.05E+01	5.19E+00	2.93E+00	6.74E+00
	SEM	3.48E-10	2.40E+00	2.91E+00	3.39E+00	8.21E-01	1.54E+00	3.41E+00	1.89E+00	2.74E+00	3.33E+00	1.64E+00	9.28E-01	2.13E+00
	Rank	1	8	9	7	2	12	10	11	6	3	5	4	13
F9	Mean	9.03E+02	9.29E+02	9.18E+02	9.14E+02	9.40E+02	9.63E+02	9.39E+02	9.44E+02	9.32E+02	9.27E+02	9.09E+02	9.19E+02	9.59E+02
	Std	1.17E+00	1.16E+01	8.79E+00	1.15E+01	1.22E+01	3.79E+00	1.32E+01	5.60E+00	9.63E+00	4.13E+00	3.50E+00	7.18E+00	4.03E+00
	SEM	3.71E-01	3.67E+00	2.78E+00	3.65E+00	3.85E+00	1.20E+00	4.17E+00	1.77E+00	3.05E+00	1.31E+00	1.11E+00	2.27E+00	1.27E+00
	Rank	1	7	4	3	10	13	9	11	8	6	2	5	12
F10	Mean	1.04E+03	1.62E+03	1.43E+03	1.81E+03	1.51E+03	2.43E+03	1.62E+03	2.00E+03	1.52E+03	1.26E+03	1.67E+03	1.21E+03	2.20E+03
	Std	8.95E+00	2.36E+02	2.56E+02	3.19E+02	3.88E+02	5.68E+01	2.78E+02	2.14E+02	2.75E+02	1.44E+02	2.87E+02	1.50E+02	1.59E+02
	SEM	2.83E+00	7.46E+01	8.09E+01	1.01E+02	1.23E+02	1.80E+01	8.79E+01	6.76E+01	8.69E+01	4.55E+01	9.09E+01	4.76E+01	5.04E+01
	Rank	1	8	4	10	5	13	7	11	6	3	9	2	12
F11	Mean	1.24E+03	1.86E+03	1.90E+03	1.71E+03	1.79E+03	2.52E+03	1.79E+03	2.45E+03	1.64E+03	1.70E+03	1.93E+03	1.88E+03	2.53E+03
	Std	1.16E+02	3.56E+02	3.89E+02	4.32E+02	3.48E+02	1.58E+02	3.37E+02	1.20E+02	2.43E+02	2.75E+02	3.13E+02	3.68E+02	1.64E+02
	SEM	3.65E+01	1.13E+02	1.23E+02	1.36E+02	1.10E+02	4.99E+01	1.07E+02	3.78E+01	7.67E+01	8.71E+01	9.88E+01	1.16E+02	5.19E+01
	Rank	1	7	9	4	5	12	6	11	2	3	10	8	13
F12	Mean	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03	1.20E+03
	Std	4.00E-02	2.06E-01	3.08E-01	8.72E-01	3.21E-01	2.64E-01	1.62E-01	2.32E-01	4.25E-01	1.22E-01	2.12E-01	1.96E-01	3.62E-01
	SEM	1.26E-02	6.50E-02	9.74E-02	2.76E-01	1.02E-01	8.35E-02	5.14E-02	7.33E-02	1.35E-01	3.86E-02	6.70E-02	6.18E-02	1.15E-01
	Rank	1	3	4	8	5	13	7	11	9	2	12	6	10
F13	Mean	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03	1.30E+03
	Std	1.65E-02	1.39E-01	7.98E-02	2.63E-02	7.14E-02	8.52E-01	7.03E-01	7.17E-02	4.62E-02	1.44E-01	4.43E-02	3.21E-02	1.03E+00
	SEM	5.22E-03	4.39E-02	2.52E-02	8.31E-03	2.26E-02	2.69E-01	2.22E-01	2.27E-02	1.46E-02	4.56E-02	1.40E-02	1.01E-02	3.26E-01
	Rank	1	8	6	2	4	13	11	10	7	9	3	5	12
F14	Mean	1.40E+03	1.40E+03	1.40E+03	1.40E+03	1.40E+03	1.44E+03	1.41E+03	1.40E+03	1.40E+03	1.40E+03	1.40E+03	1.40E+03	1.41E+03
	Std	3.81E-02	2.61E-01	2.23E-01	7.08E-02	2.03E-01	7.27E+00	6.22E+00	2.62E-01	2.08E-01	5.18E-01	4.14E-02	7.54E-02	5.84E+00
	SEM	1.20E-02	8.24E-02	7.06E-02	2.24E-02	6.42E-02	2.30E+00	1.97E+00	8.29E-02	6.57E-02	1.64E-01	1.31E-02	2.38E-02	1.85E+00
	Rank	3	7	6	2	4	13	12	10	8	9	1	5	11
F15	Mean	1.50E+03	1.50E+03	1.50E+03	1.50E+03	1.50E+03	5.76E+03	1.66E+03	1.51E+03	1.51E+03	1.50E+03	1.50E+03	1.50E+03	3.73E+03
	Std	1.10E-01	1.36E+00	2.53E+00	8.91E-01	1.41E-01	2.21E+03	6.72E+01	1.13E+01	1.13E+01	9.32E-01	3.84E-01	1.05E+00	1.55E+03
	SEM	3.48E-02	4.30E-01	8.01E-01	2.82E-01	4.45E-02	7.00E+02	2.13E+01	3.57E+00	3.58E+00	2.95E-01	1.21E-01	3.31E-01	4.89E+02
	Rank	1	6	8	3	2	13	11	10	9	7	4	5	12

and

Table 18. Statistical results of HPDE with 51 independent runs on the CEC2014 (F16–F30) 30 Agent with 1000 FEs (second set of optimizers).

Function	Analysis	HPDE	RTH	SHIO	ChOA	COA	OHO	ChOA	SCA	PSO	SHO	SDE	WOA	RSA
F16	Mean	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03	1.60E+03
	Std	4.41E-01	4.33E-01	6.41E-01	3.78E-01	5.77E-01	1.69E-01	2.10E-01	2.61E-01	4.40E-01	4.82E-01	3.98E-01	3.32E-01	1.95E-01
	SEM	1.39E-01	1.37E-01	2.03E-01	1.20E-01	1.82E-01	5.36E-02	6.65E-02	8.25E-02	1.39E-01	1.53E-01	1.26E-01	1.05E-01	6.18E-02
	Rank	1	8	9	4	2	13	11	10	5	6	3	7	12
F17	Mean	1.72E+03	2.28E+03	4.11E+04	3.49E+03	4.56E+03	3.93E+05	1.52E+05	2.70E+04	1.06E+05	4.63E+04	1.87E+03	2.19E+03	5.07E+05
	Std	1.46E+01	3.20E+02	1.10E+05	8.49E+02	1.70E+03	4.23E+04	1.03E+05	2.35E+04	1.60E+05	5.13E+04	7.15E+01	2.47E+02	1.04E+05
	SEM	4.62E+00	1.01E+02	3.49E+04	2.68E+02	5.36E+02	1.34E+04	3.27E+04	7.44E+03	5.06E+04	1.62E+04	2.26E+01	7.81E+01	3.30E+04
	Rank	1	4	8	5	6	12	11	7	10	9	2	3	13
F18	Mean	1.80E+03	1.86E+03	1.28E+04	9.56E+03	3.27E+03	2.20E+04	9.17E+03	1.43E+04	1.30E+04	9.19E+03	1.83E+03	1.87E+03	1.67E+05
	Std	3.50E-01	2.82E+01	9.23E+03	2.91E+03	1.72E+03	1.30E+04	6.91E+03	4.12E+03	8.36E+03	3.16E+03	1.18E+01	5.96E+01	4.38E+05
	SEM	1.11E-01	8.93E+00	2.92E+03	9.19E+02	5.45E+02	4.10E+03	2.19E+03	1.30E+03	2.64E+03	1.00E+03	3.73E+00	1.88E+01	1.38E+05
	Rank	1	3	9	8	5	12	6	11	10	7	2	4	13
F19	Mean	1.90E+03	1.90E+03	1.90E+03	1.90E+03	1.90E+03	1.94E+03	1.92E+03	1.91E+03	1.90E+03	1.90E+03	1.90E+03	1.90E+03	1.92E+03
	Std	4.37E-01	1.27E+00	1.37E+00	7.02E-01	1.66E+00	2.19E+01	1.90E+01	4.86E-01	7.96E-01	1.24E+00	5.94E-01	1.07E+00	1.50E+01
	SEM	1.38E-01	4.01E-01	4.33E-01	2.22E-01	5.26E-01	6.94E+00	6.00E+00	1.54E-01	2.52E-01	3.92E-01	1.88E-01	3.37E-01	4.75E+00
	Rank	1	4	8	6	7	13	12	10	9	5	2	3	11
F20	Mean	2.00E+03	2.05E+03	6.33E+03	5.79E+03	2.43E+03	9.61E+03	6.93E+03	4.64E+03	3.74E+03	5.87E+03	2.01E+03	2.06E+03	1.05E+04
	Std	3.86E-01	4.74E+01	3.12E+03	1.58E+03	2.60E+02	1.12E+03	5.61E+03	2.81E+03	1.87E+03	7.97E+02	5.27E+00	4.25E+01	4.33E+03
	SEM	1.22E-01	1.50E+01	9.87E+02	4.99E+02	8.24E+01	3.55E+02	1.77E+03	8.89E+02	5.92E+02	2.52E+02	1.67E+00	1.34E+01	1.37E+03
	Rank	1	3	10	8	5	12	11	7	6	9	2	4	13
F21	Mean	2.10E+03	2.50E+03	8.02E+03	6.15E+03	3.23E+03	1.05E+06	1.12E+04	1.13E+04	9.42E+03	9.39E+03	2.18E+03	2.27E+03	2.25E+05
	Std	1.90E-01	2.22E+02	4.44E+03	1.56E+03	8.57E+02	8.64E+05	1.07E+04	6.09E+03	4.05E+03	3.99E+03	3.75E+01	1.77E+02	1.99E+05
	SEM	6.02E-02	7.01E+01	1.40E+03	4.94E+02	2.71E+02	2.73E+05	3.38E+03	1.92E+03	1.28E+03	1.26E+03	1.18E+01	5.61E+01	6.30E+04
	Rank	1	4	7	6	5	13	10	11	9	8	2	3	12
F22	Mean	2.20E+03	2.31E+03	2.30E+03	2.37E+03	2.23E+03	2.66E+03	2.40E+03	2.28E+03	2.30E+03	2.28E+03	2.23E+03	2.23E+03	2.36E+03
	Std	6.27E+00	6.30E+01	6.62E+01	1.41E+01	3.47E+01	8.42E+01	1.27E+02	4.14E+01	5.77E+01	5.95E+01	1.60E+00	1.56E+01	5.62E+01
	SEM	1.98E+00	1.99E+01	2.09E+01	4.45E+00	1.10E+01	2.66E+01	4.01E+01	1.31E+01	1.82E+01	1.88E+01	5.07E-01	4.92E+00	1.78E+01
	Rank	1	9	8	11	4	13	12	5	7	6	3	2	10
F23	Mean	2.62E+03	2.50E+03	2.63E+03	2.63E+03	2.50E+03	2.52E+03	2.50E+03	2.64E+03	2.64E+03	2.61E+03	2.63E+03	2.63E+03	2.50E+03
	Std	4.79E-13	0.00E+00	4.96E+00	1.29E-04	0.00E+00	2.47E+00	0.00E+00	2.77E+00	5.07E+00	5.68E+01	2.82E-05	4.79E-13	0.00E+00
	SEM	1.52E-13	0.00E+00	1.57E+00	4.07E-05	0.00E+00	7.81E-01	0.00E+00	8.76E-01	1.60E+00	1.80E+01	8.91E-06	1.52E-13	0.00E+00
		7	1	11	10	1	5	1	13	12	6	9	8	1
F24	Mean	2.51E+03	2.56E+03	2.55E+03	2.52E+03	2.59E+03	2.60E+03	2.57E+03	2.55E+03	2.54E+03	2.56E+03	2.52E+03	2.53E+03	2.60E+03
	Std	1.71E+00	2.45E+01	3.80E+01	4.48E+00	2.93E+01	1.59E-01	2.39E+01	7.29E+00	2.54E+01	2.97E+01	3.67E+00	1.19E+01	8.25E+00
	SEM	5.41E-01	7.75E+00	1.20E+01	1.42E+00	9.25E+00	5.01E-02	7.57E+00	2.31E+00	8.03E+00	9.41E+00	1.16E+00	3.75E+00	2.61E+00
	Rank	1	8	6	3	11	13	10	7	5	9	2	4	12
F25	Mean	2.64E+03	2.69E+03	2.69E+03	2.68E+03	2.70E+03	2.70E+03	2.70E+03	2.69E+03	2.69E+03	2.69E+03	2.64E+03	2.69E+03	2.70E+03
	Std	3.75E+01	1.64E+01	1.71E+01	3.56E+01	0.00E+00	7.48E-02	7.64E-01	1.84E+01	9.81E+00	1.53E+01	1.56E+01	1.56E+01	0.00E+00
	SEM	1.19E+01	5.18E+00	5.41E+00	1.12E+01	0.00E+00	2.36E-02	2.42E-01	5.83E+00	3.10E+00	4.84E+00	4.92E+00	4.92E+00	0.00E+00
	Rank	1	8	4	3	11	13	10	5	9	6	2	7	11
F26	Mean	2.70E+03	2.70E+03	2.70E+03	2.70E+03	2.70E+03	2.72E+03	2.70E+03	2.70E+03	2.70E+03	2.70E+03	2.70E+03	2.70E+03	2.70E+03
	Std	2.17E-02	2.72E-01	4.99E-02	4.01E-02	2.73E-02	3.18E+01	9.31E-01	1.32E-01	8.18E-02	1.61E-01	3.68E-02	5.40E-02	1.58E+00
	SEM	6.86E-03	8.59E-02	1.58E-02	1.27E-02	8.62E-03	1.00E+01	2.94E-01	4.16E-02	2.59E-02	5.10E-02	1.17E-02	1.71E-02	5.00E-01
	Rank	1	9	6	2	3	13	11	10	7	8	5	4	12
F27	Mean	2.85E+03	2.90E+03	3.01E+03	3.02E+03	2.90E+03	2.95E+03	2.89E+03	3.10E+03	3.01E+03	3.03E+03	2.82E+03	3.01E+03	2.97E+03
	Std	1.59E+02	0.00E+00	1.66E+02	1.52E+01	0.00E+00	1.06E+01	3.76E+01	1.23E+00	1.61E+02	1.66E+02	1.92E+02	1.65E+02	1.18E+02
	SEM	5.04E+01	0.00E+00	5.23E+01	4.79E+00	0.00E+00	3.37E+00	1.19E+01	3.89E-01	5.10E+01	5.25E+01	6.07E+01	5.21E+01	3.72E+01
	Rank	2	4	9	11	4	6	3	13	8	12	1	10	7
F28	Mean	3.18E+03	3.00E+03	3.36E+03	3.34E+03	3.00E+03	3.06E+03	3.00E+03	3.25E+03	3.28E+03	3.33E+03	3.18E+03	3.18E+03	3.23E+03
	Std	3.50E+00	0.00E+00	1.17E+02	3.58E+02	0.00E+00	1.50E+01	0.00E+00	4.11E+01	9.32E+01	7.77E+01	1.14E+01	8.27E+00	1.86E+02
	SEM	1.11E+00	0.00E+00	3.69E+01	1.13E+02	0.00E+00	4.75E+00	0.00E+00	1.30E+01	2.95E+01	2.46E+01	3.61E+00	2.62E+00	5.88E+01
	Rank	5	1	13	12	1	4	1	9	10	11	6	7	8
F29	Mean	3.13E+03	3.18E+03	1.76E+05	3.88E+03	3.15E+05	1.47E+07	4.75E+04	1.43E+04	1.77E+05	3.69E+03	3.13E+03	3.18E+03	1.96E+05
	Std	5.71E+00	4.30E+01	5.45E+05	4.84E+02	9.86E+05	1.59E+06	1.37E+05	1.01E+04	5.45E+05	7.67E+02	4.32E+00	5.45E+01	4.02E+05
	SEM	1.81E+00	1.36E+01	1.72E+05	1.53E+02	3.12E+05	5.04E+05	4.32E+04	3.18E+03	1.72E+05	2.43E+02	1.37E+00	1.72E+01	1.27E+05
	Rank	1	3	9	6	12	13	8	7	10	5	2	4	11
F30	Mean	3.48E+03	3.93E+03	4.90E+03	5.33E+03	3.63E+03	1.51E+05	1.77E+04	4.53E+03	4.43E+03	5.02E+03	3.50E+03	3.77E+03	1.71E+04
	Std	2.26E+01	2.13E+02	8.42E+02	2.23E+02	5.10E+02	1.80E+05	2.70E+04	4.68E+02	6.13E+02	4.39E+02	4.16E+01	2.92E+02	1.91E+04
	SEM	7.15E+00	6.75E+01	2.66E+02	7.06E+01	1.61E+02	5.68E+04	8.53E+03	1.48E+02	1.94E+02	1.39E+02	1.32E+01	9.24E+01	6.03E+03
	Rank	1	5	8	10	3	13	12	7	6	9	2	4	11

. In both groups of competitors, HPDE consistently ranks at or near the top, emphasizing its robustness and effectiveness in handling complex optimization tasks. In the first group (F1–F15), HPDE demonstrates significant efficiency by maintaining competitive rankings and low mean errors against algorithms like SHIO and ChOA, particularly in simpler functions such as F5 and F6, where its precision and exploitation capabilities shine. In the second set of functions (F16–F30), HPDE continues to excel, notably outperforming other optimizers in complex scenarios like F17 and F21, where it deals adeptly with multimodal and high-dimensional challenges. These consistent top rankings across a variety of function types underline HPDE’s versatility and reliability as a potent tool for both academic research and practical applications, positioning it as a good choice for demanding optimization environments.

The results of the HPDE optimizer with a second group of optimizers across the CEC2014 benchmark functions (F1–F30), as shown in

Table 19. Error measurement comparison results of HPDE with the second group of optimizers over CEC2014 (F1–F30) benchmark functions (second set of optimizers).

HPDE	RTH	SHIO	ChOA	COA	OHO	ChOA	SCA	PSO	SHO	SDE	WOA	RSA
1.32E+01	1.12E-01	6.91E+06	4.55E+05	9.17E+04	3.94E+08	1.76E+07	8.59E+06	4.83E+06	1.17E+07	8.23E+00	1.73E+03	1.17E+08
3.38E-11	1.01E-11	9.30E+07	1.76E+02	1.97E+03	7.14E+09	4.08E+09	5.90E+08	3.48E+08	3.19E+06	3.28E+02	7.59E+00	6.46E+09
0.00E+00	7.13E-11	1.07E+04	7.04E+03	4.87E+03	1.21E+04	1.98E+04	2.98E+03	6.52E+03	5.84E+03	1.16E-01	1.11E-02	8.45E+03
2.13E+01	2.48E+01	4.44E+01	3.51E+01	1.84E+01	2.83E+03	8.22E+02	5.84E+01	2.11E+01	5.92E+01	4.80E+00	2.78E+01	7.78E+02
1.80E+01	2.01E+01	2.04E+01	2.03E+01	2.01E+01	2.04E+01	2.01E+01	2.04E+01	2.04E+01	2.00E+01	2.04E+01	2.01E+01	2.04E+01
3.67E-01	5.69E+00	4.48E+00	4.74E-01	4.66E+00	1.06E+01	8.78E+00	6.54E+00	4.63E+00	5.27E+00	2.43E+00	3.68E+00	9.61E+00
1.03E-02	2.57E-01	1.39E+00	3.59E-01	8.87E-02	1.98E+02	8.60E+01	1.08E+01	6.06E+00	7.22E+00	2.72E-01	3.56E-01	8.39E+01
3.93E-10	1.93E+01	2.66E+01	1.87E+01	2.69E+00	7.02E+01	3.28E+01	4.21E+01	1.79E+01	8.17E+00	9.13E+00	8.39E+00	7.23E+01
2.57E+00	2.85E+01	1.81E+01	1.38E+01	4.02E+01	6.29E+01	3.88E+01	4.38E+01	3.17E+01	2.66E+01	9.48E+00	1.93E+01	5.87E+01
3.57E+01	6.25E+02	4.35E+02	8.08E+02	5.14E+02	1.43E+03	6.19E+02	1.00E+03	5.15E+02	2.63E+02	6.73E+02	2.14E+02	1.20E+03
1.43E+02	7.58E+02	7.97E+02	6.09E+02	6.85E+02	1.42E+03	6.93E+02	1.35E+03	5.43E+02	5.98E+02	8.31E+02	7.84E+02	1.43E+03
1.68E-01	2.92E-01	3.29E-01	8.60E-01	3.80E-01	1.38E+00	4.50E-01	1.20E+00	1.06E+00	2.57E-01	1.22E+00	4.42E-01	1.09E+00
4.30E-02	3.39E-01	2.10E-01	9.07E-02	1.82E-01	4.61E+00	1.91E+00	6.04E-01	2.69E-01	3.92E-01	1.82E-01	1.99E-01	3.46E+00
2.17E-01	4.40E-01	3.65E-01	1.93E-01	2.62E-01	3.89E+01	1.50E+01	9.65E-01	5.34E-01	7.13E-01	1.35E-01	2.68E-01	1.37E+01
5.13E-01	2.45E+00	3.39E+00	1.25E+00	1.20E+00	4.26E+03	1.56E+02	1.05E+01	6.36E+00	3.34E+00	1.87E+00	1.94E+00	2.23E+03
1.72E+00	3.15E+00	3.24E+00	2.85E+00	2.52E+00	3.84E+00	3.50E+00	3.28E+00	2.90E+00	3.05E+00	2.68E+00	3.10E+00	3.78E+00
1.75E+01	5.79E+02	3.94E+04	1.79E+03	2.86E+03	3.92E+05	1.50E+05	2.53E+04	1.04E+05	4.46E+04	1.69E+02	4.94E+02	5.05E+05
4.88E-01	6.34E+01	1.10E+04	7.76E+03	1.47E+03	2.02E+04	7.37E+03	1.25E+04	1.12E+04	7.39E+03	3.31E+01	7.39E+01	1.65E+05
8.42E-01	2.98E+00	3.63E+00	3.13E+00	3.21E+00	3.58E+01	2.42E+01	5.16E+00	3.65E+00	3.13E+00	1.51E+00	2.22E+00	1.68E+01
4.87E-01	4.63E+01	4.33E+03	3.79E+03	4.25E+02	7.61E+03	4.93E+03	2.64E+03	1.74E+03	3.87E+03	1.36E+01	5.96E+01	8.49E+03
6.36E-01	3.98E+02	5.92E+03	4.05E+03	1.13E+03	1.04E+06	9.09E+03	9.20E+03	7.32E+03	7.29E+03	8.49E+01	1.71E+02	2.22E+05
2.81E+00	1.08E+02	1.02E+02	1.65E+02	3.17E+01	4.56E+02	1.99E+02	7.56E+01	9.98E+01	7.75E+01	2.77E+01	2.65E+01	1.56E+02
3.20E+02	2.00E+02	3.34E+02	3.29E+02	2.00E+02	2.18E+02	2.00E+02	3.40E+02	3.40E+02	3.07E+02	3.29E+02	3.29E+02	2.00E+02
1.09E+02	1.57E+02	1.49E+02	1.16E+02	1.86E+02	2.01E+02	1.75E+02	1.54E+02	1.38E+02	1.58E+02	1.15E+02	1.34E+02	1.96E+02
1.43E+02	1.95E+02	1.91E+02	1.76E+02	2.00E+02	2.00E+02	2.00E+02	1.94E+02	1.95E+02	1.95E+02	1.44E+02	1.95E+02	2.00E+02
1.00E+02	1.00E+02	1.00E+02	1.00E+02	1.00E+02	1.19E+02	1.02E+02	1.01E+02	1.00E+02	1.00E+02	1.00E+02	1.00E+02	1.03E+02
1.53E+02	2.00E+02	3.11E+02	3.15E+02	2.00E+02	2.47E+02	1.88E+02	4.04E+02	3.08E+02	3.28E+02	1.22E+02	3.14E+02	2.73E+02
3.79E+02	2.00E+02	5.61E+02	5.38E+02	2.00E+02	2.56E+02	2.00E+02	4.48E+02	4.76E+02	5.30E+02	3.80E+02	3.81E+02	4.31E+02
2.25E+02	2.75E+02	1.73E+05	9.82E+02	3.12E+05	1.47E+07	4.46E+04	1.14E+04	1.74E+05	7.85E+02	2.29E+02	2.77E+02	1.94E+05
4.81E+02	9.27E+02	1.90E+03	2.33E+03	6.27E+02	1.48E+05	1.47E+04	1.53E+03	1.43E+03	2.02E+03	4.96E+02	7.73E+02	1.41E+04

, illustrate HPDE’s dominant performance in a broad range of scenarios. Remarkably, HPDE maintains an impressively low mean error across nearly all functions, often outperforming other algorithms by orders of magnitude. For instance, in direct comparisons, HPDE’s error rates are significantly lower than those of SHIO, ChOA, and even advanced algorithms like COA and OHO across various functions. This is particularly evident in functions where other optimizers, such as RTH and SCA, show vulnerability in handling more complex optimization landscapes, where HPDE exhibits robustness and precision.

The consistency of HPDE is highlighted in functions with extremely challenging environments, such as F17 and F30, where it achieves minimal errors compared with competitors that struggle with much higher errors, showcasing its effectiveness in dealing with high-dimensional and multimodal problems. Even in the simpler functions, HPDE’s performance is consistently good, maintaining top rankings and showing an unparalleled ability to navigate the solution space efficiently.

Table 20. Average ranking of 12 optimizers using Friedman’s test based on CEC2014.

Algorithm	Rank
APO	3.12654239
ChOA	4.74218367
GWO	5.36894271
FOX	2.84519632
WOA	6.95126748
MVO	7.41652389
DOA	8.02756324
MFO	5.84372916
RIME	4.21537894
DBO	6.32481905
WSO	9.53762143

offers a concise illustration of the average rankings produced by Friedman’s test when HPDE is compared with other methods for a 10-dimensional setting on the CEC-2014 functions. On the CEC-2014 test suite, each method’s mean error values across all dimensions are first collected, and then their ranks in each dimension are determined. By taking the average of these ranks, one can compare the algorithms via Friedman’s test. Under Friedman’s framework, if the p-value is found to be less than or equal to the chosen $\mathsf{\alpha }$ = 0.05, the null hypothesis (asserting no significant performance difference) is rejected, implying that at least one algorithm is better or worse than the others. Within this ranking approach, the method with the lowest rank is considered the best, whereas the highest rank denotes the worst. The algorithm identified as the best is then used as a reference (control) for additional analyses.

6.2. HPDE Accuracy

The accuracy of the HPDE algorithm for each benchmark function was calculated using a comparative error analysis relative to the worst-performing algorithm on that function. For each function $F_i$, we first determined the error of the HPDE algorithm, denoted as $E_{\mathrm{HPDE},i}$, and the worst error among all compared algorithms, denoted as $E_{\mathrm{worst},i}$.

The error for the HPDE algorithm on function $F_i$ is shown in Equation (49):

$$E_{\mathrm{HPDE},i}=\left|M_{\mathrm{HPDE},i}-O_i\right|,$$

(49)

where $M_{\mathrm{HPDE},i}$ is the mean result obtained by the HPDE algorithm on $F_i$, and $O_i$ is the known optimal value of the function $F_i$.

Similarly, the worst error among all algorithms for function $F_i$ is calculated as shown in Equation (50):

$$E_{\mathrm{worst},i}=max\left\{\left|M_{\mathrm{alg},i}-O_i\right||\mathrm{for}\mathrm{all}\mathrm{algorithms}\right\},$$

(50)

where $M_{\mathrm{alg},i}$ represents the mean result of each algorithm on $F_i$.

The accuracy $A_i$ of the HPDE algorithm on function $F_i$ is then computed using Equation (51), which quantifies the performance of HPDE relative to the worst-performing algorithm as shown in Equation (51):

$$A_i=1-{\displaystyle \frac{E_{\mathrm{HPDE},i}}{E_{\mathrm{worst},i}}}.$$

(51)

By applying Equation (51) to each benchmark function, we obtained the accuracy values presented in

Table 21. Accuracy comparison of HPDE over CEC2014 (F1–F30) benchmark functions.

Function	HPDE Error	Worst Error	HPDE Accuracy
F1	$1.32\times {10}^1$	$4.76\times {10}^7$	$1-{\displaystyle \frac{1.32\times {10}^1}{4.76\times {10}^7}}=0.999999723$
F2	$3.38\times {10}^{-11}$	$7.95\times {10}^9$	$1-{\displaystyle \frac{3.38\times {10}^{-11}}{7.95\times {10}^9}}=1.0$
F3	$0.0$	$4.14\times {10}^4$	$1-{\displaystyle \frac{0.0}{4.14\times {10}^4}}=1.0$
F4	$2.13\times {10}^1$	$1.46\times {10}^3$	$1-{\displaystyle \frac{2.13\times {10}^1}{1.46\times {10}^3}}=0.985$
F5	$1.80\times {10}^1$	$2.04\times {10}^1$	$1-{\displaystyle \frac{1.80\times {10}^1}{2.04\times {10}^1}}=0.1176$
F6	$3.67\times {10}^{-1}$	$1.01\times {10}^1$	$1-{\displaystyle \frac{3.67\times {10}^{-1}}{1.01\times {10}^1}}=0.9636$
F7	$1.03\times {10}^{-2}$	$1.88\times {10}^2$	$1-{\displaystyle \frac{1.03\times {10}^{-2}}{1.88\times {10}^2}}=0.9999451$
F8	$3.93\times {10}^{-10}$	$9.17\times {10}^1$	$1-{\displaystyle \frac{3.93\times {10}^{-10}}{9.17\times {10}^1}}=1.0$
F9	$2.57\times {10}^0$	$7.11\times {10}^1$	$1-{\displaystyle \frac{2.57\times {10}^0}{7.11\times {10}^1}}=0.9638$
F10	$3.57\times {10}^1$	$1.55\times {10}^3$	$1-{\displaystyle \frac{3.57\times {10}^1}{1.55\times {10}^3}}=0.9769$
F11	$1.43\times {10}^2$	$1.56\times {10}^3$	$1-{\displaystyle \frac{1.43\times {10}^2}{1.56\times {10}^3}}=0.9083$
F12	$1.68\times {10}^{-1}$	$1.55\times {10}^0$	$1-{\displaystyle \frac{1.68\times {10}^{-1}}{1.55\times {10}^0}}=0.8916$
F13	$4.30\times {10}^{-2}$	$4.14\times {10}^0$	$1-{\displaystyle \frac{4.30\times {10}^{-2}}{4.14\times {10}^0}}=0.9896$
F14	$2.17\times {10}^{-1}$	$3.84\times {10}^1$	$1-{\displaystyle \frac{2.17\times {10}^{-1}}{3.84\times {10}^1}}=0.9943$
F15	$5.13\times {10}^{-1}$	$1.53\times {10}^4$	$1-{\displaystyle \frac{5.13\times {10}^{-1}}{1.53\times {10}^4}}=0.9999664$
F16	$1.72\times {10}^0$	$3.82\times {10}^0$	$1-{\displaystyle \frac{1.72\times {10}^0}{3.82\times {10}^0}}=0.5497$
F17	$1.75\times {10}^1$	$6.90\times {10}^5$	$1-{\displaystyle \frac{1.75\times {10}^1}{6.90\times {10}^5}}=0.9999746$
F18	$4.88\times {10}^{-1}$	$2.79\times {10}^5$	$1-{\displaystyle \frac{4.88\times {10}^{-1}}{2.79\times {10}^5}}=0.99999825$
F19	$8.42\times {10}^{-1}$	$3.56\times {10}^1$	$1-{\displaystyle \frac{8.42\times {10}^{-1}}{3.56\times {10}^1}}=0.9763$
F20	$4.87\times {10}^{-1}$	$2.50\times {10}^5$	$1-{\displaystyle \frac{4.87\times {10}^{-1}}{2.50\times {10}^5}}=0.99999805$
F21	$6.36\times {10}^{-1}$	$7.94\times {10}^5$	$1-{\displaystyle \frac{6.36\times {10}^{-1}}{7.94\times {10}^5}}=0.99999920$
F22	$2.81\times {10}^0$	$3.68\times {10}^2$	$1-{\displaystyle \frac{2.81\times {10}^0}{3.68\times {10}^2}}=0.9924$
F23	$3.20\times {10}^2$	$4.10\times {10}^2$	$1-{\displaystyle \frac{3.20\times {10}^2}{4.10\times {10}^2}}=0.2195$
F24	$1.09\times {10}^2$	$2.01\times {10}^2$	$1-{\displaystyle \frac{1.09\times {10}^2}{2.01\times {10}^2}}=0.4567$
F25	$1.43\times {10}^2$	$2.00\times {10}^2$	$1-{\displaystyle \frac{1.43\times {10}^2}{2.00\times {10}^2}}=0.2850$
F26	$1.00\times {10}^2$	$1.06\times {10}^2$	$1-{\displaystyle \frac{1.00\times {10}^2}{1.06\times {10}^2}}=0.0566$
F27	$1.53\times {10}^2$	$4.81\times {10}^2$	$1-{\displaystyle \frac{1.53\times {10}^2}{4.81\times {10}^2}}=0.6820$
F28	$3.79\times {10}^2$	$1.06\times {10}^3$	$1-{\displaystyle \frac{3.79\times {10}^2}{1.06\times {10}^3}}=0.6435$
F29	$2.25\times {10}^2$	$1.22\times {10}^6$	$1-{\displaystyle \frac{2.25\times {10}^2}{1.22\times {10}^6}}=0.9998157$
F30	$4.81\times {10}^2$	$3.14\times {10}^4$	$1-{\displaystyle \frac{4.81\times {10}^2}{3.14\times {10}^4}}=0.9847$

. The accuracy metric ranges between 0 and 1, where values closer to 1 indicate higher accuracy and better performance of the HPDE algorithm. It effectively normalizes the HPDE’s error, providing a relative measure of its optimization capability compared with the least effective algorithm for each function.

The accuracy table (Table 21) showcases the performance of the HPDE algorithm across the CEC2014 benchmark functions F1 to F30. The HPDE algorithm demonstrates exceptionally high accuracy on numerous functions, achieving values very close to 1.0. Specifically, for functions F1, F2, F3, F7, F8, F15, F17, F18, F20, F21, and F29, the accuracy exceeds 0.9999, indicating good optimization performance with minimal error relative to the worst-performing algorithms on these functions. For instance, on function F1, the HPDE accuracy is approximately 0.999999723, highlighting its effectiveness in finding solutions close to the optimal.

On functions F4, F6, F9, F10, F11, F13, F14, F19, F22, F27, F28, and F30, the HPDE algorithm maintains high accuracy levels ranging from approximately 0.89 to 0.99. This suggests consistent and reliable performance, although there is slightly more room for improvement compared with the functions where near-perfect accuracy was achieved. These results indicate that while HPDE effectively minimizes the error in these functions, some inherent challenges may prevent it from reaching the highest possible accuracy.

Conversely, the HPDE algorithm exhibits lower accuracy on functions F5, F16, F23, F24, F25, and F26, with values ranging from approximately 0.0566 to 0.5497. These lower accuracy scores suggest that the algorithm encounters difficulties in optimizing these specific functions, possibly due to factors such as complex landscapes, multimodality, or high-dimensional search spaces that challenge the convergence capabilities of the algorithm.

7. Qualitative Diagram Analysis

The convergence figures displayed for various benchmark functions from the CEC2014 set exhibit distinct behaviors, as shown in Figure 7, highlighting the efficiency and optimization capabilities of the algorithms in question. Most figures start with a steep drop, indicating a rapid improvement in optimization scores in initial iterations, which is typical in scenarios where gross optimization potentials are quickly exploited. The plateauing or flattening of the figures in the later stages, visible across most functions, suggests that finer improvements are more challenging, requiring more iterations for minimal gains. This trend is especially pronounced in the diagram that levels off at around iteration 20 and maintains a nearly constant score, emphasizing the algorithm’s early saturation in optimization potential. Conversely, the gradual declines in some figures without abrupt plateaus suggest a more consistent optimization process, potentially indicative of more complex problem spaces where incremental gains continue to be possible across iterations. The variance in the rate of convergence across these functions underscores the importance of algorithm selection based on specific problem characteristics and desired optimization precision within computational constraints.

7.1. Results of Convergence Diagram

The convergence figures, as shown in Figure 7 and Figure 8, illustrate the convergence behavior of the algorithm across different functions over 200 iterations, with each diagram showing the best solution found so far. In the F2 chart, the best score starts around 3600, exhibiting a stepwise decline with sharp drops, eventually converging to a score below 2000, indicating significant progress within the first 100 iterations, followed by smaller refinements. In the F2 chart, the algorithm starts with an initial score of 1060, dropping rapidly in the first 30 iterations to 940 and eventually converging just below 900, suggesting quick identification of a good solution with minor refinements afterward. F3 shows an initial score of 930, which quickly drops to around 820 with incremental improvements in subsequent iterations, reflecting a stepwise refinement process. In F4 4, the score starts at 880 and sharply declines to 720 in the first 30 iterations, indicating rapid convergence with minimal improvements thereafter. F5 begins at 613 and steadily declines to 606 over the 200 iterations, reflecting consistent but gradual improvement throughout the process. In F6, the algorithm starts at 521.2 and shows a stepwise decline, eventually converging around 520.6 after early significant drops, while in F7, the score starts at 2600 and drops dramatically to below 600 within the first 20 iterations, indicating effective early convergence. Figure 8 starts at ${10}^7$ and drops drastically within the first 30 iterations, stabilizing near ${10}^4$, showing the algorithm’s ability to handle large-scale problems with rapid early progress. F9 begins at ${10}^9$ and quickly drops to ${10}^8$ within the first 50 iterations, with further incremental improvements indicating a stepwise convergence pattern. In F10, the initial score of 1204 decreases incrementally, reaching around 1201.5 after 200 iterations, with improvements occurring at specific points. F11 shows an initial score of 3800, which declines sharply early on, stabilizing around 2900 after iteration 50, indicating that the algorithm converges early with minimal further improvements. Finally, in F12, the score starts at 1204 and steadily improves to 1201.5 over 200 iterations, with the stepwise pattern suggesting the gradual identification of better solutions. These figures highlight the algorithm’s efficiency in handling different functions, with early significant improvements followed by slower refinements as it approaches optimal or near-optimal solutions. The stepwise patterns reflect a balance between exploration and exploitation, with key improvements made throughout the optimization process.

7.2. Results of Search History Diagram

The search history, as shown in Figure 9 and Figure 10 for the CEC2014 benchmark functions, showcases the HPDE optimizer’s exploration and exploitation processes across different dimensions, illustrating convergence towards global minima. Each plot vividly displays the scatter of search points where the red dot signifies the optimal or near-optimal solution found by the optimizer.

The search history Figure 9 and Figure 10 illustrate the search behavior of the HPDE over CEC2014 (F1 to F12). Each diagram plots the search points in a 2D plane, with the x-axis representing $x1$ and the y-axis representing $x2$. The red dot in each diagram signifies the optimal solution found by the optimizer. In the first diagram (F1), the search points are densely clustered around the origin with a few scattered points further out, indicating effective concentration around the global optimum. The second diagram (F2) shows a more dispersed search pattern with a larger spread of points, suggesting a larger search space or multiple local minima. The third diagram (F3) displays a dense cluster of search points around a slightly off-center position, indicating focused refinement within a localized region. For function F4, the diagram shows a broad horizontal spread with a dense cluster near the optimal solution, suggesting significant variation in one dimension. In the fifth diagram (F5), the search points are concentrated in a narrow band along the x-axis with the optimal solution near the center, indicating a steep gradient in the x-direction. The sixth diagram (F6) reveals a dense cluster near the origin with a few scattered points, indicating efficient convergence with some exploration. For function F7, the search points form a dense cluster around the optimal solution, suggesting effective local search and quick convergence. The eighth diagram (F8) shows similar dense clustering, indicating effective local search and convergence. The ninth diagram (F9) shows a more even spread with concentration around the optimal solution, suggesting a complex landscape with multiple local minima. The tenth diagram (F10) exhibits a narrow vertical spread with a dense cluster near the optimal solution, indicating significant variation in the y-direction.

7.3. HPDE Trajectory Diagram

The trajectory figures of the first particle are shown in Figure 11 for selected benchmark functions (F1, F11, F12, F13, F15, F17) of the CEC2014 suite using the HPDE optimizer’s, exhibiting unique behaviors across different optimization landscapes. For function F1, the trajectory stabilizes quickly, showing rapid convergence to a low error value, which is indicative of a smoother landscape or a function where HPDE quickly finds the region of global optimum. Contrastingly, functions like F11 and F13 display more fluctuation initially, which suggests a more rugged landscape or the presence of local optima that challenge the optimizer before reaching a stable region. Functions F12 and F15 show a steady decrease in value before leveling off, which could indicate a need for fine-tuning the algorithm’s parameters to escape potential plateaus more efficiently. Function F17 shows a very gradual descent, potentially illustrating a complex optimization surface or a function that benefits from prolonged exploration to escape local minima. These trajectories collectively highlight the adaptive nature of the HPDE algorithm and its varying performance across different types of functions within the CEC2014 benchmark suite, underscoring its effectiveness and areas for potential optimization strategy enhancements.

7.4. Results of Average Fitness Diagram

The average fitness figures, as shown in Figure 12 of selected functions F13, F14, F15, F19, F22, and F23 from the CEC2014 benchmark set, show a rapid initial improvement in fitness, which stabilizes after the first few iterations. This behavior is typical in optimization scenarios where early explorations quickly lead to areas near the optimum, followed by a more gradual and challenging fine-tuning phase. The sharp decline in the fitness value at the start of the iterations suggests that the optimizer effectively identifies promising regions of the search space. Subsequent iterations, showing less dramatic changes, indicate the HPDE optimizer’s transition from exploration to exploitation, refining solutions within the identified promising regions. The differences in scale and stabilization points across these functions likely reflect varying landscape complexities and the HPDE optimizer’s ability to handle them. While functions with smoother landscapes might show a steady convergence and near-optimal values, more rugged landscapes could lead to more fluctuations and a higher settling point.

7.5. Results of Heat Map Diagram

The sensitivity analysis heatmaps for the HPDE optimizer applied to CEC2014 benchmark functions (F1 to F12), as shown in Figure 13 and Figure 14, demonstrate varied response characteristics across different settings of search agents and maximum iterations. The heatmaps encapsulate the sensitivity of the optimizer’s performance to these parameters. For simpler functions like F1 and F2, increasing the number of search agents generally results in a lower spread of final fitness values, indicating that a larger population provides a more robust search capability. However, for more complex functions like F3, the influence of increasing the number of agents is not as linear, suggesting the complexity and ruggedness of the landscape affect the optimizer’s efficiency. Interestingly, the maximum number of iterations does not always correspond with improved performance, which could be due to early convergence or the optimizer getting stuck in local minima. These heatmaps are crucial for understanding the behavior of the HPDE optimizer and can guide the tuning of its parameters to enhance optimization performance on specific types of functions.

8. Conclusions

In this paper, we presented the Hybrid Artificial Protozoa Optimizer with Differential Evolution (HPDE) algorithm, which integrates the biologically inspired principles of the Artificial Protozoa Optimizer (APO) with the robust optimization mechanisms of Differential Evolution (DE). The HPDE algorithm is meticulously designed to balance exploration and exploitation through innovative mechanisms such as autotrophic and heterotrophic foraging behaviors, dormancy and reproduction processes, and DE operations. The experimental evaluation of the CEC2014 benchmark functions demonstrates the good optimization performance of the HPDE algorithm when compared with 23 state-of-the-art optimizers. HPDE has a strong tendency to outperform competitors, achieving the best results in 24 out of 30 benchmark functions in one comparison set and in 23 functions in another, underscoring its robustness and versatility across diverse optimization challenges. Moreover, HPDE has been applied to several complex engineering design problems where it ranked top among all optimizers compared.

Beyond benchmark tests, the potential impact of HPDE in real-world applications is significant. The algorithm’s ability to adaptively explore diverse search landscapes and maintain strong convergence makes it suitable for a range of complex, large-scale problems. Potential areas of application include industrial process optimization, supply-chain management, energy systems planning, and various engineering design tasks, where HPDE can offer a robust means of navigating highly multimodal solution spaces under real-world constraints. As decision-making processes in these domains often demand reliable solutions under uncertainty, the biologically inspired core of HPDE, coupled with DE’s proven track record, provides a powerful synergy that can lead to both high-quality and efficient solutions.

Furthermore, an important direction for future work is extending HPDE to tackle multi-objective optimization problems. Many real-world challenges, including those encountered in engineering, economics, and environmental management, involve competing objectives that need to be optimized simultaneously. Accordingly, future research will focus on adapting the HPDE algorithm to handle multiple objectives, for instance, by integrating Pareto dominance or decomposition-based strategies to maintain a diverse set of optimal trade-off solutions. Another promising avenue is to investigate HPDE’s performance under real-time and dynamic conditions, such as when problem constraints or objectives evolve over time. Additionally, exploring hybridization with machine-learning techniques for parameter tuning and incorporating uncertainty quantification methods could further expand HPDE’s applicability.