Source-linked AI summary
Artificial Protozoa Optimizer (APO): A novel bio-inspired metaheuristic algorithm for engineering optimization
Xiaopeng Wang, Vaclav Snasel, Seyedali Mirjalili, Jeng-Shyang Pan, Lingping Kong, Hisham A. Shehadeh
TL;DR
The paper addresses optimization problems that are difficult for exact methods because practical objectives may be multimodal, nonlinear, or nondifferentiable. It proposes APO, which models protozoan foraging, dormancy, and reproduction, and evaluates it across standardized benchmarks and practical tasks. The experiments report competitive performance, including significant superiority over compared state-of-the-art algorithms, while identifying broader problem classes for future study.
Problem
Practical optimization problems can be multimodal, nonlinear, and nondifferentiable, limiting the suitability of exact and gradient-based methods.
Method
APO mathematically models protozoan foraging, dormancy, and reproduction as optimization search behaviors.
Results
The experiments confirmed that APO significantly surpasses the compared state-of-the-art algorithms across CEC2022, five engineering designs, and multilevel image segmentation.
Takeaways & Limitations
APO provides highly competitive results for optimization problems across continuous and discrete constrained applications.
Takeaways & Limitations
APO remains at a preliminary stage for diverse and complex optimization problems, including binary, large-scale, expensive, multi-objective, and multitask settings.
Abstract
from arXiv · showhide
This study proposes a novel artificial protozoa optimizer (APO) that is inspired by protozoa in nature. The APO mimics the survival mechanisms of protozoa by simulating their foraging, dormancy, and reproductive behaviors. The APO was mathematically modeled and implemented to perform the optimization processes of metaheuristic algorithms. The performance of the APO was verified via experimental simulations and compared with 32 state-of-the-art algorithms. Wilcoxon signed-rank test was performed for pairwise comparisons of the proposed APO with the state-of-the-art algorithms, and Friedman test was used for multiple comparisons. First, the APO was tested using 12 functions of the 2022 IEEE Congress on Evolutionary Computation benchmark. Considering practicality, the proposed APO was used to solve five popular engineering design problems in a continuous space with constraints. Moreover, the APO was applied to solve a multilevel image segmentation task in a discrete space with constraints. The experiments confirmed that the APO could provide highly competitive results for optimization problems. The source codes of Artificial Protozoa Optimizer are publicly available at https://seyedalimirjalili.com/projects and https://ww2.mathworks.cn/matlabcentral/fileexchange/162656-artificial-protozoa-optimizer.
1. Introduction
Optimization problems span many scientific and industrial fields and often challenge exact mathematical methods because they are multimodal, nonlinear, or nondifferentiable. The paper introduces APO, a bio-inspired metaheuristic that models protozoan survival behaviors and evaluates them on benchmark and practical problems.
- Motivation: Optimization searches decision spaces for solutions that maximize or minimize objectives subject to constraints across diverse application domains.Examples include signal processing, image processing, pattern recognition, mechanical design, production scheduling, and automatic control.
- Motivation: Exact mathematical optimization can suffer premature convergence, derivative and convexity requirements, and high computational costs on practical problems.These limitations are especially relevant for multimodal, nonlinear, and nondifferentiable problems.
- Metaheuristics: Metaheuristics use approximate, flexible, black-box methods to obtain acceptable solutions for optimization problems that challenge conventional techniques.They consider optimization inputs and outputs without requiring derivative information.
- Metaheuristics: Exploration diversifies candidate solutions, whereas exploitation searches locally around promising solutions, creating a central trade-off in metaheuristic design.The paper motivates new algorithms partly because no heuristic is expected to outperform others across all possible problems.
- Paper contribution: APO models protozoan foraging, dormancy, and reproduction, assigning exploration to autotrophic foraging and dormancy and exploitation to heterotrophic foraging and reproduction.Its evaluation includes the CEC2022 benchmark, five engineering designs, and multilevel image segmentation.
2. Artificial protozoa optimizer
The paper introduces APO by explaining its biological inspiration, mathematically modeling protozoan behavior, and analyzing the resulting optimization algorithm.
- Artificial protozoa optimizer: APO development proceeds from protozoan inspiration to mathematical survival models and comprehensive algorithm analysis.The algorithm is introduced as a metaheuristic optimization method.
2.1. Inspiration
Euglena provides the biological model for APO because it combines autotrophic and heterotrophic nutrition, stress-induced dormancy, and binary-fission reproduction. These behaviors supply distinct mechanisms for modeling optimization search and population renewal.
- Euglena: The paper uses euglena, a single-celled algal flagellate, as its representative protozoan model.More than 250 euglena species have been identified, and they primarily inhabit freshwater environments.
- Foraging: Euglena survives through both autotrophic photosynthesis and heterotrophic absorption of organic matter.Its phototaxis changes with light intensity, helping it locate suitable conditions for photosynthesis.
- Dormancy: Under unfavorable conditions, euglena forms a protective cyst and enters dormancy until the surrounding environment improves.This behavior is presented as a response to environmental stress.
- Reproduction: Euglena reproduces asexually through binary fission, splitting into two identical individuals along the cell’s longitudinal axis.The described process begins with nuclear mitosis and replication of cellular structures.
2.2. Mathematical models
APO represents candidate solutions as protozoa and integrates foraging, dormancy, and reproduction models into its population update process. The models use neighboring solutions, bounds, random perturbations, and mapping vectors to regulate search behavior.
- Representation: In APO, each protozoan represents a candidate solution whose position contains dim variables.The solution set is therefore represented as a population of protozoan positions.
- Foraging: Foraging models internal protozoan characteristics and external environmental influences, including species collisions and competition.The resulting operators support the algorithm’s exploration and exploitation phases.
- Foraging: Autotrophic foraging responds to light conditions, while heterotrophic foraging moves a protozoan toward a nearby food-rich location.Neighboring protozoa and random factors contribute to the position updates.
- Dormancy: Dormancy replaces a stressed protozoan with a newly generated one to maintain a constant population.The dormancy model operates within lower and upper variable bounds.
- Reproduction: Reproduction generates a duplicate protozoan and applies a forward or reverse perturbation through a binary mapping vector.The reproduction mapping vector has one element for each dimension, with elements equal to 0 or 1.
- Integration: APO integrates its mathematical models using parameters for neighbor pairs, behavioral probabilities, and the maximum dormancy-reproduction proportion.The algorithm has two special parameters: np, the number of neighbor pairs, and pfmax, the maximum proportion fraction.
2.3. Algorithm analysis
The APO analysis describes its population-search framework, bio-inspired operators, exploration–exploitation mechanisms, parameter roles, and computational complexity.
- Evaluation context: The analysis evaluates the APO framework, search operators, exploration–exploitation behavior, and computational complexity alongside benchmark and comparison-algorithm settings.The cited experimental setup references CEC2022 benchmark functions and comparison-algorithm parameter settings.
- Algorithm design: Metaheuristic optimization proceeds through population initialization, guidance, search operators, and update mechanisms that support exploration and exploitation.Initialization uses random or Latin hypercube sampling; iterative search selects reference positions, applies operators, and forms the next population.
- Algorithm design: APO uses random initialization, ordinal guidance selection, fitness-based updating, and specialized operators for exploration and exploitation.The algorithm’s two special parameters are the number of neighbor pairs and maximum proportion fraction.
- Exploration and exploitation: Autotrophic foraging and dormancy perform exploration, whereas heterotrophic foraging and reproduction perform exploitation.Autotrophs and dormant protozoa search broadly, while heterotrophs and reproducing protozoa search promising nearby regions.
- Exploration and exploitation: Mapping vectors control updated dimensions, with rank-dependent foraging changes directing inferior protozoa toward exploration and superior protozoa toward exploitation.The foraging mapping vector updates selected dimensions according to protozoan ranking, while the reproduction mapping vector is random.
- Computational complexity: The APO combines sorting, foraging, dormancy, reproduction, and fitness evaluation, with total complexity O(iter_max·ps·(log(ps)+dim+f(·))) when np=1.Its component costs include sorting at O(ps·log(ps)) and fitness evaluation at O(ps·f(·)).
3. Experimental analysis
The APO was evaluated on the CEC2022 benchmark through parameter studies, comparisons with state-of-the-art algorithms, statistical tests, and convergence analyses. Results indicate strong benchmark performance and a shifting balance from exploration toward exploitation.
- Benchmark setup: The CEC2022 evaluation used 20 dimensions, 1,000,000 maximum fitness evaluations, and 30 executions per algorithm.The APO parameter n_p was set to 1, while p_fmax was tested from 0 to 1 and performed best at 0.1 according to the Friedman test.
- Population-size analysis: Large populations outperformed medium and small populations across the tested function categories.For F1, population size 100 ranked first and population size 80 ranked second; the ranking pattern favored larger populations overall.
- Comparative evaluation: The APO was compared with 32 state-of-the-art algorithms using mean and standard deviation of best-so-far results.The best mean and standard-deviation results were highlighted in Table 5 for comparison.
- Non-parametric statistical analysis: The Wilcoxon analysis reported 12 wins, 0 draws, and 0 losses against GA, Jaya, GWO, WOA, PPE, SCA, AOA, SPO, and SPBO.Against DE and BSA, APO won 7 comparisons, drew 3, and lost 2.
- Non-parametric statistical analysis: The Friedman ranking placed APO first among 17 algorithms across the 12 benchmark functions, followed by BSA and DE.The reported ranking concluded that APO outperformed the compared state-of-the-art algorithms on CEC2022.
- Convergence analysis: APO emphasized intensive exploration early and heightened exploitation later, with sparse points far from optima and dense points near promising areas.The average and best fitness values gradually decreased during iterations, indicating improving candidate quality and convergence.
4. Application
The APO was applied to practical engineering design and multilevel image segmentation problems.
- Application: The application section covers engineering design and multilevel image segmentation problems.These applications involve practical optimization settings described for continuous and discrete spaces with constraints in the study context.
4.1. Engineering design
APO was evaluated on five constrained continuous engineering designs and achieved the overall highest Friedman ranking, with competitive feasibility and efficiency results.
- APO addressed constrained continuous optimization of tension/compression springs, pressure vessels, welded beams, speed reducers, and three-bar trusses.
- Tension/compression spring: 0.01266529 was the lowest reported cost for the tension/compression spring design, outperforming eight comparison algorithms.
- Pressure vessel: 5887.614 was APO’s reported optimal pressure-vessel cost, although Jaya and PSO achieved more competitive results.
- 1.724854 was APO’s optimal welded-beam cost, while its speed-reducer and three-bar-truss results were 2994.471 and 263.8958 in weight.
- APO ranked first overall across the five designs in the Friedman test, ahead of PSO, Jaya, DE, GWO, BSA, PPE, WOA, and GA.
- Stability and computational efficiency: APO achieved a 100% success rate on four designs, with 9900 average fitness evaluations and 0.13 s average duration for the spring problem.
- The experiments concluded that APO outperformed eight state-of-the-art algorithms on the five engineering design problems.
4.2. Multilevel image segmentation
APO was applied to constrained discrete multilevel segmentation of a 512 × 512 Lena image, with performance improving relative to competitors as threshold count increased.
- Multilevel segmentation divides an image into multiple regions using n thresholds that assign pixels to n + 1 classes.
- APO solved the 512 × 512 color Lena segmentation task using 100 individuals, 100 iterations, and 31 runs.
- Each RGB channel was thresholded separately from its pixel histogram, then the three segmented channels were concatenated.
- Segmentation quality was evaluated with PSNR, SSIM, and FSIM, where higher values indicate better image quality.
- Algorithm comparison: All nine algorithms performed identically with two thresholds, but APO showed an advantage as the threshold count increased.
- Algorithm comparison: APO ranked first overall by Friedman averaging of the three indicators, ahead of MVO, SPO, AOA, SPBO, SDO, SCA, TLBO, and GSA.
5. Conclusion
The paper introduces APO as a protozoa-inspired optimizer and validates it across benchmark, engineering, and image-segmentation tasks. The authors report that it significantly surpasses the compared state-of-the-art algorithms, while identifying broader optimization settings for future study.
- APO models protozoan foraging, dormancy, and reproduction, including a mapping vector that varies dimensional crossover between candidate solutions.
- The algorithm was evaluated on CEC2022 unimodal, multimodal, hybrid, and composition functions, plus five engineering designs and multilevel image segmentation.
- Experimental data confirmed that APO significantly surpasses the compared state-of-the-art algorithms.
- The authors identify binary, large-scale, expensive, multi-objective, and multitask optimization as future application areas.