Artificial Protozoa Optimizer (APO): A novel bio-inspired metaheuristic algorithm for engineering optimization

This paper addresses a core problem in engineering and computer science: how to efficiently find high-quality solutions to optimization tasks that are often nonlinear, non-differentiable, multimodal, and constrained. The authors propose a new population-based metaheuristic, the Artificial Protozoa Optimizer (APO), inspired by the survival behaviors of euglena protozoa—specifically foraging (with autotrophic and heterotrophic modes), dormancy, and binary fission reproduction. This matters because many practical engineering design problems are difficult for exact methods (e.g., due to NP-hardness and the need for gradients/convexity), and because metaheuristics must balance exploration (global search) and exploitation (local refinement) to avoid premature convergence.

Within the broader field of swarm and evolutionary computation, APO is positioned as a “new optimizer” that aims to improve that exploration–exploitation balance using biologically motivated operators. The paper’s main contributions are: (1) a mathematical modeling of protozoa behaviors into algorithmic update rules; (2) a foraging mechanism that uses a mapping vector to control which dimensions are updated, designed so that better candidates change less while worse candidates change more; (3) extensive evaluation on the CEC2022 single-objective constrained and unconstrained benchmark suite; and (4) demonstrations on five constrained continuous engineering design problems and a constrained multilevel image segmentation task.

Methodologically, the study uses a standard metaheuristic evaluation protocol. For benchmark testing, APO is run on 12 CEC2022 functions (20-dimensional, search range $[-100,100{]}^{\text{dim}}$), with a maximum of 1,000,000 fitness evaluations per run. Each algorithm is executed 30 independent times. The authors compare APO against 32 state-of-the-art metaheuristics (covering evolutionary, swarm, physics-based, and human-based families). APO’s key internal parameterization includes setting the number of neighbor pairs to $n_p=1$ for complexity reasons, and tuning the maximum dormancy/reproduction proportion $p_f^{\max }$ over $\{0,0.1,0.2,\dots ,1\}$; a Friedman test indicates best performance at $p_f^{\max }=0.1$. For the engineering and image segmentation applications, the authors use penalty functions for constraints and run APO with $p_s=100$ and specified iteration budgets.

APO’s algorithmic design is built around four update modes. Foraging is split into autotrophic and heterotrophic behaviors. Autotrophic foraging uses a dynamic foraging factor $f$ that varies with iteration via a cosine schedule, and it incorporates an “internal” and “external” influence term using ranked neighbor pairs. Heterotrophic foraging moves individuals toward a nearby food-rich location with stochastic directionality. Dormancy replaces inferior individuals by sampling within the variable bounds using $X_{\min }$ and $X_{\max }$, promoting exploration. Reproduction duplicates a protozoan and applies a perturbation (with forward or reverse sign), promoting exploitation. A central mechanism is the mapping vector $M_f$ in foraging, which selects which dimensions to update; the paper argues this yields exploitation for better candidates and exploration for worse candidates.

The experimental results on CEC2022 provide the strongest evidence of competitiveness. On function F1 (shifted/rotated Zakharov, global optimum at 0), APO achieves a mean of $1.5158\times 10^{-14}$ with standard deviation $2.5567\times 10^{-14}$, while GA has mean $1.7863\times 10^3$ and DE has mean $6.3541\times 10^1$. On F3 (expanded Schaffer’s f6), APO’s mean is $7.5791\times 10^{-14}$ (std $5.4509\times 10^{-14}$), outperforming DE (mean $1.1369\times 10^{-13}$) and others. On F4 (non-continuous rotated Rastrigin), APO’s mean is $5.8080$ (std $1.9764$), substantially better than GA (mean $89.475$) and PSO (mean $59.100$). On F6 (hybrid function 1), APO’s mean is $35.179$ (std $22.61$), far below GA ($1.6627\times 10^3$) and PSO ($4.1492\times 10^2$). Across the 12 benchmark functions, the authors report that APO wins 12 times and draws 0 times and loses 0 times against several baselines in Wilcoxon signed-rank pairwise comparisons (at 5% significance), and that APO is ranked first overall by Friedman test among 17 algorithms (APO, 16 competitors). The paper also reports that APO’s overall ranking is first, followed by BSA and DE, with some algorithms tied at lower ranks (e.g., MVO and SPBO both ranked fifth; PSO and TLBO both ranked ninth).

The authors further analyze convergence and exploration–exploitation behavior using a dimension-wise diversity metric. They report that exploration rate decreases over iterations while exploitation rate increases, indicating a controlled transition from global search to local refinement. They also show convergence curves and search trajectories (e.g., for five protozoa in 20 dimensions over 150 iterations) that demonstrate decreasing best fitness over time.

Limitations are not deeply quantified in the provided text, but several constraints are apparent from the methodology. First, APO is evaluated primarily in the single-objective setting (CEC2022 single-objective functions) and does not address multi-objective or expensive evaluation regimes beyond future work. Second, the paper’s parameter tuning is limited to $p_f^{\max }$ and $n_p$ (set to 1), so generalization to other problem types or different dimensionalities may require retuning. Third, while the authors include constrained engineering problems and constrained image segmentation, the constraint handling is done via penalty functions, which can be sensitive to penalty scaling and may affect fairness relative to competitors using different constraint-handling strategies.

Practical implications are demonstrated through five constrained engineering design problems: tension/compression spring, pressure vessel, welded beam, speed reducer, and three-bar truss. APO achieves the best reported optimum for the tension/compression spring with cost $0.01266529$ at variables $d=0.0516521$, $D=0.355829$, $N=11.3413$. For the pressure vessel, APO achieves $5887.614$ at $T_s=0.77916$, $T_h=0.38516$, $R=40.3707$, $L=199.3144$, outperforming most competitors except Jaya and PSO. For the welded beam, APO reaches $1.724854$ with variables $h=0.20573$, $l=3.4705$, $t=9.0366$, $b=0.20573$, matching Jaya and PSO. For speed reducer, APO achieves weight $2994.471$ (best among compared methods). For three-bar truss, APO achieves weight $263.8958$ at $A_1=0.78868$, $A_2=0.40825$. The authors also report stability/efficiency metrics (success rate, average number of fitness evaluations, and average calculation duration). For example, on the tension/compression spring, APO attains success rate $100\%$ with AFEs $9900$ and ACDs $0.13$ s, outperforming other algorithms on feasibility and efficiency.

Finally, APO is applied to multilevel image segmentation using minimum cross-entropy thresholding (MCET) with $n$ thresholds producing $n+1$ classes. On a $512\times 512$ Lena color image, APO is evaluated for 2, 4, 6, 8, and 10 thresholds using PSNR, SSIM, and FSIM. The reported results show APO leading in mean rank across the three metrics, with the paper’s Friedman-based overall ranking for segmentation indicating APO first, followed by MVO and SPO, and then AOA/SPBO tied, with SDO, SCA, TLBO, and GSA lower.

Overall, the paper concludes that APO is a highly competitive metaheuristic for both benchmark optimization and practical constrained engineering and image segmentation tasks. Who should care? Researchers developing new metaheuristics will find APO’s protozoa-inspired operator design and dimension-mapping mechanism relevant, while practitioners in engineering design and image processing may care about the reported improvements in constrained optimization quality and feasibility success rates, especially where gradient information is unavailable.