The study, published in the journal Results in Engineering by Yu-Feng Sun, Jie-Sheng Wang and colleagues, describes an enhanced version of the Manta Ray Foraging Optimization algorithm, or MRFO. The original algorithm, first introduced a few years ago, mimics the cooperative hunting strategies that manta rays use to harvest plankton. Because plankton drifts unpredictably with seasons and weather, manta rays have evolved three distinct foraging behaviors: chain foraging, in which animals line up in arcs so that each one catches food missed by the one ahead; spiral foraging, where the group coils around a dense patch of food; and somersaulting, in which individuals flip back and forth around a promising spot. Translated into mathematics, these behaviors become rules that move a population of candidate solutions around a search space, gradually converging on the best answer.

What makes the CEED problem so stubborn is its mathematical character. The relationship between a generator’s output and its fuel consumption follows a cubic curve, and the emissions of carbon dioxide, nitrogen oxides and sulfur dioxide each follow their own cubic functions. The combined objective is nonlinear, nonconvex and discontinuous, riddled with local minima that can trap classical techniques such as lambda iteration or gradient-based methods. To make matters harder, the problem is genuinely multi-objective, so the researchers converted it into a single objective by applying price penalty factors that fold the cost of each pollutant into one total figure in dollars per hour.

Constraint handling is where many algorithms stumble. Traditional approaches add a fixed penalty whenever a solution violates the power balance requirement, the equation demanding that total generation minus transmission losses equals demand. But a fixed penalty is a blunt instrument. Set it too low and the algorithm happily returns infeasible schedules; set it too high and the search becomes so restricted that it cannot explore properly. Finding the right value through trial and simulation is time-consuming and inefficient. The team’s answer was to make the penalty itself dynamic, drawing on five different mathematical functions: sinusoidal, hyperbolic tangent, arctangent, linear and hyperbolic secant. Each function is raised to an order that changes during the run, so the penalty strength adapts to both the size of the violation and the stage of the optimization. Early on, mild penalties preserve exploration; later, penalties tighten to push solutions toward feasibility.

The second major innovation targets the heart of the MRFO algorithm. In the original design, a parameter called C governs the switch between exploration, casting the net wide across the search space, and exploitation, homing in on the best solution found so far. This parameter rises monotonically from zero to one, a predictable march that can leave the algorithm poorly balanced at certain stages. The researchers replaced it with a chaotic switching factor, drawn from nine well-known chaotic maps including Chebyshev, logistic, tent, sine, Gauss and singer maps, combined with an exponential adjustment function. The result is a parameter that still trends upward but oscillates along the way, periodically reopening the search space and preventing the population from settling prematurely into a local optimum.

To find out which chaotic map worked best, the team ran all nine enhanced variants, named C1MRFO through C9MRFO, through the CEC-2022 benchmark suite of twelve test functions, using a population of thirty and five hundred iterations averaged over thirty independent runs. The Chebyshev-based variant, C1MRFO, emerged as the overall winner, ranking first among the nine and beating the original MRFO, which finished last. Wilcoxon rank-sum tests confirmed that most of the improvements were statistically significant, with p-values below the conventional 0.05 threshold. When C1MRFO was pitted against a roster of state-of-the-art metaheuristics, including the dhole optimization algorithm, evolutionary mating algorithm, dung beetle optimizer, snow geese algorithm, particle swarm optimization, grey wolf optimizer, sparrow search algorithm and marine predators algorithm, it achieved the best final rank, taking first place on many of the test functions while keeping computational times competitive.

The real test, however, came when the algorithm was applied to actual dispatch problems. The researchers simulated a six-unit power system under four load demands of 150, 175, 200 and 225 megawatts. Across all four scenarios, C1MRFO produced the lowest comprehensive total cost, which combines fuel cost with the weighted penalties for carbon dioxide, nitrogen oxide and sulfur dioxide emissions. At 150 megawatts, for example, it reached a total cost of about 10,137 dollars per hour, edging out the grey wolf optimizer and clearly beating algorithms such as the arithmetic optimization algorithm, Harris hawks optimization, whale optimization algorithm, sine cosine algorithm, particle swarm optimization and the lambda iteration method. The pattern held at every load level, with the margin over the weakest classical method growing to thousands of dollars per hour at 225 megawatts.

The dynamic penalty functions delivered a further, striking gain. When the sinusoidal penalty function was applied with the parameter order set to dy5, the total cost at 150 megawatts dropped from roughly 10,137 to 9,065.71 dollars per hour, and similar reductions appeared at the higher loads, reaching 14,621.45 dollars per hour at 225 megawatts. Comparing the five penalty shapes, the sinusoidal function consistently produced the lowest minimum costs, beating the hyperbolic tangent, arctangent, linear and hyperbolic secant alternatives by margins ranging from hundreds to nearly two thousand dollars per hour depending on the load. Notably, when the penalty order was fixed rather than dynamic, performance was worst across the board, confirming that the adaptivity of the penalty, not merely its presence, is what matters.

To verify that each improvement contributes on its own, the team conducted an ablation study on a much larger twenty-unit system with a total demand of 2,500 megawatts. The results formed a clear hierarchy: the original MRFO achieved 61,835.16 dollars per hour, the chaos-enhanced C1MRFO reached 61,826.11, and the fully improved version combining chaos with the optimal sinusoidal dynamic penalty, sin(dy5)MRFO, achieved the best figure of 61,807.36 dollars per hour. All three outperformed the phasor particle swarm optimization benchmark, which came in at 61,895.57 dollars per hour. The stepwise improvement demonstrates that the two enhancements are not redundant but superimpose their benefits, and that the framework scales to systems far larger than the six-unit case.

The authors are careful to note where the work goes next. The current approach converts the multi-objective problem into a single objective through price penalty factors, but future versions could adopt Pareto-based multi-objective optimization to generate full sets of non-dominated trade-off solutions. The test cases also assume deterministic demand, so extending the algorithm to stochastic dispatch with wind and solar integration is another goal, as is combining the dynamic penalty with feasibility-first constraint-handling strategies. For now, the study offers a compelling demonstration that borrowing the foraging logic of a filter-feeding fish, perturbing it with chaos, and teaching it to calibrate its own punishments can squeeze meaningful savings and emission reductions out of one of power engineering’s hardest scheduling problems.

Subject of Research: An improved manta ray foraging optimization algorithm with chaotic switching factors and dynamic penalty functions for solving the combined economic emission dispatch problem in power systems.

Article Title: Manta ray foraging optimization algorithm with dynamic penalty function and chaos switching factor to solve combined economic emission dispatch problem

Article References: Sun, Y.-F., Wang, J.-S., Zhang, X.-Y., Li, Y.-X., Guan, X.-Y., & Liu, X. (2026). Manta ray foraging optimization algorithm with dynamic penalty function and chaos switching factor to solve combined economic emission dispatch problem. Results in Engineering, 32, Article 113383. https://doi.org/10.1016/j.rineng.2026.113383

Image Credits: AI Generated

DOI: Not provided

Keywords: manta ray foraging optimization, combined economic emission dispatch, metaheuristic algorithms, chaotic maps, dynamic penalty function, power system optimization, emission reduction, fuel cost minimization, CEC-2022 benchmarks, swarm intelligence, constraint handling, Results in Engineering

DNA microarray technology has transformed modern medicine by allowing scientists to measure the activity levels of thousands of genes simultaneously. In cancer research, this means clinicians can potentially distinguish tumor subtypes by their molecular fingerprints rather than by appearance under a microscope alone. Yet the very richness that makes microarrays powerful also makes them difficult to analyze. A typical dataset may contain tens of thousands of gene expression values for only a few dozen or a few hundred patients, a classic case of high dimensionality paired with small sample size. The authors note that large-scale data, a lack of reliable instances, and complex correlations among genes have all complicated the analysis process, making it easy for learning algorithms to latch onto noise instead of genuine biological signal.

The standard remedy is gene selection, a form of feature selection in which an algorithm searches for the subset of genes that best supports accurate classification. Over the past two decades, researchers have deployed filters, wrappers, and embedded methods, often hybridized with metaheuristic optimizers such as genetic algorithms, moth flame optimization, and intelligent water drop algorithms. These techniques have produced steady gains, but they typically assume that all the data sits in a single repository. In real-world medicine, that assumption is increasingly untenable. Hospitals, research centers, and laboratories hold their own patient records, and privacy regulations, institutional policies, and ethical concerns often prevent them from sharing raw genomic data with outside parties.

Federated learning has emerged as a compelling answer to this dilemma. In a federated setup, multiple participants, called federates, train models collaboratively while keeping their raw data local; only model updates or intermediate results travel across the network. The approach has proven its worth in mobile keyboard prediction and medical imaging, but the authors of the new study argue that many existing federated gene selection methods lack an optimal, hierarchical mechanism for feature reduction. In other words, they may coordinate model training across institutions but do not systematically narrow down which genes should be considered at each level of the process, leaving accuracy on the table when datasets are as extreme as microarrays.

The new framework, described as a vertical federated stacking deep learning system, tackles that gap with a two-stage distributed feature selection strategy that operates at both the local and central levels. The word vertical refers to how the data is partitioned: rather than splitting patients among federates, the framework splits the genes. Using a weight-based distribution, the full set of gene features is divided among scalable federates, so each participant holds expression values for a different slice of the genome while all of them see the same patients. This design keeps the collaborative process scalable, because adding more federates distributes the gene space more thinly rather than demanding more data from any single institution.

At the first stage, each federate independently searches its own slice of genes using a wrapper-based method built on the Binary Manta Ray Foraging Optimization algorithm, abbreviated BMRFO. The manta ray foraging optimization algorithm is a relatively recent nature-inspired optimizer that mimics the foraging behavior of manta rays, including their characteristic cyclone feeding patterns, to explore a search space efficiently. In its binary form, each candidate solution is a string of on-off decisions indicating whether a given gene should be included or excluded. A wrapper approach means the optimizer evaluates each candidate gene subset by actually training and testing a classifier on it, so the selection is directly tied to predictive performance rather than to statistical correlations alone. Previous studies have shown the manta ray optimizer to be competitive in feature selection tasks ranging from network intrusion detection to medical diagnosis, which motivated the team to adapt it for the federated setting.

Once the local searches are complete, the genes selected by every federate are merged and sent to a central server, which performs a final refinement step to identify the near-optimal gene set. This hierarchical structure is the core novelty of the method: a coarse-grained distributed search that trims the gene space in parallel, followed by a centralized fine-grained selection that reconciles the candidates and eliminates redundancy. The design mirrors the way a large research consortium might operate, with each member institution pre-screening its own portion of the genome before a coordinating body consolidates the findings. Crucially, because the raw expression data never leaves the federates, the framework preserves the privacy benefits that make federated learning attractive in the first place.

The final selected genes then feed into a robust deep stacking classification model that combines four different neural architectures: a multilayer perceptron, a convolutional neural network, a recurrent neural network, and a long short-term memory network, known as LSTM. Stacking, or stacked generalization, is an ensemble technique in which the outputs of multiple base learners are combined, often by a higher-level model, so that the strengths of one architecture can compensate for the weaknesses of another. The multilayer perceptron captures nonlinear relationships among the selected genes, the convolutional network detects local patterns in the expression profiles, and the recurrent and LSTM networks, which are designed to handle sequential data, can model dependencies across the ordered gene features. By fusing these complementary perspectives, the stacked model is better equipped to generalize from the small, high-dimensional datasets that dominate microarray research.

The evaluation spanned five benchmark datasets that have become standard proving grounds for gene selection algorithms. On the Leukemia dataset, the framework achieved an accuracy of 97.78 percent in predicting test samples, while on the Lymphoma, MLL, Ovarian, and SRBCT datasets it reached 99.99 percent. The authors report that these figures represent a significant improvement over previous methods, a notable claim in a field where benchmark accuracies have been climbing for years and where differences of even a fraction of a percent can be hard to earn. The datasets themselves are publicly available through repositories such as Kaggle and the biolab cancer projection pages, and the authors state that raw processed data supporting the findings are available from the corresponding author upon reasonable request from editors or reviewers.

Beyond the headline numbers, the study carries broader implications for how machine learning and genomics intersect. As personalized medicine pushes toward models trained on data scattered across hospitals and countries, frameworks that can perform sophisticated feature selection without centralizing sensitive records become increasingly valuable. The combination of a nature-inspired optimizer with a hierarchical federated architecture suggests a template that could extend beyond microarrays to other high-dimensional biomedical data, from single-cell RNA sequencing to proteomics. The work also adds to a growing literature on hybrid metaheuristic-deep learning systems, in which optimizers inspired by animal behavior, swarm intelligence, or physical processes are used to tame search spaces that would overwhelm brute-force methods. For a discipline where the goal is to find a handful of diagnostic genes among tens of thousands of candidates, and to do so across institutional boundaries, the Iranian team’s results indicate that distributing the problem, both the data and the search, may be exactly the right way to solve it.

Subject of Research: Vertical federated learning with bio-inspired optimization for gene selection in DNA microarray cancer classification

Article Title: Vertical federated deep stacking learning gene selection based on BMRFO algorithm

Article References: Vertical federated deep stacking learning gene selection based on BMRFO algorithm. (n.d.). https://doi.org/10.1007/s10586-026-06626-4

Image Credits: AI Generated

DOI: 10.1007/s10586-026-06626-4

Keywords: federated learning, gene selection, DNA microarray, cancer classification, manta ray foraging optimization, deep learning, stacked ensemble, feature selection, bioinformatics, neural networks, LSTM, privacy-preserving machine learning