Grey Wolf Optimizer Research Articles

Neutrosophic goal programming technique with bio inspired algorithms for crop land allocation problem

In agriculture, crop planning and land distribution have been important research subjects. The distribution of land involves several multi-functional tasks, such as maximizing output and profit and minimizing costs. These functions are influenced by a variety of uncertain elements, including yield, crop price, and indeterminate factors like seed growth and suitable fertilizer. In order to address this problem, other researchers have used fuzzy and intuitionistic fuzzy optimization approaches, which did not include the indeterminacy membership functions. However, the neutrosophic optimization technique addresses the problem by using individual truth, falsity, and indeterminacy membership functions. So, to improve the optimal solution, the Neutrosophic Goal Programming (NGP) problem with hexagonal intuitionistic parameters is employed in this study. The membership functions for truth, indeterminacy, and falsity are constructed using hyperbolic, exponential, and linear membership functions. Minimizing the under deviations of truth, over deviations of indeterminacy, and falsity yields the NGP achievement function, which is used to attain optimal expenditure, production, and profit under the constraints of labour, land, food requirements, and water. Bio-inspired computing has been a major research topic in recent years. Optimization is mostly accomplished through the use of bio-inspired algorithms, which draw inspiration from natural behaviour. Bio-inspired algorithms are highly efficient in exploring large solution spaces, and helps to manage trade-offs between various goals, and providing the global optimal solution. Consequently, bio-inspired algorithms such as Grey Wolf Optimization (GWO), Social Group Optimization (SGO), and Particle Swarm Optimization (PSO) are employed in the current work to determine the global optimal solutions for the NGP achievement function. The data for the study was collected from the medium-sized farmers in Ariyalur District, Tamil Nadu, India. To illustrate the uniqueness and application of the developed method, the optimal solutions of the suggested method are compared with Zimmermann, Angelov, and Torabi techniques. The proposed technique demonstrates that the bioinspired algorithms’ optimal solution to the neutrosophic goal is superior to the existing approaches.

Large-scale magnetic target state estimation using model parameter optimization

Abstract Aiming at the problem of non-negligible target scale in the state estimation of magnetic targets, a state estimation method that requires only the measurement data of a single three-axis magnetic sensor and is insensitive to the initial value is proposed. Firstly, the state estimation model of a single three-component magnetic sensor is established based on a magnetic dipole array. A signal preprocessing method is designed to reject useless signals and keep the waveform intact. Five swarm intelligent optimization algorithms such as Particle Swarm Optimization (PSO), Artificial Hummingbird Algorithm (AHA), Artificial Rabbits Optimization (ARO), Grey Wolf Optimizer (GWO), and Improved GWO (IGWO) are introduced to perform state optimization, and two methods, such as Conjugate Gradient Least Squares (CGLS), and Stepwise Regression (SR), are introduced to calculate the magnetic moment parameters, and PSO-CGLS, AHA-CGLS, AHA-CGLS, ARO-CGLS, ARO-CGLS, ARO-CGLS, ARO-CGLS, ARO-CGLS, ARO-CGLS, ARO-CGLS, ARO-CGLS, and ARO-CGLS are designed in combination. CGLS, ARO-CGLS, GWO-CGLS, IGWO-SR, IGWO-CGLS, and six state estimation methods. In order to address the problem that the parameter optimization range is too large, the state to be estimated is adjusted from the initial state to the state at the moment of the maximum value of the magnetic field modulus, and the method of positive transverse distance estimation is used to narrow the range of positive transverse position optimization. Finally, numerical simulation tests and ship model tests are used for verification, and the results show that the proposed methods are accurate and can meet the needs of precise target state estimation.

An effective binary dynamic grey wolf optimization algorithm for the 0-1 knapsack problem

An effective binary dynamic grey wolf optimization algorithm for the 0-1 knapsack problem

A crack identification scheme based on neural network surrogate model and XFEM

Crack detection and identification is of great significance to the safety issues of engineering structures. In this paper, an intelligent crack identification scheme based on extended finite element and neural network surrogate model is proposed to realize the accurate identification of crack parameters. The method firstly employs extended finite element forward analysis to obtain the displacement data of measurement points on geometric models with different crack lengths, and inputs them as sample data to train the agent model, establishes a neural network-based inverse analysis model for crack identification, and automatically updates the threshold and weight of the neural network by using the Gray Wolf optimization algorithm to finally compute the globally optimal results. In the screening of the surrogate model, this paper verifies the advantages of the neural network surrogate model in data fitting and crack information extraction by comparing and analyzing the characteristics of neural network, support vector machine and other surrogate models, and optimizing the neural network surrogate model by adopting the Gray Wolf optimization algorithm. Finally, several numerical examples of different types of cracks are given to verify the validity of the proposed method, and the results show that the proposed method can accurately invert the geometric information of cracks.

Revolutionizing PV grid integration: Metaheuristic optimization of fractional PI controllers in T-type neutral point piloted inverters for enhanced performance

This study investigates the efficacy of FOPI regulators as a substitute for conventional proportional integral (PI) controllers in grid-connected PV inverters. The research's objective is to improve the dynamic efficiency of these systems by integrating intelligent optimization techniques that utilize both PI and FOPI controllers and employing different metaheuristic optimization procedures. The study introduces four new optimization algorithms: the Tyrannosaurus Optimization Algorithm (TROA), the Nutcracker Optimization Algorithm (NOA), the Golden Eagle Optimizer (GEO), and the Jellyfish Search Optimizer (JSO). These are made to meet the needs of multi-objective optimization. The proposal suggests using two PI/FOPI regulators to regulate both voltage and amperage provided by the multilayer inverter. The proposal employs a T-type three-level inverter, known for its superior conversion efficiency over conventional inverters. Matlab-Simulink simulations demonstrate that the Nutcracker optimization algorithm (NOA) outperforms other metaheuristic techniques for optimizing dynamic behavior, such as overshoot, settling time, execution, and rising time. Comparisons with existing methods, such as the Manta Rays Foraging Optimization (MRFO) and Grey Wolf Optimizer (GWO), show that all four new algorithms consistently outperform these existing algorithms. The data suggest that using NOA improves stability, efficiency, and power factor while reducing the inverter's THD.

A Novel Short-Term PM2.5 Forecasting Approach Using Secondary Decomposition and a Hybrid Deep Learning Model

PM2.5 pollution poses an important threat to the atmospheric environment and human health. To precisely forecast PM2.5 concentration, this study presents an innovative combined model: EMD-SE-GWO-VMD-ZCR-CNN-LSTM. First, empirical mode decomposition (EMD) is used to decompose PM2.5, and sample entropy (SE) is used to assess the subsequence complexity. Secondly, the hyperparameters of variational mode decomposition (VMD) are optimized by Gray Wolf Optimization (GWO) algorithm, and the complex subsequences are decomposed twice. Next, the sequences are divided into high-frequency and low-frequency parts by using the zero crossing rate (ZCR); the high-frequency sequences are predicted by a convolutional neural network (CNN), and the low-frequency sequences are predicted by a long short-term memory network (LSTM). Finally, the predicted values of the high-frequency and low-frequency sequences are reconstructed to obtain the final results. The experiment was conducted based on the data of 1009A, 1010A, and 1011A from three air quality monitoring stations in the Beijing area. The results indicate that the R2 value of the designed model increased by 2.63%, 0.59%, and 1.88% on average in the three air quality monitoring stations, respectively, compared with the other single model and the mixed model, which verified the significant advantages of the proposed model.

Exploring the potential of 5G uplink communication: Synergistic integration of joint power control, user grouping, and multi-learning Grey Wolf Optimizer

Non-orthogonal Multiple Access (NOMA) techniques offer potential enhancements in spectral efficiency for 5G and 6G wireless networks, facilitating broader network access. Central to realizing optimal system performance are factors like joint power control, user grouping, and decoding order. This study investigates power control and user grouping to optimize spectral efficiency in NOMA uplink systems, aiming to reduce computational difficulty. While previous research on this integrated optimization has identified several near-optimal solutions, they often come with considerable system and computational overheads. To address this, this study employed an improved Grey Wolf Optimizer (GWO), a nature-inspired metaheuristic optimization method. Although GWO is effective, it can sometimes converge prematurely and might lack diversity. To enhance its performance, this study introduces a new version of GWO, integrating Competitive Learning, Q-learning, and Greedy Selection. Competitive learning adopts agent competition, balancing exploration and exploitation and preserving diversity. Q-learning guides the search based on past experiences, enhancing adaptability and preventing redundant exploration of sub-optimal regions. Greedy selection ensures the retention of the best solutions after each iteration. The synergistic integration of these three components substantially enhances the performance of the standard GWO. This algorithm was used to manage power and user-grouping in NOMA systems, aiming to strengthen system performance while restricting computational demands. The effectiveness of the proposed algorithm was validated through numerical evaluations. Simulated outcomes revealed that when applied to the joint challenge in NOMA uplink systems, it surpasses the spectral efficiency of conventional orthogonal multiple access. Moreover, the proposed approach demonstrated superior performance compared to the standard GWO and other state-of-the-art algorithms, achieving reduced system complexity under identical constraints.

Hybrid nonlinear probabilistic model using Monte Carlo simulation and hybrid support vector regression for evaporation predictions

This study presents a hybrid model combining Support Vector Regression (SVR) with the Gray Wolf Optimizer (GWO) for evaluating evaporation dynamics. The model was tested using 50 years of meteorological data from Zabol and Chabahar stations in Iran, and compared against standard SVR. Results show that the SVR-GWO significantly outperforms SVR, reducing predictive errors by over 41% on average. For Zabol, the SVR-GWO markedly improved performance, particularly in July where R2 increased from 0.867 to 0.973. The model more closely matched observed statistics, especially for mean and standard deviation, though challenges remain in reproducing higher-order statistics. Interestingly, 5,000 functions evaluation proved nearly as effective as 500,000 suggesting computational efficiency. This integrated approach, incorporating Monte Carlo simulations and kernel density estimation, provides valuable probabilistic insights into evaporation dynamics, offering a promising tool for drought vulnerability assessment and water resources planning in regions with complex climatic patterns.

A novel hybrid deep learning approach with GWO–WOA optimization technique for human activity recognition

The effectiveness of Human Activity Recognition (HAR) models can be largely attributed to the components derived from domain expertise. The classification system swiftly and effectively categorizes human physical activity by utilizing a comprehensive collection of variables. To construct HAR models and categorize different activities, deep learning algorithms have recently seen increased application in academic research for autonomously extracting features from raw sensory data. The primary goal of this study is to develop a strong and accurate activity classification model by first extracting features from an input dataset and then using deep learning models. In this research, we look at how well the model does in the field of HAR. Analyzing time-series data from smartphones and wearable sensors, algorithms based on deep learning (DL) have been used to forecast various human actions. Applying DL-based methods to time-series data presents a variety of challenges, notwithstanding their usage for activity identification. If the suggested approach is put into practice, the concerns mentioned before may be alleviated. To construct HAR classification techniques using data collected from wearable sensors, this study presents two Hybrid Learning Algorithms (HLA). This research employs Convolutional Neural Network (CNN) and Recurrent Neural Network (RNN) methods to teach the model sequential and spatial patterns. The process of selecting characteristics has been enhanced through the use of optimization techniques. Together, the Whale Optimization Algorithm and the Grey Wolf Optimizer form a Hybrid Optimization approach, which is introduced in this study. By combining and capitalizing the benefits of both approaches, the hybrid technique enhances the optimization process as a whole. This research may also assist individuals who have undergone amputations or experienced strokes in selecting an appropriate model and developing a personalized gait reference trajectory.

RSM integrated GWO, Driving Training, and Election-Based Algorithms for optimising ethylic biodiesel from ternary oil of neem, animal fat, and jatropha

The worldwide exploration of the ethanolysis protocol (EP) has decreased despite the multifaceted benefits of ethanol, such as lower toxicity, higher oxygen content, higher renewability, and fewer emission tail compared to methanol, and the enhanced fuel properties with improved engine characteristics of multiple-oily feedstocks (MOFs) compared to single-oily feedstocks. The study first proposed a strategy for the optimisation of ethylic biodiesel synthesis from MOFs: neem, animal fat, and jatropha oil (NFJO) on a batch reactor. The project's goals were to ensure environmental benignity and encourage the use of totally biobased products. This was made possible by the introduction of novel population based algorithms such as Driving Training-Based Optimization (DTBO) and Election-Based Optimization (EBOA), which were compared with the widely used Grey Wolf Optimizer (GWO) combined with Response Surface Methodology (RSM). The yield of NFJO ethyl ester (NFJOEE) was predicted using the RSM technique, and the ideal transesterification conditions were determined using the DTBO, EBOA, and GWO algorithms. Reaction time showed a strong linear relationship with ethylic biodiesel yield, while ethanol-to-NFJO molar ratio, catalyst dosage, and reaction temperature showed nonlinear effects. Reaction time was the most significant contributor to NFJOEE yield.The important fundamental characteristics of the fuel categories were investigated using the ASTM test procedures. The maximum NFJOEE yield (86.3%) was obtained at an ethanol/NFJO molar ratio of 5.99, KOH content of 0.915 wt.%, ethylic duration of 67.43 min, and reaction temperature of 61.55 °C. EBOA outperforms DTBO and GWO regarding iteration and computation time, converging towards a global fitness value equal to 7 for 4 s, 20 for 5 s and 985 for 34 s. The key fuel properties conformed to the standards outlined by ASTMD6751 and EN 14,214 specifications. The NFJOEE fuel processing cost is 0.9328 USD, and is comparatively lesser than that of conventional diesel. The new postulated population based algorithm models can be a prospective approach for enhancing biodiesel production from numerous MOFs and ensuring a balanced ecosystem and fulfilling enviromental benignity when adopted.

Soil water content estimation by using ground penetrating radar data full waveform inversion with grey wolf optimizer algorithm

AbstractSoil water content (SWC) estimation is important for many areas including hydrology, agriculture, soil science, and environmental science. Ground penetrating radar (GPR) is a promising geophysical method for SWC estimation. However, at present, most of the studies are based on partial information of GPR, like travel time or amplitude information, to invert the SWC. Full waveform inversion (FWI) can use the information of the entire waveform, which can improve the accuracy of parameter estimation. This study proposes a novel SWC estimation scheme by using the FWI of GPR, optimized by the grey wolf optimizer (GWO) algorithm. The proposed scheme includes a petrophysical relationship to link the SWC with the relative dielectric permittivity, 1D GPR forward modeling, and a GWO optimization algorithm. First, numerical modeling was carried out, and the proposed scheme was applied to both noise‐free and noisy data to verify its applicability. Then, the proposed method was applied to data collected from a field experimental site. These results, derived from both synthetic and real datasets, show that the proposed inversion scheme resulted in a good match between the observed and calculated GPR data. In the numerical modeling, it was observed that the SWC could be inverted accurately, even when noise was present in the data. These demonstrate that the GWO method can be applied for the quantitative interpretation of GPR data. The proposed scheme shows potential for SWC estimation by using GPR full waveform data.

Enhanced Malware Detection Using Grey Wolf Optimization and Deep Belief Neural Networks

Standard identification methods are flattering and less effective as attacks from malware get increasingly sophisticated. Considering current malware outbreaks employ tactics such as polymorphism, obfuscation and encryption, to avert identification, growing complicated approaches must be developed. This paper deals with a mixed model utilizing Deep Belief Neural Network (DBNN) for classifying and Grey Wolf Optimization (GWO) for choosing features. Whereas DBNN encodes complicated patterns by hierarchical learning, GWO optimizes the choosing of the more essential features, lowering the cost of computing and dataset complexity. Investigations reveal that the suggested GWO-DBNN model beats existing machine learning procedures in terms of detection accuracy, recall, precision, and false positive rate (FPR). These mixed tactics offer dependable and scalable solutions to the challenges faced by modern malware threats.

Enhanced Malware Detection Using Grey Wolf Optimization and Deep Belief Neural Networks

Standard identification methods are flattering and less effective as attacks from malware get increasingly sophisticated. Considering current malware outbreaks employ tactics such as polymorphism, obfuscation and encryption, to avert identification, growing complicated approaches must be developed. This paper deals with a mixed model utilizing Deep Belief Neural Network (DBNN) for classifying and Grey Wolf Optimization (GWO) for choosing features. Whereas DBNN encodes complicated patterns by hierarchical learning, GWO optimizes the choosing of the more essential features, lowering the cost of computing and dataset complexity. Investigations reveal that the suggested GWO-DBNN model beats existing machine learning procedures in terms of detection accuracy, recall, precision, and false positive rate (FPR). These mixed tactics offer dependable and scalable solutions to the challenges faced by modern malware threats.

Unleashing the power of Manta Rays Foraging Optimizer: A novel approach for hyper-parameter optimization in skin cancer classification

Optimizing hyperparameters is crucial for improving the performance of deep learning (DL) models, especially in complex applications like skin cancer classification from dermoscopic images. This study introduces a novel hyperparameter optimization strategy using the Manta Rays Foraging Optimizer (MRFO). A model tailored for skin cancer classification is created by fine-tuning a Convolutional Neural Network (CNN) with MRFO, coupled with in-depth image preprocessing. Empirical evaluations on diverse datasets (ISIC, PH2, HAM10000) showcase the significant superiority of the MRFO-based model over conventional optimization algorithms. The model achieves impressive accuracy and loss metrics (ISIC: 99.43 %, 0.0250; PH2: 99.96 %, 0.0033; HAM10000: 97.70 %, 0.0626), outperforming alternative optimization algorithms such as the Grey Wolf Optimizer (98.33 % accuracy, 0.17 loss), Whale Optimization Algorithm (96 % accuracy), Grasshopper Optimization Algorithm (97.2 % accuracy), Densnet121-MRFO (99.26 % accuracy), InceptionV3 with GA (99.9 % accuracy), and African Vulture Optimization Algorithm (92.7 % accuracy). The novel approach demonstrates superior accuracy and loss metrics, underscoring its potential for precise and efficient skin cancer detection. Additionally, narrow confidence intervals and balanced precision-recall confirm the model’s generalizability and effectiveness, paving the way for early and accurate skin cancer detection and potentially improving patient outcomes.

UAV swarm path planning approach based on integration of multi-population strategy and adaptive evolutionary optimizer

Abstract Addressing the optimal path planning problem encountered by swarm of Unmanned Aerial Vehicle (UAV) in three-dimensional space under multiple constraints, the Multi-population Adaptive Cuckoo Search and Grey Wolf Optimizer (MACSGWO) integrates Multi-Population (MP) strategies and adaptive evolutionary optimizer including the enhanced Adaptive Grey Wolf Optimizer (AGWO) and Adaptive Cuckoo Search (ACS).
The optimizer strategically divides the initial population into multiple sub-groups, enabling each sub-group to independently iterate. 
During the iteration process, the algorithm adaptively adjusts parameters based on the optimal fitness values obtained by each sub-group after each iteration. The iteration cycle is divided into two stages: during the global exploration phase, each sub-group autonomously executes AGWO and periodically shares the fitness information of the Alpha wolf with other sub-groups, accelerating convergence.
In the subsequent local optimization phase, MACSGWO dynamically decides whether to initiate ACS based on the disparity in the best fitness of each sub-group after each iteration, assisting the algorithm in escaping local optima. 
In experiments involving various complex benchmark functions and swarm path planning scenarios, MACSGWO demonstrated significant superiority in solution stability, convergence speed, and optimal convergence value compared to multiple existing variant algorithms. The integration of MACSGWO with the best relay UAV selection strategy further optimized the communication efficiency within the swarm. MACSGWO ensures the efficient resolution of UAV swarm path planning problems, providing robust support for optimization challenges in complex, multi-constraint scenarios.

Hybrid Optimization Path Planning Method for AGV Based on KGWO.

To address the path planning problem for automated guided vehicles (AGVs) in challenging and complex industrial environments, a hybrid optimization approach is proposed, integrating a Kalman filter with grey wolf optimization (GWO), as well as incorporating partially matched crossover (PMX) mutation operations and roulette wheel selection. Paths are first optimized using GWO, then refined with Kalman filter corrections every ten iterations. Moreover, roulette wheel selection guides robust parent path selection, while an elite strategy and partially matched crossover (PMX) with mutation generate diverse offspring. Extensive simulations and experiments were carried out under a densely packed goods scenario and complex indoor layout scenario, within a fully automated warehouse environment. The results showed that this hybrid method not only enhanced the various optimization metrics but also ensured more predictable and collision-free navigation paths, particularly in environments with complex obstacles. These improvements lead to increased operational efficiency and safety, highlighting the method's potential in real-world applications.

Advancements in IoT Routing and Energy Efficiency: A Comprehensive Review of Algorithms and Technologies

The rapid expansion of the Internet of Things (IoT) has led to a significant increase in connected devices, resulting in growing challenges related to efficient data transmission, energy consumption, and network scalability. Routing protocols and energy-efficient algorithms play a pivotal role in addressing these challenges, enabling IoT systems to optimize performance while minimizing power usage. This review provides a comprehensive analysis of recent advancements in IoT routing techniques and energy-efficient solutions, focusing on multi-objective optimization algorithms such as fractional gravitational search, grey wolf optimization, and hybrid salp swarm-differential evolution algorithms. Additionally, we explore context-aware routing, cluster-based methods, and the integration of machine learning to enhance decision-making in dynamic IoT environments. The review also highlights the critical role of energy-efficient routing in IoT applications, such as smart agriculture, healthcare, and smart cities, emphasizing its importance in prolonging network lifetime and reducing overall operational costs. The potential of blockchain technology and AI-driven algorithms to enhance security and energy efficiency in IoT networks is discussed, alongside future directions in green routing solutions. By addressing the complexities of IoT routing and energy management, this review aims to provide insights into the latest research trends and guide future innovations in IoT systems.

Predicting the thickness loss and recovery of acrylic carpet pile imposed to UV exposure using artificial intelligence

One of the metrics that establishes the carpet quality is its compress and decompress behaviors. In a research, addressed later in the paper, carpet samples with the acrylic pile were exposed to the ultraviolet (UV) exposure for 0, 8, 12, 16, and 20 h. The wear test was then carried out using drum revolutions of 0, 4000, 6000, and 8000. Following that, the samples were subjected to a static stress of 700 KPa for 2 h. After removing the load, the sample thickness was measured every 2 min. Finally, the thickness loss and recovery features were determined. Now in this study, the statistical analysis demonstrated the significance of recovery time, UV exposure, and drum revolutions on the features mentioned above, and in the next step, they were modeled using an artificial neural network (ANN) structure. The genetic algorithm, particle swarm optimization, and gray wolf optimization (GWO) were applied as the alternatives to the backpropagation (BP) algorithm for updating the weights and biases of ANN. In the end, it was found that the thickness loss and recovery could be predicted with 6.92 and 9.14% errors, using ANN-GWO and ANN-BP-GWO, respectively.

Parameter extraction of proton exchange membrane fuel cell based on artificial rabbits’ optimization algorithm and conducting laboratory tests

Proton exchange membrane fuel cell (PEMFC) parameter extraction is an important issue in modeling and control of renewable energies. The PEMFC problem’s main objective is to estimate the optimal value of unknown parameters of the electrochemical model. The main objective function of the optimization problem is the sum of the square errors between the measured voltages and output voltages of the proposed electrochemical optimized model at various loading conditions. Natural rabbit survival strategies such as detour foraging and random hiding are influenced by Artificial rabbit optimization (ARO). Meanwhile, rabbit energy shrink is mimicked to control the smooth switching from detour foraging to random hiding. In this work, the ARO algorithm is proposed to find the parameters of PEMFC. The ARO performance is verified using experimental results obtained from conducting laboratory tests on the fuel cell test system (SCRIBNER 850e, LLC). The simulation results are assessed with four competitive algorithms: Grey Wolf Optimization Algorithm, Particle Swarm Optimizer, Salp Swarm Algorithm, and Sine Cosine Algorithm. The comparison aims to prove the superior performance of the proposed ARO compared with the other well-known competitive algorithms.

Constrained Model Predictive Control for Generation Power Distribution on Aircraft Engines

Aiming at the increasing demand for electric energy in aircraft in the future, a multi-objective optimization aircraft engine constrained model predictive control method based on generation power distribution is proposed. Firstly, based on the aircraft engine component level model and the equilibrium manifold theory, the aircraft engine equilibrium manifold expansion model is established. Secondly, the influence of the power generation is modeled, and the influence of the low- and high-pressure shaft generators on the normal operation of the aircraft engine is studied and compared. The control variables such as fuel flow and total generation power are taken as the constraint conditions to design the constraint model predictive controller. Furthermore, the multi-objective grey wolf optimization algorithm is introduced to intelligently optimize the parameters of the designed controller. At last, the simulation based on the component level model shows that the high-pressure shaft generator has less influence on the state quantity, including engine thrust, than the low-pressure shaft generator. The proposed control method using the multi-objective gray wolf optimization (MOGWO) algorithm has rapid response and no steady-state error.