Recurrent Neural Network Model Research Articles

Tensor Recurrent Neural Network With Differential Privacy

Recurrent neural network (RNN), a branch of deep learning, is a powerful model for sequential data that has outstanding performance on a wide range of important Internet of Things (IoT) tasks. This unprecedented growth of RNN model has however encountered both heterogeneous IoT data and privacy issues. Existing RNN model can not deal with heterogeneous sequential data; often the larger datasets used in training of RNN model contain sensitive information. To tackle these challenges and for the first time, this research proposes a novel differentially private tensor-based RNN (DPTRNN) that can be applied in many challenging deep learning sequence tasks for IoT systems. Specifically, to process heterogeneous sequential data, we propose a tensor-based RNN model. To guarantee privacy, we develop a tensor-based back-propagation through time algorithm with perturbation to avoid exposing the sensitive information for training the tensor-based RNN model within the framework of differential privacy. Thorough security analysis shows that the differential private tensor-based RNN efficiently protects the confidentiality of sensitive user information for IoT. Our results from extensive experiments on two challenging large video datasets suggest that our proposed scheme is practical with guarantee of data privacy preservation and acceptable accuracy loss.

Predicting vocal tract shape information from tongue contours and audio using neural networks

Midsagittal ultrasound imaging of the tongue is a portable and inexpensive way to provide articulatory information. However, although ultrasound images show a portion of the tongue surface, other vocal tract structures (e.g., palate) are not typically visible. This missing information may be useful for speech therapy and other applications, e.g., by characterizing vocal tract constrictions and informing how morphological variations affect speech patterns. Prediction of the vocal tract shape from information available during ultrasound imaging (e.g., tongue contours and audio recordings) is, thus, potentially valuable. Recent advancements in articulatory prediction from audio recordings (i.e., acoustic inversion) and speech recognition using combined articulatory and acoustic data have used neural network models. Inspired by these models, this study investigates how well fusion of articulatory and acoustic features in speaker-independent models can predict expanded articulatory information. Specifically, recurrent neural network models will be trained to predict the vocal tract shape based on partial tongue contours and acoustic features, during production of vowels and central approximants. Features will be extracted from simultaneously recorded audio and 2D MRI (USC 75-Speaker Database). Different acoustic features and network architectures will be compared, with the goal of refining future models to predict vocal tract shapes during ultrasound imaging.

Short-term PV power prediction study based on FCM-BiGRU

Addressing the issue of large prediction error in short-term load prediction of existing neural network models, this study proposes a short-term load forecasting approach that combines fuzzy C-mean clustering and a two-way gated recurrent neural network model. Fuzzy C-mean clustering is first applied to cluster the original data into three typical days, and the grouped data are trained using a bidirectional gated recurrent neural network model for load prediction. The conclusive experiment demonstrates that the proposed approach introduced in this study exhibits a high prediction accuracy in the context of short-term photovoltaic output forecasting, and there is also a substantial error reduction compared with the existing neural network methods.

ALGORITHM FOR DETECTION OF ABUSIVE CONTENT IN AUDIO CONTENT FOR IMPLEMENTATION IN OBJECT-ORIENTED INFORMATION SYSTEM

The paper proposes the basic principles of developing an object-oriented information system for detecting abusive content in Ukrainian-language audio content based on a new algorithm that uses statistical and neural network approaches to detect abusive content. Detecting offensive content in text and audio content is an urgent task, as it helps to create a safe and healthy environment for communication, especially in online platforms. Offensive content can harm the people who hear or read it and violate their rights. It can also have a negative impact on society, contributing to the spread of hatred and violence. To detect abusive speech in audio content, the proposed approach uses two key components: the use of dictionary methods and the analysis of the emotional tonality of utterances. A set of reviews was used as a dataset to determine the abusive component of the content, which was expanded by the authors by adding words of abuse. An object-oriented information system architecture written in the Python programming language in the PyCharm programming environment is proposed. The information system consists of a software module for training recurrent neural network models and further saving trained instances, and a software module for detecting abusive content in Ukrainian-language audio content using trained RNN models. Since the recurrent neural network is trained on a short text data set, the system is less efficient at identifying texts that have a larger number of words. In the example of the proposed approach, the accuracy of detecting offensive content is more than 90%. This means that the algorithm works correctly in the absence of offending highlights in the test data set. The results of the analysis of the effectiveness of the proposed approach show that in the vast majority of cases the conclusions regarding the acceptability of audio content based on the level of abuse are correct.

Deep learning–based short-term water demand forecasting in urban areas: A hybrid multichannel model

Abstract Forecasting short-term water demands is one of the most critical needs of operating companies of urban water distribution networks. Water demands have a time series nature, and various factors affect their variations and patterns, which make it difficult to forecast. In this study, we first implemented a hybrid model of convolutional neural networks (CNNs) and recurrent neural networks (RNNs) to forecast urban water demand. These models include a combination of CNN with simple RNN (CNN-Simple RNN), CNN with the gate recurrent unit (CNN-GRU), and CNN with the long short-term memory (CNN-LSTM). Then, we increased the number of CNN channels to achieve higher accuracy. The accuracy of the models increased with the number of CNN channels up to four. The evaluation metrics show that the CNN-GRU model is superior to other models. Ultimately, the four-channel CNN-GRU model demonstrated the highest accuracy, achieving a mean absolute percentage error (MAPE) of 1.65% for a 24-h forecasting horizon. The effects of the forecast horizon on the accuracy of the results were also investigated. The results show that the MAPE for a 1-h forecast horizon is 1.06% in four-channel CNN-GRU, and its value decreases with the amount of the forecast horizon.

Video quality adaptation using CNN and RNN models for cost-effective and scalable video streaming Services

Video streaming services require adaptive bit rate strategies that optimize video quality based on network conditions and user preferences to provide a cost-effective and scalable solution. In this manuscript, we present a novel architecture that utilizes a combination of Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs) to extract features from the video stream and predict optimal bit rates for each frame. The CNN feature extractor extracts relevant features from the input video stream, which are then passed on to the RNN predictor, where the optimal bit rate is predicted to ensure a personalized viewing experience tailored to the current network conditions and specific user preferences. Our experimental results demonstrate that the proposed architecture achieves significant improvements in video quality while minimizing bandwidth usage and providing a better user experience. Specifically, our results show a 37.1% improvement in average bit rate, indicating that the proposed architecture optimizes the video quality and reduces bandwidth usage by 37.1% on average. We also observed a 16.6% improvement in Quality of Experience (QoE), meaning that users will perceive the video quality as better. Additionally, rebuffering was reduced by 87.5%, indicating that users enjoy smoother video playback without interruptions. Importantly, our architecture is optimized for Dynamic Adaptive Streaming over HTTP (DASH), addressing the need for efficient streaming over the most widely used protocol in the industry.

Combining Recurrent Neural Network and Sigmoid Growth Models for Short-Term Temperature Forecasting and Tomato Growth Prediction in a Plastic Greenhouse

Compared with open-field cultivation, greenhouses can provide favorable conditions for crops to grow through environmental control. The prediction of greenhouse microclimates is a way to reduce environmental monitoring costs. This study used several recurrent neural network models, including long short-term memory (LSTM), gated recurrent unit, and bi-directional LSTM, with varying numbers of hidden layers and units, to establish a temperature forecasting model for a plastic greenhouse. To assess the generalizability of the proposed model, the most accurate forecasting model was used to predict the temperature in a greenhouse with different specifications. During a test period of four months, the best proposed model’s R2, MAPE, and RMSE values were 0.962, 3.216%, and 1.196 °C, respectively. Subsequently, the outputs of the temperature forecasting model were used to calculate growing degree days (GDDs), and the predicted GDDs were used as an input variable for the sigmoid growth models to simulate the leaf area index, fresh fruit weight, and aboveground dry matter of tomatoes. The R2 values of the growth model for the three growth traits were all higher than 0.80. Moreover, the fitted values and the parameter estimates of the growth models were similar, irrespective of whether the observed GDD (calculated using the actual observed data) or the predicted GDD (calculated using the temperature forecasting model output) was used. These results indicated that the proposed temperature forecasting model could accurately predict the temperature changes inside a greenhouse and could subsequently be used for the growth prediction of greenhouse tomatoes.

Edible Mushroom Greenhouse Environment Prediction Model Based on Attention CNN-LSTM

The large-scale production of edible mushrooms typically requires the use of greenhouses, as the greenhouse environment significantly affects the growth of edible mushrooms. It is crucial to effectively predict the temperature, humidity, and carbon dioxide fluctuations within the mushroom greenhouse for determining the environmental stress and pre-regulation of edible mushrooms. To address the nonlinearity, temporal dynamics, and strong coupling of the edible mushroom greenhouse environment, a temperature, humidity, and carbon dioxide prediction model based on the combination of the attention mechanism, the convolutional neural network, and the long short-term memory neural network (A-CNN-LSTM) is proposed. Experimental data were collected from both the inside and outside of the greenhouse, including environmental data and the on–off data of environmental control devices. After completing missing data using linear interpolation, denoising with Kalman filtering, and normalization, the recurrent neural network (RNN) model, long short-term memory (LSTM) model, and A-CNN-LSTM model were trained and tested on the time series data. These models were used to predict the environmental changes in temperature, humidity, and carbon dioxide inside the greenhouse. The results indicate that the A-CNN-LSTM model outperforms the other two models in terms of denoising, non-denoising, and different prediction time steps. The proposed method accurately predicts temperature, humidity, and carbon dioxide levels with errors of 0.17 °C (R2 = 0.974), 2.06% (R2 = 0.804), and 8.367 ppm (R2 = 0.993), respectively. These results indicate improved prediction accuracy for temperature, humidity, and carbon dioxide values inside the edible mushroom greenhouse. The findings provide a decision basis for the precise control of the greenhouse environment.

Multimodal Deep Learning for Estimating Lane-Level Urban Traffic by Fusing Closed-Circuit Television and Dedicated Short-Range Communication Data

Seamless, accurate, and reliable traffic information is critical for planning and operational strategies in traffic management. Urban networks are increasingly being equipped with various types of detectors to collect continuous traffic data. However, the widespread installation and management of these detectors at all intersections and road sections are limited by cost constraints and various problems. Moreover, the direct integration and utilization of heterogeneous data sources are challenging because of their distinct and interconnected characteristics. This study introduces a multimodal deep learning model, combining closed-circuit television (CCTV) and dedicated short-range communication (DSRC) data, to estimate lane-level urban traffic volumes. The proposed model employs a multilayer perceptron to extract features from each modality. These features are then fused and used as input into a recurrent neural network model that estimates the lane-level traffic volume. We present the multimodal deep learning model in three forms: (1) fusion of traffic volume, occupancy, and queue length from CCTV data; (2) fusion of traffic volume from CCTV data with travel time from DSRC data; and (3) fusion of different attributes from CCTV and heterogeneous DSRC data. In addition, we develop a single-modality model that solely utilizes CCTV data on traffic volume to compare the performance with the proposed multimodal model and identify scenarios where the multimodal approach is essential. The proposed model demonstrates significant improvements over the single-modality model, providing enhanced accuracy for higher temporal resolutions and lanes that permit left or right turns.

Proxy-Based Sliding Mode Force Control for Compliant Grinding via Diagonal Recurrent Neural Network and Prandtl-Ishlinskii Hysteresis Compensation Model

Traditional industrial robots often face challenges in achieving a perfectly polished surface on a workpiece because of their high mechanical rigidity. The active compliance force control device installed at the robotic arm’s end enables high-precision contact force control between the grinding tool and the workpiece. However, the complex hysteresis nonlinearity between cylinder air pressure and output force, as well as various random disturbances during the grinding process, can affect the accuracy of the contact force and potentially impact the grinding effect of the workpiece, even causing irreversible damage to the surface of the workpiece. Given the complex random variation of cylinder output force in the actual grinding process, a rate-dependent hysteresis model based on diagonal recurrent neural network and Pradtl–Ishlinskii models named dRNN-PI is designed to compensate for the complex nonlinear hysteresis of the cylinder and calculate the desired air pressure to maintain a steady contact force on the workpiece. The proxy-based sliding mode control (PSMC) is utilized to quickly track the desired air pressure without overshooting. This paper also proves the controller’s stability using the Lyapunov-based methods. Finally, the accuracy of the proposed hysteresis compensation model and the effectiveness and robustness of the PSMC are verified by experiment results.

LSTM-based deformation forecasting for additional stress estimation of existing tunnel structure induced by adjacent shield tunneling

A redistribution of stress is caused by the excavation of a new tunnel within the soil surrounding the existing shield tunnel structure. This stress redistribution results in alterations to the soil and water loads around the existing tunnel structure, ultimately impacting its stress and deformation. The deformation response of existing tunnels is primarily focused on in most existing studies, with limited research conducted on the coupling system of new tunnel excavation, soil stress state, and deformation and stress of existing tunnels. This paper proposed an innovative methodology to predict deformation in existing tunnel structures and estimate additional stress during the excavation of a new shield tunnel. The reliability of the suggested calculation method was verified using measured internal force and convergence data of the structures. The results demonstrated that the Long Short-Term Memory (LSTM) model could more effectively establish the temporal correlation between the excavation parameters of new tunnels and the deformation of existing tunnels. Compared to the Recurrent Neural Network (RNN) model, the R2 of the LSTM model increased by 11.81% in the prediction of the tunnel waist and 9.38% in the prediction of the tunnel crown. The Light gradient boosting machine (LightGBM) algorithm was used to analyze the crucial factors of deformation of the existing tunnel structures during shield tunneling, and the results showed that the relative position and the average earth pressure were the essential factors. Furthermore, a calculation method for the additional stress of existing tunnel structures was proposed based on the load structure model. The method assumed that the additional load was distributed rectangularly at the top and bottom of the existing tunnel, and triangularly at the waist of the tunnel. The comparison between the calculated results and the measured results indicated that the phased characteristics of the internal force changes in the existing tunnel structure during the construction of a new tunnel could be accurately reflected by this method.

Adversarial Attacks with Defense Mechanisms on Convolutional Neural Networks and Recurrent Neural Networks for Malware Classification

In the field of behavioral detection, deep learning has been extensively utilized. For example, deep learning models have been utilized to detect and classify malware. Deep learning, however, has vulnerabilities that can be exploited with crafted inputs, resulting in malicious files being misclassified. Cyber-Physical Systems (CPS) may be compromised by malicious files, which can have catastrophic consequences. This paper presents a method for classifying Windows portable executables (PEs) using Recurrent Neural Networks (RNNs) and Convolutional Neural Networks (CNNs). To generate malware executable adversarial examples of PE, we conduct two white-box attacks, Jacobian-based Saliency Map Attack (JSMA) and Carlini and Wagner attack (C&W). An adversarial payload was injected into the DOS header, and a section was added to the file to preserve the PE functionality. The attacks successfully evaded the CNN model with a 91% evasion rate, whereas the RNN model evaded attacks at an 84.6% rate. Two defense mechanisms based on distillation and training techniques are examined in this study for overcoming adversarial example challenges. Distillation and training against JSMA resulted in the highest reductions in the evasion rates of 48.1% and 41.49%, respectively. Distillation and training against C&W resulted in the highest decrease in evasion rates, at 48.1% and 49.9%, respectively.

Enhancing Airway Assessment with a Secure Hybrid Network-Blockchain System for CT & CBCT Image Evaluation

Our investigation explored the intricacies of airway evaluation through Cone-Beam Computed Tomography (CBCT) and Computed Tomography (CT) images. By employing innovative data augmentation strategies, we expanded our dataset significantly, enabling a more comprehensive analysis of airway characteristics. The utility of these techniques was evident in their ability to yield a diverse array of synthetic images, each representing different airway scenarios with high fidelity. A notable outcome of our study was the effective categorization of the initial image as "Class II" under the Mallampati Classification system. The augmented images further enhanced our understanding by exhibiting a spectrum of airway parameters. Moreover, our approach included training a Recurrent Neural Network (RNN) model on a dataset of CT images. This model, fortified with pseudo-labels created via K-means clustering, showcased its proficiency by accurately predicting airway assessment categories in various test scenarios. These results underscore the model's potential as a tool for swift and precise airway evaluation in clinical settings, marking a significant advancement in medical imaging technologies.

Eciton: Very Low-power Recurrent Neural Network Accelerator for Real-time Inference at the Edge

This article presents Eciton, a very low-power recurrent neural network accelerator for time series data within low-power edge sensor nodes, achieving real-time inference with a power consumption of 17 mW under load. Eciton reduces memory and chip resource requirements via 8-bit quantization and hard sigmoid activation, allowing the accelerator as well as the recurrent neural network model parameters to fit in a low-cost, low-power Lattice iCE40 UP5K FPGA. We evaluate Eciton on multiple, established time-series classification applications including predictive maintenance of mechanical systems, sound classification, and intrusion detection for IoT nodes. Binary and multi-class classification edge models are explored, demonstrating that Eciton can adapt to a variety of deployable environments and remote use cases. Eciton demonstrates real-time processing at a very low power consumption with minimal loss of accuracy on multiple inference scenarios with differing characteristics, while achieving competitive power efficiency against the state-of-the-art of similar scale. We show that the addition of this accelerator actually reduces the power budget of the sensor node by reducing power-hungry wireless transmission. The resulting power budget of the sensor node is small enough to be powered by a power harvester, potentially allowing it to run indefinitely without a battery or periodic maintenance.

RNN-Based Approach for Broccoli Harvest Time Forecast

This article investigates approaches for broccoli harvest time prediction through the application of various machine learning models. This study’s experiment is conducted on a commercial farm in Ecuador, and it integrates in situ weather and broccoli growing cycle observations made over seven years. This research incorporates models such as the persistence, thermal, and calendar models, demonstrating their strengths and limitations in calculating the optimal broccoli harvest day. Additionally, Recurrent Neural Network (RNN) models with Long Short-term Memory (LSTM) layers were developed, showcasing enhanced accuracy with an error of less than 2.5 days on average when combined with outputs from the calendar model. In the final comparison, the RNN models outperformed both the thermal and calendar models, with an error of 3.14 and 2.5 days, respectively. Furthermore, this article explores the impact of utilizing Global Ensemble Forecast System forecast weather data as a supplementary source to the in situ observations on model accuracy. The analysis revealed the limited effect of extension with a 9-day forecast on the experimental field, reaching an error reduction of up to 0.04 days. The findings provide insights into the effectiveness of different modeling approaches for optimizing broccoli harvest times, emphasizing the potential of RNN techniques in agricultural decision making.

Prediction of emergency department revisits among child and youth mental health outpatients using deep learning techniques

BackgroundThe proportion of Canadian youth seeking mental health support from an emergency department (ED) has risen in recent years. As EDs typically address urgent mental health crises, revisiting an ED may represent unmet mental health needs. Accurate ED revisit prediction could aid early intervention and ensure efficient healthcare resource allocation. We examine the potential increased accuracy and performance of graph neural network (GNN) machine learning models compared to recurrent neural network (RNN), and baseline conventional machine learning and regression models for predicting ED revisit in electronic health record (EHR) data.MethodsThis study used EHR data for children and youth aged 4–17 seeking services at McMaster Children’s Hospital’s Child and Youth Mental Health Program outpatient service to develop and evaluate GNN and RNN models to predict whether a child/youth with an ED visit had an ED revisit within 30 days. GNN and RNN models were developed and compared against conventional baseline models. Model performance for GNN, RNN, XGBoost, decision tree and logistic regression models was evaluated using F1 scores.ResultsThe GNN model outperformed the RNN model by an F1-score increase of 0.0511 and the best performing conventional machine learning model by an F1-score increase of 0.0470. Precision, recall, receiver operating characteristic (ROC) curves, and positive and negative predictive values showed that the GNN model performed the best, and the RNN model performed similarly to the XGBoost model. Performance increases were most noticeable for recall and negative predictive value than for precision and positive predictive value.ConclusionsThis study demonstrates the improved accuracy and potential utility of GNN models in predicting ED revisits among children and youth, although model performance may not be sufficient for clinical implementation. Given the improvements in recall and negative predictive value, GNN models should be further explored to develop algorithms that can inform clinical decision-making in ways that facilitate targeted interventions, optimize resource allocation, and improve outcomes for children and youth.

Handwritten digit recognition based on deep learning techniques

The identification of handwritten digits in images recognition and machine learning is a prominent research area. In order to create a handwritten digit recognition model for this investigation, deep learning is introduced. The proposed approach integrates Convolutional Neural Networks (CNN), Recurrent Neural Networks (RNN), and EfficientNetB0, three separate deep learning models. Specifically, the CNN model utilizes pooling layers and data augmentation techniques to enhance its classification ability, the RNN model takes advantage of its ability to process sequential data, and the EfficientNetB0 model benefits from a deeper and more complex network structure. These models are trained and evaluated using the Modified National Institute of Standards and Technology (MNIST) dataset. The experimental results demonstrate the efficacy of the proposed approach: the CNN model attains a remarkable accuracy of 98.9% on the test set, thereby showcasing its exceptional classification performance. Similarly, the RNN model achieves an accuracy of 96.7%, underscoring its suitability for analyzing sequential data. Furthermore, the EfficientNetB0 model attains an accuracy of 98.1%, thereby elucidating the benefits of the deeper network architecture. The models constructed in this study have significant real-world implications, such as improved object recognition systems, medical diagnostics, and autonomous driving. The EfficientNetB0 model produced an accuracy of 98.1% with its complex network architecture when applied to recognise handwritten digits.

Revealing the structural behaviour of Brunelleschi’s Dome with machine learning techniques

The Brunelleschi’s Dome is one of the most iconic symbols of the Renaissance and is among the largest masonry domes ever constructed. Since the late 17th century, first masonry cracks appeared on the Dome, giving the start to a monitoring activity. In modern times, since 1988 a monitoring system comprised of 166 electronic sensors, including deformometers and thermometers, has been in operation, providing a valuable source of real-time data on the monument’s health status. With the deformometers taking measurements at least four times per day, a vast amount of data is now available to explore the potential of the latest Artificial Intelligence and Machine Learning techniques in the field of historical-architectural heritage conservation. The objective of this contribution is twofold. Firstly, for the first time ever, we aim to unveil the overall structural behaviour of the Dome as a whole, as well as that of its specific sections (known as webs). We achieve this by evaluating the effectiveness of certain dimensionality reduction techniques on the extensive daily detections generated by the monitoring system, while also accounting for fluctuations in temperature over time. Secondly, we estimate a number of recurrent and convolutional neural network models to verify their capability for medium- and long-term prediction of the structural evolution of the Dome. We believe this contribution is an important step forward in the protection and preservation of historical buildings, showing the utility of machine learning in a context in which these are still little used.

Multi-Dimensional Wi-Fi Received Signal Strength Indicator Data Augmentation Based on Multi-Output Gaussian Process for Large-Scale Indoor Localization

Location fingerprinting using Received Signal Strength Indicators (RSSIs) has become a popular technique for indoor localization due to its use of existing Wi-Fi infrastructure and Wi-Fi-enabled devices. Artificial intelligence/machine learning techniques such as Deep Neural Networks (DNNs) have been adopted to make location fingerprinting more accurate and reliable for large-scale indoor localization applications. However, the success of DNNs for indoor localization depends on the availability of a large amount of pre-processed and labeled data for training, the collection of which could be time-consuming in large-scale indoor environments and even challenging during a pandemic situation like COVID-19. To address these issues in data collection, we investigate multi-dimensional RSSI data augmentation based on the Multi-Output Gaussian Process (MOGP), which, unlike the Single-Output Gaussian Process (SOGP), can exploit the correlation among the RSSIs from multiple access points in a single floor, neighboring floors, or a single building by collectively processing them. The feasibility of MOGP-based multi-dimensional RSSI data augmentation is demonstrated through experiments using the hierarchical indoor localization model based on a Recurrent Neural Network (RNN)—i.e., one of the state-of-the-art multi-building and multi-floor localization models—and the publicly available UJIIndoorLoc multi-building and multi-floor indoor localization database. The RNN model trained with the UJIIndoorLoc database augmented with the augmentation mode of “by a single building”, where an MOGP model is fitted based on the entire RSSI data of a building, outperforms the other two augmentation modes and results in the three-dimensional localization error of 8.42 m.

Min–max adaptive dynamic programming for zero-sum differential games

In this paper, a min–max adaptive dynamic programming approach is studied for a zero-sum differential game problem with unknown nonlinear dynamics. The unknown nonlinear dynamics is learned through the online Recurrent Neural Network (RNN). The error between the learned RNN model and the dynamics of the system is proved to converge to zero. The policy iteration algorithm is utilised to solve the Hamilton–Jacobi–Isaacs (HJI) equation associated with the differential game. Furthermore, the value function in the HJI equation is approximated through a critical Neural Network with its weights and activation functions updated online iteratively. The Uniform Ultimate Bounded stability of the closed-loop system is proved based on the Lyapunov theory. Finally, the effectiveness of the proposed solution method is demonstrated by applying it to three examples.