Prognostics And Health Management System Research Articles

Prognostics and Health Management of Unmanned Surface Vessels: Past, Present, and Future

Abstract With the increasing popularity and deployment of unmanned surface vessels (USVs) all over the world, prognostics and health management (PHM) has become an indispensable tool for health monitoring, fault diagnosis, health prognosis, and maintenance of marine equipment on USVs. USVs are designed to undertake critical and extended missions, often in extreme conditions, without human intervention. This makes the USVs susceptible to equipment malfunction, which increases the probability of system failure during mission execution. In fact, in the absence of any crew onboard, system failure during a mission can create a great inconvenience for the concerned stakeholders, which compels them to design highly reliable USVs that must have integrated intelligent PHM systems onboard. To improve mission reliability and health management of USVs, researchers have been investigating and proposing PHM-based tools or frameworks that are claimed to operate in real time. This paper presents a comprehensive review of the existing literature on recent developments in PHM-related studies in the context of USVs. It covers a broad perspective of PHM on USVs, including system simulation, sensor data, data assimilation, data fusion, advancements in diagnosis and prognosis studies, and health management. After reviewing the literature, this study summarizes the lessons learned, identifies current gaps, and proposes a new system-level framework for developing a hybrid (offline-online) optimization-based PHM system for USVs in order to overcome some of the existing challenges.

The seasonal-trend disentangle based prognostic framework for PEM fuel cells

Prognostics and Health Management (PHM) offer a potential solution to improve the durability of Proton Exchange Membrane Fuel Cells (PEMFCs), where degradation prognostic is the key component of the PHM system. However, limited by the dynamic operating environment, accurate and efficient prognostic algorithms still required for real applications. In this paper, the Seasonal-Trend Disentangle (STD) based prognostic framework is proposed to enhance the Remaining Useful Life (RUL) prediction of PEMFC system. Specifically, a lightweight parameter estimation neural network is introduced to estimate the aging parameters of the FC degradation model directly from PEMFC measurements. Besides, the integral of the FC degradation model at the specified current density range serves as the comprehensive Health Index (HI), which could combine different aging parameters and mitigate the effect of load dynamics. The STD-based prognostic model effectively captures the time-invariant features in the HI sequence from trend and seasonal phases, improving Remaining Useful Life (RUL) prediction performance. The proposed prognostic model is compared with RNN-family, ESN, and Dilated CNN methods using three distinct durability datasets from FCs operating under static, quasi-dynamic, and cyclically varying dynamic current loads. Experimental results demonstrate that the proposed algorithm outperforms the reference methods in terms of RMSE and MAPE metrics in the aging prognostic, and the proposed method could obtain satisfactory RUL prediction results 100–150 h earlier than the reference methods.

Structured Data Ontology for AI in Industrial Asset Condition Monitoring

This paper proposes an ontology for prognostics and health management (PHM) applications involving sensor networks monitoring industrial machinery. Deep learning methods show promise for the development of autonomous PHM systems but require vast quantities of structured and representative data to realize their potential. PHM systems involve unique and specialized data characterized by time and context, and thus benefit from tailored data management systems. Furthermore, the use of dissimilar standards and practices with respect to database structure and data organization is a hinderance to interoperability. To address this, this paper presents a robust, structured data ontology and schema that is designed to accommodate a wide breadth of PHM applications. The inclusion of contextual and temporal data increases its value for developing and deploying enhanced ML-driven PHM systems. Challenges around balancing the competing priorities of structure and flexibility are discussed. The proposed schema provides the benefits of a relational schema with some provisions for noSQL-like flexibility in areas where PMH applications demand it. The selection of a database engine for implementation is also discussed, and the proposed ontology is demonstrated using a Postgres database. An instance of the database was loaded with large auto-generated fictitious data via multiple Python scripts. CRUD (create, read, update, delete) operations are demonstrated with several queries that answer common PHM questions.

Spar Cap/Shear Web Debonding Under Fatigue Loading Studied On The DTU 12.6m Wind Turbine Blade

Understanding fatigue damage initiation and propagation of composite wind turbine blades is an essential step towards the design of reliable prognostic and health management systems (PHM) and to improve lifetime predictions. This study presents a numerical investigation of the debonding process in the spar cap/shear web region of a full-scale composite wind turbine blade under fatigue loading and presents the manufacturing approach for creating initial debondings. For this purpose, numerical simulations have been performed to design and specify the size and locations of artificial defects to provoke steady, controllable damage growth in the adhesive bondline between the rear shear web and cap region of the DTU 12.6m wind turbine blade under fatigue loading. Subsequently, these artificial defects are embedded into the manufacturing process of the composite wind turbine blade with the help of slip foils and partially perforated slip foils to create regions with no bonding, 30% and 50% bonding of the contact surfaces of the adhesive joint, respectively.The numerical predictions as well as the manufactured and instrumented blade sets the framework to subsequently execute experimental tests of damage growth in the adhesive joint of the bondline in order to compare numerical and experimental data. The experimental demonstration will allow to better understand the complexity of damage growth of composite structures under fatigue loading and the necessity of accurate numerical prediction tools for reliable prognostic health management systems.

OSPtk: Cost-aware Optimal Sensor Placement Toolkit Enabling Design-for-PHM in Critical Industrial Systems

Performance of a Prognostics and Health Management (PHM) system in a fielded application depends on observability from existing monitoring equipment and sensing, which get determined at the design phase. Although various technologies have been proposed in the literature, there is currently a lack of known generic tools specifically designed for performing design stage sensor placement analyses from a PHM perspective. This leads to PHM observability being an afterthought and resulting PHM designs being sub-optimal. This paper describes a new Optimal Sensor Placement (OSP) framework, its implementation as a toolkit and the experience with applying it to a new product design in the context of a Small Modular Reactor (SMR). The formulation adds multiple important features that are critical to PHM applications. Firstly, it establishes a direct link to PHM performance requirements with intent to reduce operational and maintenance costs. Moreover, it acknowledges and accounts for the costs and risks of errors that PHM system will incur, and simultaneously considers operational requirements on sensing for performance, control and/or regulatory requirements. The toolkit described here implements formulations of a large number of requirements scenarios applicable in a generic industrial product development setting.

Physics-informed machine learning in prognostics and health management: State of the art and challenges

Prognostics and health management (PHM) plays a constructive role in the equipment’s entire life health service. It has long benefited from intensive research into physics modeling and machine learning methods. However, in practice, the existing solutions often encounter difficulties caused by sparse data & incomplete system failure knowledge. Pure machine learning or physics-based methods can sometimes be infeasible in such situations. As a result, there has been a growing interest in developing physics-informed machine learning (PIML) models which allow incorporating different forms of physics knowledge at different positions of the machine learning pipeline. This combination provides significant assistance for detection, diagnostic, and prognostics. However, to the best of our knowledge, the bibliometrics analyses and the comprehensive review of the existing research concerning PIML in PHM remain vacant. Our review is therefore dedicated to filling these gaps. We synthesize the concept of PIML in PHM, and propose a taxonomy of PIML approaches from the perspective of “Expression forms of informed knowledge” and “Knowledge informed methods”. The findings and discussions presented in this paper enable us to clarify the current state of the art and the emerging opportunities of PIML approaches, especially for building PHM systems that can work under the “small data and scarce physics knowledge” paradigm.

Criticality of Prognostics in the Operations of Autonomous Aircraft

<div>This article addresses the design, testing, and evaluation of rigorous and verifiable prognostic and health management (PHM) functions applied to autonomous aircraft systems. These PHM functions—many deployed as algorithms—are integrated into a holistic framework for integrity management of aircraft components and systems that are subject to both operational degradation and incipient failure modes. The designer of a comprehensive and verifiable prognostics system is faced with significant challenges. Data (both baseline and faulted) that are correlated, time stamped, and appropriately sampled are not always readily available. Quantifying uncertainty, and its propagation and management, which are inherent in prognosis, can be difficult. High-fidelity modeling of critical components/systems can consume precious resources. Data mining tools for feature extraction and selection are not easy to develop and maintain. And finally, diagnostic and prognostic algorithms that address accurately the designer’s specifications are not easy to develop, verify, deploy, and sustain. These are just the technical challenges. On top of these are business challenges, for example, demonstrating that the PHM functionality will be economically beneficial to the system stakeholders, and finally, there are regulatory challenges, such as, assuring the authorities that the PHM system will have the necessary safety assurance levels while delivering its performance goals. This article tackles all three aspects of the use of PHM systems in autonomous systems. It outlines how some of the technical challenges have been overcome and demonstrates why PHM could be essential in this ecosystem and why regulatory authorities are increasingly open to the use of PHM systems even in the most safety-critical areas of aviation.</div>

A lifetime estimation method for multi-component degrading systems with deteriorating spare parts

As one of the most important aspects in prognostics and system health management (PHM), spare parts play significant roles in enhancing reliability, shortening downtime, and reducing operating cost. However, the performance degradation of spare parts, though frequently encountered in industrial practice, has been rarely concerned during lifetime estimation and the according inventory optimization. Based on an amenable Wiener-process-based degradation model, this paper integrates the performance degradation of spare parts into lifetime estimation for stochastic multi-component degrading systems with congeneric key parts. Under two prevalent replacement policies, i.e., unit-by-unit replacement policy and batch-by-batch replacement policy, the lifetime distribution of the system considering the performance degradation of spare parts in storage, has been derived and the corresponding cost function for inventory management optimization has been constructed. Simulations and case study of an inertial navigation platform with three gyros are conducted to illustrate and validate the proposed method. The results show that the performance degradation of spare parts should be considered during inventory management, and the presented method can achieve an accurate lifetime estimation of the concerned system.

Design and Preliminary Performance Assessment of a PHM System for Electromechanical Flight Control Actuators

The evolution toward “more electric” aircraft has seen a decisive push in the last decade due to growing environmental concerns and the development of new market segments (flying taxis). Such a push has involved both the propulsion components and the aircraft systems, with the latter seeing a progressive trend in replacing traditional solutions based on hydraulic power with electrical or electromechanical devices. Flight Control Systems (FCSs) are one of the aircraft systems affected the most since the adoption of Electromechanical Actuators (EMAs) would provide several advantages over traditional electrohydraulic or mechanical solutions, but their application is still limited due to their sensitivity to certain single points of failure that can lead to mechanical jams. The development of an effective and reliable Prognostics and Health Management (PHM) system for EMAs could help in mitigating the risk of a sudden critical failure by properly recognizing and tracking the ongoing fault and anticipating its evolution, thus boosting the acceptance of EMAs as the primary flight-control actuators in commercial aircraft. The paper is focused on the results of the preliminary activities performed within the CleanSky 2/Astib research program, dedicated to the definition of the iron bird of a new regional-transport aircraft able to provide some prognostic capabilities and act as a technological demonstrator for new PHM strategies for EMAs employed in-flight control systems. The paper is organized as follows. At first, a proper introduction to the research program is provided, along with a brief description of the employed approach. Hence the simulation models adopted for the study are presented and used to build synthetic databases to inform the definition of the PHM algorithm. The prognostic framework is then presented, and a preliminary assessment of its expected performance is discussed.

Evaluation of Different PHM Strategies on the Performances of a Prognostic Framework for Electro-Hydraulic Actuators for Stability Control Augmentation Systems

Stability Control Augmentation Systems (SCAS) are widely adopted to enhance the flight stability of rotary-wing aircraft operating in difficult aerodynamic conditions, such as low altitude missions, stationary flight nearby vertical walls or in presence of heavy gusts. Such systems are based upon small electro-hydraulic servosystems controlled in position through a dedicated servovalve. The SCAS operates with limited authority over the main control linkage translating the pilot input in the movement of the main flight control actuator. Being critical for the operability of the helicopter, the definition of a Prognostics and Health Management (PHM) framework for the SCAS systems would provide significant advantages, such as better risk mitigation, improved availability, and a reduction in the occurrences of unpredicted failures which still represent one of the most known downsides of helicopters due to their very severe operational environment. Since SCAS actuators are usually equipped with a low number of sensors, it is at the present time unclear whether a fully realized PHM system can be prepared without resorting to the introduction of additional sensors. This paper deals with this subject evaluating the performances of a fault diagnosis tool operating considering different sensors suite (traditional and with additional sensors), and different PHM strategies, using in-flight data or their combination with dedicated pre-flight checks to cover the most common failure modes. The analysis is then completed with an evaluation of the prognostic capabilities of the proposed strategies, highlighting benefits and limitations of the proposed solutions.

Predicting the Remaining Useful Life of Landing Gear with Prognostics and Health Management (PHM)

Landing gear is an essential part of an aircraft. However, the components of landing gear are susceptible to degradation over the life of their operation, which can result in the shimmy effect occurring during take-off and landing. In order to reduce unplanned flight disruptions and increase the availability of aircraft, the predictive maintenance (PdM) technique is investigated in this study. This paper presents a case study on the implementation of a health assessment and prediction workflow for remaining useful life (RUL) based on the prognostics and health management (PHM) framework of currently in-service aircraft, which could significantly benefit fleet operators and aircraft maintenance. Machine learning is utilized to develop a health indicator (HI) for landing gear using a data-driven approach, whereas a time-series analysis (TSA) is used to predict its degradation. The degradation models are evaluated using large volumes of real sensor data from in-service aircraft. Finally, the challenges of implementing a built-in PHM system for next-generation aircraft are outlined.

Holistic Fault Detection and Diagnosis System in Imbalanced, Scarce, Multi-Domain (ISMD) Data Setting for Component-Level Prognostics and Health Management (PHM)

In the current Industry 4.0 revolution, prognostics and health management (PHM) is an emerging field of research. The difficulty of obtaining data from electromechanical systems in an industrial setting increases proportionally with the scale and accessibility of the automated industry, resulting in a less interpolated PHM system. To put it another way, the development of an accurate PHM system for each industrial system necessitates a unique dataset acquired under specified conditions. In most circumstances, obtaining this one-of-a-kind dataset is difficult, and the resulting dataset has a significant imbalance, a lack of certain useful information, and contains multi-domain knowledge. To address those issues, this paper provides a fault detection and diagnosis system that evaluates and preprocesses imbalanced, scarce, multi-domain (ISMD) data acquired from an industrial robot, utilizing signal processing (SP) techniques and deep learning-based (DL) domain knowledge transfer. The domain knowledge transfer is used to produce a synthetic dataset with a high interpolation rate that contains all the useful information about each domain. For domain knowledge transfer and data generation, continuous wavelet transform (CWT) with a generative adversarial network (GAN) was used, as well as a convolutional neural network (CNN), to test the suggested methodology using transfer learning and categorize several faults. The proposed methodology was tested on a real experimental bench that included an industrial robot created by Hyundai Robotics. This test had a satisfactory outcome with a 99.7% (highest) classification accuracy achieved by transfer learning on several CNN benchmark models.

A Semantic Ontology-Based Approach to Support Model-Based Systems Engineering Design for an Aircraft Prognostic Health Management System

The Prognostic and Health Management (PHM) system of an aircraft has complex structures and diverse functions. It is highly coupled with other systems, such as the avionics system and flight management system. The Model-based Systems Engineering (MBSE) method is effective to support the design and verification of the aircraft PHM system. As a powerful semantic web construction method, ontology has been widely used to express design information, such as the concepts and the relationships between them. However, traditional graphical MBSE models have a natural weakness in transforming into ontology. In this paper, a semantic MBSE method is proposed to support the transform of the ontology model. Firstly, according to the design characteristics of the aircraft PHM system, a meta-model library of the aircraft PHM system is developed to support the design and evaluation. An MBSE modeling method based on requirement analysis, function analysis, logical architecture design, and physic architecture design is applied in the PHM design process. Secondly, the semantic system modeling language KARMA based on “graph, object, property, point, relationship, role, and extension” (GOPPRRE) is used to transfer the graphical MBSE model to the semantic MBSE model, which can be easily transformed to an ontology model. Finally, an ontology based on semantic modeling is developed to describe the MBSE entities and to support MBSE design. In this paper, a case study of an aircraft fuel PHM system is carried out to validate the proposed method. Based on the developed meta-model library, a complete MBSE design process for the aircraft PHM system is realized. And then an ontology model supporting PHM system design is generated from the semantic MBSE model. The MBSE ontology provides a shareable capability to help designers communicate effectively. Quantitative analysis based on ontology is also provided to verify the complexity and scale of the MBSE design process. Moreover, logical reasoning ability can also be provided to support the early requirement traceability for MBSE design. In general, the case study results show the feasibility and effectiveness of the proposed method for the aircraft PHM system design.

A Weighted Representation Learning Based Feature Extraction Model for Multivariate Time Series Data in the Wind Turbine PHM System

Feature extraction is essential for accurate knowledge-based prognostics and health management (PHM) of the wind turbine system. As a classic solution of knowledge-based PHM, representation learning faces the challenges of long sequence length, missing values, and insufficient labeling when dealing with multivariate time series wind power data. This paper proposes a weighted representation learning-based feature extraction model to address the above problems simultaneously. Firstly, we design a novel self-attentive convolutional autoencoder (SACAE) model which can automatically extract high-level compressed feature matrix representations from long-length wind power time series, thereby retaining more necessary information. Then a missing-weighted dependent wild bootstrapped maximum mean discrepancy (MDMMD) loss function is proposed to capture the temporal dependence, numerical similarity, and distribution similarity of samples and avoid the influence of these missing values. Finally, we present the wind turbine PHM framework to improve the utilization of unlabeled data by using them to train the MDMMD-SACAE model and then the limited labeled data can get more accurate feature representation through the trained MDMMD-SACAE model. Experimental studies on three PHM cases verify the effectiveness and robustness of the proposed approach in the wind turbine system.

Harmonic Drive Gear Failures in Industrial Robots Applications: An Overview

Industrial robots play a key role in production lines. As a direct consequence, even a slight degradation of their operating conditions could negatively affect the entire manufacturing process. To minimize economical losses, preventive measures such as planned maintenance or stand-by working stations are adopted and, in some cases, additional manipulators are installed to ensure line availability. However, since each robot performs a specific task, it degrades at a different rate from another one, so Planned Preventive Maintenance (PPM) should be replaced with Condition Based Maintenance (CBM) for a more efficient and cost-effective approach. In addition, collaborative robots (cobots) have become more and more popular in the past years increasing the level of automation, in particular in small and medium size companies. Therefore, a failure of an industrial manipulator could not only cause unexpected downtimes, but also jeopardize the safety of the personnel with whom it shares its workspace. A cobot, in fact, is labeled as safe as long as it works in nominal conditions, but this cannot be guaranteed otherwise. Within this framework, Prognostics and Health Management (PHM) techniques could be used for both a customized maintenance on a specific machine and for preventing undesired events like the aforementioned ones. An ongoing research activity at Politecnico di Torino is focused on a model-based approach able to better describe the behavior of an industrial robot in non-nominal operating conditions. In order to properly simulate faults and failures of a manipulator, it has been first necessary to analyze its failure modes and their impact on the entire system. A robot, in fact, is usually equipped with internal sensors and algorithms able to monitor its health status. However, this is only true for software or electrical failures (i.e. power supply, motor or sensors related), but it is not as effective in case of mechanical ones like bearings or gearbox wear. This paper firstly presents an overview of the possible failures related to harmonic drives which are often used in robotics due to their compact and light-weight design and the high reduction ratio. Then, it outlines the work done with the objective of providing an accurate, physics-based description of faults progression of harmonic drives used in industrial robots and the related high-fidelity models in the framework of a PHM system for fault and failure detection, isolation and remaining useful life (RUL) prediction.

Prognostics and Health Management System for Electric Vehicles with a Hierarchy Fusion Framework: Concepts, Architectures, and Methods

The prognostics and health management (PHM) of electric vehicles is an important guarantee for their safety and long-term development. At present, there are few studies researching about life cycle PHM system of electric vehicles. In this paper, we first summarize the research progress and key methods of PHM. Then, we propose a three-level PHM system with a hierarchy fusion architecture for electric vehicles based on the structure, data source of them. In the PHM system, we introduce a database consisting of the factory data, real-time data, and detection data. The electric vehicle's factory parameters are used for determining the life curve of the electric vehicle and its components, the real-time data are used for predicting the remaining useful lifetime (RUL) of the electric vehicle and its components, and the detection data are used for fault diagnosis. This health management database is established to help make condition-based maintenance decisions for electric vehicles. In this way, a complete electric vehicle PHM system is formed, which can realize the whole-life-cycle life prediction and fault diagnosis of electric vehicles.

Research on the Requirement Analysis and Management Method of Aircraft PHM System

Recent years, the forward design based on requirements has become the mainstream mode of aircraft and its component systems development. Requirements analysis and management not only runs through the whole process of system development and validation activities, but also plays an important role in these activities. However, there are some practical problems in the requirement analysis and management of prognostics and health management (PHM) system of aircraft, such as lack of method and process guidance, incomplete stakeholder analysis, and nonstandard requirement description. Therefore, based on the theory of system engineering, this paper puts forward the basic methods and suggestions of PHM system stakeholder analysis, formulates the technical process and key points of PHM system requirement analysis in demonstration and scheme stage, and clarifies the basic principles of PHM system requirement specification. And the related research of this paper aims to improve the quality and efficiency of requirement analysis and management of aircraft PHM system, and to lay the foundation for the development of system architecture and function as well.

Prognostic systems representation in a function-based Bayesian model during engineering design

Prognostics and Health Management (PHM) systems are usually only considered and set up in the late stage of design or even during the system’s lifetime, after the major design decision have been made. However, considering the PHM system’s impact on the system failure probabilities can benefit the system design early on and subsequently reduce costs. The identification of failure paths in the early phases of engineering design can guide the designer toward a safer, more reliable and cost-efficient design. Several functional failure modeling methods have been developed recently. One of their advantages is to allow for risk assessment in the early stages of the design. Risk and reliability functional failure analysis methods currently developed do not explicitly model the PHM equipment used to identify and prevent potential system failures. This paper proposes a framework to optimize prognostic systems selection and positioning during the early stages of a complex system design. A Bayesian network, incorporating the PHM systems, is used to analyze the functional model and failure propagation. The algorithm developed within the proposed framework returns the optimized placement of PHM hardware in the complex system, allowing the designer to evaluate the need for system improvement. A design tool was developed to automatically apply the proposed method. A generic pressurized water nuclear reactor primary coolant loop system is used to present a case study illustrating the proposed framework. The results obtained for this particular case study demonstrate the promise of the method introduced in this paper. The case study notably exhibits how the proposed framework can be used to support engineering design teams in making better informed decisions early in the design phase.

Windings Fault Detection and Prognosis in Electro-Mechanical Flight Control Actuators Operating in Active-Active Configuration

One of the most significant research trends in the last decades of the aeronautic industry is the effort to move towards the design and the production of “more electric aircraft”. Within this framework, the application of the electrical technology to flight control systems has seen a progressive, although slow, increase: starting with the introduction of fly-by-wire and proceeding with the partial replacement of the traditional hydraulic/electro-hydraulic actuators with purely electro-mechanical ones. This evolution allowed to obtain more flexible solutions, reduced installation issues and enhanced aircraft control capability. Electro-Mechanical Actuators (EMAs) are however far from being a mature technology and still suffer from several safety issues, which can be partially limited by increasing the complexity of their design and hence their production costs. The development of a robust Prognostics and Health Management (PHM) system could provide a way to prevent the occurrence of a critical failure without resorting to complex device design. This paper deals with the first part of the study of a comprehensive PHM system for EMAs employed as primary flight control actuators; the peculiarities of the application are presented and discussed, while a novel approach, based on short pre-flight/post-flight health monitoring tests, is proposed. Turn-to-turn short in the electric motor windings is identified as the most common electrical degradation and a particle filtering framework for anomaly detection and prognosis featuring a self-tuning non-linear model is proposed. Features, anomaly detection and a prognostic algorithm are hence evaluated through state-of-the art performance metrics and their results discussed.

Defining Optimal Maintenance Scope for Multiple k-out-of-n Load-Sharing Production Systems Connected in Series Based on RUL Predictions

This paper presents a method to define the optimal maintenance scope of a production system consisting of multiple k-out-of-n systems connected in series. Maintenance recommendations are based on Remaining Useful Life (RUL) predictions obtained from a Prognostics and Health Management (PHM) system for each production unit within the production system. Defining the techniques applied in order to estimate the degradation level of production units is out of the scope of this paper. It is assumed here that a PHM system is available and provides the degradation level and RUL estimates for each production unit. The goal is to find the maintenance scope that minimizes the expected total cost per cycle until the next maintenance activity. A k-out-of-n load-sharing system is assumed, which means that the failure of a production unit results in a higher load (and consequently a higher degradation rate) on the surviving production units. The total cost comprises the production cost and the maintenance cost. Production cost of each k-out-of-n system is also affected by the number of surviving production units. A preventive maintenance cost is incurred to maintain a degraded but still functional production unit. A corrective maintenance cost is incurredto maintain a failed production unit. An Ant Colony Optimization (ACO) approach is adopted, which allows the proposed method to deal with large instances of the problem. A numerical example is presented to illustrate the application of the proposed method.